id sid tid token lemma pos cana-5910 1 1 communications communication NOUN cana-5910 1 2 on on ADP cana-5910 1 3 applied apply VERB cana-5910 1 4 nonlinear nonlinear ADJ cana-5910 1 5 analysis analysis NOUN cana-5910 1 6 issn issn NOUN cana-5910 1 7 : : PUNCT cana-5910 1 8 1074 1074 NUM cana-5910 1 9 - - PUNCT cana-5910 1 10 133x 133x NUM cana-5910 1 11 vol vol NOUN cana-5910 1 12 31no 31no NOUN cana-5910 1 13 . . PUNCT cana-5910 2 1 8s 8s NUM cana-5910 2 2 ( ( PUNCT cana-5910 2 3 2024 2024 NUM cana-5910 2 4 1110 1110 NUM cana-5910 2 5 https://internationalpubls.com https://internationalpubls.com X cana-5910 2 6 lie lie NOUN cana-5910 2 7 symmetry symmetry NOUN cana-5910 2 8 analysis analysis NOUN cana-5910 2 9 and and CCONJ cana-5910 2 10 similarity similarity NOUN cana-5910 2 11 solutions solution NOUN cana-5910 2 12 for for ADP cana-5910 2 13 two two NUM cana-5910 2 14 - - PUNCT cana-5910 2 15 dimensional dimensional ADJ cana-5910 2 16 heat heat NOUN cana-5910 2 17 and and CCONJ cana-5910 2 18 wave wave NOUN cana-5910 2 19 equations equation NOUN cana-5910 2 20 1yatin 1yatin NUM cana-5910 2 21 adhana adhana NOUN cana-5910 2 22 , , PUNCT cana-5910 2 23 2gaurav 2gaurav NUM cana-5910 2 24 kumar kumar PROPN cana-5910 2 25 1,2department 1,2department NUM cana-5910 2 26 of of ADP cana-5910 2 27 mathematics mathematic NOUN cana-5910 2 28 , , PUNCT cana-5910 2 29 n n PRON cana-5910 2 30 a a DET cana-5910 2 31 s s PROPN cana-5910 2 32 college college NOUN cana-5910 2 33 , , PUNCT cana-5910 2 34 meerut meerut PROPN cana-5910 2 35 , , PUNCT cana-5910 2 36 india india PROPN cana-5910 2 37 email email NOUN cana-5910 2 38 : : PUNCT cana-5910 2 39 * * PUNCT cana-5910 2 40 corresponding correspond VERB cana-5910 2 41 author author NOUN cana-5910 2 42 yatinadhana@gmail.com yatinadhana@gmail.com X cana-5910 3 1 gauravkgv@gmail.com gauravkgv@gmail.com X cana-5910 3 2 article article PROPN cana-5910 3 3 history history NOUN cana-5910 3 4 received receive VERB cana-5910 3 5 : : PUNCT cana-5910 3 6 02 02 NUM cana-5910 3 7 - - SYM cana-5910 3 8 10 10 NUM cana-5910 3 9 - - PUNCT cana-5910 3 10 2024 2024 NUM cana-5910 3 11 revised revise VERB cana-5910 3 12 : : PUNCT cana-5910 3 13 25 25 NUM cana-5910 3 14 - - SYM cana-5910 3 15 11 11 NUM cana-5910 3 16 - - PUNCT cana-5910 3 17 2024 2024 NUM cana-5910 3 18 accepted accept VERB cana-5910 3 19 : : PUNCT cana-5910 3 20 20 20 NUM cana-5910 3 21 - - SYM cana-5910 3 22 12 12 NUM cana-5910 3 23 - - PUNCT cana-5910 3 24 2024 2024 NUM cana-5910 3 25 a a DET cana-5910 3 26 b b PROPN cana-5910 3 27 s s ADP cana-5910 3 28 t t PROPN cana-5910 3 29 r r NOUN cana-5910 3 30 a a DET cana-5910 3 31 c c NOUN cana-5910 3 32 t t NOUN cana-5910 3 33 this this DET cana-5910 3 34 paper paper NOUN cana-5910 3 35 employs employ VERB cana-5910 3 36 lie lie NOUN cana-5910 3 37 symmetry symmetry NOUN cana-5910 3 38 theory theory NOUN cana-5910 3 39 to to PART cana-5910 3 40 derive derive VERB cana-5910 3 41 similarity similarity NOUN cana-5910 3 42 solutions solution NOUN cana-5910 3 43 for for ADP cana-5910 3 44 the the DET cana-5910 3 45 twodimensional twodimensional ADJ cana-5910 3 46 heat heat NOUN cana-5910 3 47 equation equation NOUN cana-5910 3 48 and and CCONJ cana-5910 3 49 wave wave NOUN cana-5910 3 50 equation equation NOUN cana-5910 3 51 . . PUNCT cana-5910 4 1 by by ADP cana-5910 4 2 identifying identify VERB cana-5910 4 3 the the DET cana-5910 4 4 lie lie NOUN cana-5910 4 5 point point NOUN cana-5910 4 6 symmetries symmetry NOUN cana-5910 4 7 of of ADP cana-5910 4 8 these these DET cana-5910 4 9 partial partial ADJ cana-5910 4 10 differential differential ADJ cana-5910 4 11 equations equation NOUN cana-5910 4 12 ( ( PUNCT cana-5910 4 13 pdes pde NOUN cana-5910 4 14 ) ) PUNCT cana-5910 4 15 , , PUNCT cana-5910 4 16 we we PRON cana-5910 4 17 perform perform VERB cana-5910 4 18 symmetry symmetry NOUN cana-5910 4 19 reductions reduction NOUN cana-5910 4 20 to to PART cana-5910 4 21 transform transform VERB cana-5910 4 22 the the DET cana-5910 4 23 pdes pde NOUN cana-5910 4 24 into into ADP cana-5910 4 25 ordinary ordinary ADJ cana-5910 4 26 differential differential ADJ cana-5910 4 27 equations equation NOUN cana-5910 4 28 ( ( PUNCT cana-5910 4 29 odes ode NOUN cana-5910 4 30 ) ) PUNCT cana-5910 4 31 . . PUNCT cana-5910 5 1 the the DET cana-5910 5 2 resulting result VERB cana-5910 5 3 odes ode NOUN cana-5910 5 4 are be AUX cana-5910 5 5 solved solve VERB cana-5910 5 6 to to PART cana-5910 5 7 obtain obtain VERB cana-5910 5 8 similarity similarity NOUN cana-5910 5 9 solutions solution NOUN cana-5910 5 10 , , PUNCT cana-5910 5 11 which which PRON cana-5910 5 12 are be AUX cana-5910 5 13 invariant invariant ADJ cana-5910 5 14 under under ADP cana-5910 5 15 specific specific ADJ cana-5910 5 16 symmetry symmetry NOUN cana-5910 5 17 transformations transformation NOUN cana-5910 5 18 . . PUNCT cana-5910 6 1 we we PRON cana-5910 6 2 present present VERB cana-5910 6 3 explicit explicit ADJ cana-5910 6 4 solutions solution NOUN cana-5910 6 5 for for ADP cana-5910 6 6 both both DET cana-5910 6 7 equations equation NOUN cana-5910 6 8 , , PUNCT cana-5910 6 9 highlighting highlight VERB cana-5910 6 10 their their PRON cana-5910 6 11 physical physical ADJ cana-5910 6 12 interpretations interpretation NOUN cana-5910 6 13 and and CCONJ cana-5910 6 14 potential potential ADJ cana-5910 6 15 applications application NOUN cana-5910 6 16 . . PUNCT cana-5910 7 1 the the DET cana-5910 7 2 methodology methodology NOUN cana-5910 7 3 demonstrates demonstrate VERB cana-5910 7 4 the the DET cana-5910 7 5 power power NOUN cana-5910 7 6 of of ADP cana-5910 7 7 lie lie NOUN cana-5910 7 8 symmetry symmetry NOUN cana-5910 7 9 analysis analysis NOUN cana-5910 7 10 in in ADP cana-5910 7 11 simplifying simplify VERB cana-5910 7 12 complex complex ADJ cana-5910 7 13 pdes pde NOUN cana-5910 7 14 and and CCONJ cana-5910 7 15 uncovering uncover VERB cana-5910 7 16 physically physically ADV cana-5910 7 17 meaningful meaningful ADJ cana-5910 7 18 solutions solution NOUN cana-5910 7 19 . . PUNCT cana-5910 8 1 keywords keyword NOUN cana-5910 8 2 : : PUNCT cana-5910 8 3 lie lie NOUN cana-5910 8 4 symmetry symmetry NOUN cana-5910 8 5 , , PUNCT cana-5910 8 6 similarity similarity NOUN cana-5910 8 7 solutions solution NOUN cana-5910 8 8 , , PUNCT cana-5910 8 9 heat heat NOUN cana-5910 8 10 equation equation NOUN cana-5910 8 11 , , PUNCT cana-5910 8 12 wave wave NOUN cana-5910 8 13 equation equation NOUN cana-5910 8 14 , , PUNCT cana-5910 8 15 twodimensional twodimensional ADJ cana-5910 8 16 pdes pde NOUN cana-5910 8 17 . . PUNCT cana-5910 9 1 1 1 X cana-5910 9 2 . . X cana-5910 9 3 introduction introduction NOUN cana-5910 9 4 the the DET cana-5910 9 5 two two NUM cana-5910 9 6 - - PUNCT cana-5910 9 7 dimensional dimensional ADJ cana-5910 9 8 heat heat NOUN cana-5910 9 9 equation equation NOUN cana-5910 9 10 and and CCONJ cana-5910 9 11 wave wave NOUN cana-5910 9 12 equation equation NOUN cana-5910 9 13 are be AUX cana-5910 9 14 cornerstone cornerstone NOUN cana-5910 9 15 models model NOUN cana-5910 9 16 in in ADP cana-5910 9 17 mathematical mathematical ADJ cana-5910 9 18 physics physics NOUN cana-5910 9 19 , , PUNCT cana-5910 9 20 governing govern VERB cana-5910 9 21 a a DET cana-5910 9 22 wide wide ADJ cana-5910 9 23 array array NOUN cana-5910 9 24 of of ADP cana-5910 9 25 physical physical ADJ cana-5910 9 26 phenomena phenomenon NOUN cana-5910 9 27 , , PUNCT cana-5910 9 28 from from ADP cana-5910 9 29 heat heat NOUN cana-5910 9 30 diffusion diffusion NOUN cana-5910 9 31 in in ADP cana-5910 9 32 materials material NOUN cana-5910 9 33 to to PART cana-5910 9 34 wave wave VERB cana-5910 9 35 propagation propagation NOUN cana-5910 9 36 in in ADP cana-5910 9 37 media medium NOUN cana-5910 9 38 such such ADJ cana-5910 9 39 as as ADP cana-5910 9 40 acoustics acoustic NOUN cana-5910 9 41 and and CCONJ cana-5910 9 42 electromagnetism electromagnetism NOUN cana-5910 9 43 . . PUNCT cana-5910 10 1 the the DET cana-5910 10 2 heat heat NOUN cana-5910 10 3 equation equation NOUN cana-5910 10 4 , , PUNCT cana-5910 10 5 characterized characterize VERB cana-5910 10 6 by by ADP cana-5910 10 7 its its PRON cana-5910 10 8 parabolic parabolic ADJ cana-5910 10 9 nature nature NOUN cana-5910 10 10 , , PUNCT cana-5910 10 11 describes describe VERB cana-5910 10 12 the the DET cana-5910 10 13 time time NOUN cana-5910 10 14 evolution evolution NOUN cana-5910 10 15 of of ADP cana-5910 10 16 temperature temperature NOUN cana-5910 10 17 in in ADP cana-5910 10 18 a a DET cana-5910 10 19 two two NUM cana-5910 10 20 - - PUNCT cana-5910 10 21 dimensional dimensional ADJ cana-5910 10 22 domain domain NOUN cana-5910 10 23 , , PUNCT cana-5910 10 24 while while SCONJ cana-5910 10 25 the the DET cana-5910 10 26 wave wave NOUN cana-5910 10 27 equation equation NOUN cana-5910 10 28 , , PUNCT cana-5910 10 29 a a DET cana-5910 10 30 hyperbolic hyperbolic ADJ cana-5910 10 31 equation equation NOUN cana-5910 10 32 , , PUNCT cana-5910 10 33 models model VERB cana-5910 10 34 the the DET cana-5910 10 35 propagation propagation NOUN cana-5910 10 36 of of ADP cana-5910 10 37 disturbances disturbance NOUN cana-5910 10 38 , , PUNCT cana-5910 10 39 such such ADJ cana-5910 10 40 as as ADP cana-5910 10 41 vibrations vibration NOUN cana-5910 10 42 or or CCONJ cana-5910 10 43 electromagnetic electromagnetic ADJ cana-5910 10 44 waves wave NOUN cana-5910 10 45 , , PUNCT cana-5910 10 46 across across ADP cana-5910 10 47 a a DET cana-5910 10 48 plane plane NOUN cana-5910 10 49 . . PUNCT cana-5910 11 1 solving solve VERB cana-5910 11 2 these these DET cana-5910 11 3 partial partial ADJ cana-5910 11 4 differential differential NOUN cana-5910 11 5 equations equation NOUN cana-5910 11 6 ( ( PUNCT cana-5910 11 7 pdes pde NOUN cana-5910 11 8 ) ) PUNCT cana-5910 11 9 in in ADP cana-5910 11 10 two two NUM cana-5910 11 11 spatial spatial ADJ cana-5910 11 12 dimensions dimension NOUN cana-5910 11 13 is be AUX cana-5910 11 14 often often ADV cana-5910 11 15 challenging challenge VERB cana-5910 11 16 due due ADJ cana-5910 11 17 to to ADP cana-5910 11 18 their their PRON cana-5910 11 19 complexity complexity NOUN cana-5910 11 20 , , PUNCT cana-5910 11 21 particularly particularly ADV cana-5910 11 22 when when SCONJ cana-5910 11 23 seeking seek VERB cana-5910 11 24 exact exact ADJ cana-5910 11 25 or or CCONJ cana-5910 11 26 analytical analytical ADJ cana-5910 11 27 solutions solution NOUN cana-5910 11 28 that that PRON cana-5910 11 29 provide provide VERB cana-5910 11 30 insight insight NOUN cana-5910 11 31 into into ADP cana-5910 11 32 the the DET cana-5910 11 33 underlying underlying ADJ cana-5910 11 34 physical physical ADJ cana-5910 11 35 processes process NOUN cana-5910 11 36 . . PUNCT cana-5910 12 1 lie lie PROPN cana-5910 12 2 symmetry symmetry NOUN cana-5910 12 3 theory theory NOUN cana-5910 12 4 , , PUNCT cana-5910 12 5 pioneered pioneer VERB cana-5910 12 6 by by ADP cana-5910 12 7 sophus sophu NOUN cana-5910 12 8 lie lie VERB cana-5910 12 9 in in ADP cana-5910 12 10 the the DET cana-5910 12 11 19th 19th ADJ cana-5910 12 12 century century NOUN cana-5910 12 13 , , PUNCT cana-5910 12 14 offers offer VERB cana-5910 12 15 a a DET cana-5910 12 16 systematic systematic ADJ cana-5910 12 17 and and CCONJ cana-5910 12 18 powerful powerful ADJ cana-5910 12 19 approach approach NOUN cana-5910 12 20 to to PART cana-5910 12 21 tackle tackle VERB cana-5910 12 22 such such ADJ cana-5910 12 23 pdes pde NOUN cana-5910 12 24 . . PUNCT cana-5910 13 1 by by ADP cana-5910 13 2 identifying identify VERB cana-5910 13 3 transformations transformation NOUN cana-5910 13 4 that that PRON cana-5910 13 5 leave leave VERB cana-5910 13 6 the the DET cana-5910 13 7 equations equation NOUN cana-5910 13 8 invariant invariant ADJ cana-5910 13 9 , , PUNCT cana-5910 13 10 lie lie NOUN cana-5910 13 11 symmetry symmetry NOUN cana-5910 13 12 analysis analysis NOUN cana-5910 13 13 enables enable VERB cana-5910 13 14 the the DET cana-5910 13 15 reduction reduction NOUN cana-5910 13 16 of of ADP cana-5910 13 17 pdes pde NOUN cana-5910 13 18 to to ADP cana-5910 13 19 simpler simple ADJ cana-5910 13 20 forms form NOUN cana-5910 13 21 , , PUNCT cana-5910 13 22 often often ADV cana-5910 13 23 ordinary ordinary ADJ cana-5910 13 24 differential differential ADJ cana-5910 13 25 equations equation NOUN cana-5910 13 26 ( ( PUNCT cana-5910 13 27 odes ode NOUN cana-5910 13 28 ) ) PUNCT cana-5910 13 29 , , PUNCT cana-5910 13 30 through through ADP cana-5910 13 31 the the DET cana-5910 13 32 construction construction NOUN cana-5910 13 33 of of ADP cana-5910 13 34 similarity similarity NOUN cana-5910 13 35 variables variable NOUN cana-5910 13 36 . . PUNCT cana-5910 14 1 these these DET cana-5910 14 2 similarity similarity NOUN cana-5910 14 3 solutions solution NOUN cana-5910 14 4 are be AUX cana-5910 14 5 particularly particularly ADV cana-5910 14 6 valuable valuable ADJ cana-5910 14 7 as as SCONJ cana-5910 14 8 they they PRON cana-5910 14 9 capture capture VERB cana-5910 14 10 invariant invariant ADJ cana-5910 14 11 behaviors behavior NOUN cana-5910 14 12 under under ADP cana-5910 14 13 specific specific ADJ cana-5910 14 14 symmetry symmetry NOUN cana-5910 14 15 groups group NOUN cana-5910 14 16 , , PUNCT cana-5910 14 17 providing provide VERB cana-5910 14 18 both both DET cana-5910 14 19 mathematical mathematical ADJ cana-5910 14 20 elegance elegance NOUN cana-5910 14 21 and and CCONJ cana-5910 14 22 physical physical ADJ cana-5910 14 23 relevance relevance NOUN cana-5910 14 24 . . PUNCT cana-5910 15 1 in in ADP cana-5910 15 2 the the DET cana-5910 15 3 context context NOUN cana-5910 15 4 of of ADP cana-5910 15 5 the the DET cana-5910 15 6 two two NUM cana-5910 15 7 - - PUNCT cana-5910 15 8 dimensional dimensional ADJ cana-5910 15 9 heat heat NOUN cana-5910 15 10 and and CCONJ cana-5910 15 11 wave wave NOUN cana-5910 15 12 equations equation NOUN cana-5910 15 13 , , PUNCT cana-5910 15 14 lie lie NOUN cana-5910 15 15 symmetry symmetry NOUN cana-5910 15 16 methods method NOUN cana-5910 15 17 can can AUX cana-5910 15 18 reveal reveal VERB cana-5910 15 19 solutions solution NOUN cana-5910 15 20 that that PRON cana-5910 15 21 describe describe VERB cana-5910 15 22 fundamental fundamental ADJ cana-5910 15 23 physical physical ADJ cana-5910 15 24 scenarios scenario NOUN cana-5910 15 25 , , PUNCT cana-5910 15 26 such such ADJ cana-5910 15 27 as as ADP cana-5910 15 28 radial radial ADJ cana-5910 15 29 heat heat NOUN cana-5910 15 30 diffusion diffusion NOUN cana-5910 15 31 from from ADP cana-5910 15 32 a a DET cana-5910 15 33 point point NOUN cana-5910 15 34 source source NOUN cana-5910 15 35 or or CCONJ cana-5910 15 36 cylindrical cylindrical ADJ cana-5910 15 37 wave wave NOUN cana-5910 15 38 propagation propagation NOUN cana-5910 15 39 . . PUNCT cana-5910 16 1 this this DET cana-5910 16 2 paper paper NOUN cana-5910 16 3 aims aim VERB cana-5910 16 4 to to PART cana-5910 16 5 apply apply VERB cana-5910 16 6 lie lie NOUN cana-5910 16 7 symmetry symmetry NOUN cana-5910 16 8 theory theory NOUN cana-5910 16 9 to to PART cana-5910 16 10 derive derive VERB cana-5910 16 11 similarity similarity NOUN cana-5910 16 12 solutions solution NOUN cana-5910 16 13 for for ADP cana-5910 16 14 the the DET cana-5910 16 15 two two NUM cana-5910 16 16 - - PUNCT cana-5910 16 17 dimensional dimensional ADJ cana-5910 16 18 heat heat NOUN cana-5910 16 19 equation equation NOUN cana-5910 16 20 , , PUNCT cana-5910 16 21 given give VERB cana-5910 16 22 by by ADP cana-5910 16 23 : : PUNCT cana-5910 16 24 πœ•π‘’ πœ•π‘’ NOUN cana-5910 16 25 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 16 26 = = NOUN cana-5910 16 27 𝛼 𝛼 PROPN cana-5910 16 28 ( ( PUNCT cana-5910 16 29 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 16 30 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 16 31 + + CCONJ cana-5910 16 32 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 16 33 πœ•π‘¦2 πœ•π‘¦2 NOUN cana-5910 16 34 ) ) PUNCT cana-5910 16 35 and and CCONJ cana-5910 16 36 the the DET cana-5910 16 37 two two NUM cana-5910 16 38 - - PUNCT cana-5910 16 39 dimensional dimensional ADJ cana-5910 16 40 wave wave NOUN cana-5910 16 41 equation equation NOUN cana-5910 16 42 : : PUNCT cana-5910 16 43 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 16 44 πœ•π‘‘2 πœ•π‘‘2 NOUN cana-5910 16 45 = = SYM cana-5910 16 46 𝑐2 𝑐2 NOUN cana-5910 16 47 ( ( PUNCT cana-5910 16 48 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 16 49 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 16 50 + + CCONJ cana-5910 16 51 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 16 52 πœ•π‘¦2 πœ•π‘¦2 NOUN cana-5910 16 53 ) ) PUNCT cana-5910 16 54 where where SCONJ cana-5910 16 55 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 16 56 , , PUNCT cana-5910 16 57 𝑦 𝑦 NOUN cana-5910 16 58 , , PUNCT cana-5910 16 59 𝑑 𝑑 X cana-5910 16 60 ) ) PUNCT cana-5910 16 61 represents represent VERB cana-5910 16 62 temperature temperature NOUN cana-5910 16 63 or or CCONJ cana-5910 16 64 displacement displacement NOUN cana-5910 16 65 , , PUNCT cana-5910 16 66 Ξ± Ξ± PROPN cana-5910 16 67 \alpha \alpha VERB cana-5910 16 68 Ξ± Ξ± PROPN cana-5910 16 69 is be AUX cana-5910 16 70 the the DET cana-5910 16 71 thermal thermal ADJ cana-5910 16 72 diffusivity diffusivity NOUN cana-5910 16 73 , , PUNCT cana-5910 16 74 and and CCONJ cana-5910 16 75 c c NOUN cana-5910 16 76 is be AUX cana-5910 16 77 the the DET cana-5910 16 78 wave wave NOUN cana-5910 16 79 speed speed NOUN cana-5910 16 80 . . PUNCT cana-5910 17 1 we we PRON cana-5910 17 2 systematically systematically ADV cana-5910 17 3 determine determine VERB cana-5910 17 4 the the DET cana-5910 17 5 lie lie NOUN cana-5910 17 6 point point NOUN cana-5910 17 7 symmetries symmetry NOUN cana-5910 17 8 of of ADP cana-5910 17 9 these these DET cana-5910 17 10 equations equation NOUN cana-5910 17 11 , , PUNCT cana-5910 17 12 use use VERB cana-5910 17 13 them they PRON cana-5910 17 14 to to PART cana-5910 17 15 perform perform VERB cana-5910 17 16 symmetry symmetry NOUN cana-5910 17 17 reductions reduction NOUN cana-5910 17 18 , , PUNCT cana-5910 17 19 and and CCONJ cana-5910 17 20 solve solve VERB cana-5910 17 21 the the DET cana-5910 17 22 resulting result VERB cana-5910 17 23 odes ode NOUN cana-5910 17 24 to to PART cana-5910 17 25 obtain obtain VERB cana-5910 17 26 similarity similarity NOUN cana-5910 17 27 solutions solution NOUN cana-5910 17 28 . . PUNCT cana-5910 18 1 the the DET cana-5910 18 2 solutions solution NOUN cana-5910 18 3 are be AUX cana-5910 18 4 analyzed analyze VERB cana-5910 18 5 for for ADP cana-5910 18 6 mailto:yatinadhana@gmail.com mailto:yatinadhana@gmail.com PROPN cana-5910 18 7 mailto:gauravkgv@gmail.com mailto:gauravkgv@gmail.com PROPN cana-5910 18 8 1111 1111 NUM cana-5910 18 9 https://internationalpubls.com https://internationalpubls.com X cana-5910 19 1 their their PRON cana-5910 19 2 physical physical ADJ cana-5910 19 3 interpretations interpretation NOUN cana-5910 19 4 , , PUNCT cana-5910 19 5 such such ADJ cana-5910 19 6 as as ADP cana-5910 19 7 the the DET cana-5910 19 8 gaussian gaussian ADJ cana-5910 19 9 heat heat NOUN cana-5910 19 10 kernel kernel NOUN cana-5910 19 11 for for ADP cana-5910 19 12 the the DET cana-5910 19 13 heat heat NOUN cana-5910 19 14 equation equation NOUN cana-5910 19 15 and and CCONJ cana-5910 19 16 cylindrical cylindrical ADJ cana-5910 19 17 wave wave NOUN cana-5910 19 18 fronts front NOUN cana-5910 19 19 for for ADP cana-5910 19 20 the the DET cana-5910 19 21 wave wave NOUN cana-5910 19 22 equation equation NOUN cana-5910 19 23 . . PUNCT cana-5910 20 1 this this DET cana-5910 20 2 work work NOUN cana-5910 20 3 underscores underscore VERB cana-5910 20 4 the the DET cana-5910 20 5 versatility versatility NOUN cana-5910 20 6 of of ADP cana-5910 20 7 lie lie NOUN cana-5910 20 8 symmetry symmetry NOUN cana-5910 20 9 analysis analysis NOUN cana-5910 20 10 in in ADP cana-5910 20 11 addressing address VERB cana-5910 20 12 multidimensional multidimensional ADJ cana-5910 20 13 pdes pde NOUN cana-5910 20 14 and and CCONJ cana-5910 20 15 provides provide VERB cana-5910 20 16 a a DET cana-5910 20 17 foundation foundation NOUN cana-5910 20 18 for for ADP cana-5910 20 19 further further ADJ cana-5910 20 20 exploration exploration NOUN cana-5910 20 21 of of ADP cana-5910 20 22 nonclassical nonclassical ADJ cana-5910 20 23 symmetries symmetry NOUN cana-5910 20 24 or or CCONJ cana-5910 20 25 numerical numerical ADJ cana-5910 20 26 validations validation NOUN cana-5910 20 27 . . PUNCT cana-5910 21 1 1.1 1.1 NUM cana-5910 21 2 lie lie NOUN cana-5910 21 3 symmetry symmetry NOUN cana-5910 21 4 analysis analysis NOUN cana-5910 21 5 lie lie NOUN cana-5910 21 6 symmetry symmetry NOUN cana-5910 21 7 analysis analysis NOUN cana-5910 21 8 is be AUX cana-5910 21 9 a a DET cana-5910 21 10 powerful powerful ADJ cana-5910 21 11 mathematical mathematical ADJ cana-5910 21 12 framework framework NOUN cana-5910 21 13 for for ADP cana-5910 21 14 studying study VERB cana-5910 21 15 differential differential ADJ cana-5910 21 16 equations equation NOUN cana-5910 21 17 by by ADP cana-5910 21 18 identifying identify VERB cana-5910 21 19 transformations transformation NOUN cana-5910 21 20 that that PRON cana-5910 21 21 leave leave VERB cana-5910 21 22 the the DET cana-5910 21 23 equations equation NOUN cana-5910 21 24 invariant invariant ADJ cana-5910 21 25 . . PUNCT cana-5910 22 1 these these DET cana-5910 22 2 transformations transformation NOUN cana-5910 22 3 , , PUNCT cana-5910 22 4 forming form VERB cana-5910 22 5 a a DET cana-5910 22 6 lie lie NOUN cana-5910 22 7 group group NOUN cana-5910 22 8 , , PUNCT cana-5910 22 9 allow allow VERB cana-5910 22 10 the the DET cana-5910 22 11 reduction reduction NOUN cana-5910 22 12 of of ADP cana-5910 22 13 partial partial ADJ cana-5910 22 14 differential differential ADJ cana-5910 22 15 equations equation NOUN cana-5910 22 16 ( ( PUNCT cana-5910 22 17 pdes pde NOUN cana-5910 22 18 ) ) PUNCT cana-5910 22 19 to to ADP cana-5910 22 20 simpler simple ADJ cana-5910 22 21 forms form NOUN cana-5910 22 22 , , PUNCT cana-5910 22 23 often often ADV cana-5910 22 24 ordinary ordinary ADJ cana-5910 22 25 differential differential ADJ cana-5910 22 26 equations equation NOUN cana-5910 22 27 ( ( PUNCT cana-5910 22 28 odes ode NOUN cana-5910 22 29 ) ) PUNCT cana-5910 22 30 , , PUNCT cana-5910 22 31 through through ADP cana-5910 22 32 the the DET cana-5910 22 33 construction construction NOUN cana-5910 22 34 of of ADP cana-5910 22 35 similarity similarity NOUN cana-5910 22 36 variables variable NOUN cana-5910 22 37 . . PUNCT cana-5910 23 1 in in ADP cana-5910 23 2 this this DET cana-5910 23 3 section section NOUN cana-5910 23 4 , , PUNCT cana-5910 23 5 we we PRON cana-5910 23 6 apply apply VERB cana-5910 23 7 lie lie NOUN cana-5910 23 8 symmetry symmetry NOUN cana-5910 23 9 analysis analysis NOUN cana-5910 23 10 to to ADP cana-5910 23 11 the the DET cana-5910 23 12 two two NUM cana-5910 23 13 - - PUNCT cana-5910 23 14 dimensional dimensional ADJ cana-5910 23 15 heat heat NOUN cana-5910 23 16 equation equation NOUN cana-5910 23 17 and and CCONJ cana-5910 23 18 wave wave NOUN cana-5910 23 19 equation equation NOUN cana-5910 23 20 to to PART cana-5910 23 21 determine determine VERB cana-5910 23 22 their their PRON cana-5910 23 23 lie lie NOUN cana-5910 23 24 point point NOUN cana-5910 23 25 symmetries symmetry NOUN cana-5910 23 26 , , PUNCT cana-5910 23 27 which which PRON cana-5910 23 28 will will AUX cana-5910 23 29 be be AUX cana-5910 23 30 used use VERB cana-5910 23 31 in in ADP cana-5910 23 32 subsequent subsequent ADJ cana-5910 23 33 sections section NOUN cana-5910 23 34 to to PART cana-5910 23 35 derive derive VERB cana-5910 23 36 similarity similarity NOUN cana-5910 23 37 solutions solution NOUN cana-5910 23 38 . . PUNCT cana-5910 24 1 2 2 X cana-5910 24 2 . . X cana-5910 24 3 general general ADJ cana-5910 24 4 methodology methodology NOUN cana-5910 24 5 lie lie NOUN cana-5910 24 6 symmetry symmetry NOUN cana-5910 24 7 analysis analysis NOUN cana-5910 24 8 provides provide VERB cana-5910 24 9 a a DET cana-5910 24 10 systematic systematic ADJ cana-5910 24 11 approach approach NOUN cana-5910 24 12 to to PART cana-5910 24 13 identify identify VERB cana-5910 24 14 transformations transformation NOUN cana-5910 24 15 that that PRON cana-5910 24 16 leave leave VERB cana-5910 24 17 differential differential ADJ cana-5910 24 18 equations equation NOUN cana-5910 24 19 invariant invariant ADJ cana-5910 24 20 , , PUNCT cana-5910 24 21 enabling enable VERB cana-5910 24 22 the the DET cana-5910 24 23 reduction reduction NOUN cana-5910 24 24 of of ADP cana-5910 24 25 partial partial ADJ cana-5910 24 26 differential differential ADJ cana-5910 24 27 equations equation NOUN cana-5910 24 28 ( ( PUNCT cana-5910 24 29 pdes pde NOUN cana-5910 24 30 ) ) PUNCT cana-5910 24 31 to to ADP cana-5910 24 32 simpler simple ADJ cana-5910 24 33 forms form NOUN cana-5910 24 34 , , PUNCT cana-5910 24 35 such such ADJ cana-5910 24 36 as as ADP cana-5910 24 37 ordinary ordinary ADJ cana-5910 24 38 differential differential ADJ cana-5910 24 39 equations equation NOUN cana-5910 24 40 ( ( PUNCT cana-5910 24 41 odes ode NOUN cana-5910 24 42 ) ) PUNCT cana-5910 24 43 , , PUNCT cana-5910 24 44 through through ADP cana-5910 24 45 similarity similarity NOUN cana-5910 24 46 variables variable NOUN cana-5910 24 47 . . PUNCT cana-5910 25 1 in in ADP cana-5910 25 2 this this DET cana-5910 25 3 section section NOUN cana-5910 25 4 , , PUNCT cana-5910 25 5 we we PRON cana-5910 25 6 apply apply VERB cana-5910 25 7 lie lie NOUN cana-5910 25 8 symmetry symmetry NOUN cana-5910 25 9 analysis analysis NOUN cana-5910 25 10 to to ADP cana-5910 25 11 the the DET cana-5910 25 12 two two NUM cana-5910 25 13 - - PUNCT cana-5910 25 14 dimensional dimensional ADJ cana-5910 25 15 heat heat NOUN cana-5910 25 16 equation equation NOUN cana-5910 25 17 and and CCONJ cana-5910 25 18 wave wave NOUN cana-5910 25 19 equation equation NOUN cana-5910 25 20 to to PART cana-5910 25 21 determine determine VERB cana-5910 25 22 their their PRON cana-5910 25 23 lie lie NOUN cana-5910 25 24 point point NOUN cana-5910 25 25 symmetries symmetry NOUN cana-5910 25 26 , , PUNCT cana-5910 25 27 which which PRON cana-5910 25 28 will will AUX cana-5910 25 29 be be AUX cana-5910 25 30 used use VERB cana-5910 25 31 to to PART cana-5910 25 32 derive derive VERB cana-5910 25 33 similarity similarity NOUN cana-5910 25 34 solutions solution NOUN cana-5910 25 35 . . PUNCT cana-5910 26 1 2.1 2.1 NUM cana-5910 26 2 general general ADJ cana-5910 26 3 methodology methodology NOUN cana-5910 26 4 consider consider VERB cana-5910 26 5 a a DET cana-5910 26 6 pde pde NOUN cana-5910 26 7 of of ADP cana-5910 26 8 the the DET cana-5910 26 9 form form NOUN cana-5910 26 10 : : PUNCT cana-5910 26 11 𝐹(π‘₯ 𝐹(π‘₯ NUM cana-5910 26 12 , , PUNCT cana-5910 26 13 𝑦 𝑦 NOUN cana-5910 26 14 , , PUNCT cana-5910 26 15 𝑑 𝑑 PROPN cana-5910 26 16 , , PUNCT cana-5910 26 17 𝑒 𝑒 X cana-5910 26 18 , , PUNCT cana-5910 26 19 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 26 20 , , PUNCT cana-5910 26 21 𝑒𝑑 𝑒𝑑 NOUN cana-5910 26 22 , , PUNCT cana-5910 26 23 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ NOUN cana-5910 26 24 , , PUNCT cana-5910 26 25 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 26 26 , , PUNCT cana-5910 26 27 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 26 28 … … PUNCT cana-5910 26 29 … … PUNCT cana-5910 26 30 . . PUNCT cana-5910 26 31 . . PUNCT cana-5910 26 32 ) ) PUNCT cana-5910 27 1 = = PUNCT cana-5910 27 2 0 0 NUM cana-5910 27 3 where where SCONJ cana-5910 27 4 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 27 5 , , PUNCT cana-5910 27 6 𝑦 𝑦 NOUN cana-5910 27 7 , , PUNCT cana-5910 27 8 𝑑 𝑑 X cana-5910 27 9 ) ) PUNCT cana-5910 27 10 is be AUX cana-5910 27 11 the the DET cana-5910 27 12 dependent dependent ADJ cana-5910 27 13 variable variable NOUN cana-5910 27 14 , , PUNCT cana-5910 27 15 and and CCONJ cana-5910 27 16 π‘₯ π‘₯ NOUN cana-5910 27 17 , , PUNCT cana-5910 27 18 𝑦 𝑦 NOUN cana-5910 27 19 , , PUNCT cana-5910 27 20 𝑑 𝑑 PROPN cana-5910 27 21 are be AUX cana-5910 27 22 independent independent ADJ cana-5910 27 23 variables variable NOUN cana-5910 27 24 . . PUNCT cana-5910 28 1 a a DET cana-5910 28 2 lie lie NOUN cana-5910 28 3 point point NOUN cana-5910 28 4 symmetry symmetry NOUN cana-5910 28 5 is be AUX cana-5910 28 6 a a DET cana-5910 28 7 oneparameter oneparameter ADJ cana-5910 28 8 group group NOUN cana-5910 28 9 of of ADP cana-5910 28 10 transformations transformation NOUN cana-5910 28 11 : : PUNCT cana-5910 28 12 π‘₯β€² π‘₯β€² PUNCT cana-5910 28 13 = = PUNCT cana-5910 29 1 π‘₯ π‘₯ PROPN cana-5910 30 1 + + NOUN cana-5910 30 2 νœ€ νœ€ PRON cana-5910 30 3 πœ‰π‘₯ πœ‰π‘₯ INTJ cana-5910 30 4 ( ( PUNCT cana-5910 30 5 π‘₯ π‘₯ PROPN cana-5910 30 6 , , PUNCT cana-5910 30 7 𝑦 𝑦 NOUN cana-5910 30 8 , , PUNCT cana-5910 30 9 𝑑 𝑑 NOUN cana-5910 30 10 , , PUNCT cana-5910 30 11 𝑒 𝑒 NOUN cana-5910 30 12 ) ) PUNCT cana-5910 30 13 + + CCONJ cana-5910 31 1 𝑂(νœ€2 𝑂(νœ€2 NOUN cana-5910 31 2 ) ) PUNCT cana-5910 31 3 , , PUNCT cana-5910 31 4 𝑦′ 𝑦′ X cana-5910 31 5 = = SYM cana-5910 31 6 𝑦 𝑦 SYM cana-5910 31 7 + + X cana-5910 31 8 νœ€ νœ€ PRON cana-5910 31 9 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 31 10 ( ( PUNCT cana-5910 31 11 π‘₯ π‘₯ NOUN cana-5910 31 12 , , PUNCT cana-5910 31 13 𝑦 𝑦 NOUN cana-5910 31 14 , , PUNCT cana-5910 31 15 𝑑 𝑑 NOUN cana-5910 31 16 , , PUNCT cana-5910 31 17 𝑒 𝑒 NOUN cana-5910 31 18 ) ) PUNCT cana-5910 31 19 + + CCONJ cana-5910 31 20 𝑂(νœ€2 𝑂(νœ€2 NOUN cana-5910 31 21 ) ) PUNCT cana-5910 31 22 , , PUNCT cana-5910 31 23 𝑑′ 𝑑′ X cana-5910 31 24 = = SYM cana-5910 31 25 𝑑 𝑑 PROPN cana-5910 31 26 + + X cana-5910 31 27 νœ€πœ(π‘₯ νœ€πœ(π‘₯ NUM cana-5910 31 28 , , PUNCT cana-5910 31 29 𝑦 𝑦 NOUN cana-5910 31 30 , , PUNCT cana-5910 31 31 𝑑 𝑑 PROPN cana-5910 31 32 , , PUNCT cana-5910 31 33 𝑒 𝑒 NOUN cana-5910 31 34 ) ) PUNCT cana-5910 31 35 + + CCONJ cana-5910 31 36 𝑂(νœ€2 𝑂(νœ€2 NOUN cana-5910 31 37 ) ) PUNCT cana-5910 31 38 , , PUNCT cana-5910 31 39 𝑒′ 𝑒′ VERB cana-5910 31 40 = = SYM cana-5910 31 41 𝑒 𝑒 PROPN cana-5910 31 42 + + X cana-5910 31 43 νœ€ νœ€ PROPN cana-5910 31 44 πœ‚(π‘₯ πœ‚(π‘₯ PROPN cana-5910 31 45 , , PUNCT cana-5910 31 46 𝑦 𝑦 PROPN cana-5910 31 47 , , PUNCT cana-5910 31 48 𝑑 𝑑 PROPN cana-5910 31 49 , , PUNCT cana-5910 31 50 𝑒 𝑒 NOUN cana-5910 31 51 ) ) PUNCT cana-5910 31 52 + + CCONJ cana-5910 32 1 𝑂(νœ€2 𝑂(νœ€2 NOUN cana-5910 32 2 ) ) PUNCT cana-5910 32 3 , , PUNCT cana-5910 32 4 that that PRON cana-5910 32 5 leaves leave VERB cana-5910 32 6 the the DET cana-5910 32 7 pde pde NOUN cana-5910 32 8 invariant invariant ADJ cana-5910 32 9 , , PUNCT cana-5910 32 10 where where SCONJ cana-5910 32 11 Ο΅ Ο΅ PROPN cana-5910 32 12 is be AUX cana-5910 32 13 a a DET cana-5910 32 14 small small ADJ cana-5910 32 15 parameter parameter NOUN cana-5910 32 16 , , PUNCT cana-5910 32 17 and and CCONJ cana-5910 32 18 πœ‰π‘₯ πœ‰π‘₯ INTJ cana-5910 32 19 , , PUNCT cana-5910 32 20 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 32 21 , , PUNCT cana-5910 32 22 𝜏 𝜏 NOUN cana-5910 32 23 , , PUNCT cana-5910 32 24 πœ‚ πœ‚ X cana-5910 32 25 are be AUX cana-5910 32 26 the the DET cana-5910 32 27 infinitesimals infinitesimal NOUN cana-5910 32 28 . . PUNCT cana-5910 33 1 the the DET cana-5910 33 2 infinitesimal infinitesimal ADJ cana-5910 33 3 generator generator NOUN cana-5910 33 4 of of ADP cana-5910 33 5 the the DET cana-5910 33 6 symmetry symmetry NOUN cana-5910 33 7 is be AUX cana-5910 33 8 : : PUNCT cana-5910 33 9 𝑉 𝑉 PROPN cana-5910 33 10 = = PUNCT cana-5910 33 11 πœ‰π‘₯ πœ‰π‘₯ PROPN cana-5910 33 12 πœ• πœ• NOUN cana-5910 33 13 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 33 14 + + CCONJ cana-5910 33 15 πœ‰π‘¦ πœ‰π‘¦ ADP cana-5910 33 16 πœ• πœ• NOUN cana-5910 33 17 πœ•π‘¦ πœ•π‘¦ NOUN cana-5910 33 18 + + CCONJ cana-5910 33 19 𝜏 𝜏 DET cana-5910 33 20 πœ•π‘¦ πœ•π‘¦ NOUN cana-5910 33 21 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 33 22 + + CCONJ cana-5910 33 23 πœ‚ πœ‚ NOUN cana-5910 33 24 πœ• πœ• NOUN cana-5910 33 25 πœ•π‘’ πœ•π‘’ NOUN cana-5910 33 26 . . PUNCT cana-5910 34 1 to to PART cana-5910 34 2 find find VERB cana-5910 34 3 the the DET cana-5910 34 4 symmetries symmetry NOUN cana-5910 34 5 , , PUNCT cana-5910 34 6 we we PRON cana-5910 34 7 apply apply VERB cana-5910 34 8 the the DET cana-5910 34 9 prolonged prolonged ADJ cana-5910 34 10 generator generator NOUN cana-5910 34 11 , , PUNCT cana-5910 34 12 which which PRON cana-5910 34 13 accounts account VERB cana-5910 34 14 for for ADP cana-5910 34 15 the the DET cana-5910 34 16 transformations transformation NOUN cana-5910 34 17 of of ADP cana-5910 34 18 derivatives derivative NOUN cana-5910 34 19 ( ( PUNCT cana-5910 34 20 e.g. e.g. ADV cana-5910 34 21 , , PUNCT cana-5910 34 22 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 34 23 , , PUNCT cana-5910 34 24 𝑒𝑑 𝑒𝑑 NOUN cana-5910 34 25 , , PUNCT cana-5910 34 26 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ PROPN cana-5910 34 27 ) ) PUNCT cana-5910 34 28 . . PUNCT cana-5910 35 1 the the DET cana-5910 35 2 invariance invariance NOUN cana-5910 35 3 condition condition NOUN cana-5910 35 4 is be AUX cana-5910 35 5 : : PUNCT cana-5910 35 6 π‘π‘Ÿ(𝑛)𝑉(𝐹 π‘π‘Ÿ(𝑛)𝑉(𝐹 X cana-5910 35 7 ) ) PUNCT cana-5910 36 1 = = PUNCT cana-5910 36 2 0 0 PUNCT cana-5910 37 1 π‘œπ‘› π‘œπ‘› PROPN cana-5910 37 2 𝐹 𝐹 PROPN cana-5910 37 3 = = NOUN cana-5910 37 4 0 0 PROPN cana-5910 37 5 , , PUNCT cana-5910 37 6 where where SCONJ cana-5910 37 7 π‘π‘Ÿ(𝑛)𝑉 π‘π‘Ÿ(𝑛)𝑉 PROPN cana-5910 37 8 is be AUX cana-5910 37 9 the the DET cana-5910 37 10 n n ADV cana-5910 37 11 - - PUNCT cana-5910 37 12 th th VERB cana-5910 37 13 prolongation prolongation NOUN cana-5910 37 14 of of ADP cana-5910 37 15 v v NOUN cana-5910 37 16 , , PUNCT cana-5910 37 17 accounting account VERB cana-5910 37 18 for for ADP cana-5910 37 19 transformations transformation NOUN cana-5910 37 20 of of ADP cana-5910 37 21 derivatives derivative NOUN cana-5910 37 22 up up ADP cana-5910 37 23 to to ADP cana-5910 37 24 the the DET cana-5910 37 25 highest high ADJ cana-5910 37 26 order order NOUN cana-5910 37 27 n n NOUN cana-5910 37 28 in in ADP cana-5910 37 29 the the DET cana-5910 37 30 pde pde NOUN cana-5910 37 31 . . PUNCT cana-5910 38 1 this this DET cana-5910 38 2 condition condition NOUN cana-5910 38 3 yields yield VERB cana-5910 38 4 a a DET cana-5910 38 5 system system NOUN cana-5910 38 6 of of ADP cana-5910 38 7 determining determine VERB cana-5910 38 8 equations equation NOUN cana-5910 38 9 for for ADP cana-5910 38 10 πœ‰π‘₯ πœ‰π‘₯ PROPN cana-5910 38 11 , , PUNCT cana-5910 38 12 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 38 13 , , PUNCT cana-5910 38 14 𝜏 𝜏 NOUN cana-5910 38 15 , , PUNCT cana-5910 38 16 πœ‚ πœ‚ NOUN cana-5910 38 17 , , PUNCT cana-5910 38 18 which which PRON cana-5910 38 19 are be AUX cana-5910 38 20 solved solve VERB cana-5910 38 21 to to PART cana-5910 38 22 obtain obtain VERB cana-5910 38 23 the the DET cana-5910 38 24 lie lie NOUN cana-5910 38 25 algebra algebra NOUN cana-5910 38 26 of of ADP cana-5910 38 27 symmetries symmetry NOUN cana-5910 38 28 . . PUNCT cana-5910 39 1 2.2 2.2 NUM cana-5910 39 2 symmetries symmetry NOUN cana-5910 39 3 of of ADP cana-5910 39 4 the the DET cana-5910 39 5 two two NUM cana-5910 39 6 - - PUNCT cana-5910 39 7 dimensional dimensional ADJ cana-5910 39 8 heat heat NOUN cana-5910 39 9 equation equation NOUN cana-5910 39 10 the the DET cana-5910 39 11 two two NUM cana-5910 39 12 - - PUNCT cana-5910 39 13 dimensional dimensional ADJ cana-5910 39 14 heat heat NOUN cana-5910 39 15 equation equation NOUN cana-5910 39 16 is be AUX cana-5910 39 17 : : PUNCT cana-5910 39 18 πœ•π‘’ πœ•π‘’ NOUN cana-5910 39 19 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 39 20 = = NOUN cana-5910 39 21 𝛼 𝛼 PROPN cana-5910 39 22 ( ( PUNCT cana-5910 39 23 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 39 24 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 39 25 + + CCONJ cana-5910 39 26 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 39 27 πœ•π‘¦2 πœ•π‘¦2 NOUN cana-5910 39 28 ) ) PUNCT cana-5910 39 29 or or CCONJ cana-5910 39 30 𝑒𝑑 𝑒𝑑 X cana-5910 39 31 = = SYM cana-5910 39 32 𝛼 𝛼 PROPN cana-5910 39 33 ( ( PUNCT cana-5910 39 34 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 39 35 + + CCONJ cana-5910 39 36 𝑒𝑦𝑦 𝑒𝑦𝑦 PROPN cana-5910 39 37 ) ) PUNCT cana-5910 39 38 , , PUNCT cana-5910 39 39 where where SCONJ cana-5910 39 40 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 39 41 , , PUNCT cana-5910 39 42 𝑦 𝑦 NOUN cana-5910 39 43 , , PUNCT cana-5910 39 44 𝑑 𝑑 X cana-5910 39 45 ) ) PUNCT cana-5910 39 46 is be AUX cana-5910 39 47 the the DET cana-5910 39 48 temperature temperature NOUN cana-5910 39 49 , , PUNCT cana-5910 39 50 and and CCONJ cana-5910 39 51 Ξ± Ξ± PRON cana-5910 39 52 is be AUX cana-5910 39 53 the the DET cana-5910 39 54 thermal thermal ADJ cana-5910 39 55 diffusivity diffusivity NOUN cana-5910 39 56 ? ? PUNCT cana-5910 40 1 as as SCONJ cana-5910 40 2 the the DET cana-5910 40 3 equation equation NOUN cana-5910 40 4 involves involve VERB cana-5910 40 5 second second ADJ cana-5910 40 6 derivatives derivative NOUN cana-5910 40 7 , , PUNCT cana-5910 40 8 we we PRON cana-5910 40 9 use use VERB cana-5910 40 10 the the DET cana-5910 40 11 second second ADJ cana-5910 40 12 prolongation prolongation NOUN cana-5910 40 13 : : PUNCT cana-5910 40 14 π‘π‘Ÿ(2 π‘π‘Ÿ(2 X cana-5910 40 15 ) ) PUNCT cana-5910 40 16 𝑉 𝑉 PROPN cana-5910 40 17 = = SYM cana-5910 40 18 𝑉 𝑉 PROPN cana-5910 40 19 + + CCONJ cana-5910 40 20 πœ‚π‘₯ πœ‚π‘₯ PROPN cana-5910 40 21 πœ• πœ• PROPN cana-5910 40 22 πœ•π‘’π‘₯ πœ•π‘’π‘₯ PRON cana-5910 40 23 + + PROPN cana-5910 40 24 πœ‚π‘¦ πœ‚π‘¦ ADP cana-5910 40 25 πœ• πœ• NOUN cana-5910 40 26 πœ•π‘’π‘¦ πœ•π‘’π‘¦ PUNCT cana-5910 41 1 + + CCONJ cana-5910 41 2 πœ‚π‘‘ πœ‚π‘‘ ADP cana-5910 41 3 πœ• πœ• NOUN cana-5910 41 4 πœ•π‘’π‘‘ πœ•π‘’π‘‘ PUNCT cana-5910 41 5 + + CCONJ cana-5910 41 6 πœ‚π‘₯π‘₯ πœ‚π‘₯π‘₯ PRON cana-5910 41 7 πœ• πœ• NOUN cana-5910 41 8 πœ•π‘’π‘₯π‘₯ πœ•π‘’π‘₯π‘₯ VERB cana-5910 41 9 + + NUM cana-5910 41 10 πœ‚π‘¦π‘¦ πœ‚π‘¦π‘¦ PROPN cana-5910 41 11 πœ• πœ• NOUN cana-5910 41 12 πœ•π‘’π‘¦π‘¦ πœ•π‘’π‘¦π‘¦ VERB cana-5910 41 13 + + CCONJ cana-5910 41 14 … … PUNCT cana-5910 41 15 , , PUNCT cana-5910 41 16 1112 1112 NUM cana-5910 41 17 https://internationalpubls.com https://internationalpubls.com X cana-5910 41 18 where where SCONJ cana-5910 41 19 πœ‚π‘₯ πœ‚π‘₯ INTJ cana-5910 41 20 , , PUNCT cana-5910 41 21 πœ‚π‘¦ πœ‚π‘¦ ADV cana-5910 41 22 , , PUNCT cana-5910 41 23 πœ‚π‘‘ πœ‚π‘‘ INTJ cana-5910 41 24 , , PUNCT cana-5910 41 25 πœ‚π‘₯π‘₯ πœ‚π‘₯π‘₯ INTJ cana-5910 41 26 , , PUNCT cana-5910 41 27 πœ‚π‘¦π‘¦ πœ‚π‘¦π‘¦ PROPN cana-5910 41 28 are be AUX cana-5910 41 29 the the DET cana-5910 41 30 prolonged prolong VERB cana-5910 41 31 infinitesimals infinitesimal NOUN cana-5910 41 32 , , PUNCT cana-5910 41 33 computed compute VERB cana-5910 41 34 as as ADP cana-5910 41 35 : : PUNCT cana-5910 41 36 πœ‚π‘₯ πœ‚π‘₯ PROPN cana-5910 41 37 = = SYM cana-5910 41 38 𝐷π‘₯ 𝐷π‘₯ PROPN cana-5910 41 39 ( ( PUNCT cana-5910 41 40 πœ‚ πœ‚ NOUN cana-5910 41 41 βˆ’ βˆ’ PROPN cana-5910 41 42 πœ‰π‘₯ πœ‰π‘₯ NOUN cana-5910 41 43 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 41 44 βˆ’ βˆ’ PROPN cana-5910 41 45 πœ‰π‘¦π‘’π‘¦ πœ‰π‘¦π‘’π‘¦ NOUN cana-5910 41 46 βˆ’ βˆ’ NOUN cana-5910 41 47 𝜏 𝜏 NOUN cana-5910 41 48 𝑒𝑑 𝑒𝑑 NOUN cana-5910 41 49 ) ) PUNCT cana-5910 42 1 + + NUM cana-5910 42 2 πœ‰π‘₯ πœ‰π‘₯ ADP cana-5910 42 3 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ NOUN cana-5910 42 4 + + CCONJ cana-5910 42 5 πœ‰π‘¦π‘’π‘₯𝑦 πœ‰π‘¦π‘’π‘₯𝑦 NOUN cana-5910 42 6 + + CCONJ cana-5910 42 7 𝜏 𝜏 DET cana-5910 42 8 𝑒π‘₯𝑑 𝑒π‘₯𝑑 NOUN cana-5910 42 9 , , PUNCT cana-5910 42 10 πœ‚π‘‘ πœ‚π‘‘ ADP cana-5910 42 11 = = SYM cana-5910 42 12 𝐷𝑑 𝐷𝑑 PROPN cana-5910 42 13 ( ( PUNCT cana-5910 42 14 πœ‚ πœ‚ NOUN cana-5910 42 15 βˆ’ βˆ’ PROPN cana-5910 42 16 πœ‰π‘₯ πœ‰π‘₯ NOUN cana-5910 42 17 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 42 18 βˆ’ βˆ’ PROPN cana-5910 42 19 πœ‰π‘¦ πœ‰π‘¦ ADP cana-5910 42 20 𝑒𝑦 𝑒𝑦 PRON cana-5910 42 21 βˆ’ βˆ’ PROPN cana-5910 42 22 𝜏 𝜏 NOUN cana-5910 42 23 𝑒𝑑 𝑒𝑑 NOUN cana-5910 42 24 ) ) PUNCT cana-5910 43 1 + + NUM cana-5910 43 2 πœ‰π‘₯ πœ‰π‘₯ VERB cana-5910 43 3 𝑒π‘₯𝑑 𝑒π‘₯𝑑 NOUN cana-5910 43 4 + + CCONJ cana-5910 43 5 πœ‰π‘¦ πœ‰π‘¦ PRON cana-5910 43 6 𝑒𝑦𝑑 𝑒𝑦𝑑 NOUN cana-5910 43 7 + + CCONJ cana-5910 43 8 𝜏 𝜏 PRON cana-5910 43 9 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 43 10 , , PUNCT cana-5910 43 11 πœ‚π‘₯π‘₯ πœ‚π‘₯π‘₯ PRON cana-5910 43 12 = = SYM cana-5910 44 1 𝐷π‘₯ 𝐷π‘₯ PROPN cana-5910 44 2 ( ( PUNCT cana-5910 44 3 πœ‚π‘₯ πœ‚π‘₯ NOUN cana-5910 44 4 βˆ’ βˆ’ PROPN cana-5910 44 5 πœ‰π‘₯𝑒π‘₯π‘₯ πœ‰π‘₯𝑒π‘₯π‘₯ NOUN cana-5910 44 6 βˆ’ βˆ’ PROPN cana-5910 44 7 πœ‰π‘¦ πœ‰π‘¦ ADP cana-5910 44 8 𝑒π‘₯𝑦 𝑒π‘₯𝑦 NOUN cana-5910 44 9 βˆ’ βˆ’ PROPN cana-5910 44 10 𝜏 𝜏 NOUN cana-5910 44 11 𝑒π‘₯𝑑 𝑒π‘₯𝑑 NOUN cana-5910 44 12 ) ) PUNCT cana-5910 45 1 + + NUM cana-5910 45 2 πœ‰π‘₯ πœ‰π‘₯ VERB cana-5910 45 3 𝑒π‘₯π‘₯π‘₯ 𝑒π‘₯π‘₯π‘₯ NOUN cana-5910 45 4 + + CCONJ cana-5910 45 5 πœ‰π‘¦ πœ‰π‘¦ ADJ cana-5910 45 6 𝑒π‘₯𝑦𝑦 𝑒π‘₯𝑦𝑦 ADJ cana-5910 45 7 + + CCONJ cana-5910 45 8 𝜏 𝜏 ADJ cana-5910 45 9 𝑒π‘₯π‘₯𝑑 𝑒π‘₯π‘₯𝑑 NOUN cana-5910 45 10 , , PUNCT cana-5910 45 11 πœ‚π‘¦π‘¦ πœ‚π‘¦π‘¦ PROPN cana-5910 45 12 = = SYM cana-5910 46 1 𝐷𝑦 𝐷𝑦 PROPN cana-5910 46 2 ( ( PUNCT cana-5910 46 3 πœ‚π‘¦ πœ‚π‘¦ ADV cana-5910 46 4 βˆ’ βˆ’ PROPN cana-5910 46 5 πœ‰π‘₯ πœ‰π‘₯ PROPN cana-5910 46 6 𝑒π‘₯𝑦 𝑒π‘₯𝑦 PROPN cana-5910 46 7 βˆ’ βˆ’ PROPN cana-5910 46 8 πœ‰π‘¦ πœ‰π‘¦ ADP cana-5910 46 9 𝑒𝑦𝑦 𝑒𝑦𝑦 ADV cana-5910 46 10 βˆ’ βˆ’ PROPN cana-5910 46 11 𝜏 𝜏 DET cana-5910 46 12 𝑒𝑦𝑑 𝑒𝑦𝑑 NOUN cana-5910 46 13 ) ) PUNCT cana-5910 47 1 + + NUM cana-5910 47 2 πœ‰π‘₯ πœ‰π‘₯ X cana-5910 47 3 𝑒π‘₯𝑦𝑦 𝑒π‘₯𝑦𝑦 ADJ cana-5910 47 4 + + CCONJ cana-5910 47 5 πœ‰π‘¦ πœ‰π‘¦ ADJ cana-5910 47 6 𝑒𝑦𝑦𝑦 𝑒𝑦𝑦𝑦 NOUN cana-5910 47 7 + + CCONJ cana-5910 47 8 𝜏 𝜏 DET cana-5910 47 9 𝑒𝑦𝑦𝑑 𝑒𝑦𝑦𝑑 NOUN cana-5910 47 10 , , PUNCT cana-5910 47 11 with with ADP cana-5910 47 12 𝐷π‘₯ 𝐷π‘₯ PROPN cana-5910 47 13 , , PUNCT cana-5910 47 14 𝐷𝑦 𝐷𝑦 PROPN cana-5910 47 15 , , PUNCT cana-5910 47 16 𝐷𝑑 𝐷𝑑 PROPN cana-5910 47 17 denoting denote VERB cana-5910 47 18 total total ADJ cana-5910 47 19 derivatives derivative NOUN cana-5910 47 20 . . PUNCT cana-5910 48 1 the the DET cana-5910 48 2 invariance invariance NOUN cana-5910 48 3 condition condition NOUN cana-5910 48 4 is be AUX cana-5910 48 5 : : PUNCT cana-5910 48 6 πœ‚π‘‘ πœ‚π‘‘ ADP cana-5910 48 7 βˆ’ βˆ’ PROPN cana-5910 48 8 𝛼 𝛼 X cana-5910 48 9 ( ( PUNCT cana-5910 48 10 πœ‚π‘₯π‘₯ πœ‚π‘₯π‘₯ X cana-5910 48 11 + + CCONJ cana-5910 48 12 πœ‚π‘¦π‘¦ πœ‚π‘¦π‘¦ PROPN cana-5910 48 13 ) ) PUNCT cana-5910 48 14 = = PUNCT cana-5910 49 1 0 0 NUM cana-5910 49 2 π‘œπ‘› π‘œπ‘› NOUN cana-5910 49 3 𝑒𝑑 𝑒𝑑 NOUN cana-5910 49 4 = = SYM cana-5910 49 5 𝛼 𝛼 PROPN cana-5910 49 6 ( ( PUNCT cana-5910 49 7 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 49 8 + + CCONJ cana-5910 49 9 𝑒𝑦𝑦 𝑒𝑦𝑦 ADJ cana-5910 49 10 ) ) PUNCT cana-5910 49 11 . . PUNCT cana-5910 50 1 substituting substitute VERB cana-5910 50 2 the the DET cana-5910 50 3 prolonged prolong VERB cana-5910 50 4 infinitesimals infinitesimal NOUN cana-5910 50 5 and and CCONJ cana-5910 50 6 the the DET cana-5910 50 7 constraint constraint NOUN cana-5910 50 8 𝑒𝑑 𝑒𝑑 NOUN cana-5910 50 9 = = SYM cana-5910 50 10 𝛼 𝛼 PROPN cana-5910 50 11 ( ( PUNCT cana-5910 50 12 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 50 13 + + CCONJ cana-5910 50 14 𝑒𝑦𝑦 𝑒𝑦𝑦 PROPN cana-5910 50 15 ) ) PUNCT cana-5910 50 16 , , PUNCT cana-5910 50 17 we we PRON cana-5910 50 18 equate equate VERB cana-5910 50 19 coefficients coefficient NOUN cana-5910 50 20 of of ADP cana-5910 50 21 independent independent ADJ cana-5910 50 22 derivative derivative ADJ cana-5910 50 23 terms term NOUN cana-5910 50 24 ( ( PUNCT cana-5910 50 25 𝑒. 𝑒. PROPN cana-5910 50 26 𝑔. 𝑔. PROPN cana-5910 50 27 , , PUNCT cana-5910 50 28 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 50 29 , , PUNCT cana-5910 50 30 𝑒𝑦 𝑒𝑦 INTJ cana-5910 50 31 , , PUNCT cana-5910 50 32 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ NOUN cana-5910 50 33 , , PUNCT cana-5910 50 34 𝑒π‘₯𝑦 𝑒π‘₯𝑦 PROPN cana-5910 50 35 ) ) PUNCT cana-5910 50 36 to to PART cana-5910 50 37 obtain obtain VERB cana-5910 50 38 the the DET cana-5910 50 39 determining determine VERB cana-5910 50 40 equations equation NOUN cana-5910 50 41 . . PUNCT cana-5910 51 1 after after ADP cana-5910 51 2 simplification simplification NOUN cana-5910 51 3 , , PUNCT cana-5910 51 4 these these PRON cana-5910 51 5 include include VERB cana-5910 51 6 : : PUNCT cana-5910 52 1 1 1 X cana-5910 52 2 . . X cana-5910 52 3 πœ‰π‘’ πœ‰π‘’ INTJ cana-5910 52 4 π‘₯ π‘₯ NOUN cana-5910 53 1 = = PUNCT cana-5910 53 2 πœ‰π‘’ πœ‰π‘’ PROPN cana-5910 53 3 𝑦 𝑦 NOUN cana-5910 53 4 = = PUNCT cana-5910 53 5 πœπ‘’ πœπ‘’ NOUN cana-5910 53 6 = = NOUN cana-5910 53 7 0 0 NUM cana-5910 53 8 : : PUNCT cana-5910 53 9 π‘‡β„Žπ‘’ π‘‡β„Žπ‘’ PROPN cana-5910 53 10 π‘–π‘›π‘“π‘–π‘›π‘–π‘‘π‘’π‘ π‘–π‘šπ‘Žπ‘™π‘  π‘–π‘›π‘“π‘–π‘›π‘–π‘‘π‘’π‘ π‘–π‘šπ‘Žπ‘™π‘  ADJ cana-5910 53 11 πœ‰π‘₯ πœ‰π‘₯ PROPN cana-5910 53 12 , , PUNCT cana-5910 53 13 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 53 14 , , PUNCT cana-5910 53 15 𝜏 𝜏 PROPN cana-5910 53 16 π‘Žπ‘Ÿπ‘’ π‘Žπ‘Ÿπ‘’ NOUN cana-5910 53 17 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑑 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑑 ADV cana-5910 53 18 π‘œπ‘“ π‘œπ‘“ INTJ cana-5910 53 19 𝑒. 𝑒. NOUN cana-5910 53 20 2 2 NUM cana-5910 53 21 . . PUNCT cana-5910 53 22 πœ‚π‘’π‘’ πœ‚π‘’π‘’ NOUN cana-5910 53 23 = = NOUN cana-5910 53 24 0 0 NUM cana-5910 53 25 : : PUNCT cana-5910 53 26 πœ‚ πœ‚ X cana-5910 53 27 𝑖𝑠 𝑖𝑠 NOUN cana-5910 53 28 π‘Žπ‘‘ π‘Žπ‘‘ INTJ cana-5910 53 29 π‘šπ‘œπ‘ π‘‘ π‘šπ‘œπ‘ π‘‘ PROPN cana-5910 53 30 π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ PROPN cana-5910 53 31 𝑖𝑛 𝑖𝑛 X cana-5910 54 1 𝑒 𝑒 PROPN cana-5910 54 2 , , PUNCT cana-5910 54 3 π‘ π‘œ π‘ π‘œ ADP cana-5910 54 4 πœ‚ πœ‚ NOUN cana-5910 54 5 = = SYM cana-5910 54 6 π‘Ž(π‘₯ π‘Ž(π‘₯ PROPN cana-5910 54 7 , , PUNCT cana-5910 54 8 𝑦 𝑦 NOUN cana-5910 54 9 , , PUNCT cana-5910 54 10 𝑑)𝑒 𝑑)𝑒 PUNCT cana-5910 54 11 + + CCONJ cana-5910 55 1 𝑏(π‘₯ 𝑏(π‘₯ NOUN cana-5910 55 2 , , PUNCT cana-5910 55 3 𝑦 𝑦 NOUN cana-5910 55 4 , , PUNCT cana-5910 55 5 𝑑 𝑑 NOUN cana-5910 55 6 ) ) PUNCT cana-5910 55 7 . . PUNCT cana-5910 56 1 3 3 X cana-5910 56 2 . . X cana-5910 56 3 πœ‰π‘¦ πœ‰π‘¦ ADP cana-5910 56 4 π‘₯ π‘₯ NOUN cana-5910 56 5 = = PUNCT cana-5910 56 6 πœ‰π‘₯ πœ‰π‘₯ PROPN cana-5910 56 7 𝑦 𝑦 NOUN cana-5910 56 8 : : PUNCT cana-5910 56 9 π‘…π‘œπ‘‘π‘Žπ‘‘π‘–π‘œπ‘›π‘Žπ‘™ π‘…π‘œπ‘‘π‘Žπ‘‘π‘–π‘œπ‘›π‘Žπ‘™ NOUN cana-5910 56 10 π‘ π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘¦ π‘ π‘¦π‘šπ‘šπ‘’π‘‘π‘Ÿπ‘¦ NOUN cana-5910 56 11 𝑖𝑛 𝑖𝑛 INTJ cana-5910 56 12 π‘‘β„Žπ‘’ π‘‘β„Žπ‘’ VERB cana-5910 57 1 π‘₯ π‘₯ INTJ cana-5910 57 2 βˆ’ βˆ’ NOUN cana-5910 57 3 𝑦 𝑦 SYM cana-5910 57 4 π‘π‘™π‘Žπ‘›π‘’. π‘π‘™π‘Žπ‘›π‘’. NOUN cana-5910 57 5 4 4 NUM cana-5910 57 6 . . X cana-5910 58 1 𝜏π‘₯ 𝜏π‘₯ ADV cana-5910 58 2 = = SYM cana-5910 58 3 πœπ‘¦ πœπ‘¦ PROPN cana-5910 58 4 = = SYM cana-5910 58 5 0 0 NUM cana-5910 58 6 : : PUNCT cana-5910 58 7 𝜏 𝜏 X cana-5910 58 8 = = PUNCT cana-5910 58 9 𝜏(𝑑 𝜏(𝑑 NOUN cana-5910 58 10 ) ) PUNCT cana-5910 58 11 . . PUNCT cana-5910 59 1 5 5 X cana-5910 59 2 . . X cana-5910 59 3 πœ‰π‘‘ πœ‰π‘‘ NOUN cana-5910 59 4 π‘₯ π‘₯ PROPN cana-5910 60 1 = = PUNCT cana-5910 60 2 πœ‰π‘‘ πœ‰π‘‘ PROPN cana-5910 60 3 𝑦 𝑦 NOUN cana-5910 60 4 = = SYM cana-5910 60 5 0 0 NUM cana-5910 60 6 : : PUNCT cana-5910 60 7 πœ‰π‘₯ πœ‰π‘₯ INTJ cana-5910 60 8 = = PUNCT cana-5910 60 9 πœ‰π‘₯ πœ‰π‘₯ PROPN cana-5910 60 10 ( ( PUNCT cana-5910 60 11 π‘₯ π‘₯ PROPN cana-5910 60 12 , , PUNCT cana-5910 60 13 𝑦 𝑦 NOUN cana-5910 60 14 ) ) PUNCT cana-5910 60 15 , , PUNCT cana-5910 60 16 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 60 17 = = SYM cana-5910 60 18 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 60 19 ( ( PUNCT cana-5910 60 20 π‘₯ π‘₯ NOUN cana-5910 60 21 , , PUNCT cana-5910 60 22 𝑦 𝑦 NOUN cana-5910 60 23 ) ) PUNCT cana-5910 60 24 . . PUNCT cana-5910 61 1 6 6 NUM cana-5910 61 2 . . X cana-5910 61 3 𝛼 𝛼 PROPN cana-5910 61 4 ( ( PUNCT cana-5910 61 5 π‘Žπ‘₯π‘₯ π‘Žπ‘₯π‘₯ VERB cana-5910 61 6 + + SYM cana-5910 61 7 π‘Žπ‘¦π‘¦ π‘Žπ‘¦π‘¦ NOUN cana-5910 61 8 ) ) PUNCT cana-5910 61 9 βˆ’ βˆ’ NOUN cana-5910 62 1 π‘Žπ‘‘ π‘Žπ‘‘ ADP cana-5910 62 2 0 0 NUM cana-5910 62 3 : : PUNCT cana-5910 62 4 the the DET cana-5910 62 5 coefficient coefficient NOUN cana-5910 62 6 π‘Ž(π‘₯ π‘Ž(π‘₯ PROPN cana-5910 62 7 , , PUNCT cana-5910 62 8 𝑦 𝑦 PROPN cana-5910 62 9 , , PUNCT cana-5910 62 10 𝑑 𝑑 NOUN cana-5910 62 11 ) ) PUNCT cana-5910 62 12 satisfies satisfy VERB cana-5910 62 13 the the DET cana-5910 62 14 heat heat NOUN cana-5910 62 15 equation equation NOUN cana-5910 62 16 . . PUNCT cana-5910 63 1 7 7 X cana-5910 63 2 . . X cana-5910 63 3 𝛼 𝛼 PROPN cana-5910 63 4 ( ( PUNCT cana-5910 63 5 𝑏π‘₯π‘₯ 𝑏π‘₯π‘₯ PROPN cana-5910 63 6 + + CCONJ cana-5910 63 7 𝑏𝑦𝑦 𝑏𝑦𝑦 PROPN cana-5910 63 8 ) ) PUNCT cana-5910 63 9 βˆ’ βˆ’ NOUN cana-5910 63 10 𝑏𝑑 𝑏𝑑 NOUN cana-5910 63 11 = = NOUN cana-5910 63 12 0 0 NUM cana-5910 63 13 : : PUNCT cana-5910 63 14 the the DET cana-5910 63 15 function function NOUN cana-5910 63 16 𝑏(π‘₯ 𝑏(π‘₯ PROPN cana-5910 63 17 , , PUNCT cana-5910 63 18 𝑦 𝑦 NOUN cana-5910 63 19 , , PUNCT cana-5910 63 20 𝑑 𝑑 NOUN cana-5910 63 21 ) ) PUNCT cana-5910 63 22 satisfies satisfy VERB cana-5910 63 23 the the DET cana-5910 63 24 heat heat NOUN cana-5910 63 25 equation equation NOUN cana-5910 63 26 . . PUNCT cana-5910 64 1 8 8 X cana-5910 64 2 . . X cana-5910 64 3 2𝛼 2𝛼 NUM cana-5910 64 4 πœ‰π‘₯ πœ‰π‘₯ PROPN cana-5910 65 1 π‘₯ π‘₯ INTJ cana-5910 65 2 βˆ’ βˆ’ NOUN cana-5910 65 3 πœπ‘‘ πœπ‘‘ PROPN cana-5910 65 4 = = SYM cana-5910 65 5 0 0 PROPN cana-5910 65 6 , , PUNCT cana-5910 65 7 2𝛼 2𝛼 PROPN cana-5910 65 8 πœ‰π‘¦ πœ‰π‘¦ ADJ cana-5910 65 9 𝑦 𝑦 NOUN cana-5910 65 10 βˆ’ βˆ’ NOUN cana-5910 65 11 πœπ‘‘ πœπ‘‘ PROPN cana-5910 65 12 = = SYM cana-5910 65 13 0 0 NUM cana-5910 65 14 : : PUNCT cana-5910 65 15 scaling scale VERB cana-5910 65 16 relations relation NOUN cana-5910 65 17 . . PUNCT cana-5910 66 1 9 9 X cana-5910 66 2 . . X cana-5910 67 1 𝛼 𝛼 X cana-5910 67 2 ( ( PUNCT cana-5910 67 3 πœ‰π‘₯π‘₯ πœ‰π‘₯π‘₯ NOUN cana-5910 67 4 π‘₯ π‘₯ NOUN cana-5910 67 5 + + CCONJ cana-5910 67 6 πœ‰π‘¦π‘¦ πœ‰π‘¦π‘¦ NOUN cana-5910 67 7 π‘₯ π‘₯ NOUN cana-5910 67 8 ) ) PUNCT cana-5910 67 9 βˆ’ βˆ’ PROPN cana-5910 67 10 πœ‰π‘‘ πœ‰π‘‘ NOUN cana-5910 67 11 π‘₯ π‘₯ PROPN cana-5910 67 12 + + NUM cana-5910 67 13 2𝛼 2𝛼 PROPN cana-5910 67 14 π‘Žπ‘₯ π‘Žπ‘₯ X cana-5910 67 15 = = SYM cana-5910 67 16 0 0 NUM cana-5910 67 17 , , PUNCT cana-5910 67 18 𝛼 𝛼 PROPN cana-5910 67 19 ( ( PUNCT cana-5910 67 20 πœ‰π‘₯π‘₯ πœ‰π‘₯π‘₯ NOUN cana-5910 67 21 𝑦 𝑦 NOUN cana-5910 67 22 + + CCONJ cana-5910 67 23 πœ‰π‘¦π‘¦ πœ‰π‘¦π‘¦ PROPN cana-5910 67 24 𝑦 𝑦 NOUN cana-5910 67 25 ) ) PUNCT cana-5910 67 26 βˆ’ βˆ’ PROPN cana-5910 67 27 πœ‰π‘‘ πœ‰π‘‘ PROPN cana-5910 67 28 𝑦 𝑦 PROPN cana-5910 67 29 + + NUM cana-5910 67 30 2𝛼 2𝛼 PROPN cana-5910 67 31 π‘Žπ‘¦ π‘Žπ‘¦ NOUN cana-5910 67 32 = = SYM cana-5910 67 33 0 0 X cana-5910 67 34 . . PUNCT cana-5910 68 1 solving solve VERB cana-5910 68 2 these these PRON cana-5910 68 3 , , PUNCT cana-5910 68 4 we we PRON cana-5910 68 5 assume assume VERB cana-5910 68 6 π‘Ž π‘Ž ADJ cana-5910 68 7 = = SYM cana-5910 68 8 π‘Ž(𝑑 π‘Ž(𝑑 NOUN cana-5910 68 9 ) ) PUNCT cana-5910 68 10 , , PUNCT cana-5910 68 11 so so ADV cana-5910 68 12 π‘Žπ‘₯π‘₯ π‘Žπ‘₯π‘₯ VERB cana-5910 68 13 = = SYM cana-5910 68 14 π‘Žπ‘¦π‘¦ π‘Žπ‘¦π‘¦ NOUN cana-5910 68 15 = = PUNCT cana-5910 68 16 0 0 NUM cana-5910 68 17 , , PUNCT cana-5910 68 18 and and CCONJ cana-5910 68 19 from from ADP cana-5910 68 20 ( ( PUNCT cana-5910 68 21 6 6 NUM cana-5910 68 22 ) ) PUNCT cana-5910 68 23 , , PUNCT cana-5910 68 24 π‘Žπ‘‘ π‘Žπ‘‘ PROPN cana-5910 68 25 = = SYM cana-5910 68 26 0 0 NUM cana-5910 68 27 , , PUNCT cana-5910 68 28 implying imply VERB cana-5910 68 29 a a DET cana-5910 68 30 = = NOUN cana-5910 68 31 𝑐1 𝑐1 NOUN cana-5910 68 32 . . PUNCT cana-5910 69 1 for for ADP cana-5910 69 2 πœ‰π‘₯ πœ‰π‘₯ PROPN cana-5910 69 3 , , PUNCT cana-5910 69 4 πœ‰π‘¦ πœ‰π‘¦ NUM cana-5910 69 5 , , PUNCT cana-5910 69 6 𝜏 𝜏 NOUN cana-5910 69 7 , , PUNCT cana-5910 69 8 assume assume VERB cana-5910 69 9 linear linear ADJ cana-5910 69 10 forms form NOUN cana-5910 69 11 : : PUNCT cana-5910 69 12 πœ‰π‘₯ πœ‰π‘₯ X cana-5910 69 13 = = SYM cana-5910 69 14 π‘˜1 π‘˜1 PROPN cana-5910 69 15 π‘₯ π‘₯ PROPN cana-5910 70 1 + + CCONJ cana-5910 70 2 π‘˜2 π‘˜2 PROPN cana-5910 70 3 𝑦 𝑦 PROPN cana-5910 70 4 + + X cana-5910 70 5 π‘˜3 π‘˜3 PROPN cana-5910 70 6 , , PUNCT cana-5910 70 7 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 70 8 = = SYM cana-5910 70 9 π‘˜4 π‘˜4 PROPN cana-5910 70 10 π‘₯ π‘₯ PROPN cana-5910 71 1 + + CCONJ cana-5910 71 2 π‘˜5 π‘˜5 PROPN cana-5910 71 3 𝑦 𝑦 PROPN cana-5910 71 4 + + CCONJ cana-5910 71 5 π‘˜6 π‘˜6 PROPN cana-5910 71 6 , , PUNCT cana-5910 71 7 𝜏 𝜏 NOUN cana-5910 71 8 = = SYM cana-5910 71 9 π‘˜7 π‘˜7 NUM cana-5910 71 10 𝑑 𝑑 PROPN cana-5910 71 11 + + X cana-5910 71 12 π‘˜8 π‘˜8 NOUN cana-5910 71 13 . . PUNCT cana-5910 72 1 from from ADP cana-5910 72 2 ( ( PUNCT cana-5910 72 3 3 3 NUM cana-5910 72 4 ) ) PUNCT cana-5910 72 5 , , PUNCT cana-5910 72 6 π‘˜2 π‘˜2 PROPN cana-5910 72 7 = = PUNCT cana-5910 72 8 βˆ’π‘˜4 βˆ’π‘˜4 PROPN cana-5910 72 9 , , PUNCT cana-5910 72 10 indicating indicate VERB cana-5910 72 11 rotational rotational ADJ cana-5910 72 12 symmetry symmetry NOUN cana-5910 72 13 . . PUNCT cana-5910 73 1 from from ADP cana-5910 73 2 ( ( PUNCT cana-5910 73 3 8) 8) NUM cana-5910 73 4 , , PUNCT cana-5910 73 5 πœ‰π‘₯ πœ‰π‘₯ ADP cana-5910 73 6 π‘₯ π‘₯ NOUN cana-5910 73 7 = = SYM cana-5910 73 8 πœ‰π‘¦ πœ‰π‘¦ PROPN cana-5910 73 9 𝑦 𝑦 NOUN cana-5910 73 10 = = PUNCT cana-5910 73 11 πœπ‘‘ πœπ‘‘ PROPN cana-5910 73 12 / / SYM cana-5910 73 13 ( ( PUNCT cana-5910 73 14 2𝛼 2𝛼 NUM cana-5910 73 15 ) ) PUNCT cana-5910 73 16 = = SYM cana-5910 74 1 π‘˜7 π‘˜7 NOUN cana-5910 74 2 / / SYM cana-5910 74 3 ( ( PUNCT cana-5910 74 4 2𝛼 2𝛼 PROPN cana-5910 74 5 ) ) PUNCT cana-5910 74 6 , , PUNCT cana-5910 74 7 π‘ π‘œ π‘ π‘œ ADP cana-5910 74 8 π‘˜1 π‘˜1 ADJ cana-5910 74 9 = = NOUN cana-5910 74 10 π‘˜5 π‘˜5 PROPN cana-5910 74 11 = = PUNCT cana-5910 75 1 π‘˜7 π‘˜7 NOUN cana-5910 75 2 / / PUNCT cana-5910 75 3 ( ( PUNCT cana-5910 75 4 2𝛼 2𝛼 PROPN cana-5910 75 5 ) ) PUNCT cana-5910 75 6 . . PUNCT cana-5910 76 1 the the DET cana-5910 76 2 function function NOUN cana-5910 76 3 𝑏(π‘₯ 𝑏(π‘₯ PROPN cana-5910 76 4 , , PUNCT cana-5910 76 5 𝑦 𝑦 NOUN cana-5910 76 6 , , PUNCT cana-5910 76 7 𝑑 𝑑 NOUN cana-5910 76 8 ) ) PUNCT cana-5910 76 9 , , PUNCT cana-5910 76 10 satisfying satisfy VERB cana-5910 76 11 the the DET cana-5910 76 12 heat heat NOUN cana-5910 76 13 equation equation NOUN cana-5910 76 14 , , PUNCT cana-5910 76 15 contributes contribute VERB cana-5910 76 16 to to ADP cana-5910 76 17 an an DET cana-5910 76 18 infinite infinite ADJ cana-5910 76 19 - - PUNCT cana-5910 76 20 dimensional dimensional ADJ cana-5910 76 21 symmetry symmetry NOUN cana-5910 76 22 . . PUNCT cana-5910 77 1 the the DET cana-5910 77 2 finite finite ADJ cana-5910 77 3 - - ADJ cana-5910 77 4 dimensional dimensional ADJ cana-5910 77 5 lie lie NOUN cana-5910 77 6 algebra algebra NOUN cana-5910 77 7 is be AUX cana-5910 77 8 spanned span VERB cana-5910 77 9 by by ADP cana-5910 77 10 : : PUNCT cana-5910 77 11 𝑉1 𝑉1 PROPN cana-5910 77 12 = = SYM cana-5910 77 13 πœ• πœ• PROPN cana-5910 77 14 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 77 15 , , PUNCT cana-5910 77 16 𝑉2 𝑉2 NOUN cana-5910 77 17 = = SYM cana-5910 77 18 πœ• πœ• PROPN cana-5910 77 19 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 77 20 , , PUNCT cana-5910 77 21 𝑉3 𝑉3 NOUN cana-5910 77 22 = = SYM cana-5910 77 23 πœ• πœ• PROPN cana-5910 77 24 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 77 25 , , PUNCT cana-5910 77 26 𝑉4 𝑉4 NOUN cana-5910 77 27 = = SYM cana-5910 77 28 𝑒 𝑒 NOUN cana-5910 77 29 πœ• πœ• PROPN cana-5910 77 30 πœ•π‘’ πœ•π‘’ NOUN cana-5910 77 31 , , PUNCT cana-5910 77 32 𝑉5 𝑉5 PROPN cana-5910 77 33 = = SYM cana-5910 77 34 π‘₯ π‘₯ PROPN cana-5910 77 35 πœ• πœ• NOUN cana-5910 77 36 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 77 37 + + CCONJ cana-5910 77 38 𝑦 𝑦 NOUN cana-5910 77 39 πœ• πœ• NOUN cana-5910 77 40 πœ•π‘¦ πœ•π‘¦ NOUN cana-5910 78 1 + + NUM cana-5910 78 2 2𝑑 2𝑑 NUM cana-5910 78 3 πœ• πœ• PROPN cana-5910 78 4 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 78 5 𝑉6 𝑉6 NOUN cana-5910 78 6 = = PRON cana-5910 78 7 ( ( PUNCT cana-5910 78 8 π‘₯ π‘₯ X cana-5910 78 9 / / SYM cana-5910 78 10 ( ( PUNCT cana-5910 78 11 2𝛼 2𝛼 PROPN cana-5910 78 12 ) ) PUNCT cana-5910 78 13 ) ) PUNCT cana-5910 78 14 πœ• πœ• PROPN cana-5910 78 15 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 78 16 + + CCONJ cana-5910 78 17 ( ( PUNCT cana-5910 78 18 𝑦 𝑦 NOUN cana-5910 78 19 / / SYM cana-5910 78 20 ( ( PUNCT cana-5910 78 21 2𝛼 2𝛼 PROPN cana-5910 78 22 ) ) PUNCT cana-5910 78 23 ) ) PUNCT cana-5910 78 24 πœ• πœ• NOUN cana-5910 78 25 πœ•π‘¦ πœ•π‘¦ NOUN cana-5910 78 26 + + CCONJ cana-5910 78 27 𝑑 𝑑 PROPN cana-5910 78 28 πœ• πœ• PROPN cana-5910 78 29 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 78 30 βˆ’ βˆ’ PROPN cana-5910 78 31 ( ( PUNCT cana-5910 78 32 ( ( PUNCT cana-5910 78 33 π‘₯2 π‘₯2 NOUN cana-5910 78 34 + + CCONJ cana-5910 78 35 𝑦2 𝑦2 NOUN cana-5910 78 36 ) ) PUNCT cana-5910 78 37 / / PUNCT cana-5910 78 38 ( ( PUNCT cana-5910 78 39 4𝛼 4𝛼 NOUN cana-5910 78 40 ) ) PUNCT cana-5910 78 41 ) ) PUNCT cana-5910 79 1 𝑒 𝑒 PROPN cana-5910 79 2 πœ• πœ• NOUN cana-5910 79 3 πœ•π‘’ πœ•π‘’ NOUN cana-5910 79 4 𝑉7 𝑉7 NOUN cana-5910 79 5 = = SYM cana-5910 79 6 𝑦 𝑦 NOUN cana-5910 79 7 πœ• πœ• NOUN cana-5910 79 8 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 79 9 βˆ’ βˆ’ ADP cana-5910 79 10 π‘₯ π‘₯ DET cana-5910 79 11 πœ• πœ• PROPN cana-5910 79 12 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 79 13 additionally additionally ADV cana-5910 79 14 , , PUNCT cana-5910 79 15 the the DET cana-5910 79 16 infinite infinite ADJ cana-5910 79 17 - - PUNCT cana-5910 79 18 dimensional dimensional ADJ cana-5910 79 19 symmetry symmetry NOUN cana-5910 79 20 is be AUX cana-5910 79 21 : : PUNCT cana-5910 79 22 1113 1113 NUM cana-5910 79 23 https://internationalpubls.com https://internationalpubls.com X cana-5910 80 1 𝑉𝑀 𝑉𝑀 PROPN cana-5910 80 2 = = SYM cana-5910 80 3 𝑀(π‘₯ 𝑀(π‘₯ PROPN cana-5910 80 4 , , PUNCT cana-5910 80 5 𝑦 𝑦 NOUN cana-5910 80 6 , , PUNCT cana-5910 80 7 𝑑 𝑑 NOUN cana-5910 80 8 ) ) PUNCT cana-5910 80 9 πœ• πœ• NOUN cana-5910 80 10 πœ•π‘’ πœ•π‘’ NOUN cana-5910 80 11 , , PUNCT cana-5910 80 12 π‘€β„Žπ‘’π‘Ÿπ‘’ π‘€β„Žπ‘’π‘Ÿπ‘’ NOUN cana-5910 80 13 𝑀𝑑 𝑀𝑑 VERB cana-5910 80 14 = = SYM cana-5910 80 15 𝛼 𝛼 PROPN cana-5910 80 16 ( ( PUNCT cana-5910 80 17 𝑀π‘₯π‘₯ 𝑀π‘₯π‘₯ NOUN cana-5910 80 18 + + SYM cana-5910 80 19 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 80 20 ) ) PUNCT cana-5910 80 21 . . PUNCT cana-5910 81 1 these these DET cana-5910 81 2 symmetries symmetry NOUN cana-5910 81 3 correspond correspond VERB cana-5910 81 4 to to ADP cana-5910 81 5 translations translation NOUN cana-5910 81 6 ( ( PUNCT cana-5910 81 7 𝑉1 𝑉1 PROPN cana-5910 81 8 , , PUNCT cana-5910 81 9 𝑉2 𝑉2 NOUN cana-5910 81 10 , , PUNCT cana-5910 81 11 𝑉3 𝑉3 NOUN cana-5910 81 12 ) ) PUNCT cana-5910 81 13 , , PUNCT cana-5910 81 14 scaling scaling NOUN cana-5910 81 15 ( ( PUNCT cana-5910 81 16 𝑉4 𝑉4 NOUN cana-5910 81 17 , , PUNCT cana-5910 81 18 𝑉5 𝑉5 PROPN cana-5910 81 19 ) ) PUNCT cana-5910 81 20 , , PUNCT cana-5910 81 21 a a DET cana-5910 81 22 special special ADJ cana-5910 81 23 conformal conformal NOUN cana-5910 81 24 - - PUNCT cana-5910 81 25 like like ADJ cana-5910 81 26 transformation transformation NOUN cana-5910 81 27 ( ( PUNCT cana-5910 81 28 𝑉6 𝑉6 NOUN cana-5910 81 29 ) ) PUNCT cana-5910 81 30 , , PUNCT cana-5910 81 31 rotation rotation NOUN cana-5910 81 32 ( ( PUNCT cana-5910 81 33 𝑉7 𝑉7 NOUN cana-5910 81 34 ) ) PUNCT cana-5910 81 35 , , PUNCT cana-5910 81 36 and and CCONJ cana-5910 81 37 linear linear PROPN cana-5910 81 38 superposition superposition NOUN cana-5910 81 39 ( ( PUNCT cana-5910 81 40 𝑉𝑀 𝑉𝑀 PROPN cana-5910 81 41 ) ) PUNCT cana-5910 81 42 2.3 2.3 NUM cana-5910 81 43 symmetries symmetry NOUN cana-5910 81 44 of of ADP cana-5910 81 45 the the DET cana-5910 81 46 two two NUM cana-5910 81 47 - - PUNCT cana-5910 81 48 dimensional dimensional ADJ cana-5910 81 49 wave wave NOUN cana-5910 81 50 equation equation NOUN cana-5910 81 51 the the DET cana-5910 81 52 two two NUM cana-5910 81 53 - - PUNCT cana-5910 81 54 dimensional dimensional ADJ cana-5910 81 55 wave wave NOUN cana-5910 81 56 equation equation NOUN cana-5910 81 57 is be AUX cana-5910 81 58 : : PUNCT cana-5910 81 59 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 81 60 πœ•π‘‘2 πœ•π‘‘2 NOUN cana-5910 81 61 = = SYM cana-5910 81 62 𝑐2 𝑐2 NOUN cana-5910 81 63 ( ( PUNCT cana-5910 81 64 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 81 65 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 81 66 + + CCONJ cana-5910 81 67 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 81 68 πœ•π‘¦2 πœ•π‘¦2 NOUN cana-5910 81 69 ) ) PUNCT cana-5910 81 70 or or CCONJ cana-5910 81 71 : : PUNCT cana-5910 81 72 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 81 73 = = NOUN cana-5910 81 74 𝑐2 𝑐2 NOUN cana-5910 81 75 ( ( PUNCT cana-5910 81 76 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 81 77 + + CCONJ cana-5910 81 78 𝑒𝑦𝑦 𝑒𝑦𝑦 PROPN cana-5910 81 79 ) ) PUNCT cana-5910 81 80 , , PUNCT cana-5910 81 81 where where SCONJ cana-5910 81 82 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 81 83 , , PUNCT cana-5910 81 84 𝑦 𝑦 NOUN cana-5910 81 85 , , PUNCT cana-5910 81 86 𝑑 𝑑 X cana-5910 81 87 ) ) PUNCT cana-5910 81 88 is be AUX cana-5910 81 89 the the DET cana-5910 81 90 displacement displacement NOUN cana-5910 81 91 , , PUNCT cana-5910 81 92 and and CCONJ cana-5910 81 93 c c NOUN cana-5910 81 94 is be AUX cana-5910 81 95 the the DET cana-5910 81 96 wave wave NOUN cana-5910 81 97 speed speed NOUN cana-5910 81 98 . . PUNCT cana-5910 82 1 the the DET cana-5910 82 2 second second ADJ cana-5910 82 3 prolongation prolongation NOUN cana-5910 82 4 is be AUX cana-5910 82 5 required require VERB cana-5910 82 6 , , PUNCT cana-5910 82 7 and and CCONJ cana-5910 82 8 the the DET cana-5910 82 9 invariance invariance NOUN cana-5910 82 10 condition condition NOUN cana-5910 82 11 is be AUX cana-5910 82 12 : : PUNCT cana-5910 82 13 πœ‚π‘‘π‘‘ πœ‚π‘‘π‘‘ ADJ cana-5910 82 14 βˆ’ βˆ’ NOUN cana-5910 82 15 𝑐2 𝑐2 NOUN cana-5910 82 16 ( ( PUNCT cana-5910 82 17 πœ‚π‘₯π‘₯ πœ‚π‘₯π‘₯ X cana-5910 82 18 + + CCONJ cana-5910 82 19 πœ‚π‘¦π‘¦ πœ‚π‘¦π‘¦ PROPN cana-5910 82 20 ) ) PUNCT cana-5910 82 21 = = PUNCT cana-5910 82 22 0 0 NUM cana-5910 82 23 π‘œπ‘› π‘œπ‘› PROPN cana-5910 82 24 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 82 25 = = NOUN cana-5910 82 26 𝑐2 𝑐2 NOUN cana-5910 82 27 ( ( PUNCT cana-5910 82 28 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 82 29 + + CCONJ cana-5910 82 30 𝑒𝑦𝑦 𝑒𝑦𝑦 ADJ cana-5910 82 31 ) ) PUNCT cana-5910 82 32 . . PUNCT cana-5910 83 1 the the DET cana-5910 83 2 determining determine VERB cana-5910 83 3 equations equation NOUN cana-5910 83 4 include include VERB cana-5910 83 5 : : PUNCT cana-5910 84 1 1 1 X cana-5910 84 2 . . X cana-5910 84 3 πœ‰π‘’ πœ‰π‘’ INTJ cana-5910 84 4 π‘₯ π‘₯ NOUN cana-5910 85 1 = = PUNCT cana-5910 85 2 πœ‰π‘’ πœ‰π‘’ PROPN cana-5910 85 3 𝑦 𝑦 NOUN cana-5910 85 4 = = PUNCT cana-5910 85 5 πœπ‘’ πœπ‘’ NOUN cana-5910 85 6 = = NOUN cana-5910 85 7 0 0 NUM cana-5910 85 8 . . NOUN cana-5910 85 9 2 2 X cana-5910 85 10 . . X cana-5910 85 11 πœ‚π‘’π‘’ πœ‚π‘’π‘’ NOUN cana-5910 85 12 = = SYM cana-5910 85 13 0 0 NUM cana-5910 85 14 , , PUNCT cana-5910 85 15 so so ADV cana-5910 85 16 Ξ· Ξ· PROPN cana-5910 85 17 = = PROPN cana-5910 85 18 π‘Ž(π‘₯ π‘Ž(π‘₯ PROPN cana-5910 85 19 , , PUNCT cana-5910 85 20 𝑦 𝑦 NOUN cana-5910 85 21 , , PUNCT cana-5910 85 22 𝑑)𝑒 𝑑)𝑒 PUNCT cana-5910 85 23 + + CCONJ cana-5910 86 1 𝑏(π‘₯ 𝑏(π‘₯ NOUN cana-5910 86 2 , , PUNCT cana-5910 86 3 𝑦 𝑦 NOUN cana-5910 86 4 , , PUNCT cana-5910 86 5 𝑑 𝑑 NOUN cana-5910 86 6 ) ) PUNCT cana-5910 86 7 3 3 NUM cana-5910 86 8 . . X cana-5910 87 1 πœ‰π‘¦ πœ‰π‘¦ ADP cana-5910 87 2 π‘₯ π‘₯ X cana-5910 87 3 = = PUNCT cana-5910 87 4 πœ‰π‘₯ πœ‰π‘₯ PROPN cana-5910 87 5 𝑦 𝑦 NOUN cana-5910 87 6 . . PUNCT cana-5910 88 1 4 4 X cana-5910 88 2 . . X cana-5910 88 3 π‘Žπ‘‘π‘‘ π‘Žπ‘‘π‘‘ NOUN cana-5910 88 4 = = SYM cana-5910 88 5 𝑐2 𝑐2 NOUN cana-5910 88 6 ( ( PUNCT cana-5910 88 7 π‘Žπ‘₯π‘₯ π‘Žπ‘₯π‘₯ VERB cana-5910 88 8 + + CCONJ cana-5910 88 9 π‘Žπ‘¦π‘¦ π‘Žπ‘¦π‘¦ NOUN cana-5910 88 10 ) ) PUNCT cana-5910 88 11 , , PUNCT cana-5910 88 12 𝑏𝑑𝑑 𝑏𝑑𝑑 NOUN cana-5910 88 13 = = SYM cana-5910 88 14 𝑐2 𝑐2 PROPN cana-5910 88 15 ( ( PUNCT cana-5910 88 16 𝑏π‘₯π‘₯ 𝑏π‘₯π‘₯ PROPN cana-5910 88 17 + + CCONJ cana-5910 88 18 𝑏𝑦𝑦 𝑏𝑦𝑦 PROPN cana-5910 88 19 ) ) PUNCT cana-5910 88 20 . . PUNCT cana-5910 89 1 5 5 X cana-5910 89 2 . . X cana-5910 89 3 πœ‰π‘‘ πœ‰π‘‘ NOUN cana-5910 89 4 π‘₯ π‘₯ PROPN cana-5910 90 1 = = PUNCT cana-5910 90 2 πœ‰π‘‘ πœ‰π‘‘ PROPN cana-5910 90 3 𝑦 𝑦 PROPN cana-5910 90 4 = = X cana-5910 90 5 𝜏π‘₯ 𝜏π‘₯ ADP cana-5910 90 6 = = PUNCT cana-5910 90 7 πœπ‘¦ πœπ‘¦ PROPN cana-5910 90 8 = = SYM cana-5910 90 9 0 0 NUM cana-5910 90 10 . . NOUN cana-5910 90 11 6 6 NUM cana-5910 91 1 . . X cana-5910 91 2 πœ‰π‘₯π‘₯ πœ‰π‘₯π‘₯ NOUN cana-5910 91 3 π‘₯ π‘₯ PROPN cana-5910 92 1 + + CCONJ cana-5910 92 2 πœ‰π‘¦π‘¦ πœ‰π‘¦π‘¦ NOUN cana-5910 92 3 π‘₯ π‘₯ NOUN cana-5910 92 4 = = PUNCT cana-5910 92 5 πœ‰π‘₯π‘₯ πœ‰π‘₯π‘₯ PROPN cana-5910 92 6 𝑦 𝑦 NOUN cana-5910 92 7 + + CCONJ cana-5910 92 8 πœ‰π‘¦π‘¦ πœ‰π‘¦π‘¦ PROPN cana-5910 92 9 𝑦 𝑦 NOUN cana-5910 92 10 = = X cana-5910 92 11 πœπ‘‘π‘‘ πœπ‘‘π‘‘ ADJ cana-5910 92 12 = = SYM cana-5910 92 13 0 0 X cana-5910 92 14 . . PUNCT cana-5910 92 15 assume assume VERB cana-5910 92 16 πœ‰π‘₯ πœ‰π‘₯ X cana-5910 93 1 = = SYM cana-5910 93 2 π‘˜1 π‘˜1 PROPN cana-5910 93 3 π‘₯ π‘₯ PROPN cana-5910 93 4 + + CCONJ cana-5910 93 5 π‘˜2 π‘˜2 PROPN cana-5910 93 6 𝑦 𝑦 PROPN cana-5910 93 7 + + X cana-5910 93 8 π‘˜3 π‘˜3 PROPN cana-5910 93 9 , , PUNCT cana-5910 93 10 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 93 11 = = SYM cana-5910 93 12 π‘˜4 π‘˜4 PROPN cana-5910 93 13 π‘₯ π‘₯ PROPN cana-5910 94 1 + + CCONJ cana-5910 94 2 π‘˜5 π‘˜5 PROPN cana-5910 94 3 𝑦 𝑦 PROPN cana-5910 94 4 + + CCONJ cana-5910 94 5 π‘˜6 π‘˜6 PROPN cana-5910 94 6 , , PUNCT cana-5910 94 7 𝜏 𝜏 NOUN cana-5910 94 8 = = SYM cana-5910 94 9 π‘˜7 π‘˜7 PROPN cana-5910 94 10 𝑑 𝑑 PROPN cana-5910 94 11 + + X cana-5910 94 12 π‘˜8 π‘˜8 NOUN cana-5910 94 13 , , PUNCT cana-5910 94 14 πœ‚ πœ‚ NOUN cana-5910 94 15 = = SYM cana-5910 94 16 π‘˜9 π‘˜9 PROPN cana-5910 94 17 𝑒 𝑒 NOUN cana-5910 94 18 + + X cana-5910 94 19 𝑏(π‘₯ 𝑏(π‘₯ PROPN cana-5910 94 20 , , PUNCT cana-5910 94 21 𝑦 𝑦 NOUN cana-5910 94 22 , , PUNCT cana-5910 94 23 𝑑 𝑑 NOUN cana-5910 94 24 ) ) PUNCT cana-5910 94 25 . . PUNCT cana-5910 95 1 from from ADP cana-5910 95 2 ( ( PUNCT cana-5910 95 3 3 3 NUM cana-5910 95 4 ) ) PUNCT cana-5910 95 5 , , PUNCT cana-5910 95 6 π‘˜2 π‘˜2 PROPN cana-5910 95 7 = = PUNCT cana-5910 95 8 βˆ’π‘˜4 βˆ’π‘˜4 PROPN cana-5910 95 9 . . PUNCT cana-5910 96 1 from from ADP cana-5910 96 2 ( ( PUNCT cana-5910 96 3 5 5 NUM cana-5910 96 4 ) ) PUNCT cana-5910 96 5 , , PUNCT cana-5910 96 6 πœ‰π‘₯ πœ‰π‘₯ PROPN cana-5910 96 7 , , PUNCT cana-5910 96 8 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 96 9 are be AUX cana-5910 96 10 independent independent ADJ cana-5910 96 11 of of ADP cana-5910 96 12 t t PROPN cana-5910 96 13 , , PUNCT cana-5910 96 14 and and CCONJ cana-5910 96 15 Ο„ Ο„ PROPN cana-5910 96 16 is be AUX cana-5910 96 17 independent independent ADJ cana-5910 96 18 of of ADP cana-5910 96 19 π‘₯ π‘₯ PROPN cana-5910 96 20 , , PUNCT cana-5910 96 21 𝑦. 𝑦. PROPN cana-5910 96 22 from from ADP cana-5910 96 23 ( ( PUNCT cana-5910 96 24 6 6 NUM cana-5910 96 25 ) ) PUNCT cana-5910 96 26 , , PUNCT cana-5910 96 27 π‘˜1 π‘˜1 PROPN cana-5910 97 1 + + CCONJ cana-5910 97 2 π‘˜5 π‘˜5 NOUN cana-5910 97 3 = = PUNCT cana-5910 97 4 0 0 X cana-5910 97 5 . . PUNCT cana-5910 98 1 the the DET cana-5910 98 2 function function NOUN cana-5910 98 3 𝑏(π‘₯ 𝑏(π‘₯ PROPN cana-5910 98 4 , , PUNCT cana-5910 98 5 𝑦 𝑦 NOUN cana-5910 98 6 , , PUNCT cana-5910 98 7 𝑑 𝑑 NOUN cana-5910 98 8 ) ) PUNCT cana-5910 98 9 satisfies satisfy VERB cana-5910 98 10 the the DET cana-5910 98 11 wave wave NOUN cana-5910 98 12 equation equation NOUN cana-5910 98 13 , , PUNCT cana-5910 98 14 contributing contribute VERB cana-5910 98 15 to to ADP cana-5910 98 16 an an DET cana-5910 98 17 infinite infinite ADJ cana-5910 98 18 - - PUNCT cana-5910 98 19 dimensional dimensional ADJ cana-5910 98 20 symmetry symmetry NOUN cana-5910 98 21 . . PUNCT cana-5910 99 1 the the DET cana-5910 99 2 finite finite ADJ cana-5910 99 3 - - ADJ cana-5910 99 4 dimensional dimensional ADJ cana-5910 99 5 lie lie NOUN cana-5910 99 6 algebra algebra NOUN cana-5910 99 7 includes include VERB cana-5910 99 8 : : PUNCT cana-5910 99 9 1 1 NUM cana-5910 99 10 . . X cana-5910 99 11 π‘Š1 π‘Š1 PROPN cana-5910 99 12 = = SYM cana-5910 100 1 πœ• πœ• PROPN cana-5910 100 2 πœ•π‘₯ πœ•π‘₯ X cana-5910 100 3 2 2 NUM cana-5910 100 4 . . X cana-5910 100 5 π‘Š2 π‘Š2 PROPN cana-5910 100 6 = = SYM cana-5910 101 1 πœ• πœ• PROPN cana-5910 101 2 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 101 3 3 3 NUM cana-5910 101 4 . . X cana-5910 101 5 π‘Š3 π‘Š3 NOUN cana-5910 101 6 = = SYM cana-5910 101 7 πœ• πœ• PROPN cana-5910 101 8 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 101 9 4 4 NUM cana-5910 101 10 . . PUNCT cana-5910 102 1 π‘Š4 π‘Š4 PROPN cana-5910 102 2 = = PUNCT cana-5910 102 3 𝑒 𝑒 PRON cana-5910 102 4 πœ• πœ• NOUN cana-5910 102 5 πœ•π‘’ πœ•π‘’ NOUN cana-5910 102 6 5 5 NUM cana-5910 102 7 . . PUNCT cana-5910 102 8 π‘Š5 π‘Š5 PROPN cana-5910 103 1 = = PUNCT cana-5910 103 2 π‘₯ π‘₯ PROPN cana-5910 103 3 πœ• πœ• NOUN cana-5910 103 4 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 103 5 + + CCONJ cana-5910 103 6 𝑦 𝑦 NOUN cana-5910 103 7 πœ• πœ• NOUN cana-5910 103 8 πœ•π‘¦ πœ•π‘¦ NOUN cana-5910 103 9 + + CCONJ cana-5910 103 10 𝑑 𝑑 PROPN cana-5910 103 11 πœ• πœ• PROPN cana-5910 103 12 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 103 13 6 6 NUM cana-5910 103 14 . . PUNCT cana-5910 103 15 π‘Š6 π‘Š6 PROPN cana-5910 103 16 = = SYM cana-5910 103 17 𝑦 𝑦 PROPN cana-5910 103 18 πœ• πœ• NOUN cana-5910 103 19 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 103 20 βˆ’ βˆ’ ADP cana-5910 103 21 π‘₯ π‘₯ DET cana-5910 103 22 πœ• πœ• NOUN cana-5910 103 23 πœ•π‘¦ πœ•π‘¦ NOUN cana-5910 103 24 7 7 NUM cana-5910 103 25 . . PUNCT cana-5910 103 26 π‘Š7 π‘Š7 PROPN cana-5910 104 1 = = PUNCT cana-5910 104 2 𝑑 𝑑 PROPN cana-5910 104 3 πœ• πœ• NOUN cana-5910 104 4 πœ•π‘₯ πœ•π‘₯ X cana-5910 104 5 + + CCONJ cana-5910 104 6 ( ( PUNCT cana-5910 104 7 π‘₯𝑐2 π‘₯𝑐2 PROPN cana-5910 104 8 ) ) PUNCT cana-5910 104 9 πœ• πœ• PROPN cana-5910 104 10 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 104 11 , , PUNCT cana-5910 104 12 8 8 NUM cana-5910 104 13 . . PUNCT cana-5910 104 14 π‘Š8 π‘Š8 PROPN cana-5910 104 15 = = SYM cana-5910 104 16 𝑑 𝑑 PROPN cana-5910 104 17 πœ• πœ• PROPN cana-5910 104 18 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 104 19 + + CCONJ cana-5910 104 20 ( ( PUNCT cana-5910 104 21 𝑦𝑐2 𝑦𝑐2 NOUN cana-5910 104 22 ) ) PUNCT cana-5910 104 23 πœ• πœ• PROPN cana-5910 104 24 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 104 25 . . PUNCT cana-5910 105 1 the the DET cana-5910 105 2 infinite infinite ADJ cana-5910 105 3 - - PUNCT cana-5910 105 4 dimensional dimensional ADJ cana-5910 105 5 symmetry symmetry NOUN cana-5910 105 6 is be AUX cana-5910 105 7 : : PUNCT cana-5910 105 8 π‘Šπ‘€ π‘Šπ‘€ PROPN cana-5910 105 9 = = SYM cana-5910 105 10 𝑀(π‘₯ 𝑀(π‘₯ PROPN cana-5910 105 11 , , PUNCT cana-5910 105 12 𝑦 𝑦 NOUN cana-5910 105 13 , , PUNCT cana-5910 105 14 𝑑 𝑑 NOUN cana-5910 105 15 ) ) PUNCT cana-5910 105 16 πœ• πœ• NOUN cana-5910 105 17 πœ•π‘’ πœ•π‘’ NOUN cana-5910 105 18 , , PUNCT cana-5910 105 19 where where SCONJ cana-5910 105 20 𝑀𝑑𝑑 𝑀𝑑𝑑 NOUN cana-5910 105 21 = = NOUN cana-5910 105 22 𝑐2 𝑐2 NOUN cana-5910 105 23 ( ( PUNCT cana-5910 105 24 𝑀π‘₯π‘₯ 𝑀π‘₯π‘₯ NOUN cana-5910 105 25 + + CCONJ cana-5910 105 26 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 105 27 ) ) PUNCT cana-5910 105 28 . . PUNCT cana-5910 106 1 these these DET cana-5910 106 2 symmetries symmetry NOUN cana-5910 106 3 represent represent VERB cana-5910 106 4 translations translation NOUN cana-5910 106 5 ( ( PUNCT cana-5910 106 6 π‘Š1 π‘Š1 PROPN cana-5910 106 7 , , PUNCT cana-5910 106 8 π‘Š2 π‘Š2 PROPN cana-5910 106 9 , , PUNCT cana-5910 106 10 π‘Š3 π‘Š3 NOUN cana-5910 106 11 ) ) PUNCT cana-5910 106 12 , , PUNCT cana-5910 106 13 scaling scale VERB cana-5910 106 14 ( ( PUNCT cana-5910 106 15 π‘Š4 π‘Š4 PROPN cana-5910 106 16 , , PUNCT cana-5910 106 17 π‘Š5 π‘Š5 PROPN cana-5910 106 18 ) ) PUNCT cana-5910 106 19 , , PUNCT cana-5910 106 20 rotation rotation NOUN cana-5910 106 21 ( ( PUNCT cana-5910 106 22 π‘Š6 π‘Š6 PROPN cana-5910 106 23 ) ) PUNCT cana-5910 106 24 , , PUNCT cana-5910 106 25 lorentz lorentz PROPN cana-5910 106 26 - - PUNCT cana-5910 106 27 like like ADJ cana-5910 106 28 transformations transformation NOUN cana-5910 106 29 ( ( PUNCT cana-5910 106 30 π‘Š7 π‘Š7 PROPN cana-5910 106 31 , , PUNCT cana-5910 106 32 π‘Š8 π‘Š8 NOUN cana-5910 106 33 ) ) PUNCT cana-5910 106 34 , , PUNCT cana-5910 106 35 and and CCONJ cana-5910 106 36 linear linear PROPN cana-5910 106 37 superposition superposition NOUN cana-5910 106 38 ( ( PUNCT cana-5910 106 39 π‘Šπ‘€ π‘Šπ‘€ PROPN cana-5910 106 40 ) ) PUNCT cana-5910 106 41 . . PUNCT cana-5910 107 1 1114 1114 NUM cana-5910 107 2 https://internationalpubls.com https://internationalpubls.com X cana-5910 107 3 3 3 X cana-5910 107 4 . . X cana-5910 107 5 similarity similarity NOUN cana-5910 107 6 solutions solution NOUN cana-5910 107 7 for for ADP cana-5910 107 8 the the DET cana-5910 107 9 heat heat NOUN cana-5910 107 10 equation equation NOUN cana-5910 107 11 the the DET cana-5910 107 12 lie lie NOUN cana-5910 107 13 point point NOUN cana-5910 107 14 symmetries symmetry NOUN cana-5910 107 15 derived derive VERB cana-5910 107 16 in in ADP cana-5910 107 17 section section NOUN cana-5910 107 18 2 2 NUM cana-5910 107 19 provide provide VERB cana-5910 107 20 a a DET cana-5910 107 21 foundation foundation NOUN cana-5910 107 22 for for ADP cana-5910 107 23 reducing reduce VERB cana-5910 107 24 the the DET cana-5910 107 25 two two NUM cana-5910 107 26 - - PUNCT cana-5910 107 27 dimensional dimensional ADJ cana-5910 107 28 heat heat NOUN cana-5910 107 29 equation equation NOUN cana-5910 107 30 to to ADP cana-5910 107 31 simpler simple ADJ cana-5910 107 32 forms form NOUN cana-5910 107 33 , , PUNCT cana-5910 107 34 yielding yield VERB cana-5910 107 35 similarity similarity NOUN cana-5910 107 36 solutions solution NOUN cana-5910 107 37 that that PRON cana-5910 107 38 are be AUX cana-5910 107 39 invariant invariant ADJ cana-5910 107 40 under under ADP cana-5910 107 41 specific specific ADJ cana-5910 107 42 transformations transformation NOUN cana-5910 107 43 . . PUNCT cana-5910 108 1 in in ADP cana-5910 108 2 this this DET cana-5910 108 3 section section NOUN cana-5910 108 4 , , PUNCT cana-5910 108 5 we we PRON cana-5910 108 6 use use VERB cana-5910 108 7 selected select VERB cana-5910 108 8 symmetries symmetry NOUN cana-5910 108 9 to to PART cana-5910 108 10 reduce reduce VERB cana-5910 108 11 the the DET cana-5910 108 12 heat heat NOUN cana-5910 108 13 equation equation NOUN cana-5910 108 14 to to ADP cana-5910 108 15 ordinary ordinary ADJ cana-5910 108 16 differential differential ADJ cana-5910 108 17 equations equation NOUN cana-5910 108 18 ( ( PUNCT cana-5910 108 19 odes ode NOUN cana-5910 108 20 ) ) PUNCT cana-5910 108 21 and and CCONJ cana-5910 108 22 solve solve VERB cana-5910 108 23 them they PRON cana-5910 108 24 to to PART cana-5910 108 25 obtain obtain VERB cana-5910 108 26 physically physically ADV cana-5910 108 27 meaningful meaningful ADJ cana-5910 108 28 solutions solution NOUN cana-5910 108 29 . . PUNCT cana-5910 109 1 we we PRON cana-5910 109 2 focus focus VERB cana-5910 109 3 on on ADP cana-5910 109 4 the the DET cana-5910 109 5 scaling scale VERB cana-5910 109 6 symmetry symmetry NOUN cana-5910 109 7 v5 v5 PROPN cana-5910 109 8 v_5 v_5 PROPN cana-5910 109 9 v5 v5 PROPN cana-5910 109 10 and and CCONJ cana-5910 109 11 the the DET cana-5910 109 12 conformal conformal NOUN cana-5910 109 13 - - PUNCT cana-5910 109 14 like like ADJ cana-5910 109 15 symmetry symmetry NOUN cana-5910 109 16 v6 v6 PROPN cana-5910 109 17 v_6 v_6 PROPN cana-5910 109 18 v6 v6 PROPN cana-5910 109 19 , , PUNCT cana-5910 109 20 deriving derive VERB cana-5910 109 21 solutions solution NOUN cana-5910 109 22 that that PRON cana-5910 109 23 describe describe VERB cana-5910 109 24 radial radial ADJ cana-5910 109 25 heat heat NOUN cana-5910 109 26 diffusion diffusion NOUN cana-5910 109 27 , , PUNCT cana-5910 109 28 including include VERB cana-5910 109 29 the the DET cana-5910 109 30 fundamental fundamental ADJ cana-5910 109 31 solution solution NOUN cana-5910 109 32 for for ADP cana-5910 109 33 a a DET cana-5910 109 34 point point NOUN cana-5910 109 35 source source NOUN cana-5910 109 36 . . PUNCT cana-5910 110 1 3.1 3.1 NUM cana-5910 110 2 reduction reduction NOUN cana-5910 110 3 using use VERB cana-5910 110 4 the the DET cana-5910 110 5 scaling scale VERB cana-5910 110 6 symmetry symmetry NOUN cana-5910 110 7 π‘½πŸ“ π‘½πŸ“ VERB cana-5910 110 8 the the DET cana-5910 110 9 two two NUM cana-5910 110 10 - - PUNCT cana-5910 110 11 dimensional dimensional ADJ cana-5910 110 12 heat heat NOUN cana-5910 110 13 equation equation NOUN cana-5910 110 14 is be AUX cana-5910 110 15 : : PUNCT cana-5910 111 1 πœ•π‘’ πœ•π‘’ NOUN cana-5910 111 2 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 111 3 = = NOUN cana-5910 111 4 𝛼 𝛼 PROPN cana-5910 111 5 ( ( PUNCT cana-5910 111 6 πœ•2 πœ•2 NOUN cana-5910 111 7 𝑒 𝑒 X cana-5910 111 8 πœ•π‘₯2 πœ•π‘₯2 PROPN cana-5910 111 9 + + CCONJ cana-5910 111 10 πœ•2 πœ•2 X cana-5910 111 11 𝑒 𝑒 PRON cana-5910 111 12 πœ•π‘¦2 πœ•π‘¦2 PROPN cana-5910 111 13 ) ) PUNCT cana-5910 111 14 , , PUNCT cana-5910 111 15 or or CCONJ cana-5910 111 16 : : PUNCT cana-5910 111 17 𝑒𝑑 𝑒𝑑 NOUN cana-5910 111 18 = = SYM cana-5910 111 19 𝛼 𝛼 PROPN cana-5910 111 20 ( ( PUNCT cana-5910 111 21 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 111 22 + + CCONJ cana-5910 111 23 𝑒𝑦𝑦 𝑒𝑦𝑦 PROPN cana-5910 111 24 ) ) PUNCT cana-5910 111 25 , , PUNCT cana-5910 111 26 where where SCONJ cana-5910 111 27 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 111 28 , , PUNCT cana-5910 111 29 𝑦 𝑦 NOUN cana-5910 111 30 , , PUNCT cana-5910 111 31 𝑑 𝑑 X cana-5910 111 32 ) ) PUNCT cana-5910 111 33 is be AUX cana-5910 111 34 the the DET cana-5910 111 35 temperature temperature NOUN cana-5910 111 36 , , PUNCT cana-5910 111 37 and and CCONJ cana-5910 111 38 Ξ± Ξ± PRON cana-5910 111 39 is be AUX cana-5910 111 40 the the DET cana-5910 111 41 thermal thermal ADJ cana-5910 111 42 diffusivity diffusivity NOUN cana-5910 111 43 . . PUNCT cana-5910 112 1 consider consider VERB cana-5910 112 2 the the DET cana-5910 112 3 scaling scale VERB cana-5910 112 4 symmetry symmetry NOUN cana-5910 112 5 : : PUNCT cana-5910 112 6 𝑉5 𝑉5 PROPN cana-5910 112 7 = = PUNCT cana-5910 112 8 π‘₯ π‘₯ PROPN cana-5910 112 9 πœ• πœ• NOUN cana-5910 112 10 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 112 11 + + CCONJ cana-5910 113 1 𝑦 𝑦 NOUN cana-5910 113 2 πœ• πœ• NOUN cana-5910 113 3 πœ•π‘¦ πœ•π‘¦ NOUN cana-5910 114 1 + + NUM cana-5910 114 2 2𝑑 2𝑑 NUM cana-5910 114 3 πœ• πœ• PROPN cana-5910 114 4 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 114 5 to to PART cana-5910 114 6 find find VERB cana-5910 114 7 invariant invariant ADJ cana-5910 114 8 solutions solution NOUN cana-5910 114 9 , , PUNCT cana-5910 114 10 we we PRON cana-5910 114 11 solve solve VERB cana-5910 114 12 the the DET cana-5910 114 13 characteristic characteristic ADJ cana-5910 114 14 equations equation NOUN cana-5910 114 15 : : PUNCT cana-5910 114 16 𝑑π‘₯ 𝑑π‘₯ VERB cana-5910 114 17 π‘₯ π‘₯ X cana-5910 114 18 = = X cana-5910 114 19 𝑑𝑦 𝑑𝑦 PROPN cana-5910 114 20 𝑦 𝑦 NOUN cana-5910 114 21 = = SYM cana-5910 114 22 𝑑𝑑 𝑑𝑑 ADP cana-5910 114 23 2𝑑 2𝑑 NUM cana-5910 114 24 = = PUNCT cana-5910 114 25 𝑑𝑒 𝑑𝑒 NOUN cana-5910 114 26 0 0 NUM cana-5910 114 27 from from ADP cana-5910 114 28 𝑑π‘₯ 𝑑π‘₯ NOUN cana-5910 114 29 π‘₯ π‘₯ NOUN cana-5910 114 30 = = SYM cana-5910 114 31 𝑑𝑑 𝑑𝑑 ADP cana-5910 114 32 2𝑑 2𝑑 NOUN cana-5910 114 33 , , PUNCT cana-5910 114 34 π‘₯ π‘₯ NOUN cana-5910 114 35 = = SYM cana-5910 114 36 π‘˜1𝑑 π‘˜1𝑑 PROPN cana-5910 114 37 1 1 NUM cana-5910 114 38 2 2 NUM cana-5910 114 39 yielding yield VERB cana-5910 114 40 the the DET cana-5910 114 41 similarity similarity NOUN cana-5910 114 42 variable variable NOUN cana-5910 114 43 πœ‰ πœ‰ NOUN cana-5910 114 44 = = PUNCT cana-5910 114 45 π‘₯𝑑 π‘₯𝑑 ADP cana-5910 114 46 1 1 NUM cana-5910 114 47 2 2 NUM cana-5910 114 48 from from ADP cana-5910 114 49 𝑑𝑦 𝑑𝑦 PROPN cana-5910 114 50 𝑦 𝑦 PROPN cana-5910 114 51 = = SYM cana-5910 114 52 𝑑𝑑 𝑑𝑑 ADP cana-5910 114 53 2𝑑 2𝑑 NOUN cana-5910 114 54 , , PUNCT cana-5910 114 55 𝑦 𝑦 NOUN cana-5910 114 56 = = SYM cana-5910 114 57 π‘˜2𝑑 π‘˜2𝑑 PROPN cana-5910 114 58 1 1 NUM cana-5910 114 59 2 2 NUM cana-5910 114 60 , , PUNCT cana-5910 114 61 yielding yield VERB cana-5910 114 62 Ξ· Ξ· PROPN cana-5910 114 63 = = PROPN cana-5910 114 64 𝑦𝑑 𝑦𝑑 PROPN cana-5910 114 65 1 1 NUM cana-5910 114 66 2 2 NUM cana-5910 114 67 . . PUNCT cana-5910 114 68 from from ADP cana-5910 114 69 𝑑𝑒 𝑑𝑒 PROPN cana-5910 114 70 0 0 NUM cana-5910 114 71 , , PUNCT cana-5910 114 72 u u NOUN cana-5910 114 73 is be AUX cana-5910 114 74 constant constant ADJ cana-5910 114 75 along along ADP cana-5910 114 76 characteristics characteristic NOUN cana-5910 114 77 , , PUNCT cana-5910 114 78 so so SCONJ cana-5910 114 79 𝑒 𝑒 X cana-5910 114 80 = = SYM cana-5910 114 81 𝑓(πœ‰ 𝑓(πœ‰ PROPN cana-5910 114 82 , , PUNCT cana-5910 114 83 πœ‚ πœ‚ NOUN cana-5910 114 84 ) ) PUNCT cana-5910 114 85 thus thus ADV cana-5910 114 86 , , PUNCT cana-5910 114 87 we we PRON cana-5910 114 88 assume assume VERB cana-5910 114 89 : : PUNCT cana-5910 114 90 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 114 91 , , PUNCT cana-5910 114 92 𝑦 𝑦 NOUN cana-5910 114 93 , , PUNCT cana-5910 114 94 𝑑 𝑑 NOUN cana-5910 114 95 ) ) PUNCT cana-5910 114 96 = = SYM cana-5910 115 1 𝑓(πœ‰ 𝑓(πœ‰ PROPN cana-5910 115 2 , , PUNCT cana-5910 115 3 πœ‚ πœ‚ NOUN cana-5910 115 4 ) ) PUNCT cana-5910 115 5 , , PUNCT cana-5910 115 6 π‘€β„Žπ‘’π‘Ÿπ‘’ π‘€β„Žπ‘’π‘Ÿπ‘’ NOUN cana-5910 115 7 πœ‰ πœ‰ X cana-5910 115 8 = = SYM cana-5910 115 9 π‘₯𝑑 π‘₯𝑑 ADP cana-5910 115 10 1 1 NUM cana-5910 115 11 2 2 NUM cana-5910 115 12 , , PUNCT cana-5910 115 13 πœ‚ πœ‚ X cana-5910 115 14 = = SYM cana-5910 115 15 𝑦𝑑 𝑦𝑑 PROPN cana-5910 115 16 1 1 NUM cana-5910 115 17 2 2 NUM cana-5910 115 18 . . PUNCT cana-5910 115 19 substitute substitute NOUN cana-5910 115 20 into into ADP cana-5910 115 21 the the DET cana-5910 115 22 heat heat NOUN cana-5910 115 23 equation equation NOUN cana-5910 115 24 . . PUNCT cana-5910 116 1 compute compute VERB cana-5910 116 2 the the DET cana-5910 116 3 derivatives derivative NOUN cana-5910 116 4 : : PUNCT cana-5910 116 5 𝑒𝑑 𝑒𝑑 NOUN cana-5910 116 6 = = SYM cana-5910 116 7 ( ( PUNCT cana-5910 116 8 πœ•π‘“ πœ•π‘“ ADJ cana-5910 116 9 πœ•πœ‰ πœ•πœ‰ NOUN cana-5910 116 10 ) ) PUNCT cana-5910 116 11 ( ( PUNCT cana-5910 116 12 πœ•πœ‰ πœ•πœ‰ NOUN cana-5910 116 13 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 116 14 ) ) PUNCT cana-5910 117 1 + + CCONJ cana-5910 117 2 ( ( PUNCT cana-5910 117 3 πœ•π‘“ πœ•π‘“ ADJ cana-5910 117 4 πœ•πœ‚ πœ•πœ‚ PROPN cana-5910 117 5 ) ) PUNCT cana-5910 117 6 ( ( PUNCT cana-5910 117 7 πœ•πœ‚ πœ•πœ‚ PROPN cana-5910 117 8 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 117 9 ) ) PUNCT cana-5910 117 10 = = SYM cana-5910 117 11 ( ( PUNCT cana-5910 117 12 πœ•π‘“ πœ•π‘“ ADJ cana-5910 117 13 πœ•πœ‰ πœ•πœ‰ NOUN cana-5910 117 14 ) ) PUNCT cana-5910 117 15 ( ( PUNCT cana-5910 117 16 βˆ’π‘₯ βˆ’π‘₯ NOUN cana-5910 117 17 2𝑑 2𝑑 NOUN cana-5910 117 18 3 3 NUM cana-5910 117 19 2 2 NUM cana-5910 117 20 ) ) PUNCT cana-5910 117 21 + + CCONJ cana-5910 118 1 πœ•π‘“ πœ•π‘“ ADJ cana-5910 118 2 πœ•πœ‚ πœ•πœ‚ PROPN cana-5910 118 3 ( ( PUNCT cana-5910 118 4 βˆ’π‘¦ βˆ’π‘¦ NOUN cana-5910 118 5 2𝑑 2𝑑 NUM cana-5910 118 6 3 3 NUM cana-5910 118 7 2 2 NUM cana-5910 118 8 ) ) PUNCT cana-5910 118 9 = = SYM cana-5910 119 1 βˆ’ βˆ’ PROPN cana-5910 119 2 ( ( PUNCT cana-5910 119 3 πœ‰ πœ‰ NOUN cana-5910 119 4 2𝑑 2𝑑 NUM cana-5910 119 5 ) ) PUNCT cana-5910 119 6 ( ( PUNCT cana-5910 119 7 πœ•π‘“ πœ•π‘“ ADJ cana-5910 119 8 πœ•πœ‰ πœ•πœ‰ NOUN cana-5910 119 9 ) ) PUNCT cana-5910 119 10 βˆ’ βˆ’ PROPN cana-5910 120 1 ( ( PUNCT cana-5910 120 2 πœ‚ πœ‚ NOUN cana-5910 120 3 2𝑑 2𝑑 NUM cana-5910 120 4 ) ) PUNCT cana-5910 120 5 ( ( PUNCT cana-5910 120 6 πœ•π‘“ πœ•π‘“ ADV cana-5910 120 7 πœ•πœ‚ πœ•πœ‚ PROPN cana-5910 120 8 ) ) PUNCT cana-5910 120 9 , , PUNCT cana-5910 120 10 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 120 11 = = SYM cana-5910 120 12 ( ( PUNCT cana-5910 120 13 πœ•π‘“ πœ•π‘“ ADJ cana-5910 120 14 πœ•πœ‰ πœ•πœ‰ NOUN cana-5910 120 15 ) ) PUNCT cana-5910 120 16 ( ( PUNCT cana-5910 120 17 πœ•πœ‰ πœ•πœ‰ X cana-5910 120 18 πœ•π‘₯ πœ•π‘₯ X cana-5910 120 19 ) ) PUNCT cana-5910 120 20 = = SYM cana-5910 120 21 ( ( PUNCT cana-5910 120 22 πœ•π‘“ πœ•π‘“ ADJ cana-5910 120 23 πœ•πœ‰ πœ•πœ‰ NOUN cana-5910 120 24 ) ) PUNCT cana-5910 120 25 ( ( PUNCT cana-5910 120 26 1 1 NUM cana-5910 120 27 𝑑 𝑑 NOUN cana-5910 120 28 1 1 NUM cana-5910 120 29 2 2 NUM cana-5910 120 30 ) ) PUNCT cana-5910 120 31 , , PUNCT cana-5910 120 32 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 120 33 = = SYM cana-5910 121 1 πœ• πœ• PROPN cana-5910 121 2 πœ•π‘₯ πœ•π‘₯ X cana-5910 121 3 [ [ X cana-5910 121 4 ( ( PUNCT cana-5910 121 5 πœ•π‘“ πœ•π‘“ ADJ cana-5910 121 6 πœ•πœ‰ πœ•πœ‰ NOUN cana-5910 121 7 ) ) PUNCT cana-5910 121 8 ( ( PUNCT cana-5910 121 9 1 1 NUM cana-5910 121 10 𝑑 𝑑 PROPN cana-5910 121 11 1 1 NUM cana-5910 121 12 2 2 NUM cana-5910 121 13 ) ) PUNCT cana-5910 121 14 ] ] PUNCT cana-5910 122 1 = = PUNCT cana-5910 122 2 ( ( PUNCT cana-5910 122 3 1 1 NUM cana-5910 122 4 𝑑 𝑑 PROPN cana-5910 122 5 1 1 NUM cana-5910 122 6 2 2 NUM cana-5910 122 7 ) ) PUNCT cana-5910 122 8 ( ( PUNCT cana-5910 122 9 πœ•2𝑓 πœ•2𝑓 PROPN cana-5910 122 10 πœ•πœ‰2 πœ•πœ‰2 NOUN cana-5910 122 11 ) ) PUNCT cana-5910 122 12 ( ( PUNCT cana-5910 122 13 πœ•πœ‰ πœ•πœ‰ X cana-5910 122 14 πœ•π‘₯ πœ•π‘₯ X cana-5910 122 15 ) ) PUNCT cana-5910 122 16 = = PUNCT cana-5910 123 1 ( ( PUNCT cana-5910 123 2 1 1 NUM cana-5910 123 3 𝑑 𝑑 NOUN cana-5910 123 4 ) ) PUNCT cana-5910 123 5 ( ( PUNCT cana-5910 123 6 πœ•2𝑓 πœ•2𝑓 ADJ cana-5910 123 7 πœ•πœ‰2 πœ•πœ‰2 NOUN cana-5910 123 8 ) ) PUNCT cana-5910 123 9 , , PUNCT cana-5910 123 10 𝑒𝑦 𝑒𝑦 PROPN cana-5910 123 11 = = PUNCT cana-5910 123 12 ( ( PUNCT cana-5910 123 13 πœ•π‘“ πœ•π‘“ ADJ cana-5910 123 14 πœ•πœ‚ πœ•πœ‚ PROPN cana-5910 123 15 ) ) PUNCT cana-5910 123 16 ( ( PUNCT cana-5910 123 17 πœ•πœ‚ πœ•πœ‚ ADJ cana-5910 123 18 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 123 19 ) ) PUNCT cana-5910 124 1 = = SYM cana-5910 124 2 ( ( PUNCT cana-5910 124 3 πœ•π‘“ πœ•π‘“ ADJ cana-5910 124 4 πœ•πœ‚ πœ•πœ‚ PROPN cana-5910 124 5 ) ) PUNCT cana-5910 124 6 ( ( PUNCT cana-5910 124 7 1 1 NUM cana-5910 124 8 𝑑 𝑑 NOUN cana-5910 124 9 1 1 NUM cana-5910 124 10 2 2 NUM cana-5910 124 11 ) ) PUNCT cana-5910 124 12 , , PUNCT cana-5910 124 13 𝑒𝑦𝑦 𝑒𝑦𝑦 ADV cana-5910 124 14 = = SYM cana-5910 124 15 πœ• πœ• PROPN cana-5910 124 16 πœ•π‘¦ πœ•π‘¦ NOUN cana-5910 124 17 [ [ X cana-5910 124 18 ( ( PUNCT cana-5910 124 19 πœ•π‘“ πœ•π‘“ ADJ cana-5910 124 20 πœ•πœ‚ πœ•πœ‚ PROPN cana-5910 124 21 ) ) PUNCT cana-5910 124 22 ( ( PUNCT cana-5910 124 23 1 1 NUM cana-5910 124 24 𝑑 𝑑 NOUN cana-5910 124 25 1 1 NUM cana-5910 124 26 2 2 NUM cana-5910 124 27 ) ) PUNCT cana-5910 124 28 ] ] PUNCT cana-5910 125 1 = = PUNCT cana-5910 125 2 ( ( PUNCT cana-5910 125 3 1 1 NUM cana-5910 125 4 / / SYM cana-5910 125 5 𝑑 𝑑 NOUN cana-5910 125 6 ) ) PUNCT cana-5910 125 7 πœ•2𝑓 πœ•2𝑓 NUM cana-5910 125 8 πœ•πœ‚2 πœ•πœ‚2 NOUN cana-5910 125 9 . . PUNCT cana-5910 126 1 substitute substitute NOUN cana-5910 126 2 into into ADP cana-5910 126 3 𝑒𝑑 𝑒𝑑 NOUN cana-5910 126 4 = = SYM cana-5910 126 5 𝛼 𝛼 PROPN cana-5910 126 6 ( ( PUNCT cana-5910 126 7 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 126 8 + + CCONJ cana-5910 126 9 𝑒𝑦𝑦 𝑒𝑦𝑦 ADV cana-5910 126 10 ): ): PUNCT cana-5910 126 11 βˆ’ βˆ’ PROPN cana-5910 126 12 πœ‰ πœ‰ NOUN cana-5910 126 13 2𝑑 2𝑑 NUM cana-5910 126 14 ( ( PUNCT cana-5910 126 15 πœ•π‘“ πœ•π‘“ ADJ cana-5910 126 16 πœ•πœ‰ πœ•πœ‰ NOUN cana-5910 126 17 ) ) PUNCT cana-5910 127 1 βˆ’ βˆ’ ADP cana-5910 127 2 πœ‚ πœ‚ NOUN cana-5910 127 3 2𝑑 2𝑑 NUM cana-5910 127 4 ( ( PUNCT cana-5910 127 5 πœ•π‘“ πœ•π‘“ ADJ cana-5910 127 6 πœ•πœ‚ πœ•πœ‚ ADJ cana-5910 127 7 ) ) PUNCT cana-5910 128 1 = = PUNCT cana-5910 128 2 𝛼 𝛼 X cana-5910 128 3 1 1 NUM cana-5910 128 4 𝑑 𝑑 NOUN cana-5910 128 5 [ [ X cana-5910 128 6 ( ( PUNCT cana-5910 128 7 πœ•2𝑓 πœ•2𝑓 ADJ cana-5910 128 8 πœ•πœ‰2 πœ•πœ‰2 NOUN cana-5910 128 9 ) ) PUNCT cana-5910 129 1 + + CCONJ cana-5910 129 2 ( ( PUNCT cana-5910 129 3 πœ•2𝑓 πœ•2𝑓 ADJ cana-5910 129 4 πœ•πœ‚2 πœ•πœ‚2 NOUN cana-5910 129 5 ) ) PUNCT cana-5910 129 6 ] ] PUNCT cana-5910 129 7 . . PUNCT cana-5910 130 1 multiply multiply VERB cana-5910 130 2 through through ADP cana-5910 130 3 by by ADP cana-5910 130 4 t t PROPN cana-5910 130 5 : : PUNCT cana-5910 130 6 1115 1115 NUM cana-5910 130 7 https://internationalpubls.com https://internationalpubls.com X cana-5910 131 1 βˆ’ βˆ’ NOUN cana-5910 131 2 πœ‰ πœ‰ SYM cana-5910 131 3 2 2 NUM cana-5910 131 4 ( ( PUNCT cana-5910 131 5 πœ•π‘“ πœ•π‘“ ADJ cana-5910 131 6 πœ•πœ‰ πœ•πœ‰ NOUN cana-5910 131 7 ) ) PUNCT cana-5910 132 1 βˆ’ βˆ’ NOUN cana-5910 132 2 πœ‚ πœ‚ NOUN cana-5910 132 3 2 2 NUM cana-5910 132 4 ( ( PUNCT cana-5910 132 5 πœ•π‘“ πœ•π‘“ ADJ cana-5910 132 6 πœ•πœ‚ πœ•πœ‚ ADJ cana-5910 132 7 ) ) PUNCT cana-5910 132 8 = = PUNCT cana-5910 133 1 𝛼 𝛼 PRON cana-5910 133 2 [ [ X cana-5910 133 3 ( ( PUNCT cana-5910 133 4 πœ•2𝑓 πœ•2𝑓 ADJ cana-5910 133 5 πœ•πœ‰2 πœ•πœ‰2 NOUN cana-5910 133 6 ) ) PUNCT cana-5910 134 1 + + CCONJ cana-5910 134 2 ( ( PUNCT cana-5910 134 3 πœ•2𝑓 πœ•2𝑓 ADJ cana-5910 134 4 πœ•πœ‚2 πœ•πœ‚2 NOUN cana-5910 134 5 ) ) PUNCT cana-5910 134 6 ] ] PUNCT cana-5910 134 7 . . PUNCT cana-5910 135 1 to to PART cana-5910 135 2 simplify simplify VERB cana-5910 135 3 , , PUNCT cana-5910 135 4 assume assume VERB cana-5910 135 5 radial radial ADJ cana-5910 135 6 symmetry symmetry NOUN cana-5910 135 7 , , PUNCT cana-5910 135 8 where where SCONJ cana-5910 135 9 f(ΞΎ f(ΞΎ PROPN cana-5910 135 10 , , PUNCT cana-5910 135 11 Ξ· Ξ· NOUN cana-5910 135 12 ) ) PUNCT cana-5910 135 13 depends depend VERB cana-5910 135 14 on on ADP cana-5910 135 15 the the DET cana-5910 135 16 radial radial ADJ cana-5910 135 17 variable variable NOUN cana-5910 135 18 π‘Ÿ π‘Ÿ NOUN cana-5910 135 19 = = SYM cana-5910 135 20 ( ( PUNCT cana-5910 135 21 πœ‰2 πœ‰2 PROPN cana-5910 135 22 + + CCONJ cana-5910 135 23 πœ‚2 πœ‚2 VERB cana-5910 135 24 ) ) PUNCT cana-5910 135 25 1 1 NUM cana-5910 135 26 2 2 NUM cana-5910 135 27 = = SYM cana-5910 135 28 ( ( PUNCT cana-5910 135 29 ( ( PUNCT cana-5910 135 30 π‘₯2 π‘₯2 NOUN cana-5910 135 31 + + CCONJ cana-5910 135 32 𝑦2 𝑦2 NOUN cana-5910 135 33 ) ) PUNCT cana-5910 135 34 𝑑 𝑑 PROPN cana-5910 135 35 ) ) PUNCT cana-5910 135 36 1 1 NUM cana-5910 135 37 2 2 NUM cana-5910 135 38 . . PUNCT cana-5910 136 1 π‘‡β„Žπ‘’π‘  π‘‡β„Žπ‘’π‘  PROPN cana-5910 136 2 , , PUNCT cana-5910 136 3 𝑓(πœ‰ 𝑓(πœ‰ PROPN cana-5910 136 4 , , PUNCT cana-5910 136 5 πœ‚ πœ‚ NOUN cana-5910 136 6 ) ) PUNCT cana-5910 136 7 = = SYM cana-5910 136 8 𝐹(π‘Ÿ 𝐹(π‘Ÿ PROPN cana-5910 136 9 ) ) PUNCT cana-5910 136 10 . . PUNCT cana-5910 137 1 in in ADP cana-5910 137 2 polar polar ADJ cana-5910 137 3 coordinates coordinate NOUN cana-5910 137 4 ( ( PUNCT cana-5910 137 5 πœ‰ πœ‰ X cana-5910 137 6 = = PUNCT cana-5910 137 7 π‘Ÿ π‘Ÿ NOUN cana-5910 137 8 π‘π‘œπ‘ (πœƒ π‘π‘œπ‘ (πœƒ NOUN cana-5910 137 9 ) ) PUNCT cana-5910 137 10 , , PUNCT cana-5910 137 11 πœ‚ πœ‚ NOUN cana-5910 137 12 = = PUNCT cana-5910 137 13 π‘Ÿ π‘Ÿ PRON cana-5910 137 14 𝑠𝑖𝑛(πœƒ 𝑠𝑖𝑛(πœƒ PROPN cana-5910 137 15 ) ) PUNCT cana-5910 137 16 ) ) PUNCT cana-5910 137 17 , , PUNCT cana-5910 137 18 compute compute NOUN cana-5910 137 19 : : PUNCT cana-5910 137 20 πœ•π‘“ πœ•π‘“ ADJ cana-5910 137 21 πœ•πœ‰ πœ•πœ‰ NOUN cana-5910 137 22 = = SYM cana-5910 137 23 𝐹′(π‘Ÿ 𝐹′(π‘Ÿ NUM cana-5910 137 24 ) ) PUNCT cana-5910 137 25 πœ‰ πœ‰ VERB cana-5910 137 26 π‘Ÿ π‘Ÿ NOUN cana-5910 137 27 , , PUNCT cana-5910 137 28 πœ•π‘“ πœ•π‘“ ADJ cana-5910 137 29 πœ•πœ‚ πœ•πœ‚ PROPN cana-5910 137 30 = = SYM cana-5910 137 31 𝐹′(π‘Ÿ 𝐹′(π‘Ÿ ADJ cana-5910 137 32 ) ) PUNCT cana-5910 137 33 πœ‚ πœ‚ NOUN cana-5910 137 34 π‘Ÿ π‘Ÿ NOUN cana-5910 137 35 , , PUNCT cana-5910 137 36 πœ•2𝑓 πœ•2𝑓 ADJ cana-5910 137 37 πœ•πœ‰2 πœ•πœ‰2 NOUN cana-5910 137 38 + + CCONJ cana-5910 137 39 πœ•2𝑓 πœ•2𝑓 ADJ cana-5910 137 40 πœ•πœ‚2 πœ•πœ‚2 NOUN cana-5910 137 41 = = SYM cana-5910 137 42 𝐹′′(π‘Ÿ 𝐹′′(π‘Ÿ ADJ cana-5910 137 43 ) ) PUNCT cana-5910 138 1 + + CCONJ cana-5910 138 2 1 1 NUM cana-5910 138 3 π‘Ÿ π‘Ÿ NOUN cana-5910 138 4 𝐹′(π‘Ÿ 𝐹′(π‘Ÿ NUM cana-5910 138 5 ) ) PUNCT cana-5910 138 6 . . PUNCT cana-5910 139 1 the the DET cana-5910 139 2 equation equation NOUN cana-5910 139 3 becomes become VERB cana-5910 139 4 : : PUNCT cana-5910 139 5 βˆ’ βˆ’ PROPN cana-5910 139 6 π‘Ÿ π‘Ÿ SYM cana-5910 139 7 2 2 NUM cana-5910 139 8 𝐹′(π‘Ÿ 𝐹′(π‘Ÿ NUM cana-5910 139 9 ) ) PUNCT cana-5910 139 10 = = SYM cana-5910 139 11 𝛼 𝛼 PROPN cana-5910 140 1 [ [ X cana-5910 140 2 𝐹′′(π‘Ÿ 𝐹′′(π‘Ÿ ADJ cana-5910 140 3 ) ) PUNCT cana-5910 140 4 + + CCONJ cana-5910 140 5 1 1 NUM cana-5910 140 6 π‘Ÿ π‘Ÿ NOUN cana-5910 140 7 𝐹′(π‘Ÿ 𝐹′(π‘Ÿ NUM cana-5910 140 8 ) ) PUNCT cana-5910 140 9 ] ] PUNCT cana-5910 140 10 . . PUNCT cana-5910 141 1 rearrange rearrange VERB cana-5910 141 2 : : PUNCT cana-5910 141 3 𝐹′′(π‘Ÿ 𝐹′′(π‘Ÿ ADJ cana-5910 141 4 ) ) PUNCT cana-5910 142 1 + + CCONJ cana-5910 142 2 ( ( PUNCT cana-5910 142 3 1 1 NUM cana-5910 142 4 π‘Ÿ π‘Ÿ NOUN cana-5910 142 5 + + NUM cana-5910 142 6 π‘Ÿ π‘Ÿ PRON cana-5910 142 7 2𝛼 2𝛼 PROPN cana-5910 142 8 ) ) PUNCT cana-5910 142 9 𝐹′(π‘Ÿ 𝐹′(π‘Ÿ PROPN cana-5910 142 10 ) ) PUNCT cana-5910 143 1 = = SYM cana-5910 143 2 0 0 X cana-5910 143 3 . . PUNCT cana-5910 144 1 let let VERB cana-5910 144 2 𝑣 𝑣 PRON cana-5910 144 3 = = SYM cana-5910 144 4 𝐹′(π‘Ÿ 𝐹′(π‘Ÿ PROPN cana-5910 144 5 ) ) PUNCT cana-5910 144 6 , , PUNCT cana-5910 144 7 π‘ π‘œ π‘ π‘œ ADP cana-5910 144 8 𝑣′ 𝑣′ NUM cana-5910 144 9 = = SYM cana-5910 144 10 𝐹′′(π‘Ÿ 𝐹′′(π‘Ÿ ADJ cana-5910 144 11 ) ) PUNCT cana-5910 144 12 , , PUNCT cana-5910 144 13 yielding yield VERB cana-5910 144 14 : : PUNCT cana-5910 144 15 𝑣′ 𝑣′ NUM cana-5910 144 16 + + CCONJ cana-5910 144 17 ( ( PUNCT cana-5910 144 18 1 1 NUM cana-5910 144 19 π‘Ÿ π‘Ÿ NOUN cana-5910 144 20 + + NUM cana-5910 144 21 π‘Ÿ π‘Ÿ DET cana-5910 144 22 2𝛼 2𝛼 NUM cana-5910 144 23 ) ) PUNCT cana-5910 145 1 𝑣 𝑣 ADP cana-5910 145 2 = = SYM cana-5910 145 3 0 0 NUM cana-5910 145 4 . . PUNCT cana-5910 146 1 𝑣′ 𝑣′ PROPN cana-5910 146 2 𝑣 𝑣 X cana-5910 146 3 = = PRON cana-5910 146 4 βˆ’ βˆ’ PROPN cana-5910 147 1 ( ( PUNCT cana-5910 147 2 1 1 NUM cana-5910 147 3 π‘Ÿ π‘Ÿ NOUN cana-5910 147 4 + + NUM cana-5910 147 5 π‘Ÿ π‘Ÿ PRON cana-5910 147 6 2𝛼 2𝛼 PROPN cana-5910 147 7 ) ) PUNCT cana-5910 147 8 , , PUNCT cana-5910 147 9 𝑣 𝑣 X cana-5910 147 10 = = PUNCT cana-5910 147 11 ( ( PUNCT cana-5910 147 12 𝑐1 𝑐1 NOUN cana-5910 147 13 π‘Ÿ π‘Ÿ NOUN cana-5910 147 14 ) ) PUNCT cana-5910 147 15 π‘’βˆ’ π‘’βˆ’ VERB cana-5910 147 16 π‘Ÿ2 π‘Ÿ2 NOUN cana-5910 147 17 4𝛼 4𝛼 NOUN cana-5910 147 18 . . PUNCT cana-5910 148 1 thus thus ADV cana-5910 148 2 : : PUNCT cana-5910 148 3 𝐹′(π‘Ÿ 𝐹′(π‘Ÿ NUM cana-5910 148 4 ) ) PUNCT cana-5910 148 5 = = PUNCT cana-5910 149 1 ( ( PUNCT cana-5910 149 2 𝑐1 𝑐1 NOUN cana-5910 149 3 π‘Ÿ π‘Ÿ NOUN cana-5910 149 4 ) ) PUNCT cana-5910 149 5 π‘’βˆ’ π‘’βˆ’ VERB cana-5910 149 6 π‘Ÿ2 π‘Ÿ2 NOUN cana-5910 149 7 4𝛼 4𝛼 NOUN cana-5910 149 8 . . PUNCT cana-5910 150 1 integrate integrate VERB cana-5910 150 2 : : PUNCT cana-5910 150 3 𝐹(π‘Ÿ 𝐹(π‘Ÿ X cana-5910 150 4 ) ) PUNCT cana-5910 150 5 = = SYM cana-5910 151 1 𝑐1 𝑐1 NOUN cana-5910 151 2 ∫ ∫ PROPN cana-5910 151 3 ( ( PUNCT cana-5910 151 4 1 1 NUM cana-5910 151 5 π‘Ÿ π‘Ÿ NOUN cana-5910 151 6 ) ) PUNCT cana-5910 151 7 π‘’βˆ’ π‘’βˆ’ X cana-5910 152 1 π‘Ÿ2 π‘Ÿ2 NOUN cana-5910 152 2 4𝛼 4𝛼 NOUN cana-5910 152 3 π‘‘π‘Ÿ π‘‘π‘Ÿ NOUN cana-5910 152 4 + + NUM cana-5910 152 5 𝑐2 𝑐2 NOUN cana-5910 152 6 . . PUNCT cana-5910 153 1 substitute substitute NOUN cana-5910 153 2 𝑠 𝑠 PROPN cana-5910 153 3 = = SYM cana-5910 153 4 π‘Ÿ2 π‘Ÿ2 PROPN cana-5910 153 5 4𝛼 4𝛼 NOUN cana-5910 153 6 , , PUNCT cana-5910 153 7 π‘ π‘œ π‘ π‘œ ADP cana-5910 153 8 π‘Ÿ π‘Ÿ NOUN cana-5910 153 9 = = SYM cana-5910 153 10 ( ( PUNCT cana-5910 153 11 4𝛼 4𝛼 NOUN cana-5910 153 12 𝑠 𝑠 PROPN cana-5910 153 13 ) ) PUNCT cana-5910 153 14 1 1 NUM cana-5910 153 15 2 2 NUM cana-5910 153 16 , , PUNCT cana-5910 153 17 π‘‘π‘Ÿ π‘‘π‘Ÿ NOUN cana-5910 153 18 = = PUNCT cana-5910 153 19 ( ( PUNCT cana-5910 153 20 𝛼 𝛼 NOUN cana-5910 153 21 𝑠 𝑠 PROPN cana-5910 153 22 ) ) PUNCT cana-5910 153 23 1 1 NUM cana-5910 153 24 2 2 NUM cana-5910 153 25 𝑑𝑠 𝑑𝑠 ADP cana-5910 153 26 2 2 NUM cana-5910 153 27 𝐹(π‘Ÿ 𝐹(π‘Ÿ NOUN cana-5910 153 28 ) ) PUNCT cana-5910 153 29 = = PUNCT cana-5910 154 1 𝑐1 𝑐1 NOUN cana-5910 154 2 ∫ ∫ PROPN cana-5910 154 3 π‘ βˆ’1 π‘ βˆ’1 PROPN cana-5910 154 4 π‘’βˆ’π‘  π‘’βˆ’π‘  ADV cana-5910 154 5 ( ( PUNCT cana-5910 154 6 𝛼 𝛼 NOUN cana-5910 154 7 𝑠 𝑠 PROPN cana-5910 154 8 ) ) PUNCT cana-5910 154 9 1 1 NUM cana-5910 154 10 2 2 NUM cana-5910 154 11 𝑑𝑠 𝑑𝑠 ADP cana-5910 154 12 2 2 NUM cana-5910 154 13 + + NUM cana-5910 154 14 𝑐2 𝑐2 NOUN cana-5910 154 15 = = SYM cana-5910 154 16 ( ( PUNCT cana-5910 154 17 𝑐1 𝑐1 NOUN cana-5910 154 18 2 2 NUM cana-5910 154 19 ) ) PUNCT cana-5910 154 20 𝛼 𝛼 SYM cana-5910 154 21 1 1 NUM cana-5910 154 22 2 2 NUM cana-5910 154 23 ∫ ∫ NOUN cana-5910 154 24 π‘ βˆ’ π‘ βˆ’ PROPN cana-5910 154 25 3 3 NUM cana-5910 154 26 2 2 NUM cana-5910 154 27 π‘’βˆ’π‘  π‘’βˆ’π‘  ADV cana-5910 154 28 𝑑𝑠 𝑑𝑠 ADP cana-5910 154 29 + + X cana-5910 154 30 𝑐2 𝑐2 NOUN cana-5910 154 31 . . PUNCT cana-5910 155 1 this this DET cana-5910 155 2 integral integral ADJ cana-5910 155 3 is be AUX cana-5910 155 4 related relate VERB cana-5910 155 5 to to ADP cana-5910 155 6 the the DET cana-5910 155 7 incomplete incomplete ADJ cana-5910 155 8 gamma gamma NOUN cana-5910 155 9 function function NOUN cana-5910 155 10 , , PUNCT cana-5910 155 11 but but CCONJ cana-5910 155 12 for for ADP cana-5910 155 13 a a DET cana-5910 155 14 physically physically ADV cana-5910 155 15 relevant relevant ADJ cana-5910 155 16 solution solution NOUN cana-5910 155 17 , , PUNCT cana-5910 155 18 consider consider VERB cana-5910 155 19 the the DET cana-5910 155 20 form form NOUN cana-5910 155 21 of of ADP cana-5910 155 22 the the DET cana-5910 155 23 fundamental fundamental ADJ cana-5910 155 24 solution solution NOUN cana-5910 155 25 . . PUNCT cana-5910 156 1 test test NOUN cana-5910 156 2 𝐹(π‘Ÿ 𝐹(π‘Ÿ NUM cana-5910 156 3 ) ) PUNCT cana-5910 156 4 = = PUNCT cana-5910 156 5 π‘’βˆ’ π‘’βˆ’ VERB cana-5910 156 6 π‘Ÿ2 π‘Ÿ2 NOUN cana-5910 156 7 4𝛼 4𝛼 NOUN cana-5910 156 8 𝐹′(π‘Ÿ 𝐹′(π‘Ÿ NUM cana-5910 156 9 ) ) PUNCT cana-5910 157 1 = = SYM cana-5910 158 1 βˆ’ βˆ’ PROPN cana-5910 159 1 π‘Ÿ π‘Ÿ NOUN cana-5910 159 2 2𝛼 2𝛼 NUM cana-5910 159 3 π‘’βˆ’ π‘’βˆ’ VERB cana-5910 159 4 π‘Ÿ2 π‘Ÿ2 NOUN cana-5910 159 5 4𝛼 4𝛼 NOUN cana-5910 159 6 , , PUNCT cana-5910 159 7 𝐹′′(π‘Ÿ 𝐹′′(π‘Ÿ ADJ cana-5910 159 8 ) ) PUNCT cana-5910 160 1 = = PUNCT cana-5910 161 1 [ [ PUNCT cana-5910 161 2 π‘Ÿ2 π‘Ÿ2 NOUN cana-5910 161 3 4𝛼2 4𝛼2 NOUN cana-5910 161 4 βˆ’ βˆ’ NUM cana-5910 161 5 1 1 NUM cana-5910 161 6 2𝛼 2𝛼 PROPN cana-5910 161 7 ] ] PUNCT cana-5910 161 8 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 161 9 π‘Ÿ2 π‘Ÿ2 NOUN cana-5910 161 10 4𝛼 4𝛼 NOUN cana-5910 161 11 1116 1116 NUM cana-5910 161 12 https://internationalpubls.com https://internationalpubls.com X cana-5910 162 1 this this PRON cana-5910 162 2 does do AUX cana-5910 162 3 not not PART cana-5910 162 4 directly directly ADV cana-5910 162 5 satisfy satisfy VERB cana-5910 162 6 the the DET cana-5910 162 7 ode ode NOUN cana-5910 162 8 , , PUNCT cana-5910 162 9 so so SCONJ cana-5910 162 10 we we PRON cana-5910 162 11 rely rely VERB cana-5910 162 12 on on ADP cana-5910 162 13 the the DET cana-5910 162 14 integrated integrate VERB cana-5910 162 15 form form NOUN cana-5910 162 16 . . PUNCT cana-5910 163 1 the the DET cana-5910 163 2 general general ADJ cana-5910 163 3 solution solution NOUN cana-5910 163 4 is be AUX cana-5910 163 5 : : PUNCT cana-5910 163 6 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 163 7 , , PUNCT cana-5910 163 8 𝑦 𝑦 NOUN cana-5910 163 9 , , PUNCT cana-5910 163 10 𝑑 𝑑 NOUN cana-5910 163 11 ) ) PUNCT cana-5910 163 12 = = PUNCT cana-5910 164 1 𝑐1 𝑐1 NOUN cana-5910 164 2 𝑑 𝑑 X cana-5910 164 3 [ [ PUNCT cana-5910 164 4 π‘’βˆ’(π‘₯2 π‘’βˆ’(π‘₯2 NOUN cana-5910 164 5 + + NUM cana-5910 164 6 𝑦2 𝑦2 NOUN cana-5910 164 7 ) ) PUNCT cana-5910 164 8 4𝛼 4𝛼 PROPN cana-5910 164 9 𝑑 𝑑 X cana-5910 164 10 ] ] X cana-5910 165 1 + + NUM cana-5910 165 2 𝑐2 𝑐2 NOUN cana-5910 165 3 . . PUNCT cana-5910 166 1 this this PRON cana-5910 166 2 is be AUX cana-5910 166 3 the the DET cana-5910 166 4 two two NUM cana-5910 166 5 - - PUNCT cana-5910 166 6 dimensional dimensional ADJ cana-5910 166 7 fundamental fundamental ADJ cana-5910 166 8 solution solution NOUN cana-5910 166 9 , , PUNCT cana-5910 166 10 representing represent VERB cana-5910 166 11 heat heat NOUN cana-5910 166 12 diffusion diffusion NOUN cana-5910 166 13 from from ADP cana-5910 166 14 a a DET cana-5910 166 15 point point NOUN cana-5910 166 16 source source NOUN cana-5910 166 17 at at ADP cana-5910 166 18 the the DET cana-5910 166 19 origin origin NOUN cana-5910 166 20 . . PUNCT cana-5910 167 1 3.2 3.2 NUM cana-5910 167 2 reduction reduction NOUN cana-5910 167 3 using use VERB cana-5910 167 4 the the DET cana-5910 167 5 conformal conformal ADJ cana-5910 167 6 - - PUNCT cana-5910 167 7 like like ADJ cana-5910 167 8 symmetry symmetry NOUN cana-5910 167 9 π‘½πŸ” π‘½πŸ” NOUN cana-5910 167 10 𝑉6 𝑉6 NOUN cana-5910 167 11 = = PUNCT cana-5910 167 12 π‘₯ π‘₯ PROPN cana-5910 167 13 2𝛼 2𝛼 PROPN cana-5910 167 14 πœ• πœ• PROPN cana-5910 167 15 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 167 16 + + CCONJ cana-5910 167 17 𝑦 𝑦 NUM cana-5910 167 18 2𝛼 2𝛼 PROPN cana-5910 167 19 πœ• πœ• PROPN cana-5910 167 20 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 167 21 + + CCONJ cana-5910 167 22 𝑑 𝑑 PROPN cana-5910 167 23 πœ• πœ• PROPN cana-5910 168 1 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 168 2 βˆ’ βˆ’ PROPN cana-5910 168 3 π‘₯2 π‘₯2 NOUN cana-5910 168 4 + + CCONJ cana-5910 168 5 𝑦2 𝑦2 PROPN cana-5910 168 6 4𝛼 4𝛼 NOUN cana-5910 168 7 𝑒 𝑒 PROPN cana-5910 168 8 πœ• πœ• PROPN cana-5910 168 9 πœ•π‘’ πœ•π‘’ NOUN cana-5910 168 10 . . PUNCT cana-5910 169 1 solve solve VERB cana-5910 169 2 the the DET cana-5910 169 3 characteristic characteristic ADJ cana-5910 169 4 equations equation NOUN cana-5910 169 5 : : PUNCT cana-5910 169 6 𝑑π‘₯ 𝑑π‘₯ VERB cana-5910 169 7 π‘₯ π‘₯ PRON cana-5910 169 8 2𝛼 2𝛼 PROPN cana-5910 169 9 = = SYM cana-5910 169 10 𝑑𝑦 𝑑𝑦 PROPN cana-5910 169 11 𝑦 𝑦 NOUN cana-5910 169 12 2𝛼 2𝛼 PROPN cana-5910 169 13 = = SYM cana-5910 169 14 𝑑𝑑 𝑑𝑑 ADP cana-5910 169 15 𝑑 𝑑 PROPN cana-5910 169 16 = = NOUN cana-5910 169 17 𝑑𝑒 𝑑𝑒 NOUN cana-5910 169 18 π‘₯2 π‘₯2 NOUN cana-5910 169 19 + + CCONJ cana-5910 169 20 𝑦2 𝑦2 PROPN cana-5910 169 21 4𝛼 4𝛼 NOUN cana-5910 169 22 𝑒 𝑒 NOUN cana-5910 169 23 from from ADP cana-5910 169 24 𝑑π‘₯ 𝑑π‘₯ NOUN cana-5910 169 25 π‘₯ π‘₯ PRON cana-5910 169 26 2𝛼 2𝛼 PROPN cana-5910 169 27 = = SYM cana-5910 169 28 𝑑𝑑 𝑑𝑑 ADP cana-5910 169 29 𝑑 𝑑 PROPN cana-5910 169 30 , , PUNCT cana-5910 169 31 π‘₯ π‘₯ NOUN cana-5910 169 32 = = SYM cana-5910 169 33 π‘˜1 π‘˜1 NOUN cana-5910 169 34 𝑑 𝑑 PROPN cana-5910 169 35 1 1 NUM cana-5910 169 36 2𝛼 2𝛼 NUM cana-5910 169 37 , , PUNCT cana-5910 169 38 yielding yield VERB cana-5910 169 39 ΞΎ ΞΎ PROPN cana-5910 169 40 = = SYM cana-5910 169 41 π‘₯ π‘₯ PROPN cana-5910 169 42 𝑑 𝑑 PROPN cana-5910 169 43 1 1 NUM cana-5910 169 44 2𝛼 2𝛼 NUM cana-5910 169 45 from from ADP cana-5910 169 46 𝑑𝑦 𝑑𝑦 PROPN cana-5910 169 47 𝑦 𝑦 PROPN cana-5910 169 48 2𝛼 2𝛼 PROPN cana-5910 169 49 = = SYM cana-5910 169 50 𝑑𝑑 𝑑𝑑 ADP cana-5910 169 51 𝑑 𝑑 PROPN cana-5910 169 52 , , PUNCT cana-5910 169 53 𝑦 𝑦 NOUN cana-5910 169 54 = = SYM cana-5910 169 55 π‘˜2 π‘˜2 PROPN cana-5910 169 56 𝑑 𝑑 PROPN cana-5910 169 57 1 1 NUM cana-5910 169 58 2𝛼 2𝛼 NUM cana-5910 169 59 , , PUNCT cana-5910 169 60 yielding yield VERB cana-5910 169 61 Ξ· Ξ· PROPN cana-5910 169 62 = = PROPN cana-5910 169 63 𝑦 𝑦 NUM cana-5910 169 64 𝑑 𝑑 PROPN cana-5910 169 65 1 1 NUM cana-5910 169 66 2𝛼 2𝛼 NUM cana-5910 169 67 . . PUNCT cana-5910 170 1 from from ADP cana-5910 170 2 𝑑𝑒 𝑑𝑒 ADP cana-5910 170 3 π‘₯2 π‘₯2 NOUN cana-5910 170 4 + + CCONJ cana-5910 170 5 𝑦2 𝑦2 PROPN cana-5910 170 6 4𝛼 4𝛼 NOUN cana-5910 170 7 𝑒 𝑒 X cana-5910 170 8 = = X cana-5910 170 9 𝑑𝑑 𝑑𝑑 ADP cana-5910 170 10 𝑑 𝑑 PROPN cana-5910 170 11 , , PUNCT cana-5910 170 12 𝑒 𝑒 X cana-5910 170 13 = = SYM cana-5910 170 14 𝑑 𝑑 PROPN cana-5910 170 15 1 1 NUM cana-5910 170 16 2 2 NUM cana-5910 170 17 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 170 18 ( ( PUNCT cana-5910 170 19 π‘₯2 π‘₯2 NOUN cana-5910 170 20 + + CCONJ cana-5910 170 21 𝑦2 𝑦2 NOUN cana-5910 170 22 ) ) PUNCT cana-5910 170 23 4𝛼 4𝛼 PROPN cana-5910 170 24 𝑑 𝑑 PROPN cana-5910 170 25 𝑓(πœ‰ 𝑓(πœ‰ PROPN cana-5910 170 26 , , PUNCT cana-5910 170 27 πœ‚ πœ‚ NOUN cana-5910 170 28 ) ) PUNCT cana-5910 170 29 . . PUNCT cana-5910 171 1 assume assume VERB cana-5910 171 2 : : PUNCT cana-5910 171 3 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 171 4 , , PUNCT cana-5910 171 5 𝑦 𝑦 NOUN cana-5910 171 6 , , PUNCT cana-5910 171 7 𝑑 𝑑 NOUN cana-5910 171 8 ) ) PUNCT cana-5910 171 9 = = SYM cana-5910 171 10 𝑑 𝑑 NOUN cana-5910 171 11 1 1 NUM cana-5910 171 12 2 2 NUM cana-5910 171 13 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 171 14 ( ( PUNCT cana-5910 171 15 π‘₯2 π‘₯2 NOUN cana-5910 171 16 + + CCONJ cana-5910 171 17 𝑦2 𝑦2 NOUN cana-5910 171 18 ) ) PUNCT cana-5910 171 19 4𝛼 4𝛼 PROPN cana-5910 171 20 𝑑 𝑑 PROPN cana-5910 171 21 𝑓(πœ‰ 𝑓(πœ‰ PROPN cana-5910 171 22 , , PUNCT cana-5910 171 23 πœ‚ πœ‚ NOUN cana-5910 171 24 ) ) PUNCT cana-5910 171 25 , , PUNCT cana-5910 171 26 π‘€β„Žπ‘’π‘Ÿπ‘’ π‘€β„Žπ‘’π‘Ÿπ‘’ NOUN cana-5910 171 27 πœ‰ πœ‰ X cana-5910 171 28 = = SYM cana-5910 171 29 π‘₯ π‘₯ PROPN cana-5910 171 30 𝑑 𝑑 PROPN cana-5910 171 31 1 1 NUM cana-5910 171 32 2𝛼 2𝛼 NUM cana-5910 171 33 , , PUNCT cana-5910 171 34 πœ‚ πœ‚ NOUN cana-5910 171 35 = = SYM cana-5910 171 36 𝑦 𝑦 SYM cana-5910 171 37 𝑑 𝑑 PROPN cana-5910 171 38 1 1 NUM cana-5910 171 39 2𝛼 2𝛼 NUM cana-5910 171 40 . . PUNCT cana-5910 172 1 this this DET cana-5910 172 2 form form NOUN cana-5910 172 3 is be AUX cana-5910 172 4 complex complex ADJ cana-5910 172 5 , , PUNCT cana-5910 172 6 so so SCONJ cana-5910 172 7 we we PRON cana-5910 172 8 test test VERB cana-5910 172 9 the the DET cana-5910 172 10 fundamental fundamental ADJ cana-5910 172 11 solution solution NOUN cana-5910 172 12 directly directly ADV cana-5910 172 13 , , PUNCT cana-5910 172 14 as as SCONJ cana-5910 172 15 𝑉6 𝑉6 NOUN cana-5910 172 16 suggests suggest VERB cana-5910 172 17 a a DET cana-5910 172 18 gaussian gaussian ADJ cana-5910 172 19 profile profile NOUN cana-5910 172 20 . . PUNCT cana-5910 173 1 substituting substitute VERB cana-5910 173 2 𝑒 𝑒 NOUN cana-5910 173 3 = = PUNCT cana-5910 173 4 ( ( PUNCT cana-5910 173 5 𝑐1 𝑐1 NOUN cana-5910 173 6 𝑑 𝑑 NOUN cana-5910 173 7 ) ) PUNCT cana-5910 173 8 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 173 9 ( ( PUNCT cana-5910 173 10 π‘₯2 π‘₯2 NOUN cana-5910 173 11 + + CCONJ cana-5910 173 12 𝑦2 𝑦2 NOUN cana-5910 173 13 ) ) PUNCT cana-5910 173 14 4𝛼 4𝛼 PROPN cana-5910 173 15 𝑑 𝑑 VERB cana-5910 173 16 into into ADP cana-5910 173 17 the the DET cana-5910 173 18 heat heat NOUN cana-5910 173 19 equation equation NOUN cana-5910 173 20 confirms confirm VERB cana-5910 173 21 it it PRON cana-5910 173 22 satisfies satisfy VERB cana-5910 173 23 : : PUNCT cana-5910 173 24 𝑒𝑑 𝑒𝑑 NOUN cana-5910 173 25 = = NOUN cana-5910 173 26 𝑐1 𝑐1 NOUN cana-5910 173 27 [ [ PUNCT cana-5910 173 28 ( ( PUNCT cana-5910 173 29 π‘₯2 π‘₯2 NOUN cana-5910 173 30 + + CCONJ cana-5910 173 31 𝑦2 𝑦2 NOUN cana-5910 173 32 ) ) PUNCT cana-5910 173 33 4𝛼 4𝛼 NOUN cana-5910 173 34 𝑑2 𝑑2 NOUN cana-5910 173 35 βˆ’ βˆ’ PROPN cana-5910 173 36 1 1 NUM cana-5910 173 37 𝑑 𝑑 NOUN cana-5910 173 38 ] ] X cana-5910 173 39 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 173 40 ( ( PUNCT cana-5910 173 41 π‘₯2 π‘₯2 NOUN cana-5910 173 42 + + CCONJ cana-5910 173 43 𝑦2 𝑦2 NOUN cana-5910 173 44 ) ) PUNCT cana-5910 173 45 4𝛼 4𝛼 PROPN cana-5910 173 46 𝑑 𝑑 PROPN cana-5910 173 47 , , PUNCT cana-5910 173 48 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 173 49 + + CCONJ cana-5910 173 50 𝑒𝑦𝑦 𝑒𝑦𝑦 NOUN cana-5910 173 51 = = SYM cana-5910 173 52 𝑐1 𝑐1 NOUN cana-5910 173 53 [ [ PUNCT cana-5910 173 54 βˆ’(π‘₯2 βˆ’(π‘₯2 NUM cana-5910 173 55 + + NUM cana-5910 173 56 𝑦2 𝑦2 NOUN cana-5910 173 57 ) ) PUNCT cana-5910 173 58 4𝛼2 4𝛼2 NOUN cana-5910 174 1 𝑑2 𝑑2 NOUN cana-5910 175 1 + + CCONJ cana-5910 176 1 1 1 NUM cana-5910 176 2 𝛼 𝛼 X cana-5910 176 3 𝑑 𝑑 NOUN cana-5910 176 4 ] ] X cana-5910 176 5 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 176 6 ( ( PUNCT cana-5910 176 7 π‘₯2 π‘₯2 NOUN cana-5910 176 8 + + CCONJ cana-5910 176 9 𝑦2 𝑦2 NOUN cana-5910 176 10 ) ) PUNCT cana-5910 176 11 4𝛼 4𝛼 PROPN cana-5910 176 12 𝑑 𝑑 PROPN cana-5910 176 13 , , PUNCT cana-5910 176 14 𝛼 𝛼 PROPN cana-5910 176 15 ( ( PUNCT cana-5910 176 16 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 176 17 + + CCONJ cana-5910 176 18 𝑒𝑦𝑦 𝑒𝑦𝑦 ADJ cana-5910 176 19 ) ) PUNCT cana-5910 176 20 = = SYM cana-5910 177 1 𝛼 𝛼 X cana-5910 177 2 𝑐1 𝑐1 NOUN cana-5910 177 3 [ [ PUNCT cana-5910 177 4 βˆ’(π‘₯2 βˆ’(π‘₯2 NUM cana-5910 177 5 + + NUM cana-5910 177 6 𝑦2 𝑦2 NOUN cana-5910 177 7 ) ) PUNCT cana-5910 177 8 4𝛼2 4𝛼2 NOUN cana-5910 178 1 𝑑2 𝑑2 NOUN cana-5910 179 1 + + CCONJ cana-5910 180 1 1 1 NUM cana-5910 180 2 𝛼 𝛼 X cana-5910 180 3 𝑑 𝑑 NOUN cana-5910 180 4 ] ] X cana-5910 180 5 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 180 6 ( ( PUNCT cana-5910 180 7 π‘₯2 π‘₯2 NOUN cana-5910 180 8 + + CCONJ cana-5910 180 9 𝑦2 𝑦2 NOUN cana-5910 180 10 ) ) PUNCT cana-5910 180 11 4𝛼 4𝛼 PROPN cana-5910 180 12 𝑑 𝑑 PROPN cana-5910 180 13 , , PUNCT cana-5910 180 14 = = SYM cana-5910 180 15 𝑒𝑑 𝑒𝑑 NOUN cana-5910 180 16 . . PUNCT cana-5910 181 1 thus thus ADV cana-5910 181 2 , , PUNCT cana-5910 181 3 the the DET cana-5910 181 4 solution solution NOUN cana-5910 181 5 is be AUX cana-5910 181 6 : : PUNCT cana-5910 181 7 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 181 8 , , PUNCT cana-5910 181 9 𝑦 𝑦 NOUN cana-5910 181 10 , , PUNCT cana-5910 181 11 𝑑 𝑑 NOUN cana-5910 181 12 ) ) PUNCT cana-5910 181 13 = = PUNCT cana-5910 182 1 𝑐1 𝑐1 NOUN cana-5910 182 2 𝑑 𝑑 X cana-5910 182 3 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 182 4 ( ( PUNCT cana-5910 182 5 π‘₯2 π‘₯2 NOUN cana-5910 182 6 + + CCONJ cana-5910 182 7 𝑦2 𝑦2 NOUN cana-5910 182 8 ) ) PUNCT cana-5910 182 9 4𝛼 4𝛼 PROPN cana-5910 182 10 𝑑 𝑑 PROPN cana-5910 182 11 . . PROPN cana-5910 182 12 3.3 3.3 NUM cana-5910 182 13 physical physical ADJ cana-5910 182 14 interpretation interpretation NOUN cana-5910 182 15 the the DET cana-5910 182 16 solution solution NOUN cana-5910 182 17 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 182 18 , , PUNCT cana-5910 182 19 𝑦 𝑦 NOUN cana-5910 182 20 , , PUNCT cana-5910 182 21 𝑑 𝑑 NOUN cana-5910 182 22 ) ) PUNCT cana-5910 182 23 = = PUNCT cana-5910 183 1 𝑐1 𝑐1 NOUN cana-5910 183 2 𝑑 𝑑 X cana-5910 183 3 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 183 4 ( ( PUNCT cana-5910 183 5 π‘₯2 π‘₯2 NOUN cana-5910 183 6 + + CCONJ cana-5910 183 7 𝑦2 𝑦2 NOUN cana-5910 183 8 ) ) PUNCT cana-5910 183 9 4𝛼 4𝛼 PROPN cana-5910 183 10 𝑑 𝑑 PROPN cana-5910 183 11 is be AUX cana-5910 183 12 the the DET cana-5910 183 13 fundamental fundamental ADJ cana-5910 183 14 solution solution NOUN cana-5910 183 15 ( ( PUNCT cana-5910 183 16 gaussian gaussian ADJ cana-5910 183 17 kernel kernel NOUN cana-5910 183 18 ) ) PUNCT cana-5910 183 19 for for ADP cana-5910 183 20 the the DET cana-5910 183 21 twodimensional twodimensional ADJ cana-5910 183 22 heat heat NOUN cana-5910 183 23 equation equation NOUN cana-5910 183 24 , , PUNCT cana-5910 183 25 describing describe VERB cana-5910 183 26 the the DET cana-5910 183 27 diffusion diffusion NOUN cana-5910 183 28 of of ADP cana-5910 183 29 heat heat NOUN cana-5910 183 30 from from ADP cana-5910 183 31 an an DET cana-5910 183 32 instantaneous instantaneous ADJ cana-5910 183 33 point point NOUN cana-5910 183 34 source source NOUN cana-5910 183 35 at at ADP cana-5910 183 36 ( ( PUNCT cana-5910 183 37 π‘₯ π‘₯ NOUN cana-5910 183 38 , , PUNCT cana-5910 183 39 𝑦 𝑦 NOUN cana-5910 183 40 ) ) PUNCT cana-5910 183 41 = = SYM cana-5910 183 42 ( ( PUNCT cana-5910 183 43 0 0 NUM cana-5910 183 44 , , PUNCT cana-5910 183 45 0 0 NUM cana-5910 183 46 ) ) PUNCT cana-5910 183 47 π‘Žπ‘‘ π‘Žπ‘‘ ADP cana-5910 183 48 𝑑 𝑑 PROPN cana-5910 183 49 = = NOUN cana-5910 183 50 0 0 X cana-5910 183 51 . . PUNCT cana-5910 184 1 the the DET cana-5910 184 2 factor factor NOUN cana-5910 184 3 1 1 NUM cana-5910 184 4 / / SYM cana-5910 184 5 t t PROPN cana-5910 184 6 reflects reflect VERB cana-5910 184 7 the the DET cana-5910 184 8 spreading spreading NOUN cana-5910 184 9 of of ADP cana-5910 184 10 heat heat NOUN cana-5910 184 11 over over ADP cana-5910 184 12 time time NOUN cana-5910 184 13 , , PUNCT cana-5910 184 14 and and CCONJ cana-5910 184 15 the the DET cana-5910 184 16 exponential exponential ADJ cana-5910 184 17 term term NOUN cana-5910 184 18 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 184 19 ( ( PUNCT cana-5910 184 20 π‘₯2 π‘₯2 NOUN cana-5910 184 21 + + CCONJ cana-5910 184 22 𝑦2 𝑦2 NOUN cana-5910 184 23 ) ) PUNCT cana-5910 184 24 4𝛼 4𝛼 PROPN cana-5910 184 25 𝑑 𝑑 PROPN cana-5910 184 26 indicates indicate VERB cana-5910 184 27 a a DET cana-5910 184 28 radially radially ADV cana-5910 184 29 symmetric symmetric ADJ cana-5910 184 30 temperature temperature NOUN cana-5910 184 31 distribution distribution NOUN cana-5910 184 32 that that PRON cana-5910 184 33 decays decay VERB cana-5910 184 34 with with ADP cana-5910 184 35 distance distance NOUN cana-5910 184 36 . . PUNCT cana-5910 185 1 this this DET cana-5910 185 2 solution solution NOUN cana-5910 185 3 is be AUX cana-5910 185 4 widely widely ADV cana-5910 185 5 used use VERB cana-5910 185 6 in in ADP cana-5910 185 7 heat heat NOUN cana-5910 185 8 conduction conduction NOUN cana-5910 185 9 problems problem NOUN cana-5910 185 10 , , PUNCT cana-5910 185 11 such such ADJ cana-5910 185 12 as as ADP cana-5910 185 13 modeling model VERB cana-5910 185 14 temperature temperature NOUN cana-5910 185 15 in in ADP cana-5910 185 16 a a DET cana-5910 185 17 plane plane NOUN cana-5910 185 18 following follow VERB cana-5910 185 19 a a DET cana-5910 185 20 localized localize VERB cana-5910 185 21 heat heat NOUN cana-5910 185 22 pulse pulse NOUN cana-5910 185 23 . . PUNCT cana-5910 186 1 the the DET cana-5910 186 2 constant constant ADJ cana-5910 186 3 𝑐1 𝑐1 NOUN cana-5910 186 4 is be AUX cana-5910 186 5 determined determine VERB cana-5910 186 6 by by ADP cana-5910 186 7 initial initial ADJ cana-5910 186 8 conditions condition NOUN cana-5910 186 9 , , PUNCT cana-5910 186 10 typically typically ADV cana-5910 186 11 normalized normalize VERB cana-5910 186 12 to to PART cana-5910 186 13 conserve conserve VERB cana-5910 186 14 total total ADJ cana-5910 186 15 heat heat NOUN cana-5910 186 16 . . PUNCT cana-5910 187 1 1117 1117 NUM cana-5910 187 2 https://internationalpubls.com https://internationalpubls.com X cana-5910 187 3 3.4 3.4 NUM cana-5910 187 4 application application NOUN cana-5910 187 5 to to ADP cana-5910 187 6 initial initial ADJ cana-5910 187 7 conditions condition NOUN cana-5910 187 8 for for ADP cana-5910 187 9 an an DET cana-5910 187 10 initial initial ADJ cana-5910 187 11 condition condition NOUN cana-5910 187 12 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 187 13 , , PUNCT cana-5910 187 14 𝑦 𝑦 NOUN cana-5910 187 15 , , PUNCT cana-5910 187 16 0 0 NUM cana-5910 187 17 ) ) PUNCT cana-5910 187 18 = = SYM cana-5910 188 1 𝛿(π‘₯ 𝛿(π‘₯ NOUN cana-5910 188 2 , , PUNCT cana-5910 188 3 𝑦 𝑦 NOUN cana-5910 188 4 ) ) PUNCT cana-5910 188 5 ( ( PUNCT cana-5910 188 6 dirac dirac NOUN cana-5910 188 7 delta delta NOUN cana-5910 188 8 function function NOUN cana-5910 188 9 at at ADP cana-5910 188 10 the the DET cana-5910 188 11 origin origin NOUN cana-5910 188 12 ) ) PUNCT cana-5910 188 13 , , PUNCT cana-5910 188 14 the the DET cana-5910 188 15 solution solution NOUN cana-5910 188 16 is be AUX cana-5910 188 17 : : PUNCT cana-5910 188 18 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 188 19 , , PUNCT cana-5910 188 20 𝑦 𝑦 NOUN cana-5910 188 21 , , PUNCT cana-5910 188 22 𝑑 𝑑 NOUN cana-5910 188 23 ) ) PUNCT cana-5910 188 24 = = SYM cana-5910 188 25 1 1 NUM cana-5910 188 26 4πœ‹ 4πœ‹ NOUN cana-5910 188 27 𝛼 𝛼 X cana-5910 188 28 𝑑 𝑑 X cana-5910 188 29 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 188 30 ( ( PUNCT cana-5910 188 31 π‘₯2 π‘₯2 NOUN cana-5910 188 32 + + CCONJ cana-5910 188 33 𝑦2 𝑦2 NOUN cana-5910 188 34 ) ) PUNCT cana-5910 188 35 4𝛼 4𝛼 PROPN cana-5910 188 36 𝑑 𝑑 NOUN cana-5910 188 37 where where SCONJ cana-5910 188 38 the the DET cana-5910 188 39 constant constant ADJ cana-5910 188 40 1 1 NUM cana-5910 188 41 4πœ‹ 4πœ‹ NOUN cana-5910 188 42 𝛼 𝛼 PRON cana-5910 188 43 𝑑 𝑑 NOUN cana-5910 188 44 ensures ensure VERB cana-5910 188 45 the the DET cana-5910 188 46 integral integral NOUN cana-5910 188 47 of of ADP cana-5910 188 48 u u NOUN cana-5910 188 49 over over ADP cana-5910 188 50 the the DET cana-5910 188 51 plane plane NOUN cana-5910 188 52 equals equal VERB cana-5910 188 53 1 1 NUM cana-5910 188 54 , , PUNCT cana-5910 188 55 conserving conserve VERB cana-5910 188 56 the the DET cana-5910 188 57 initial initial ADJ cana-5910 188 58 heat heat NOUN cana-5910 188 59 . . PUNCT cana-5910 189 1 this this DET cana-5910 189 2 solution solution NOUN cana-5910 189 3 is be AUX cana-5910 189 4 verified verify VERB cana-5910 189 5 by by ADP cana-5910 189 6 checking check VERB cana-5910 189 7 the the DET cana-5910 189 8 heat heat NOUN cana-5910 189 9 equation equation NOUN cana-5910 189 10 and and CCONJ cana-5910 189 11 the the DET cana-5910 189 12 initial initial ADJ cana-5910 189 13 condition condition NOUN cana-5910 189 14 as as ADP cana-5910 189 15 𝑑 𝑑 PROPN cana-5910 189 16 β†’ β†’ SYM cana-5910 189 17 0 0 NUM cana-5910 189 18 + + NOUN cana-5910 189 19 . . NOUN cana-5910 189 20 4 4 X cana-5910 189 21 . . PUNCT cana-5910 190 1 examples example NOUN cana-5910 190 2 4.1 4.1 NUM cana-5910 190 3 solve solve VERB cana-5910 190 4 the the DET cana-5910 190 5 heat heat NOUN cana-5910 190 6 equation equation NOUN cana-5910 191 1 πœ•π‘’ πœ•π‘’ NOUN cana-5910 191 2 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 191 3 = = NOUN cana-5910 191 4 2 2 NUM cana-5910 191 5 ( ( PUNCT cana-5910 191 6 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 191 7 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 191 8 + + CCONJ cana-5910 191 9 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 191 10 πœ•π‘¦2 πœ•π‘¦2 NUM cana-5910 191 11 ) ) PUNCT cana-5910 191 12 by by ADP cana-5910 191 13 lie lie NOUN cana-5910 191 14 symmetry symmetry NOUN cana-5910 191 15 theory theory NOUN cana-5910 191 16 when when SCONJ cana-5910 191 17 𝑒 𝑒 X cana-5910 191 18 = = SYM cana-5910 191 19 0 0 NUM cana-5910 191 20 π‘€β„Žπ‘’π‘› π‘€β„Žπ‘’π‘› VERB cana-5910 191 21 𝑑 𝑑 PROPN cana-5910 191 22 = = SYM cana-5910 191 23 ∞ ∞ PROPN cana-5910 191 24 , , PUNCT cana-5910 191 25 π‘₯ π‘₯ PROPN cana-5910 191 26 = = SYM cana-5910 191 27 0 0 NUM cana-5910 191 28 π‘œπ‘Ÿ π‘œπ‘Ÿ ADP cana-5910 191 29 𝑙 𝑙 X cana-5910 191 30 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-5910 191 31 𝑦 𝑦 PROPN cana-5910 191 32 = = SYM cana-5910 191 33 0 0 NUM cana-5910 191 34 π‘œπ‘Ÿ π‘œπ‘Ÿ ADP cana-5910 191 35 𝑙 𝑙 DET cana-5910 191 36 solution solution NOUN cana-5910 191 37 to to PART cana-5910 191 38 solve solve VERB cana-5910 191 39 the the DET cana-5910 191 40 given give VERB cana-5910 191 41 partial partial ADJ cana-5910 191 42 differential differential NOUN cana-5910 191 43 equation equation NOUN cana-5910 191 44 ( ( PUNCT cana-5910 191 45 pde pde NOUN cana-5910 191 46 ) ) PUNCT cana-5910 191 47 using use VERB cana-5910 191 48 lie lie NOUN cana-5910 191 49 symmetry symmetry NOUN cana-5910 191 50 theory theory NOUN cana-5910 191 51 , , PUNCT cana-5910 191 52 we we PRON cana-5910 191 53 need need VERB cana-5910 191 54 to to PART cana-5910 191 55 carefully carefully ADV cana-5910 191 56 analyse analyse VERB cana-5910 191 57 the the DET cana-5910 191 58 equation equation NOUN cana-5910 191 59 , , PUNCT cana-5910 191 60 boundary boundary ADJ cana-5910 191 61 conditions condition NOUN cana-5910 191 62 , , PUNCT cana-5910 191 63 and and CCONJ cana-5910 191 64 apply apply VERB cana-5910 191 65 the the DET cana-5910 191 66 lie lie NOUN cana-5910 191 67 group group NOUN cana-5910 191 68 method method NOUN cana-5910 191 69 systematically systematically ADV cana-5910 191 70 problem problem NOUN cana-5910 191 71 statement statement NOUN cana-5910 191 72 we we PRON cana-5910 191 73 are be AUX cana-5910 191 74 tasked task VERB cana-5910 191 75 with with ADP cana-5910 191 76 solving solve VERB cana-5910 191 77 the the DET cana-5910 191 78 pde pde NOUN cana-5910 191 79 : : PUNCT cana-5910 192 1 πœ•π‘’ πœ•π‘’ NOUN cana-5910 192 2 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 192 3 = = NOUN cana-5910 192 4 2 2 NUM cana-5910 192 5 ( ( PUNCT cana-5910 192 6 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 192 7 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 192 8 + + CCONJ cana-5910 192 9 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 192 10 πœ•π‘¦2 πœ•π‘¦2 NOUN cana-5910 192 11 ) ) PUNCT cana-5910 192 12 with with ADP cana-5910 192 13 boundary boundary ADJ cana-5910 192 14 conditions condition NOUN cana-5910 192 15 : : PUNCT cana-5910 192 16 𝑒 𝑒 X cana-5910 192 17 = = SYM cana-5910 192 18 0 0 NUM cana-5910 192 19 π‘€β„Žπ‘’π‘› π‘€β„Žπ‘’π‘› VERB cana-5910 192 20 𝑑 𝑑 PROPN cana-5910 192 21 β†’ β†’ SYM cana-5910 192 22 ∞ ∞ PROPN cana-5910 192 23 , , PUNCT cana-5910 192 24 π‘₯ π‘₯ PROPN cana-5910 192 25 = = SYM cana-5910 192 26 0 0 NUM cana-5910 192 27 π‘œπ‘Ÿ π‘œπ‘Ÿ PRON cana-5910 192 28 π‘₯ π‘₯ NOUN cana-5910 192 29 = = SYM cana-5910 192 30 𝑙 𝑙 PROPN cana-5910 192 31 , , PUNCT cana-5910 192 32 𝑦 𝑦 NOUN cana-5910 192 33 = = SYM cana-5910 192 34 0 0 NUM cana-5910 192 35 π‘œπ‘Ÿ π‘œπ‘Ÿ PROPN cana-5910 192 36 𝑦 𝑦 NOUN cana-5910 192 37 = = PUNCT cana-5910 193 1 𝑙. 𝑙. NOUN cana-5910 194 1 this this PRON cana-5910 194 2 is be AUX cana-5910 194 3 a a DET cana-5910 194 4 two two NUM cana-5910 194 5 - - PUNCT cana-5910 194 6 dimensional dimensional ADJ cana-5910 194 7 heat heat NOUN cana-5910 194 8 equation equation NOUN cana-5910 194 9 with with ADP cana-5910 194 10 a a DET cana-5910 194 11 diffusion diffusion NOUN cana-5910 194 12 coefficient coefficient NOUN cana-5910 194 13 of of ADP cana-5910 194 14 2 2 NUM cana-5910 194 15 , , PUNCT cana-5910 194 16 defined define VERB cana-5910 194 17 on on ADP cana-5910 194 18 the the DET cana-5910 194 19 domain domain NOUN cana-5910 194 20 0 0 PUNCT cana-5910 194 21 < < X cana-5910 194 22 π‘₯ π‘₯ X cana-5910 194 23 < < X cana-5910 194 24 𝑙 𝑙 X cana-5910 194 25 , , PUNCT cana-5910 194 26 0 0 PUNCT cana-5910 194 27 < < X cana-5910 194 28 𝑦 𝑦 X cana-5910 194 29 < < X cana-5910 194 30 𝑙 𝑙 X cana-5910 194 31 , , PUNCT cana-5910 194 32 with with ADP cana-5910 194 33 homogeneous homogeneous ADJ cana-5910 194 34 dirichlet dirichlet PROPN cana-5910 194 35 boundary boundary ADJ cana-5910 194 36 conditions condition NOUN cana-5910 194 37 and and CCONJ cana-5910 194 38 a a DET cana-5910 194 39 condition condition NOUN cana-5910 194 40 at at ADP cana-5910 194 41 infinite infinite ADJ cana-5910 194 42 time time NOUN cana-5910 194 43 . . PUNCT cana-5910 195 1 we we PRON cana-5910 195 2 will will AUX cana-5910 195 3 use use VERB cana-5910 195 4 lie lie NOUN cana-5910 195 5 symmetry symmetry NOUN cana-5910 195 6 theory theory NOUN cana-5910 195 7 to to PART cana-5910 195 8 find find VERB cana-5910 195 9 symmetry symmetry NOUN cana-5910 195 10 reductions reduction NOUN cana-5910 195 11 and and CCONJ cana-5910 195 12 seek seek VERB cana-5910 195 13 solutions solution NOUN cana-5910 195 14 . . PUNCT cana-5910 196 1 step step NOUN cana-5910 196 2 1 1 NUM cana-5910 196 3 : : PUNCT cana-5910 196 4 formulate formulate VERB cana-5910 196 5 the the DET cana-5910 196 6 pde pde NOUN cana-5910 196 7 the the DET cana-5910 196 8 given give VERB cana-5910 196 9 pde pde NOUN cana-5910 196 10 is be AUX cana-5910 196 11 : : PUNCT cana-5910 196 12 πœ•π‘’ πœ•π‘’ NOUN cana-5910 196 13 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 196 14 = = NOUN cana-5910 196 15 2 2 NUM cana-5910 196 16 ( ( PUNCT cana-5910 196 17 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 196 18 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 196 19 + + CCONJ cana-5910 196 20 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 196 21 πœ•π‘¦2 πœ•π‘¦2 NOUN cana-5910 196 22 ) ) PUNCT cana-5910 196 23 this this PRON cana-5910 196 24 can can AUX cana-5910 196 25 be be AUX cana-5910 196 26 written write VERB cana-5910 196 27 as as ADP cana-5910 196 28 : : PUNCT cana-5910 196 29 𝑒𝑑 𝑒𝑑 NOUN cana-5910 196 30 βˆ’ βˆ’ PROPN cana-5910 196 31 2(𝑒π‘₯π‘₯ 2(𝑒π‘₯π‘₯ NOUN cana-5910 196 32 + + CCONJ cana-5910 196 33 𝑒𝑦𝑦 𝑒𝑦𝑦 ADJ cana-5910 196 34 ) ) PUNCT cana-5910 196 35 = = SYM cana-5910 196 36 0 0 NUM cana-5910 196 37 , , PUNCT cana-5910 196 38 where where SCONJ cana-5910 196 39 subscripts subscript NOUN cana-5910 196 40 denote denote VERB cana-5910 196 41 partial partial ADJ cana-5910 196 42 derivatives derivative NOUN cana-5910 196 43 : : PUNCT cana-5910 196 44 𝑒𝑑 𝑒𝑑 NOUN cana-5910 196 45 = = PUNCT cana-5910 196 46 πœ•π‘’ πœ•π‘’ PROPN cana-5910 196 47 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 196 48 , , PUNCT cana-5910 196 49 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 196 50 = = SYM cana-5910 196 51 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 196 52 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 196 53 , , PUNCT cana-5910 196 54 𝑒𝑦𝑦 𝑒𝑦𝑦 ADV cana-5910 196 55 = = SYM cana-5910 196 56 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 196 57 πœ•π‘¦2 πœ•π‘¦2 PROPN cana-5910 196 58 . . PUNCT cana-5910 197 1 the the DET cana-5910 197 2 boundary boundary ADJ cana-5910 197 3 conditions condition NOUN cana-5910 197 4 are be AUX cana-5910 197 5 : : PUNCT cana-5910 197 6 β€’ β€’ ADP cana-5910 197 7 𝑒(𝑑 𝑒(𝑑 NOUN cana-5910 197 8 , , PUNCT cana-5910 197 9 π‘₯ π‘₯ NOUN cana-5910 197 10 , , PUNCT cana-5910 197 11 𝑦 𝑦 NOUN cana-5910 197 12 ) ) PUNCT cana-5910 197 13 = = SYM cana-5910 198 1 0 0 NUM cana-5910 199 1 π‘Žπ‘‘ π‘Žπ‘‘ PRON cana-5910 199 2 π‘₯ π‘₯ NOUN cana-5910 199 3 = = SYM cana-5910 199 4 0 0 NUM cana-5910 199 5 , , PUNCT cana-5910 199 6 π‘₯ π‘₯ NOUN cana-5910 199 7 = = SYM cana-5910 199 8 𝑙 𝑙 PROPN cana-5910 199 9 , , PUNCT cana-5910 199 10 𝑦 𝑦 NOUN cana-5910 199 11 = = SYM cana-5910 199 12 0 0 NUM cana-5910 199 13 , , PUNCT cana-5910 199 14 𝑦 𝑦 NOUN cana-5910 199 15 = = SYM cana-5910 199 16 𝑙 𝑙 PRON cana-5910 200 1 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ NOUN cana-5910 200 2 π‘Žπ‘™π‘™ π‘Žπ‘™π‘™ NOUN cana-5910 200 3 𝑑. 𝑑. NOUN cana-5910 200 4 β€’ β€’ PRON cana-5910 200 5 𝑒(𝑑 𝑒(𝑑 NOUN cana-5910 200 6 , , PUNCT cana-5910 200 7 π‘₯ π‘₯ NOUN cana-5910 200 8 , , PUNCT cana-5910 200 9 𝑦 𝑦 NOUN cana-5910 200 10 ) ) PUNCT cana-5910 200 11 β†’ β†’ SYM cana-5910 200 12 0 0 NUM cana-5910 200 13 π‘Žπ‘  π‘Žπ‘  PROPN cana-5910 200 14 𝑑 𝑑 PROPN cana-5910 200 15 β†’ β†’ SYM cana-5910 200 16 ∞ ∞ NUM cana-5910 200 17 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-5910 200 18 0 0 NUM cana-5910 200 19 < < X cana-5910 200 20 π‘₯ π‘₯ X cana-5910 200 21 < < X cana-5910 200 22 𝑙 𝑙 X cana-5910 200 23 , , PUNCT cana-5910 200 24 0 0 PUNCT cana-5910 200 25 < < X cana-5910 200 26 𝑦 𝑦 X cana-5910 200 27 < < X cana-5910 200 28 𝑙. 𝑙. NOUN cana-5910 200 29 our our PRON cana-5910 200 30 goal goal NOUN cana-5910 200 31 is be AUX cana-5910 200 32 to to PART cana-5910 200 33 find find VERB cana-5910 200 34 lie lie NOUN cana-5910 200 35 point point NOUN cana-5910 200 36 symmetries symmetry NOUN cana-5910 200 37 of of ADP cana-5910 200 38 the the DET cana-5910 200 39 pde pde NOUN cana-5910 200 40 , , PUNCT cana-5910 200 41 use use VERB cana-5910 200 42 them they PRON cana-5910 200 43 to to PART cana-5910 200 44 reduce reduce VERB cana-5910 200 45 the the DET cana-5910 200 46 pde pde NOUN cana-5910 200 47 to to ADP cana-5910 200 48 an an DET cana-5910 200 49 ordinary ordinary ADJ cana-5910 200 50 differential differential ADJ cana-5910 200 51 equation equation NOUN cana-5910 200 52 ( ( PUNCT cana-5910 200 53 ode ode PROPN cana-5910 200 54 ) ) PUNCT cana-5910 200 55 or or CCONJ cana-5910 200 56 simpler simple ADJ cana-5910 200 57 pde pde NOUN cana-5910 200 58 , , PUNCT cana-5910 200 59 and and CCONJ cana-5910 200 60 solve solve VERB cana-5910 200 61 while while SCONJ cana-5910 200 62 respecting respect VERB cana-5910 200 63 the the DET cana-5910 200 64 boundary boundary ADJ cana-5910 200 65 conditions condition NOUN cana-5910 200 66 . . PUNCT cana-5910 201 1 step step NOUN cana-5910 201 2 2 2 NUM cana-5910 201 3 : : PUNCT cana-5910 201 4 lie lie NOUN cana-5910 201 5 symmetry symmetry NOUN cana-5910 201 6 analysis analysis NOUN cana-5910 201 7 lie lie NOUN cana-5910 201 8 symmetry symmetry NOUN cana-5910 201 9 theory theory NOUN cana-5910 201 10 involves involve VERB cana-5910 201 11 finding find VERB cana-5910 201 12 infinitesimal infinitesimal ADJ cana-5910 201 13 transformations transformation NOUN cana-5910 201 14 that that PRON cana-5910 201 15 leave leave VERB cana-5910 201 16 the the DET cana-5910 201 17 pde pde NOUN cana-5910 201 18 invariant invariant ADJ cana-5910 201 19 . . PUNCT cana-5910 202 1 consider consider VERB cana-5910 202 2 a a DET cana-5910 202 3 oneparameter oneparameter ADJ cana-5910 202 4 lie lie NOUN cana-5910 202 5 group group NOUN cana-5910 202 6 of of ADP cana-5910 202 7 transformations transformation NOUN cana-5910 202 8 : : PUNCT cana-5910 202 9 𝑑 𝑑 X cana-5910 202 10 βˆ— βˆ— NOUN cana-5910 202 11 = = SYM cana-5910 202 12 𝑑 𝑑 PROPN cana-5910 202 13 + + NOUN cana-5910 202 14 νœ€πœ(𝑑 νœ€πœ(𝑑 NUM cana-5910 202 15 , , PUNCT cana-5910 202 16 π‘₯ π‘₯ NOUN cana-5910 202 17 , , PUNCT cana-5910 202 18 𝑦 𝑦 NOUN cana-5910 202 19 , , PUNCT cana-5910 202 20 𝑒 𝑒 X cana-5910 202 21 ) ) PUNCT cana-5910 202 22 + + CCONJ cana-5910 202 23 𝑂(νœ€Β² 𝑂(νœ€Β² NOUN cana-5910 202 24 ) ) PUNCT cana-5910 202 25 , , PUNCT cana-5910 203 1 π‘₯ π‘₯ NOUN cana-5910 203 2 βˆ— βˆ— NOUN cana-5910 203 3 = = PUNCT cana-5910 204 1 π‘₯ π‘₯ NOUN cana-5910 205 1 + + NOUN cana-5910 205 2 νœ€πœ‰(𝑑 νœ€πœ‰(𝑑 NUM cana-5910 205 3 , , PUNCT cana-5910 205 4 π‘₯ π‘₯ X cana-5910 205 5 , , PUNCT cana-5910 205 6 𝑦 𝑦 NOUN cana-5910 205 7 , , PUNCT cana-5910 205 8 𝑒 𝑒 X cana-5910 205 9 ) ) PUNCT cana-5910 205 10 + + CCONJ cana-5910 205 11 𝑂(νœ€Β² 𝑂(νœ€Β² NOUN cana-5910 205 12 ) ) PUNCT cana-5910 205 13 , , PUNCT cana-5910 205 14 𝑦 𝑦 NOUN cana-5910 205 15 βˆ— βˆ— NOUN cana-5910 205 16 = = SYM cana-5910 205 17 𝑦 𝑦 NOUN cana-5910 205 18 + + NUM cana-5910 205 19 νœ€πœ‚(𝑑 νœ€πœ‚(𝑑 NUM cana-5910 205 20 , , PUNCT cana-5910 205 21 π‘₯ π‘₯ PRON cana-5910 205 22 , , PUNCT cana-5910 205 23 𝑦 𝑦 NOUN cana-5910 205 24 , , PUNCT cana-5910 205 25 𝑒 𝑒 X cana-5910 205 26 ) ) PUNCT cana-5910 205 27 + + CCONJ cana-5910 205 28 𝑂(νœ€Β² 𝑂(νœ€Β² NOUN cana-5910 205 29 ) ) PUNCT cana-5910 205 30 , , PUNCT cana-5910 205 31 1118 1118 NUM cana-5910 205 32 https://internationalpubls.com https://internationalpubls.com X cana-5910 205 33 𝑒 𝑒 ADP cana-5910 205 34 βˆ— βˆ— NOUN cana-5910 205 35 = = SYM cana-5910 205 36 𝑒 𝑒 NOUN cana-5910 205 37 + + X cana-5910 205 38 νœ€πœ‘(𝑑 νœ€πœ‘(𝑑 NUM cana-5910 205 39 , , PUNCT cana-5910 205 40 π‘₯ π‘₯ NOUN cana-5910 205 41 , , PUNCT cana-5910 205 42 𝑦 𝑦 NOUN cana-5910 205 43 , , PUNCT cana-5910 205 44 𝑒 𝑒 X cana-5910 205 45 ) ) PUNCT cana-5910 205 46 + + CCONJ cana-5910 205 47 𝑂(νœ€Β² 𝑂(νœ€Β² NOUN cana-5910 205 48 ) ) PUNCT cana-5910 205 49 , , PUNCT cana-5910 205 50 where where SCONJ cana-5910 205 51 𝜏 𝜏 X cana-5910 205 52 , , PUNCT cana-5910 205 53 πœ‰ πœ‰ X cana-5910 205 54 , , PUNCT cana-5910 205 55 πœ‚ πœ‚ NOUN cana-5910 205 56 , , PUNCT cana-5910 205 57 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-5910 205 58 πœ‘ πœ‘ PROPN cana-5910 205 59 are be AUX cana-5910 205 60 the the DET cana-5910 205 61 infinitesimals infinitesimal NOUN cana-5910 205 62 corresponding correspond VERB cana-5910 205 63 to to ADP cana-5910 205 64 𝑑 𝑑 PROPN cana-5910 205 65 , , PUNCT cana-5910 205 66 π‘₯ π‘₯ NOUN cana-5910 205 67 , , PUNCT cana-5910 205 68 𝑦 𝑦 NOUN cana-5910 205 69 , , PUNCT cana-5910 205 70 and and CCONJ cana-5910 205 71 𝑒 𝑒 NOUN cana-5910 205 72 , , PUNCT cana-5910 205 73 and and CCONJ cana-5910 205 74 νœ€ νœ€ PRON cana-5910 205 75 is be AUX cana-5910 205 76 a a DET cana-5910 205 77 small small ADJ cana-5910 205 78 parameter parameter NOUN cana-5910 205 79 . . PUNCT cana-5910 206 1 the the DET cana-5910 206 2 infinitesimal infinitesimal ADJ cana-5910 206 3 generator generator NOUN cana-5910 206 4 is be AUX cana-5910 206 5 : : PUNCT cana-5910 206 6 𝑋 𝑋 NOUN cana-5910 206 7 = = SYM cana-5910 206 8 𝜏 𝜏 PROPN cana-5910 206 9 πœ• πœ• PROPN cana-5910 206 10 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 206 11 + + CCONJ cana-5910 206 12 πœ‰ πœ‰ PROPN cana-5910 206 13 πœ• πœ• NOUN cana-5910 206 14 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 206 15 + + CCONJ cana-5910 206 16 πœ‚ πœ‚ DET cana-5910 206 17 πœ• πœ• NOUN cana-5910 206 18 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 206 19 + + CCONJ cana-5910 206 20 πœ‘ πœ‘ PROPN cana-5910 206 21 πœ• πœ• NOUN cana-5910 206 22 πœ•π‘’ πœ•π‘’ NOUN cana-5910 206 23 . . PUNCT cana-5910 207 1 to to PART cana-5910 207 2 find find VERB cana-5910 207 3 the the DET cana-5910 207 4 symmetries symmetry NOUN cana-5910 207 5 , , PUNCT cana-5910 207 6 we we PRON cana-5910 207 7 need need VERB cana-5910 207 8 the the DET cana-5910 207 9 pde pde NOUN cana-5910 207 10 to to PART cana-5910 207 11 be be AUX cana-5910 207 12 invariant invariant ADJ cana-5910 207 13 under under ADP cana-5910 207 14 these these DET cana-5910 207 15 transformations transformation NOUN cana-5910 207 16 . . PUNCT cana-5910 208 1 this this PRON cana-5910 208 2 requires require VERB cana-5910 208 3 computing compute VERB cana-5910 208 4 the the DET cana-5910 208 5 prolonged prolonged ADJ cana-5910 208 6 generator generator NOUN cana-5910 208 7 to to PART cana-5910 208 8 include include VERB cana-5910 208 9 derivatives derivative NOUN cana-5910 208 10 up up ADP cana-5910 208 11 to to ADP cana-5910 208 12 the the DET cana-5910 208 13 second second ADJ cana-5910 208 14 order order NOUN cana-5910 208 15 , , PUNCT cana-5910 208 16 since since SCONJ cana-5910 208 17 the the DET cana-5910 208 18 pde pde NOUN cana-5910 208 19 involves involve VERB cana-5910 208 20 𝑒𝑑 𝑒𝑑 NOUN cana-5910 208 21 , , PUNCT cana-5910 208 22 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 208 23 , , PUNCT cana-5910 208 24 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ X cana-5910 208 25 𝑒𝑦𝑦. 𝑒𝑦𝑦. X cana-5910 208 26 the the DET cana-5910 208 27 prolonged prolonged ADJ cana-5910 208 28 generator generator NOUN cana-5910 208 29 is be AUX cana-5910 208 30 : : PUNCT cana-5910 208 31 𝑋2 𝑋2 ADJ cana-5910 208 32 = = SYM cana-5910 208 33 𝑋 𝑋 NOUN cana-5910 208 34 + + CCONJ cana-5910 208 35 πœ‘π‘‘ πœ‘π‘‘ ADP cana-5910 208 36 πœ• πœ• NOUN cana-5910 208 37 πœ•π‘’π‘‘ πœ•π‘’π‘‘ PUNCT cana-5910 208 38 + + CCONJ cana-5910 208 39 πœ‘π‘₯ πœ‘π‘₯ PROPN cana-5910 208 40 πœ• πœ• PROPN cana-5910 208 41 πœ•π‘’π‘₯ πœ•π‘’π‘₯ PRON cana-5910 208 42 + + PROPN cana-5910 208 43 πœ‘π‘¦ πœ‘π‘¦ PROPN cana-5910 208 44 πœ• πœ• NOUN cana-5910 208 45 πœ•π‘’π‘¦ πœ•π‘’π‘¦ PUNCT cana-5910 209 1 + + CCONJ cana-5910 209 2 πœ‘π‘₯π‘₯ πœ‘π‘₯π‘₯ PROPN cana-5910 209 3 πœ• πœ• NOUN cana-5910 209 4 πœ•π‘’π‘₯π‘₯ πœ•π‘’π‘₯π‘₯ VERB cana-5910 209 5 + + CCONJ cana-5910 209 6 πœ‘π‘¦π‘¦ πœ‘π‘¦π‘¦ ADP cana-5910 209 7 πœ• πœ• NOUN cana-5910 209 8 πœ•π‘’π‘¦π‘¦ πœ•π‘’π‘¦π‘¦ VERB cana-5910 209 9 + + PUNCT cana-5910 209 10 . . PUNCT cana-5910 209 11 . . PUNCT cana-5910 210 1 . . PUNCT cana-5910 211 1 , , PUNCT cana-5910 211 2 where where SCONJ cana-5910 211 3 πœ‘π‘‘ πœ‘π‘‘ ADP cana-5910 211 4 , , PUNCT cana-5910 211 5 πœ‘π‘₯ πœ‘π‘₯ INTJ cana-5910 211 6 , , PUNCT cana-5910 211 7 πœ‘π‘¦ πœ‘π‘¦ PROPN cana-5910 211 8 , , PUNCT cana-5910 211 9 πœ‘π‘₯π‘₯ πœ‘π‘₯π‘₯ NOUN cana-5910 211 10 , , PUNCT cana-5910 211 11 πœ‘π‘¦π‘¦ πœ‘π‘¦π‘¦ X cana-5910 211 12 are be AUX cana-5910 211 13 the the DET cana-5910 211 14 extended extended ADJ cana-5910 211 15 infinitesimals infinitesimal NOUN cana-5910 211 16 . . PUNCT cana-5910 212 1 the the DET cana-5910 212 2 invariance invariance NOUN cana-5910 212 3 condition condition NOUN cana-5910 212 4 is be AUX cana-5910 212 5 applied apply VERB cana-5910 212 6 to to ADP cana-5910 212 7 the the DET cana-5910 212 8 pde pde NOUN cana-5910 212 9 : : PUNCT cana-5910 212 10 𝑒𝑑 𝑒𝑑 NOUN cana-5910 212 11 βˆ’ βˆ’ PROPN cana-5910 212 12 2(𝑒π‘₯π‘₯ 2(𝑒π‘₯π‘₯ NOUN cana-5910 212 13 + + CCONJ cana-5910 212 14 𝑒𝑦𝑦 𝑒𝑦𝑦 X cana-5910 212 15 ) ) PUNCT cana-5910 212 16 = = SYM cana-5910 213 1 0 0 X cana-5910 213 2 . . PUNCT cana-5910 213 3 applying apply VERB cana-5910 213 4 the the DET cana-5910 213 5 second second ADJ cana-5910 213 6 prolongation prolongation NOUN cana-5910 213 7 𝑋2 𝑋2 VERB cana-5910 213 8 to to ADP cana-5910 213 9 the the DET cana-5910 213 10 pde pde NOUN cana-5910 213 11 gives give VERB cana-5910 213 12 : : PUNCT cana-5910 213 13 𝑋2 𝑋2 PROPN cana-5910 213 14 ( ( PUNCT cana-5910 213 15 𝑒𝑑 𝑒𝑑 NOUN cana-5910 213 16 βˆ’ βˆ’ PROPN cana-5910 213 17 2𝑒π‘₯π‘₯ 2𝑒π‘₯π‘₯ NUM cana-5910 213 18 βˆ’ βˆ’ PROPN cana-5910 213 19 2𝑒𝑦𝑦 2𝑒𝑦𝑦 NUM cana-5910 213 20 ) ) PUNCT cana-5910 213 21 |𝑒𝑑= |𝑒𝑑= NOUN cana-5910 213 22 2𝑒π‘₯π‘₯+ 2𝑒π‘₯π‘₯+ NUM cana-5910 213 23 2𝑒𝑦𝑦 2𝑒𝑦𝑦 NUM cana-5910 213 24 = = SYM cana-5910 213 25 0 0 NUM cana-5910 213 26 . . PUNCT cana-5910 214 1 this this PRON cana-5910 214 2 results result VERB cana-5910 214 3 in in ADP cana-5910 214 4 : : PUNCT cana-5910 214 5 πœ‘π‘‘ πœ‘π‘‘ ADP cana-5910 214 6 βˆ’ βˆ’ PROPN cana-5910 215 1 2πœ‘π‘₯π‘₯ 2πœ‘π‘₯π‘₯ NUM cana-5910 215 2 βˆ’ βˆ’ PROPN cana-5910 215 3 2πœ‘π‘¦π‘¦ 2πœ‘π‘¦π‘¦ NUM cana-5910 215 4 = = SYM cana-5910 215 5 0 0 NUM cana-5910 215 6 , , PUNCT cana-5910 215 7 whenever whenever SCONJ cana-5910 215 8 𝑒𝑑 𝑒𝑑 NOUN cana-5910 215 9 = = PUNCT cana-5910 215 10 2𝑒π‘₯π‘₯ 2𝑒π‘₯π‘₯ PROPN cana-5910 215 11 + + CCONJ cana-5910 215 12 2𝑒𝑦𝑦. 2𝑒𝑦𝑦. NUM cana-5910 215 13 the the DET cana-5910 215 14 expressions expression NOUN cana-5910 215 15 for for ADP cana-5910 215 16 the the DET cana-5910 215 17 extended extend VERB cana-5910 215 18 infinitesimals infinitesimal NOUN cana-5910 215 19 are be AUX cana-5910 215 20 : : PUNCT cana-5910 215 21 πœ‘^𝑑 πœ‘^𝑑 NOUN cana-5910 215 22 = = SYM cana-5910 215 23 𝐷𝑑 𝐷𝑑 PROPN cana-5910 215 24 ( ( PUNCT cana-5910 215 25 πœ‘ πœ‘ NOUN cana-5910 215 26 βˆ’ βˆ’ PROPN cana-5910 215 27 π‘’π‘‘πœ π‘’π‘‘πœ ADV cana-5910 215 28 βˆ’ βˆ’ PRON cana-5910 215 29 𝑒π‘₯πœ‰ 𝑒π‘₯πœ‰ NOUN cana-5910 215 30 βˆ’ βˆ’ PROPN cana-5910 215 31 π‘’π‘¦πœ‚ π‘’π‘¦πœ‚ NOUN cana-5910 215 32 ) ) PUNCT cana-5910 216 1 + + CCONJ cana-5910 216 2 𝑒𝑑 𝑒𝑑 VERB cana-5910 216 3 πœπ‘‘ πœπ‘‘ NOUN cana-5910 216 4 + + CCONJ cana-5910 216 5 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 216 6 πœ‰π‘‘ πœ‰π‘‘ NOUN cana-5910 216 7 + + CCONJ cana-5910 216 8 𝑒𝑦 𝑒𝑦 PROPN cana-5910 216 9 πœ‚π‘‘ πœ‚π‘‘ NOUN cana-5910 216 10 , , PUNCT cana-5910 216 11 πœ‘π‘₯π‘₯ πœ‘π‘₯π‘₯ NOUN cana-5910 216 12 = = SYM cana-5910 216 13 𝐷π‘₯ 𝐷π‘₯ PROPN cana-5910 216 14 ( ( PUNCT cana-5910 216 15 πœ‘π‘₯ πœ‘π‘₯ NOUN cana-5910 216 16 βˆ’ βˆ’ PROPN cana-5910 216 17 𝑒π‘₯π‘₯πœ‰ 𝑒π‘₯π‘₯πœ‰ PROPN cana-5910 216 18 βˆ’ βˆ’ PROPN cana-5910 216 19 𝑒π‘₯π‘¦πœ‚ 𝑒π‘₯π‘¦πœ‚ NOUN cana-5910 216 20 βˆ’ βˆ’ PROPN cana-5910 216 21 𝑒π‘₯π‘‘πœ 𝑒π‘₯π‘‘πœ NOUN cana-5910 216 22 ) ) PUNCT cana-5910 217 1 + + CCONJ cana-5910 217 2 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 217 3 πœ‰π‘₯ πœ‰π‘₯ X cana-5910 217 4 + + CCONJ cana-5910 217 5 𝑒π‘₯𝑦 𝑒π‘₯𝑦 PROPN cana-5910 217 6 πœ‚π‘₯ πœ‚π‘₯ PROPN cana-5910 217 7 + + CCONJ cana-5910 217 8 𝑒π‘₯𝑑 𝑒π‘₯𝑑 NOUN cana-5910 217 9 𝜏π‘₯ 𝜏π‘₯ ADV cana-5910 217 10 , , PUNCT cana-5910 217 11 πœ‘π‘¦π‘¦ πœ‘π‘¦π‘¦ PROPN cana-5910 217 12 = = PUNCT cana-5910 218 1 𝐷𝑦 𝐷𝑦 PROPN cana-5910 218 2 ( ( PUNCT cana-5910 218 3 πœ‘π‘¦ πœ‘π‘¦ PROPN cana-5910 218 4 βˆ’ βˆ’ PROPN cana-5910 218 5 𝑒𝑦π‘₯ 𝑒𝑦π‘₯ NOUN cana-5910 218 6 πœ‰ πœ‰ PROPN cana-5910 218 7 βˆ’ βˆ’ PROPN cana-5910 218 8 𝑒𝑦𝑦 𝑒𝑦𝑦 NOUN cana-5910 218 9 πœ‚ πœ‚ NOUN cana-5910 218 10 βˆ’ βˆ’ PROPN cana-5910 218 11 𝑒_𝑦𝑑 𝑒_𝑦𝑑 NOUN cana-5910 218 12 𝜏 𝜏 NUM cana-5910 218 13 ) ) PUNCT cana-5910 218 14 + + CCONJ cana-5910 218 15 𝑒𝑦π‘₯ 𝑒𝑦π‘₯ VERB cana-5910 218 16 πœ‰π‘¦ πœ‰π‘¦ NOUN cana-5910 218 17 + + CCONJ cana-5910 218 18 𝑒𝑦𝑦 𝑒𝑦𝑦 VERB cana-5910 218 19 πœ‚π‘¦ πœ‚π‘¦ ADV cana-5910 218 20 + + CCONJ cana-5910 218 21 𝑒𝑦𝑑 𝑒𝑦𝑑 VERB cana-5910 218 22 πœπ‘¦ πœπ‘¦ PRON cana-5910 218 23 , , PUNCT cana-5910 218 24 where where SCONJ cana-5910 218 25 𝐷𝑑 𝐷𝑑 PROPN cana-5910 218 26 , , PUNCT cana-5910 218 27 𝐷π‘₯ 𝐷π‘₯ PROPN cana-5910 218 28 , , PUNCT cana-5910 218 29 𝐷𝑦 𝐷𝑦 PROPN cana-5910 218 30 are be AUX cana-5910 218 31 total total ADJ cana-5910 218 32 derivatives derivative NOUN cana-5910 218 33 , , PUNCT cana-5910 218 34 and and CCONJ cana-5910 218 35 πœ‘π‘₯ πœ‘π‘₯ INTJ cana-5910 218 36 , , PUNCT cana-5910 218 37 πœ‘π‘¦ πœ‘π‘¦ AUX cana-5910 218 38 involve involve VERB cana-5910 218 39 first first ADJ cana-5910 218 40 prolongations prolongation NOUN cana-5910 218 41 . . PUNCT cana-5910 219 1 substituting substitute VERB cana-5910 219 2 these these PRON cana-5910 219 3 into into ADP cana-5910 219 4 the the DET cana-5910 219 5 invariance invariance NOUN cana-5910 219 6 condition condition NOUN cana-5910 219 7 produces produce VERB cana-5910 219 8 a a DET cana-5910 219 9 determining determine VERB cana-5910 219 10 equation equation NOUN cana-5910 219 11 , , PUNCT cana-5910 219 12 which which PRON cana-5910 219 13 is be AUX cana-5910 219 14 a a DET cana-5910 219 15 pde pde NOUN cana-5910 219 16 in in ADP cana-5910 219 17 𝜏 𝜏 PROPN cana-5910 219 18 , , PUNCT cana-5910 219 19 πœ‰ πœ‰ X cana-5910 219 20 , , PUNCT cana-5910 219 21 πœ‚ πœ‚ NOUN cana-5910 219 22 , , PUNCT cana-5910 219 23 πœ‘. πœ‘. ADJ cana-5910 219 24 step step NOUN cana-5910 219 25 3 3 NUM cana-5910 219 26 : : PUNCT cana-5910 219 27 determining determine VERB cana-5910 219 28 equations equation NOUN cana-5910 219 29 to to PART cana-5910 219 30 simplify simplify VERB cana-5910 219 31 , , PUNCT cana-5910 219 32 assume assume VERB cana-5910 219 33 the the DET cana-5910 219 34 infinitesimals infinitesimal NOUN cana-5910 219 35 are be AUX cana-5910 219 36 of of ADP cana-5910 219 37 the the DET cana-5910 219 38 form form NOUN cana-5910 219 39 𝜏 𝜏 X cana-5910 219 40 = = SYM cana-5910 219 41 𝜏(𝑑 𝜏(𝑑 PROPN cana-5910 219 42 , , PUNCT cana-5910 219 43 π‘₯ π‘₯ X cana-5910 219 44 , , PUNCT cana-5910 219 45 𝑦 𝑦 NOUN cana-5910 219 46 ) ) PUNCT cana-5910 219 47 , , PUNCT cana-5910 219 48 πœ‰ πœ‰ X cana-5910 219 49 = = PUNCT cana-5910 219 50 πœ‰(𝑑 πœ‰(𝑑 PROPN cana-5910 219 51 , , PUNCT cana-5910 219 52 π‘₯ π‘₯ PROPN cana-5910 219 53 , , PUNCT cana-5910 219 54 𝑦 𝑦 NOUN cana-5910 219 55 ) ) PUNCT cana-5910 219 56 , , PUNCT cana-5910 219 57 πœ‚ πœ‚ X cana-5910 219 58 = = PUNCT cana-5910 219 59 πœ‚(𝑑 πœ‚(𝑑 NOUN cana-5910 219 60 , , PUNCT cana-5910 219 61 π‘₯ π‘₯ X cana-5910 219 62 , , PUNCT cana-5910 219 63 𝑦 𝑦 NOUN cana-5910 219 64 ) ) PUNCT cana-5910 219 65 , , PUNCT cana-5910 219 66 πœ‘ πœ‘ X cana-5910 219 67 = = SYM cana-5910 219 68 πœ‘(𝑑 πœ‘(𝑑 PROPN cana-5910 219 69 , , PUNCT cana-5910 219 70 π‘₯ π‘₯ NOUN cana-5910 219 71 , , PUNCT cana-5910 219 72 𝑦 𝑦 NOUN cana-5910 219 73 , , PUNCT cana-5910 219 74 𝑒 𝑒 NOUN cana-5910 219 75 ) ) PUNCT cana-5910 219 76 . . PUNCT cana-5910 220 1 for for ADP cana-5910 220 2 the the DET cana-5910 220 3 heat heat NOUN cana-5910 220 4 equation equation NOUN cana-5910 220 5 , , PUNCT cana-5910 220 6 it it PRON cana-5910 220 7 ’s ’ VERB cana-5910 220 8 common common ADJ cana-5910 220 9 to to PART cana-5910 220 10 find find VERB cana-5910 220 11 that that SCONJ cana-5910 220 12 Ο† Ο† PROPN cana-5910 220 13 is be AUX cana-5910 220 14 linear linear ADJ cana-5910 220 15 in in ADP cana-5910 220 16 u u NOUN cana-5910 220 17 due due ADP cana-5910 220 18 to to ADP cana-5910 220 19 the the DET cana-5910 220 20 linearity linearity NOUN cana-5910 220 21 of of ADP cana-5910 220 22 the the DET cana-5910 220 23 pde pde NOUN cana-5910 220 24 : : PUNCT cana-5910 220 25 πœ‘ πœ‘ X cana-5910 220 26 = = PUNCT cana-5910 220 27 𝛼(𝑑 𝛼(𝑑 PROPN cana-5910 220 28 , , PUNCT cana-5910 220 29 π‘₯ π‘₯ NOUN cana-5910 220 30 , , PUNCT cana-5910 220 31 𝑦)𝑒 𝑦)𝑒 ADJ cana-5910 220 32 + + CCONJ cana-5910 220 33 𝛽(𝑑 𝛽(𝑑 PROPN cana-5910 220 34 , , PUNCT cana-5910 220 35 π‘₯ π‘₯ PROPN cana-5910 220 36 , , PUNCT cana-5910 220 37 𝑦 𝑦 NOUN cana-5910 220 38 ) ) PUNCT cana-5910 220 39 . . PUNCT cana-5910 221 1 for for ADP cana-5910 221 2 simplicity simplicity NOUN cana-5910 221 3 , , PUNCT cana-5910 221 4 let let VERB cana-5910 221 5 ’s ’s NOUN cana-5910 221 6 try try VERB cana-5910 221 7 πœ‘ πœ‘ NOUN cana-5910 221 8 = = SYM cana-5910 221 9 𝛼(𝑑 𝛼(𝑑 X cana-5910 221 10 , , PUNCT cana-5910 221 11 π‘₯ π‘₯ NOUN cana-5910 221 12 , , PUNCT cana-5910 221 13 𝑦)𝑒 𝑦)𝑒 ADJ cana-5910 221 14 ( ( PUNCT cana-5910 221 15 setting set VERB cana-5910 221 16 𝛽 𝛽 NOUN cana-5910 221 17 = = SYM cana-5910 221 18 0 0 NUM cana-5910 221 19 , , PUNCT cana-5910 221 20 as as SCONJ cana-5910 221 21 𝛽 𝛽 NOUN cana-5910 221 22 corresponds correspond VERB cana-5910 221 23 to to ADP cana-5910 221 24 the the DET cana-5910 221 25 trivial trivial ADJ cana-5910 221 26 symmetry symmetry NOUN cana-5910 221 27 𝑒 𝑒 PROPN cana-5910 221 28 β†’ β†’ SYM cana-5910 221 29 𝑒 𝑒 X cana-5910 221 30 + + CCONJ cana-5910 221 31 constant constant ADJ cana-5910 221 32 for for ADP cana-5910 221 33 linear linear ADJ cana-5910 221 34 homogeneous homogeneous ADJ cana-5910 221 35 pdes pde NOUN cana-5910 221 36 ) ) PUNCT cana-5910 221 37 . . PUNCT cana-5910 222 1 the the DET cana-5910 222 2 determining determine VERB cana-5910 222 3 equations equation NOUN cana-5910 222 4 are be AUX cana-5910 222 5 complex complex ADJ cana-5910 222 6 , , PUNCT cana-5910 222 7 so so SCONJ cana-5910 222 8 we we PRON cana-5910 222 9 compute compute VERB cana-5910 222 10 key key ADJ cana-5910 222 11 terms term NOUN cana-5910 222 12 . . PUNCT cana-5910 223 1 the the DET cana-5910 223 2 invariance invariance NOUN cana-5910 223 3 condition condition NOUN cana-5910 223 4 leads lead VERB cana-5910 223 5 to to ADP cana-5910 223 6 a a DET cana-5910 223 7 system system NOUN cana-5910 223 8 of of ADP cana-5910 223 9 pdes pde NOUN cana-5910 223 10 for for ADP cana-5910 223 11 𝜏 𝜏 NOUN cana-5910 223 12 , , PUNCT cana-5910 223 13 πœ‰ πœ‰ X cana-5910 223 14 , , PUNCT cana-5910 223 15 πœ‚ πœ‚ NOUN cana-5910 223 16 , , PUNCT cana-5910 223 17 𝛼. 𝛼. NOUN cana-5910 223 18 after after ADP cana-5910 223 19 applying apply VERB cana-5910 223 20 the the DET cana-5910 223 21 prolongation prolongation NOUN cana-5910 223 22 and and CCONJ cana-5910 223 23 collecting collect VERB cana-5910 223 24 coefficients coefficient NOUN cana-5910 223 25 of of ADP cana-5910 223 26 𝑒𝑑 𝑒𝑑 NOUN cana-5910 223 27 , , PUNCT cana-5910 223 28 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 223 29 , , PUNCT cana-5910 223 30 𝑒𝑦𝑦 𝑒𝑦𝑦 INTJ cana-5910 223 31 , , PUNCT cana-5910 223 32 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 223 33 , , PUNCT cana-5910 223 34 𝑒𝑦 𝑒𝑦 PROPN cana-5910 223 35 , , PUNCT cana-5910 223 36 𝑒 𝑒 X cana-5910 223 37 , , PUNCT cana-5910 223 38 and and CCONJ cana-5910 223 39 independent independent ADJ cana-5910 223 40 terms term NOUN cana-5910 223 41 , , PUNCT cana-5910 223 42 we we PRON cana-5910 223 43 get get VERB cana-5910 223 44 equations equation NOUN cana-5910 223 45 such such ADJ cana-5910 223 46 as as ADP cana-5910 223 47 : : PUNCT cana-5910 223 48 β€’ β€’ NUM cana-5910 223 49 coefficient coefficient NOUN cana-5910 223 50 of of ADP cana-5910 223 51 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ PROPN cana-5910 223 52 : : PUNCT cana-5910 223 53 πœ‰π‘‘ πœ‰π‘‘ PROPN cana-5910 223 54 = = PROPN cana-5910 223 55 0 0 PROPN cana-5910 223 56 , , PUNCT cana-5910 223 57 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 223 58 = = SYM cana-5910 223 59 0 0 NUM cana-5910 223 60 , , PUNCT cana-5910 223 61 𝜏π‘₯ 𝜏π‘₯ ADV cana-5910 223 62 = = SYM cana-5910 223 63 0 0 NUM cana-5910 223 64 , , PUNCT cana-5910 223 65 πœ‚π‘₯ πœ‚π‘₯ NOUN cana-5910 223 66 = = SYM cana-5910 223 67 0 0 NUM cana-5910 223 68 , , PUNCT cana-5910 223 69 𝛼π‘₯ 𝛼π‘₯ ADV cana-5910 223 70 = = SYM cana-5910 223 71 2πœ‰π‘₯. 2πœ‰π‘₯. NUM cana-5910 223 72 β€’ β€’ NOUN cana-5910 223 73 coefficient coefficient NOUN cana-5910 223 74 of of ADP cana-5910 223 75 𝑒𝑦𝑦 𝑒𝑦𝑦 ADV cana-5910 223 76 : : PUNCT cana-5910 223 77 πœ‚π‘‘ πœ‚π‘‘ ADP cana-5910 223 78 = = SYM cana-5910 223 79 0 0 NUM cana-5910 223 80 , , PUNCT cana-5910 223 81 πœ‚π‘₯ πœ‚π‘₯ NOUN cana-5910 223 82 = = SYM cana-5910 223 83 0 0 NUM cana-5910 223 84 , , PUNCT cana-5910 223 85 πœπ‘¦ πœπ‘¦ PROPN cana-5910 223 86 = = SYM cana-5910 223 87 0 0 NUM cana-5910 223 88 , , PUNCT cana-5910 223 89 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 223 90 = = SYM cana-5910 223 91 0 0 NUM cana-5910 223 92 , , PUNCT cana-5910 223 93 𝛼𝑦 𝛼𝑦 NOUN cana-5910 223 94 = = SYM cana-5910 223 95 2πœ‚π‘¦ 2πœ‚π‘¦ NOUN cana-5910 223 96 . . PUNCT cana-5910 224 1 β€’ β€’ NUM cana-5910 224 2 coefficient coefficient NOUN cana-5910 224 3 of of ADP cana-5910 224 4 𝑒𝑑 𝑒𝑑 NOUN cana-5910 224 5 : : PUNCT cana-5910 224 6 πœπ‘’ πœπ‘’ X cana-5910 224 7 = = SYM cana-5910 224 8 0 0 PROPN cana-5910 224 9 , , PUNCT cana-5910 224 10 πœ‰π‘’ πœ‰π‘’ PROPN cana-5910 224 11 = = NOUN cana-5910 224 12 0 0 NUM cana-5910 224 13 , , PUNCT cana-5910 224 14 πœ‚π‘’ πœ‚π‘’ ADP cana-5910 224 15 = = SYM cana-5910 224 16 0 0 NUM cana-5910 224 17 , , PUNCT cana-5910 224 18 𝛼𝑑 𝛼𝑑 NOUN cana-5910 224 19 = = SYM cana-5910 224 20 2(𝛼π‘₯π‘₯ 2(𝛼π‘₯π‘₯ NUM cana-5910 224 21 + + CCONJ cana-5910 224 22 𝛼𝑦𝑦 𝛼𝑦𝑦 VERB cana-5910 224 23 ) ) PUNCT cana-5910 224 24 . . PUNCT cana-5910 225 1 β€’ β€’ NUM cana-5910 225 2 mixed mixed ADJ cana-5910 225 3 terms term NOUN cana-5910 225 4 and and CCONJ cana-5910 225 5 others other NOUN cana-5910 225 6 lead lead VERB cana-5910 225 7 to to ADP cana-5910 225 8 : : PUNCT cana-5910 225 9 𝜏π‘₯π‘₯ 𝜏π‘₯π‘₯ PROPN cana-5910 225 10 = = SYM cana-5910 225 11 0 0 NUM cana-5910 225 12 , , PUNCT cana-5910 225 13 πœπ‘¦π‘¦ πœπ‘¦π‘¦ NOUN cana-5910 225 14 = = SYM cana-5910 225 15 0 0 NUM cana-5910 225 16 , , PUNCT cana-5910 225 17 πœ‰π‘₯π‘₯ πœ‰π‘₯π‘₯ NOUN cana-5910 225 18 = = SYM cana-5910 225 19 0 0 NUM cana-5910 225 20 , , PUNCT cana-5910 225 21 πœ‚π‘¦π‘¦ πœ‚π‘¦π‘¦ X cana-5910 226 1 = = SYM cana-5910 226 2 0 0 NUM cana-5910 226 3 , , PUNCT cana-5910 226 4 etc etc X cana-5910 226 5 . . X cana-5910 227 1 solving solve VERB cana-5910 227 2 these these PRON cana-5910 227 3 , , PUNCT cana-5910 227 4 we we PRON cana-5910 227 5 find find VERB cana-5910 227 6 : : PUNCT cana-5910 227 7 β€’ β€’ NUM cana-5910 227 8 𝜏 𝜏 X cana-5910 227 9 = = PUNCT cana-5910 227 10 𝜏(𝑑 𝜏(𝑑 NOUN cana-5910 227 11 ) ) PUNCT cana-5910 227 12 , , PUNCT cana-5910 227 13 πœ‰ πœ‰ NOUN cana-5910 227 14 = = SYM cana-5910 227 15 πœ‰(π‘₯ πœ‰(π‘₯ PROPN cana-5910 227 16 ) ) PUNCT cana-5910 227 17 , , PUNCT cana-5910 227 18 πœ‚ πœ‚ X cana-5910 227 19 = = SYM cana-5910 227 20 πœ‚(𝑦 πœ‚(𝑦 NOUN cana-5910 227 21 ) ) PUNCT cana-5910 227 22 ( ( PUNCT cana-5910 227 23 π‘“π‘Ÿπ‘œπ‘š π‘“π‘Ÿπ‘œπ‘š VERB cana-5910 227 24 𝜏π‘₯ 𝜏π‘₯ NOUN cana-5910 227 25 = = SYM cana-5910 227 26 0 0 PROPN cana-5910 227 27 , , PUNCT cana-5910 227 28 πœπ‘¦ πœπ‘¦ PROPN cana-5910 227 29 = = SYM cana-5910 227 30 0 0 PROPN cana-5910 227 31 , , PUNCT cana-5910 227 32 πœ‰π‘‘ πœ‰π‘‘ NOUN cana-5910 227 33 = = PROPN cana-5910 227 34 0 0 PROPN cana-5910 227 35 , , PUNCT cana-5910 227 36 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 227 37 = = SYM cana-5910 227 38 0 0 NUM cana-5910 227 39 , , PUNCT cana-5910 227 40 πœ‚π‘‘ πœ‚π‘‘ ADP cana-5910 227 41 = = SYM cana-5910 227 42 0 0 NUM cana-5910 227 43 , , PUNCT cana-5910 227 44 πœ‚π‘₯ πœ‚π‘₯ NOUN cana-5910 227 45 = = NOUN cana-5910 227 46 0 0 NUM cana-5910 227 47 ) ) PUNCT cana-5910 227 48 . . PUNCT cana-5910 228 1 β€’ β€’ NUM cana-5910 228 2 𝜏π‘₯π‘₯ 𝜏π‘₯π‘₯ ADJ cana-5910 228 3 = = NOUN cana-5910 228 4 0 0 NUM cana-5910 228 5 , , PUNCT cana-5910 228 6 πœπ‘¦π‘¦ πœπ‘¦π‘¦ NOUN cana-5910 228 7 = = SYM cana-5910 228 8 0 0 NUM cana-5910 228 9 imply imply ADV cana-5910 228 10 Ο„ Ο„ PROPN cana-5910 228 11 is be AUX cana-5910 228 12 linear linear ADJ cana-5910 228 13 in in ADP cana-5910 228 14 t t PROPN cana-5910 228 15 , , PUNCT cana-5910 228 16 but but CCONJ cana-5910 228 17 since since SCONJ cana-5910 228 18 𝜏 𝜏 PROPN cana-5910 228 19 = = SYM cana-5910 228 20 𝜏(𝑑 𝜏(𝑑 PROPN cana-5910 228 21 ) ) PUNCT cana-5910 228 22 , , PUNCT cana-5910 229 1 𝜏 𝜏 X cana-5910 229 2 = = PUNCT cana-5910 229 3 π‘Ž1 π‘Ž1 ADP cana-5910 229 4 𝑑 𝑑 PROPN cana-5910 229 5 + + PROPN cana-5910 229 6 π‘Ž2 π‘Ž2 PROPN cana-5910 229 7 . . PUNCT cana-5910 229 8 β€’ β€’ NUM cana-5910 229 9 πœ‰π‘₯π‘₯ πœ‰π‘₯π‘₯ NOUN cana-5910 229 10 = = SYM cana-5910 229 11 0 0 NUM cana-5910 229 12 implies imply VERB cana-5910 229 13 πœ‰ πœ‰ ADJ cana-5910 229 14 = = SYM cana-5910 229 15 𝑏1 𝑏1 VERB cana-5910 229 16 π‘₯ π‘₯ PROPN cana-5910 229 17 + + CCONJ cana-5910 229 18 𝑏2 𝑏2 PROPN cana-5910 229 19 , , PUNCT cana-5910 229 20 πœ‚π‘¦π‘¦ πœ‚π‘¦π‘¦ X cana-5910 229 21 = = SYM cana-5910 229 22 0 0 NUM cana-5910 229 23 implies imply VERB cana-5910 229 24 πœ‚ πœ‚ X cana-5910 229 25 = = SYM cana-5910 229 26 𝑐1 𝑐1 NOUN cana-5910 229 27 𝑦 𝑦 NOUN cana-5910 229 28 + + NUM cana-5910 229 29 𝑐2 𝑐2 NOUN cana-5910 229 30 . . PUNCT cana-5910 229 31 1119 1119 NUM cana-5910 229 32 https://internationalpubls.com https://internationalpubls.com X cana-5910 229 33 β€’ β€’ NOUN cana-5910 229 34 from from ADP cana-5910 229 35 𝛼π‘₯ 𝛼π‘₯ ADV cana-5910 229 36 = = SYM cana-5910 229 37 2πœ‰π‘₯ 2πœ‰π‘₯ NOUN cana-5910 229 38 , , PUNCT cana-5910 229 39 𝛼𝑦 𝛼𝑦 NOUN cana-5910 229 40 = = SYM cana-5910 229 41 2πœ‚π‘¦ 2πœ‚π‘¦ PROPN cana-5910 229 42 , , PUNCT cana-5910 229 43 we we PRON cana-5910 229 44 get get VERB cana-5910 229 45 𝛼π‘₯ 𝛼π‘₯ ADV cana-5910 229 46 = = SYM cana-5910 229 47 2𝑏1 2𝑏1 NUM cana-5910 229 48 , , PUNCT cana-5910 229 49 𝛼𝑦 𝛼𝑦 PROPN cana-5910 229 50 = = SYM cana-5910 229 51 2𝑐1 2𝑐1 PROPN cana-5910 229 52 , , PUNCT cana-5910 229 53 π‘ π‘œ π‘ π‘œ ADP cana-5910 229 54 𝛼 𝛼 NOUN cana-5910 229 55 = = NOUN cana-5910 229 56 2𝑏1 2𝑏1 NUM cana-5910 229 57 π‘₯ π‘₯ PROPN cana-5910 230 1 + + NUM cana-5910 230 2 2𝑐1 2𝑐1 NUM cana-5910 230 3 𝑦 𝑦 NOUN cana-5910 230 4 + + CCONJ cana-5910 230 5 𝑓(𝑑 𝑓(𝑑 NOUN cana-5910 230 6 ) ) PUNCT cana-5910 230 7 . . PUNCT cana-5910 231 1 β€’ β€’ NOUN cana-5910 231 2 the the DET cana-5910 231 3 equation equation NOUN cana-5910 231 4 𝛼𝑑 𝛼𝑑 NOUN cana-5910 231 5 = = SYM cana-5910 231 6 2(𝛼π‘₯π‘₯ 2(𝛼π‘₯π‘₯ NUM cana-5910 231 7 + + CCONJ cana-5910 231 8 𝛼𝑦𝑦 𝛼𝑦𝑦 VERB cana-5910 231 9 ) ) PUNCT cana-5910 231 10 𝑔𝑖𝑣𝑒𝑠 𝑔𝑖𝑣𝑒𝑠 NOUN cana-5910 231 11 𝑓′(𝑑 𝑓′(𝑑 PROPN cana-5910 231 12 ) ) PUNCT cana-5910 231 13 = = SYM cana-5910 231 14 0 0 NUM cana-5910 231 15 , , PUNCT cana-5910 231 16 π‘ π‘œ π‘ π‘œ ADP cana-5910 231 17 𝑓(𝑑 𝑓(𝑑 NOUN cana-5910 231 18 ) ) PUNCT cana-5910 231 19 = = VERB cana-5910 232 1 π‘˜. π‘˜. NOUN cana-5910 232 2 β€’ β€’ NUM cana-5910 232 3 other other ADJ cana-5910 232 4 equations equation NOUN cana-5910 232 5 constrain constrain VERB cana-5910 232 6 constants constant NOUN cana-5910 232 7 , , PUNCT cana-5910 232 8 leading lead VERB cana-5910 232 9 to to ADP cana-5910 232 10 symmetries symmetry NOUN cana-5910 232 11 . . PUNCT cana-5910 233 1 for for ADP cana-5910 233 2 the the DET cana-5910 233 3 heat heat NOUN cana-5910 233 4 equation equation NOUN cana-5910 233 5 𝑒𝑑 𝑒𝑑 NOUN cana-5910 233 6 = = SYM cana-5910 233 7 π‘˜(𝑒π‘₯π‘₯ π‘˜(𝑒π‘₯π‘₯ PROPN cana-5910 233 8 + + CCONJ cana-5910 233 9 𝑒𝑦𝑦 𝑒𝑦𝑦 PROPN cana-5910 233 10 ) ) PUNCT cana-5910 233 11 , , PUNCT cana-5910 233 12 standard standard ADJ cana-5910 233 13 symmetries symmetry NOUN cana-5910 233 14 include include VERB cana-5910 233 15 : : PUNCT cana-5910 233 16 1 1 X cana-5910 233 17 . . NOUN cana-5910 233 18 time time NOUN cana-5910 233 19 translation translation NOUN cana-5910 233 20 : : PUNCT cana-5910 234 1 𝑋1 𝑋1 PROPN cana-5910 234 2 = = SYM cana-5910 234 3 πœ• πœ• PROPN cana-5910 234 4 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 234 5 , , PUNCT cana-5910 234 6 ( ( PUNCT cana-5910 234 7 𝜏 𝜏 NOUN cana-5910 234 8 = = SYM cana-5910 234 9 1 1 NUM cana-5910 234 10 , , PUNCT cana-5910 234 11 πœ‰ πœ‰ NOUN cana-5910 234 12 = = SYM cana-5910 234 13 0 0 NUM cana-5910 234 14 , , PUNCT cana-5910 234 15 πœ‚ πœ‚ NOUN cana-5910 234 16 = = SYM cana-5910 234 17 0 0 NUM cana-5910 234 18 , , PUNCT cana-5910 234 19 πœ‘ πœ‘ NOUN cana-5910 234 20 = = NOUN cana-5910 234 21 0 0 NUM cana-5910 234 22 ) ) PUNCT cana-5910 234 23 . . PUNCT cana-5910 235 1 2 2 X cana-5910 235 2 . . NOUN cana-5910 235 3 space space NOUN cana-5910 235 4 translations translation NOUN cana-5910 235 5 : : PUNCT cana-5910 235 6 𝑋2 𝑋2 VERB cana-5910 235 7 = = SYM cana-5910 235 8 πœ• πœ• NOUN cana-5910 235 9 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 235 10 , , PUNCT cana-5910 235 11 𝑋3 𝑋3 NOUN cana-5910 235 12 = = SYM cana-5910 235 13 πœ• πœ• PROPN cana-5910 235 14 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 235 15 . . PUNCT cana-5910 236 1 3 3 X cana-5910 236 2 . . X cana-5910 236 3 scaling scaling NOUN cana-5910 236 4 : : PUNCT cana-5910 236 5 𝑋4 𝑋4 VERB cana-5910 236 6 = = PUNCT cana-5910 237 1 2𝑑 2𝑑 NUM cana-5910 237 2 πœ• πœ• PROPN cana-5910 237 3 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 237 4 + + CCONJ cana-5910 237 5 π‘₯ π‘₯ DET cana-5910 237 6 πœ• πœ• NOUN cana-5910 237 7 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 237 8 + + CCONJ cana-5910 237 9 𝑦 𝑦 NOUN cana-5910 237 10 πœ• πœ• NOUN cana-5910 237 11 πœ•π‘¦ πœ•π‘¦ NOUN cana-5910 237 12 , , PUNCT cana-5910 237 13 ( ( PUNCT cana-5910 237 14 πœ‘ πœ‘ NOUN cana-5910 237 15 = = SYM cana-5910 237 16 0 0 NUM cana-5910 237 17 ) ) PUNCT cana-5910 237 18 . . PUNCT cana-5910 238 1 4 4 X cana-5910 238 2 . . PUNCT cana-5910 238 3 solution solution NOUN cana-5910 238 4 scaling scaling NOUN cana-5910 238 5 : : PUNCT cana-5910 238 6 𝑋5 𝑋5 PROPN cana-5910 238 7 = = SYM cana-5910 238 8 𝑒 𝑒 PROPN cana-5910 238 9 πœ• πœ• PROPN cana-5910 238 10 πœ•π‘’ πœ•π‘’ NOUN cana-5910 238 11 , , PUNCT cana-5910 238 12 ( ( PUNCT cana-5910 238 13 πœ‘ πœ‘ NOUN cana-5910 238 14 = = SYM cana-5910 238 15 𝑒 𝑒 X cana-5910 238 16 ) ) PUNCT cana-5910 238 17 . . PUNCT cana-5910 239 1 5 5 X cana-5910 239 2 . . X cana-5910 239 3 galilean galilean PROPN cana-5910 239 4 boosts boost NOUN cana-5910 239 5 , , PUNCT cana-5910 239 6 rotations rotation NOUN cana-5910 239 7 , , PUNCT cana-5910 239 8 and and CCONJ cana-5910 239 9 infinite infinite ADJ cana-5910 239 10 - - PUNCT cana-5910 239 11 dimensional dimensional ADJ cana-5910 239 12 symmetries symmetry NOUN cana-5910 239 13 ( ( PUNCT cana-5910 239 14 for for ADP cana-5910 239 15 unbounded unbounded ADJ cana-5910 239 16 domains domain NOUN cana-5910 239 17 ) ) PUNCT cana-5910 239 18 . . PUNCT cana-5910 240 1 given give VERB cana-5910 240 2 our our PRON cana-5910 240 3 coefficient coefficient NOUN cana-5910 240 4 2 2 NUM cana-5910 240 5 , , PUNCT cana-5910 240 6 we we PRON cana-5910 240 7 adjust adjust VERB cana-5910 240 8 the the DET cana-5910 240 9 scaling scale VERB cana-5910 240 10 symmetry symmetry NOUN cana-5910 240 11 . . PUNCT cana-5910 241 1 testing test VERB cana-5910 241 2 the the DET cana-5910 241 3 scaling scaling ADJ cana-5910 241 4 symmetry symmetry NOUN cana-5910 241 5 : : PUNCT cana-5910 241 6 𝜏 𝜏 NOUN cana-5910 241 7 = = SYM cana-5910 241 8 2π‘Ž 2π‘Ž NUM cana-5910 241 9 𝑑 𝑑 NOUN cana-5910 241 10 , , PUNCT cana-5910 241 11 πœ‰ πœ‰ NOUN cana-5910 241 12 = = SYM cana-5910 241 13 π‘Ž π‘Ž PRON cana-5910 241 14 π‘₯ π‘₯ NOUN cana-5910 241 15 , , PUNCT cana-5910 241 16 πœ‚ πœ‚ NOUN cana-5910 241 17 = = SYM cana-5910 241 18 π‘Ž π‘Ž SYM cana-5910 241 19 𝑦 𝑦 NOUN cana-5910 241 20 , , PUNCT cana-5910 241 21 πœ‘ πœ‘ NOUN cana-5910 241 22 = = SYM cana-5910 241 23 𝑏 𝑏 NOUN cana-5910 241 24 𝑒 𝑒 PROPN cana-5910 241 25 , , PUNCT cana-5910 241 26 substitute substitute NOUN cana-5910 241 27 into into ADP cana-5910 241 28 determining determine VERB cana-5910 241 29 equations equation NOUN cana-5910 241 30 . . PUNCT cana-5910 242 1 the the DET cana-5910 242 2 key key ADJ cana-5910 242 3 equation equation NOUN cana-5910 242 4 becomes become VERB cana-5910 242 5 : : PUNCT cana-5910 242 6 πœ‘π‘‘ πœ‘π‘‘ ADP cana-5910 242 7 βˆ’ βˆ’ PROPN cana-5910 243 1 2πœ‘π‘₯π‘₯ 2πœ‘π‘₯π‘₯ NUM cana-5910 243 2 βˆ’ βˆ’ PROPN cana-5910 243 3 2πœ‘π‘¦π‘¦ 2πœ‘π‘¦π‘¦ NOUN cana-5910 243 4 + + SYM cana-5910 243 5 𝑒𝑑(𝛼 𝑒𝑑(𝛼 NUM cana-5910 243 6 βˆ’ βˆ’ NOUN cana-5910 243 7 πœπ‘‘ πœπ‘‘ PROPN cana-5910 243 8 ) ) PUNCT cana-5910 243 9 βˆ’ βˆ’ PROPN cana-5910 244 1 2𝑒π‘₯π‘₯ 2𝑒π‘₯π‘₯ PROPN cana-5910 244 2 ( ( PUNCT cana-5910 244 3 𝛼 𝛼 NOUN cana-5910 244 4 βˆ’ βˆ’ NOUN cana-5910 244 5 2πœ‰π‘₯ 2πœ‰π‘₯ NUM cana-5910 244 6 ) ) PUNCT cana-5910 245 1 βˆ’ βˆ’ PROPN cana-5910 245 2 2𝑒𝑦𝑦 2𝑒𝑦𝑦 NUM cana-5910 245 3 ( ( PUNCT cana-5910 245 4 𝛼 𝛼 PROPN cana-5910 245 5 βˆ’ βˆ’ PROPN cana-5910 245 6 2πœ‚π‘¦ 2πœ‚π‘¦ NOUN cana-5910 245 7 ) ) PUNCT cana-5910 246 1 + + CCONJ cana-5910 246 2 . . PUNCT cana-5910 246 3 . . PUNCT cana-5910 246 4 . . PUNCT cana-5910 247 1 = = PUNCT cana-5910 247 2 0 0 X cana-5910 247 3 . . PUNCT cana-5910 248 1 this this PRON cana-5910 248 2 confirms confirm VERB cana-5910 248 3 symmetries symmetry NOUN cana-5910 248 4 like like ADP cana-5910 248 5 : : PUNCT cana-5910 248 6 𝑋 𝑋 PROPN cana-5910 248 7 = = PUNCT cana-5910 248 8 2𝑑 2𝑑 NUM cana-5910 248 9 πœ• πœ• PROPN cana-5910 248 10 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 248 11 + + CCONJ cana-5910 248 12 π‘₯ π‘₯ DET cana-5910 248 13 πœ• πœ• NOUN cana-5910 248 14 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 248 15 + + CCONJ cana-5910 248 16 𝑦 𝑦 NOUN cana-5910 248 17 πœ• πœ• NOUN cana-5910 248 18 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 248 19 βˆ’ βˆ’ NOUN cana-5910 248 20 𝑒 𝑒 PROPN cana-5910 248 21 πœ• πœ• PROPN cana-5910 248 22 πœ•π‘’ πœ•π‘’ NOUN cana-5910 248 23 , , PUNCT cana-5910 248 24 corresponding correspond VERB cana-5910 248 25 to to ADP cana-5910 248 26 𝜏 𝜏 NOUN cana-5910 248 27 = = PUNCT cana-5910 248 28 2𝑑 2𝑑 NOUN cana-5910 248 29 , , PUNCT cana-5910 248 30 πœ‰ πœ‰ PROPN cana-5910 248 31 = = SYM cana-5910 248 32 π‘₯ π‘₯ PROPN cana-5910 248 33 , , PUNCT cana-5910 248 34 πœ‚ πœ‚ X cana-5910 248 35 = = SYM cana-5910 248 36 𝑦 𝑦 PROPN cana-5910 248 37 , , PUNCT cana-5910 248 38 πœ‘ πœ‘ PROPN cana-5910 248 39 = = SYM cana-5910 248 40 βˆ’π‘’ βˆ’π‘’ NOUN cana-5910 248 41 , , PUNCT cana-5910 248 42 which which PRON cana-5910 248 43 is be AUX cana-5910 248 44 typical typical ADJ cana-5910 248 45 for for ADP cana-5910 248 46 the the DET cana-5910 248 47 heat heat NOUN cana-5910 248 48 equation equation NOUN cana-5910 248 49 with with ADP cana-5910 248 50 a a DET cana-5910 248 51 modified modify VERB cana-5910 248 52 coefficient coefficient NOUN cana-5910 248 53 . . PUNCT cana-5910 249 1 step step NOUN cana-5910 249 2 4 4 NUM cana-5910 249 3 : : PUNCT cana-5910 249 4 symmetry symmetry NOUN cana-5910 249 5 reduction reduction NOUN cana-5910 249 6 choose choose VERB cana-5910 249 7 the the DET cana-5910 249 8 scaling scale VERB cana-5910 249 9 symmetry symmetry NOUN cana-5910 249 10 : : PUNCT cana-5910 249 11 𝑋 𝑋 PROPN cana-5910 249 12 = = PUNCT cana-5910 249 13 2𝑑 2𝑑 NUM cana-5910 249 14 πœ• πœ• PROPN cana-5910 249 15 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 249 16 + + CCONJ cana-5910 249 17 π‘₯ π‘₯ PRON cana-5910 249 18 πœ• πœ• NOUN cana-5910 249 19 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 249 20 + + CCONJ cana-5910 249 21 𝑦 𝑦 NOUN cana-5910 249 22 πœ• πœ• NOUN cana-5910 249 23 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 249 24 βˆ’ βˆ’ NOUN cana-5910 249 25 𝑒 𝑒 PRON cana-5910 249 26 πœ• πœ• PROPN cana-5910 249 27 πœ•π‘’ πœ•π‘’ NOUN cana-5910 249 28 . . PUNCT cana-5910 250 1 the the DET cana-5910 250 2 invariants invariant NOUN cana-5910 250 3 are be AUX cana-5910 250 4 found find VERB cana-5910 250 5 by by ADP cana-5910 250 6 solving solve VERB cana-5910 250 7 : : PUNCT cana-5910 250 8 𝑑π‘₯ 𝑑π‘₯ ADP cana-5910 250 9 π‘₯ π‘₯ NOUN cana-5910 250 10 = = X cana-5910 250 11 𝑑𝑦 𝑑𝑦 PROPN cana-5910 250 12 𝑦 𝑦 NOUN cana-5910 250 13 = = SYM cana-5910 250 14 𝑑𝑑 𝑑𝑑 ADP cana-5910 250 15 2𝑑 2𝑑 NUM cana-5910 250 16 = = PUNCT cana-5910 250 17 𝑑𝑒 𝑑𝑒 PROPN cana-5910 250 18 βˆ’π‘’ βˆ’π‘’ NOUN cana-5910 250 19 . . PUNCT cana-5910 251 1 from from ADP cana-5910 251 2 𝑑π‘₯ 𝑑π‘₯ NOUN cana-5910 251 3 π‘₯ π‘₯ NOUN cana-5910 251 4 = = PUNCT cana-5910 251 5 𝑑𝑦 𝑑𝑦 PROPN cana-5910 251 6 𝑦 𝑦 PROPN cana-5910 251 7 , , PUNCT cana-5910 251 8 𝑀𝑒 𝑀𝑒 INTJ cana-5910 251 9 𝑔𝑒𝑑 𝑔𝑒𝑑 NOUN cana-5910 251 10 π‘₯ π‘₯ X cana-5910 251 11 𝑦 𝑦 PROPN cana-5910 251 12 = = SYM cana-5910 251 13 𝑐1 𝑐1 NOUN cana-5910 251 14 , , PUNCT cana-5910 251 15 π‘ π‘œ π‘ π‘œ PROPN cana-5910 251 16 πœ‰1 πœ‰1 PROPN cana-5910 251 17 = = SYM cana-5910 251 18 π‘₯ π‘₯ PROPN cana-5910 251 19 𝑦 𝑦 NOUN cana-5910 251 20 . . PUNCT cana-5910 252 1 from from ADP cana-5910 252 2 𝑑π‘₯ 𝑑π‘₯ NOUN cana-5910 252 3 π‘₯ π‘₯ NOUN cana-5910 252 4 = = SYM cana-5910 252 5 𝑑𝑑 𝑑𝑑 ADP cana-5910 252 6 2𝑑 2𝑑 NOUN cana-5910 252 7 , , PUNCT cana-5910 252 8 𝑀𝑒 𝑀𝑒 INTJ cana-5910 252 9 𝑔𝑒𝑑 𝑔𝑒𝑑 NOUN cana-5910 252 10 π‘₯2 π‘₯2 PROPN cana-5910 252 11 𝑑 𝑑 PROPN cana-5910 252 12 = = PROPN cana-5910 252 13 𝑐2 𝑐2 NOUN cana-5910 252 14 , , PUNCT cana-5910 252 15 π‘ π‘œ π‘ π‘œ ADP cana-5910 252 16 πœ‰2 πœ‰2 ADJ cana-5910 252 17 = = SYM cana-5910 252 18 π‘₯2 π‘₯2 PROPN cana-5910 252 19 𝑑 𝑑 PROPN cana-5910 252 20 . . PUNCT cana-5910 253 1 from from ADP cana-5910 253 2 𝑑π‘₯ 𝑑π‘₯ NOUN cana-5910 253 3 π‘₯ π‘₯ NOUN cana-5910 253 4 = = PUNCT cana-5910 253 5 𝑑𝑒 𝑑𝑒 X cana-5910 253 6 βˆ’π‘’ βˆ’π‘’ NOUN cana-5910 253 7 , , PUNCT cana-5910 253 8 𝑀𝑒 𝑀𝑒 INTJ cana-5910 253 9 𝑔𝑒𝑑 𝑔𝑒𝑑 NOUN cana-5910 253 10 𝑒 𝑒 PROPN cana-5910 253 11 π‘₯ π‘₯ X cana-5910 253 12 = = SYM cana-5910 253 13 𝑐3 𝑐3 NOUN cana-5910 253 14 , , PUNCT cana-5910 253 15 π‘ π‘œ π‘ π‘œ ADP cana-5910 253 16 𝑒 𝑒 X cana-5910 253 17 = = NOUN cana-5910 253 18 π‘˜ π‘˜ PROPN cana-5910 253 19 π‘₯ π‘₯ X cana-5910 253 20 . . PUNCT cana-5910 254 1 however however ADV cana-5910 254 2 , , PUNCT cana-5910 254 3 a a DET cana-5910 254 4 more more ADV cana-5910 254 5 useful useful ADJ cana-5910 254 6 form form NOUN cana-5910 254 7 is be AUX cana-5910 254 8 : : PUNCT cana-5910 254 9 πœ‰1 πœ‰1 PROPN cana-5910 254 10 = = SYM cana-5910 254 11 π‘₯ π‘₯ PROPN cana-5910 254 12 βˆšπ‘‘ βˆšπ‘‘ VERB cana-5910 254 13 , , PUNCT cana-5910 254 14 πœ‰2 πœ‰2 PROPN cana-5910 254 15 = = SYM cana-5910 254 16 𝑦 𝑦 X cana-5910 254 17 βˆšπ‘‘ βˆšπ‘‘ VERB cana-5910 254 18 , , PUNCT cana-5910 254 19 𝑒 𝑒 X cana-5910 254 20 = = SYM cana-5910 254 21 𝑣(πœ‰1 𝑣(πœ‰1 NUM cana-5910 254 22 , , PUNCT cana-5910 254 23 πœ‰2 πœ‰2 PROPN cana-5910 254 24 ) ) PUNCT cana-5910 254 25 βˆšπ‘‘ βˆšπ‘‘ VERB cana-5910 254 26 . . PUNCT cana-5910 255 1 let let VERB cana-5910 255 2 : : PUNCT cana-5910 255 3 πœ‰ πœ‰ X cana-5910 255 4 = = SYM cana-5910 255 5 π‘₯ π‘₯ X cana-5910 255 6 βˆšπ‘‘ βˆšπ‘‘ VERB cana-5910 255 7 , , PUNCT cana-5910 255 8 πœ‚ πœ‚ X cana-5910 255 9 = = SYM cana-5910 255 10 𝑦 𝑦 NOUN cana-5910 255 11 βˆšπ‘‘ βˆšπ‘‘ VERB cana-5910 255 12 , , PUNCT cana-5910 255 13 𝑒 𝑒 X cana-5910 255 14 = = SYM cana-5910 255 15 𝑣(πœ‰ 𝑣(πœ‰ NOUN cana-5910 255 16 , , PUNCT cana-5910 255 17 πœ‚ πœ‚ NOUN cana-5910 255 18 ) ) PUNCT cana-5910 255 19 βˆšπ‘‘ βˆšπ‘‘ VERB cana-5910 255 20 . . PUNCT cana-5910 256 1 transform transform VERB cana-5910 256 2 the the DET cana-5910 256 3 pde pde NOUN cana-5910 256 4 . . PUNCT cana-5910 257 1 compute compute NOUN cana-5910 257 2 derivatives derivative NOUN cana-5910 257 3 : : PUNCT cana-5910 257 4 1120 1120 NUM cana-5910 257 5 https://internationalpubls.com https://internationalpubls.com X cana-5910 257 6 𝑒𝑑 𝑒𝑑 NOUN cana-5910 257 7 = = NOUN cana-5910 257 8 βˆ’ βˆ’ PROPN cana-5910 257 9 𝑣 𝑣 ADP cana-5910 257 10 2𝑑 2𝑑 NUM cana-5910 257 11 3 3 NUM cana-5910 257 12 2 2 NUM cana-5910 257 13 + + CCONJ cana-5910 257 14 ( ( PUNCT cana-5910 257 15 π‘£πœ‰πœ‰π‘‘+ π‘£πœ‰πœ‰π‘‘+ X cana-5910 257 16 π‘£πœ‚πœ‚π‘‘ π‘£πœ‚πœ‚π‘‘ NOUN cana-5910 257 17 ) ) PUNCT cana-5910 257 18 βˆšπ‘‘ βˆšπ‘‘ VERB cana-5910 257 19 , , PUNCT cana-5910 257 20 π‘€β„Žπ‘’π‘Ÿπ‘’ π‘€β„Žπ‘’π‘Ÿπ‘’ NOUN cana-5910 257 21 πœ‰π‘‘ πœ‰π‘‘ NOUN cana-5910 257 22 = = PROPN cana-5910 258 1 βˆ’ βˆ’ PROPN cana-5910 258 2 π‘₯ π‘₯ NOUN cana-5910 258 3 2𝑑 2𝑑 NUM cana-5910 258 4 3 3 NUM cana-5910 258 5 2 2 NUM cana-5910 258 6 , , PUNCT cana-5910 258 7 πœ‚π‘‘ πœ‚π‘‘ NOUN cana-5910 258 8 = = SYM cana-5910 258 9 βˆ’ βˆ’ PROPN cana-5910 258 10 𝑦 𝑦 NUM cana-5910 258 11 2𝑑 2𝑑 NUM cana-5910 258 12 3 3 NUM cana-5910 258 13 2 2 NUM cana-5910 258 14 , , PUNCT cana-5910 258 15 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 258 16 = = PRON cana-5910 258 17 π‘£πœ‰ π‘£πœ‰ PART cana-5910 258 18 βˆšπ‘‘ βˆšπ‘‘ VERB cana-5910 258 19 Β· Β· PUNCT cana-5910 258 20 ( ( PUNCT cana-5910 258 21 1 1 NUM cana-5910 258 22 βˆšπ‘‘ βˆšπ‘‘ PRON cana-5910 258 23 ) ) PUNCT cana-5910 259 1 = = PUNCT cana-5910 259 2 π‘£πœ‰ π‘£πœ‰ NUM cana-5910 259 3 𝑑 𝑑 PROPN cana-5910 259 4 , , PUNCT cana-5910 259 5 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 259 6 = = SYM cana-5910 259 7 πœ• πœ• PROPN cana-5910 259 8 πœ•π‘₯ πœ•π‘₯ X cana-5910 259 9 ( ( PUNCT cana-5910 259 10 π‘£πœ‰ π‘£πœ‰ INTJ cana-5910 259 11 𝑑 𝑑 PROPN cana-5910 259 12 ) ) PUNCT cana-5910 259 13 = = SYM cana-5910 259 14 ( ( PUNCT cana-5910 259 15 π‘£πœ‰πœ‰ π‘£πœ‰πœ‰ PROPN cana-5910 259 16 𝑑 𝑑 NOUN cana-5910 259 17 ) ) PUNCT cana-5910 259 18 Β· Β· PUNCT cana-5910 259 19 ( ( PUNCT cana-5910 259 20 1 1 NUM cana-5910 259 21 βˆšπ‘‘ βˆšπ‘‘ ADJ cana-5910 259 22 ) ) PUNCT cana-5910 260 1 = = PUNCT cana-5910 260 2 π‘£πœ‰πœ‰/𝑑 π‘£πœ‰πœ‰/𝑑 DET cana-5910 260 3 3 3 NUM cana-5910 260 4 2 2 NUM cana-5910 260 5 , , PUNCT cana-5910 260 6 𝑒𝑦𝑦 𝑒𝑦𝑦 ADV cana-5910 260 7 = = SYM cana-5910 260 8 π‘£πœ‚πœ‚ π‘£πœ‚πœ‚ X cana-5910 260 9 𝑑 𝑑 PROPN cana-5910 260 10 3 3 NUM cana-5910 260 11 2 2 NUM cana-5910 260 12 . . PUNCT cana-5910 261 1 substitute substitute NOUN cana-5910 261 2 into into ADP cana-5910 261 3 the the DET cana-5910 261 4 pde pde NOUN cana-5910 261 5 : : PUNCT cana-5910 262 1 βˆ’ βˆ’ ADP cana-5910 262 2 𝑣 𝑣 ADP cana-5910 262 3 2𝑑 2𝑑 NUM cana-5910 262 4 3 3 NUM cana-5910 262 5 2 2 NUM cana-5910 262 6 + + CCONJ cana-5910 262 7 π‘£πœ‰(βˆ’ π‘£πœ‰(βˆ’ PROPN cana-5910 262 8 π‘₯ π‘₯ PROPN cana-5910 262 9 2𝑑 2𝑑 NUM cana-5910 262 10 3 3 NUM cana-5910 262 11 2 2 NUM cana-5910 262 12 ) ) PUNCT cana-5910 263 1 + + CCONJ cana-5910 263 2 π‘£πœ‚(βˆ’ π‘£πœ‚(βˆ’ NUM cana-5910 263 3 𝑦 𝑦 NUM cana-5910 263 4 2𝑑 2𝑑 NUM cana-5910 263 5 3 3 NUM cana-5910 263 6 2 2 NUM cana-5910 263 7 ) ) PUNCT cana-5910 263 8 βˆšπ‘‘ βˆšπ‘‘ VERB cana-5910 263 9 = = SYM cana-5910 264 1 2 2 NUM cana-5910 264 2 ( ( PUNCT cana-5910 264 3 π‘£πœ‰πœ‰ π‘£πœ‰πœ‰ X cana-5910 264 4 𝑑 𝑑 PROPN cana-5910 264 5 3 3 NUM cana-5910 264 6 2 2 NUM cana-5910 264 7 + + NUM cana-5910 264 8 π‘£πœ‚πœ‚ π‘£πœ‚πœ‚ NOUN cana-5910 264 9 𝑑 𝑑 NOUN cana-5910 264 10 3 3 NUM cana-5910 264 11 2 2 NUM cana-5910 264 12 ) ) PUNCT cana-5910 264 13 . . PUNCT cana-5910 265 1 multiply multiply VERB cana-5910 265 2 through through ADP cana-5910 265 3 by by ADP cana-5910 265 4 𝑑 𝑑 PROPN cana-5910 265 5 3 3 NUM cana-5910 265 6 2 2 NUM cana-5910 265 7 : : PUNCT cana-5910 265 8 βˆ’ βˆ’ PROPN cana-5910 265 9 𝑣 𝑣 DET cana-5910 265 10 2 2 NUM cana-5910 265 11 βˆ’ βˆ’ NOUN cana-5910 265 12 π‘₯ π‘₯ PRON cana-5910 265 13 π‘£πœ‰ π‘£πœ‰ PRON cana-5910 265 14 2𝑑 2𝑑 NOUN cana-5910 265 15 βˆ’ βˆ’ PROPN cana-5910 265 16 𝑦 𝑦 NOUN cana-5910 265 17 π‘£πœ‚ π‘£πœ‚ VERB cana-5910 265 18 2𝑑 2𝑑 NOUN cana-5910 265 19 = = SYM cana-5910 265 20 2(π‘£πœ‰πœ‰ 2(π‘£πœ‰πœ‰ NUM cana-5910 265 21 + + CCONJ cana-5910 265 22 π‘£πœ‚πœ‚ π‘£πœ‚πœ‚ NOUN cana-5910 265 23 ) ) PUNCT cana-5910 265 24 . . PUNCT cana-5910 266 1 since since SCONJ cana-5910 266 2 πœ‰ πœ‰ PROPN cana-5910 266 3 = = VERB cana-5910 266 4 π‘₯ π‘₯ X cana-5910 266 5 βˆšπ‘‘ βˆšπ‘‘ VERB cana-5910 266 6 , , PUNCT cana-5910 266 7 πœ‚ πœ‚ X cana-5910 266 8 = = SYM cana-5910 266 9 𝑦 𝑦 NOUN cana-5910 266 10 βˆšπ‘‘ βˆšπ‘‘ VERB cana-5910 266 11 , , PUNCT cana-5910 266 12 we we PRON cana-5910 266 13 need need VERB cana-5910 266 14 consistency consistency NOUN cana-5910 266 15 . . PUNCT cana-5910 267 1 try try VERB cana-5910 267 2 a a DET cana-5910 267 3 different different ADJ cana-5910 267 4 reduction reduction NOUN cana-5910 267 5 or or CCONJ cana-5910 267 6 adjust adjust NOUN cana-5910 267 7 . . PUNCT cana-5910 268 1 alternatively alternatively ADV cana-5910 268 2 , , PUNCT cana-5910 268 3 use use VERB cana-5910 268 4 : : PUNCT cana-5910 268 5 𝑒 𝑒 X cana-5910 268 6 = = SYM cana-5910 268 7 π‘’βˆ’πœ†π‘‘ π‘’βˆ’πœ†π‘‘ NUM cana-5910 268 8 𝑀(π‘₯ 𝑀(π‘₯ PROPN cana-5910 268 9 , , PUNCT cana-5910 268 10 𝑦 𝑦 NOUN cana-5910 268 11 ) ) PUNCT cana-5910 268 12 , , PUNCT cana-5910 268 13 which which PRON cana-5910 268 14 respects respect VERB cana-5910 268 15 𝑒 𝑒 PRON cana-5910 268 16 β†’ β†’ SYM cana-5910 268 17 0 0 NUM cana-5910 268 18 π‘Žπ‘  π‘Žπ‘  PROPN cana-5910 268 19 𝑑 𝑑 PROPN cana-5910 268 20 β†’ β†’ SYM cana-5910 268 21 ∞. ∞. PROPN cana-5910 268 22 substitute substitute NOUN cana-5910 268 23 : : PUNCT cana-5910 268 24 𝑒𝑑 𝑒𝑑 NOUN cana-5910 268 25 = = PUNCT cana-5910 268 26 βˆ’πœ† βˆ’πœ† PROPN cana-5910 268 27 π‘’βˆ’πœ†π‘‘ π‘’βˆ’πœ†π‘‘ PROPN cana-5910 268 28 𝑀 𝑀 ADP cana-5910 268 29 , , PUNCT cana-5910 268 30 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 268 31 = = SYM cana-5910 268 32 π‘’βˆ’πœ†π‘‘ π‘’βˆ’πœ†π‘‘ NOUN cana-5910 268 33 𝑀π‘₯π‘₯ 𝑀π‘₯π‘₯ NOUN cana-5910 268 34 , , PUNCT cana-5910 268 35 𝑒𝑦𝑦 𝑒𝑦𝑦 ADV cana-5910 268 36 = = PUNCT cana-5910 268 37 π‘’βˆ’πœ†π‘‘ π‘’βˆ’πœ†π‘‘ NOUN cana-5910 268 38 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 268 39 , , PUNCT cana-5910 268 40 βˆ’πœ† βˆ’πœ† ADV cana-5910 268 41 π‘’βˆ’πœ†π‘‘ π‘’βˆ’πœ†π‘‘ NOUN cana-5910 268 42 𝑀 𝑀 ADP cana-5910 268 43 = = SYM cana-5910 268 44 2 2 NUM cana-5910 268 45 π‘’βˆ’πœ†π‘‘ π‘’βˆ’πœ†π‘‘ NOUN cana-5910 268 46 ( ( PUNCT cana-5910 268 47 𝑀π‘₯π‘₯ 𝑀π‘₯π‘₯ NOUN cana-5910 268 48 + + CCONJ cana-5910 268 49 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 268 50 ) ) PUNCT cana-5910 268 51 , , PUNCT cana-5910 268 52 βˆ’πœ† βˆ’πœ† ADJ cana-5910 268 53 𝑀 𝑀 ADP cana-5910 268 54 = = SYM cana-5910 268 55 2 2 NUM cana-5910 268 56 ( ( PUNCT cana-5910 268 57 𝑀π‘₯π‘₯ 𝑀π‘₯π‘₯ NOUN cana-5910 268 58 + + CCONJ cana-5910 268 59 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 268 60 ) ) PUNCT cana-5910 268 61 . . PUNCT cana-5910 269 1 this this PRON cana-5910 269 2 is be AUX cana-5910 269 3 the the DET cana-5910 269 4 helmholtz helmholtz NOUN cana-5910 269 5 equation equation NOUN cana-5910 269 6 : : PUNCT cana-5910 269 7 𝑀π‘₯π‘₯ 𝑀π‘₯π‘₯ NOUN cana-5910 269 8 + + CCONJ cana-5910 269 9 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 269 10 + + CCONJ cana-5910 269 11 ( ( PUNCT cana-5910 269 12 βˆ’ βˆ’ PROPN cana-5910 269 13 πœ† πœ† SYM cana-5910 269 14 2 2 NUM cana-5910 269 15 ) ) PUNCT cana-5910 269 16 𝑀 𝑀 ADP cana-5910 269 17 = = SYM cana-5910 269 18 0 0 X cana-5910 269 19 . . PUNCT cana-5910 269 20 boundary boundary ADJ cana-5910 269 21 conditions condition NOUN cana-5910 269 22 : : PUNCT cana-5910 269 23 𝑀 𝑀 X cana-5910 269 24 = = SYM cana-5910 269 25 0 0 NUM cana-5910 270 1 π‘Žπ‘‘ π‘Žπ‘‘ PRON cana-5910 270 2 π‘₯ π‘₯ NOUN cana-5910 270 3 = = SYM cana-5910 270 4 0 0 NUM cana-5910 270 5 , , PUNCT cana-5910 270 6 𝑙 𝑙 X cana-5910 270 7 , , PUNCT cana-5910 270 8 𝑦 𝑦 NOUN cana-5910 270 9 = = SYM cana-5910 270 10 0 0 NUM cana-5910 270 11 , , PUNCT cana-5910 270 12 𝑙. 𝑙. ADJ cana-5910 270 13 step step NOUN cana-5910 270 14 5 5 NUM cana-5910 270 15 : : PUNCT cana-5910 270 16 solve solve VERB cana-5910 270 17 the the DET cana-5910 270 18 reduced reduce VERB cana-5910 270 19 equation equation NOUN cana-5910 270 20 solve solve NOUN cana-5910 270 21 : : PUNCT cana-5910 270 22 𝑀π‘₯π‘₯ 𝑀π‘₯π‘₯ NOUN cana-5910 270 23 + + CCONJ cana-5910 270 24 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 270 25 βˆ’ βˆ’ X cana-5910 270 26 ( ( PUNCT cana-5910 270 27 πœ† πœ† PROPN cana-5910 270 28 2 2 X cana-5910 270 29 ) ) PUNCT cana-5910 270 30 𝑀 𝑀 ADP cana-5910 270 31 = = SYM cana-5910 270 32 0 0 NUM cana-5910 270 33 , , PUNCT cana-5910 270 34 with with ADP cana-5910 270 35 𝑀(0 𝑀(0 NOUN cana-5910 270 36 , , PUNCT cana-5910 270 37 𝑦 𝑦 NOUN cana-5910 270 38 ) ) PUNCT cana-5910 270 39 = = SYM cana-5910 270 40 𝑀(𝑙 𝑀(𝑙 NOUN cana-5910 270 41 , , PUNCT cana-5910 270 42 𝑦 𝑦 NOUN cana-5910 270 43 ) ) PUNCT cana-5910 270 44 = = SYM cana-5910 270 45 𝑀(π‘₯ 𝑀(π‘₯ NOUN cana-5910 270 46 , , PUNCT cana-5910 270 47 0 0 NUM cana-5910 270 48 ) ) PUNCT cana-5910 270 49 = = SYM cana-5910 271 1 𝑀(π‘₯ 𝑀(π‘₯ NOUN cana-5910 271 2 , , PUNCT cana-5910 271 3 𝑙 𝑙 NUM cana-5910 271 4 ) ) PUNCT cana-5910 271 5 = = SYM cana-5910 271 6 0 0 X cana-5910 271 7 . . X cana-5910 271 8 use use NOUN cana-5910 271 9 separation separation NOUN cana-5910 271 10 of of ADP cana-5910 271 11 variables variable NOUN cana-5910 271 12 : : PUNCT cana-5910 272 1 𝑀(π‘₯ 𝑀(π‘₯ NOUN cana-5910 272 2 , , PUNCT cana-5910 272 3 𝑦 𝑦 NOUN cana-5910 272 4 ) ) PUNCT cana-5910 272 5 = = SYM cana-5910 272 6 𝑋(π‘₯)π‘Œ(𝑦 𝑋(π‘₯)π‘Œ(𝑦 NOUN cana-5910 272 7 ) ) PUNCT cana-5910 272 8 . . PUNCT cana-5910 273 1 substitute substitute NOUN cana-5910 273 2 : : PUNCT cana-5910 273 3 π‘‹β€²β€²π‘Œ π‘‹β€²β€²π‘Œ PROPN cana-5910 274 1 + + CCONJ cana-5910 274 2 𝑋 𝑋 PROPN cana-5910 274 3 π‘Œβ€²β€² π‘Œβ€²β€² X cana-5910 274 4 βˆ’ βˆ’ PROPN cana-5910 275 1 ( ( PUNCT cana-5910 275 2 πœ† πœ† NOUN cana-5910 275 3 2 2 X cana-5910 275 4 ) ) PUNCT cana-5910 275 5 𝑋 𝑋 PROPN cana-5910 275 6 π‘Œ π‘Œ PROPN cana-5910 275 7 = = SYM cana-5910 275 8 0 0 NUM cana-5910 275 9 , , PUNCT cana-5910 275 10 ( ( PUNCT cana-5910 275 11 𝑋′′ 𝑋′′ NOUN cana-5910 275 12 𝑋 𝑋 PROPN cana-5910 275 13 ) ) PUNCT cana-5910 276 1 + + CCONJ cana-5910 276 2 ( ( PUNCT cana-5910 276 3 π‘Œβ€²β€² π‘Œβ€²β€² X cana-5910 276 4 π‘Œ π‘Œ NOUN cana-5910 276 5 ) ) PUNCT cana-5910 276 6 = = PUNCT cana-5910 276 7 πœ† πœ† PRON cana-5910 276 8 2 2 NUM cana-5910 276 9 . . PUNCT cana-5910 277 1 set set NOUN cana-5910 277 2 : : PUNCT cana-5910 277 3 𝑋′′ 𝑋′′ PROPN cana-5910 277 4 𝑋 𝑋 PROPN cana-5910 277 5 = = SYM cana-5910 277 6 βˆ’πœ‡ βˆ’πœ‡ PROPN cana-5910 277 7 , , PUNCT cana-5910 277 8 π‘Œβ€²β€² π‘Œβ€²β€² X cana-5910 277 9 π‘Œ π‘Œ NOUN cana-5910 277 10 = = SYM cana-5910 277 11 βˆ’πœˆ βˆ’πœˆ NOUN cana-5910 277 12 , , PUNCT cana-5910 277 13 πœ‡ πœ‡ ADP cana-5910 277 14 + + CCONJ cana-5910 277 15 𝜈 𝜈 X cana-5910 277 16 = = SYM cana-5910 277 17 πœ† πœ† ADP cana-5910 277 18 2 2 NUM cana-5910 277 19 . . PUNCT cana-5910 278 1 solve solve NOUN cana-5910 278 2 : : PUNCT cana-5910 278 3 𝑋′′ 𝑋′′ X cana-5910 278 4 + + CCONJ cana-5910 278 5 πœ‡ πœ‡ ADP cana-5910 278 6 𝑋 𝑋 NOUN cana-5910 278 7 = = SYM cana-5910 278 8 0 0 NUM cana-5910 278 9 , , PUNCT cana-5910 278 10 𝑋(0 𝑋(0 X cana-5910 278 11 ) ) PUNCT cana-5910 278 12 = = SYM cana-5910 279 1 𝑋(𝑙 𝑋(𝑙 X cana-5910 279 2 ) ) PUNCT cana-5910 279 3 = = SYM cana-5910 279 4 0 0 NUM cana-5910 279 5 , , PUNCT cana-5910 279 6 1121 1121 NUM cana-5910 279 7 https://internationalpubls.com https://internationalpubls.com X cana-5910 279 8 𝑋(π‘₯ 𝑋(π‘₯ NUM cana-5910 279 9 ) ) PUNCT cana-5910 279 10 = = PRON cana-5910 279 11 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 279 12 ( ( PUNCT cana-5910 279 13 π‘›πœ‹π‘₯ π‘›πœ‹π‘₯ NOUN cana-5910 279 14 𝑙 𝑙 NOUN cana-5910 279 15 ) ) PUNCT cana-5910 279 16 , , PUNCT cana-5910 279 17 πœ‡π‘› πœ‡π‘› NOUN cana-5910 279 18 = = PUNCT cana-5910 279 19 ( ( PUNCT cana-5910 279 20 π‘›πœ‹ π‘›πœ‹ INTJ cana-5910 279 21 𝑙 𝑙 NUM cana-5910 279 22 ) ) PUNCT cana-5910 279 23 2 2 NUM cana-5910 279 24 , , PUNCT cana-5910 279 25 𝑛 𝑛 NOUN cana-5910 279 26 = = SYM cana-5910 279 27 1 1 NUM cana-5910 279 28 , , PUNCT cana-5910 279 29 2 2 NUM cana-5910 279 30 , , PUNCT cana-5910 279 31 . . PUNCT cana-5910 279 32 .. .. PUNCT cana-5910 280 1 π‘Œβ€²β€² π‘Œβ€²β€² X cana-5910 281 1 + + CCONJ cana-5910 281 2 𝜈 𝜈 X cana-5910 281 3 π‘Œ π‘Œ PROPN cana-5910 281 4 = = SYM cana-5910 281 5 0 0 NUM cana-5910 281 6 , , PUNCT cana-5910 281 7 π‘Œ(0 π‘Œ(0 PRON cana-5910 281 8 ) ) PUNCT cana-5910 281 9 = = SYM cana-5910 282 1 π‘Œ(𝑙 π‘Œ(𝑙 NOUN cana-5910 282 2 ) ) PUNCT cana-5910 282 3 = = SYM cana-5910 282 4 0 0 NUM cana-5910 282 5 , , PUNCT cana-5910 282 6 π‘Œ(𝑦 π‘Œ(𝑦 NUM cana-5910 282 7 ) ) PUNCT cana-5910 282 8 = = SYM cana-5910 282 9 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 282 10 ( ( PUNCT cana-5910 282 11 π‘šπœ‹π‘¦ π‘šπœ‹π‘¦ NOUN cana-5910 282 12 𝑙 𝑙 NOUN cana-5910 282 13 ) ) PUNCT cana-5910 282 14 , , PUNCT cana-5910 282 15 𝜈_π‘š 𝜈_π‘š PUNCT cana-5910 283 1 = = PUNCT cana-5910 283 2 ( ( PUNCT cana-5910 283 3 π‘šπœ‹ π‘šπœ‹ ADP cana-5910 283 4 𝑙 𝑙 NUM cana-5910 283 5 ) ) PUNCT cana-5910 283 6 2 2 NUM cana-5910 283 7 , , PUNCT cana-5910 283 8 π‘š π‘š X cana-5910 283 9 = = SYM cana-5910 283 10 1 1 NUM cana-5910 283 11 , , PUNCT cana-5910 283 12 2 2 NUM cana-5910 283 13 , , PUNCT cana-5910 283 14 ... ... PUNCT cana-5910 283 15 then then ADV cana-5910 283 16 : : PUNCT cana-5910 283 17 πœ†/2 πœ†/2 PROPN cana-5910 283 18 = = SYM cana-5910 283 19 ( ( PUNCT cana-5910 283 20 π‘›πœ‹ π‘›πœ‹ INTJ cana-5910 283 21 𝑙 𝑙 NUM cana-5910 283 22 ) ) PUNCT cana-5910 283 23 2 2 NUM cana-5910 284 1 + + CCONJ cana-5910 284 2 ( ( PUNCT cana-5910 284 3 π‘šπœ‹ π‘šπœ‹ ADP cana-5910 284 4 𝑙 𝑙 NUM cana-5910 284 5 ) ) PUNCT cana-5910 284 6 2 2 NUM cana-5910 284 7 , , PUNCT cana-5910 284 8 πœ†π‘›π‘š πœ†π‘›π‘š NOUN cana-5910 284 9 = = SYM cana-5910 284 10 2πœ‹2(𝑛2 2πœ‹2(𝑛2 NUM cana-5910 284 11 + + NUM cana-5910 284 12 π‘š2 π‘š2 NOUN cana-5910 284 13 ) ) PUNCT cana-5910 284 14 𝑙2 𝑙2 PROPN cana-5910 284 15 . . PUNCT cana-5910 285 1 thus thus ADV cana-5910 285 2 : : PUNCT cana-5910 285 3 π‘€π‘›π‘š(π‘₯,𝑦 π‘€π‘›π‘š(π‘₯,𝑦 X cana-5910 285 4 ) ) PUNCT cana-5910 286 1 = = SYM cana-5910 286 2 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 286 3 ( ( PUNCT cana-5910 286 4 π‘›πœ‹π‘₯ π‘›πœ‹π‘₯ NOUN cana-5910 286 5 𝑙 𝑙 NOUN cana-5910 286 6 ) ) PUNCT cana-5910 286 7 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 286 8 ( ( PUNCT cana-5910 286 9 π‘šπœ‹π‘¦ π‘šπœ‹π‘¦ NOUN cana-5910 286 10 𝑙 𝑙 NOUN cana-5910 286 11 ) ) PUNCT cana-5910 286 12 , , PUNCT cana-5910 286 13 π‘’π‘›π‘š(𝑑,π‘₯,𝑦 π‘’π‘›π‘š(𝑑,π‘₯,𝑦 NOUN cana-5910 286 14 ) ) PUNCT cana-5910 286 15 = = PUNCT cana-5910 286 16 π‘’βˆ’πœ†π‘›π‘šπ‘‘ π‘’βˆ’πœ†π‘›π‘šπ‘‘ ADP cana-5910 286 17 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 286 18 ( ( PUNCT cana-5910 286 19 π‘›πœ‹π‘₯ π‘›πœ‹π‘₯ NOUN cana-5910 286 20 𝑙 𝑙 NOUN cana-5910 286 21 ) ) PUNCT cana-5910 286 22 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 286 23 ( ( PUNCT cana-5910 286 24 π‘šπœ‹π‘¦ π‘šπœ‹π‘¦ NOUN cana-5910 286 25 𝑙 𝑙 NOUN cana-5910 286 26 ) ) PUNCT cana-5910 286 27 , , PUNCT cana-5910 286 28 πœ†π‘›π‘š πœ†π‘›π‘š NOUN cana-5910 286 29 = = SYM cana-5910 286 30 2πœ‹2(𝑛2 2πœ‹2(𝑛2 NUM cana-5910 286 31 + + NUM cana-5910 286 32 π‘š2 π‘š2 NOUN cana-5910 286 33 ) ) PUNCT cana-5910 286 34 𝑙2 𝑙2 PROPN cana-5910 286 35 . . PUNCT cana-5910 287 1 the the DET cana-5910 287 2 general general ADJ cana-5910 287 3 solution solution NOUN cana-5910 287 4 is be AUX cana-5910 287 5 : : PUNCT cana-5910 287 6 𝑒(𝑑 𝑒(𝑑 NOUN cana-5910 287 7 , , PUNCT cana-5910 287 8 π‘₯ π‘₯ NOUN cana-5910 287 9 , , PUNCT cana-5910 287 10 𝑦 𝑦 NOUN cana-5910 287 11 ) ) PUNCT cana-5910 287 12 = = SYM cana-5910 287 13 𝛴{𝑛=1 𝛴{𝑛=1 NOUN cana-5910 287 14 } } PUNCT cana-5910 287 15 ∞ ∞ NUM cana-5910 287 16 𝛴{π‘š=1 𝛴{π‘š=1 PROPN cana-5910 287 17 } } PUNCT cana-5910 287 18 ∞ ∞ NUM cana-5910 288 1 π΄π‘›π‘š π΄π‘›π‘š PROPN cana-5910 288 2 𝑒 𝑒 PROPN cana-5910 288 3 βˆ’ βˆ’ PROPN cana-5910 288 4 2πœ‹2(𝑛2 2πœ‹2(𝑛2 NUM cana-5910 288 5 + + NUM cana-5910 288 6 π‘š2)𝑑 π‘š2)𝑑 NOUN cana-5910 288 7 𝑙2 𝑙2 NOUN cana-5910 288 8 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 288 9 ( ( PUNCT cana-5910 288 10 π‘›πœ‹π‘₯ π‘›πœ‹π‘₯ NOUN cana-5910 288 11 𝑙 𝑙 NOUN cana-5910 288 12 ) ) PUNCT cana-5910 288 13 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 288 14 ( ( PUNCT cana-5910 288 15 π‘šπœ‹π‘¦ π‘šπœ‹π‘¦ NOUN cana-5910 288 16 𝑙 𝑙 NOUN cana-5910 288 17 ) ) PUNCT cana-5910 288 18 . . PUNCT cana-5910 289 1 step step NOUN cana-5910 289 2 6 6 NUM cana-5910 289 3 : : PUNCT cana-5910 289 4 apply apply VERB cana-5910 289 5 boundary boundary ADJ cana-5910 289 6 and and CCONJ cana-5910 289 7 initial initial ADJ cana-5910 289 8 conditions condition NOUN cana-5910 289 9 β€’ β€’ ADP cana-5910 289 10 boundary boundary ADJ cana-5910 289 11 conditions condition NOUN cana-5910 289 12 𝑒 𝑒 NOUN cana-5910 289 13 = = SYM cana-5910 289 14 0 0 NUM cana-5910 289 15 π‘Žπ‘‘ π‘Žπ‘‘ PRON cana-5910 290 1 π‘₯ π‘₯ NOUN cana-5910 290 2 = = SYM cana-5910 290 3 0 0 NUM cana-5910 290 4 , , PUNCT cana-5910 290 5 𝑙 𝑙 X cana-5910 290 6 , , PUNCT cana-5910 290 7 𝑦 𝑦 NOUN cana-5910 290 8 = = SYM cana-5910 290 9 0 0 NUM cana-5910 290 10 , , PUNCT cana-5910 290 11 𝑙 𝑙 PRON cana-5910 290 12 are be AUX cana-5910 290 13 satisfied satisfied ADJ cana-5910 290 14 , , PUNCT cana-5910 290 15 as as ADP cana-5910 290 16 𝑠𝑖𝑛 𝑠𝑖𝑛 ADJ cana-5910 290 17 ( ( PUNCT cana-5910 290 18 π‘›πœ‹ π‘›πœ‹ PART cana-5910 290 19 Β· Β· PUNCT cana-5910 290 20 0 0 NUM cana-5910 291 1 𝑙 𝑙 NOUN cana-5910 291 2 ) ) PUNCT cana-5910 291 3 = = SYM cana-5910 291 4 𝑠𝑖𝑛(π‘›πœ‹ 𝑠𝑖𝑛(π‘›πœ‹ NOUN cana-5910 291 5 ) ) PUNCT cana-5910 291 6 = = SYM cana-5910 291 7 0 0 NUM cana-5910 291 8 , , PUNCT cana-5910 291 9 etc etc X cana-5910 291 10 . . X cana-5910 291 11 β€’ β€’ ADV cana-5910 291 12 as as ADP cana-5910 291 13 𝑑 𝑑 PROPN cana-5910 291 14 β†’ β†’ SYM cana-5910 291 15 ∞ ∞ PROPN cana-5910 291 16 , , PUNCT cana-5910 291 17 𝑒 𝑒 PROPN cana-5910 291 18 βˆ’ βˆ’ PROPN cana-5910 291 19 2πœ‹2(𝑛2 2πœ‹2(𝑛2 NUM cana-5910 291 20 + + NUM cana-5910 291 21 π‘š2)𝑑 π‘š2)𝑑 NOUN cana-5910 291 22 𝑙2 𝑙2 NOUN cana-5910 291 23 β†’ β†’ SYM cana-5910 291 24 0 0 NUM cana-5910 291 25 , , PUNCT cana-5910 291 26 π‘ π‘œ π‘ π‘œ ADP cana-5910 291 27 𝑒 𝑒 NOUN cana-5910 291 28 β†’ β†’ SYM cana-5910 291 29 0 0 NUM cana-5910 291 30 , , PUNCT cana-5910 291 31 satisfying satisfy VERB cana-5910 291 32 the the DET cana-5910 291 33 condition condition NOUN cana-5910 291 34 . . PUNCT cana-5910 292 1 the the DET cana-5910 292 2 coefficients coefficient NOUN cana-5910 292 3 π΄π‘›π‘š π΄π‘›π‘š PROPN cana-5910 292 4 depend depend VERB cana-5910 292 5 on on ADP cana-5910 292 6 an an DET cana-5910 292 7 initial initial ADJ cana-5910 292 8 condition condition NOUN cana-5910 292 9 𝑒(0 𝑒(0 PROPN cana-5910 292 10 , , PUNCT cana-5910 292 11 π‘₯ π‘₯ PROPN cana-5910 292 12 , , PUNCT cana-5910 292 13 𝑦 𝑦 NOUN cana-5910 292 14 ) ) PUNCT cana-5910 292 15 = = SYM cana-5910 292 16 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-5910 292 17 , , PUNCT cana-5910 292 18 𝑦 𝑦 NOUN cana-5910 292 19 ) ) PUNCT cana-5910 292 20 , , PUNCT cana-5910 292 21 which which PRON cana-5910 292 22 is be AUX cana-5910 292 23 not not PART cana-5910 292 24 provided provide VERB cana-5910 292 25 . . PUNCT cana-5910 293 1 if if SCONJ cana-5910 293 2 no no DET cana-5910 293 3 initial initial ADJ cana-5910 293 4 condition condition NOUN cana-5910 293 5 is be AUX cana-5910 293 6 given give VERB cana-5910 293 7 , , PUNCT cana-5910 293 8 the the DET cana-5910 293 9 solution solution NOUN cana-5910 293 10 is be AUX cana-5910 293 11 : : PUNCT cana-5910 293 12 𝑒(𝑑 𝑒(𝑑 NOUN cana-5910 293 13 , , PUNCT cana-5910 293 14 π‘₯ π‘₯ NOUN cana-5910 293 15 , , PUNCT cana-5910 293 16 𝑦 𝑦 NOUN cana-5910 293 17 ) ) PUNCT cana-5910 293 18 = = SYM cana-5910 293 19 𝛴{𝑛=1 𝛴{𝑛=1 NOUN cana-5910 293 20 } } PUNCT cana-5910 293 21 ∞ ∞ NUM cana-5910 293 22 𝛴{π‘š=1 𝛴{π‘š=1 PROPN cana-5910 293 23 } } PUNCT cana-5910 293 24 ∞ ∞ NUM cana-5910 294 1 π΄π‘›π‘š π΄π‘›π‘š PROPN cana-5910 294 2 𝑒 𝑒 PROPN cana-5910 294 3 βˆ’ βˆ’ PROPN cana-5910 294 4 2πœ‹2(𝑛2 2πœ‹2(𝑛2 NUM cana-5910 294 5 + + NUM cana-5910 294 6 π‘š2)𝑑 π‘š2)𝑑 NOUN cana-5910 294 7 𝑙2 𝑙2 NOUN cana-5910 294 8 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 294 9 ( ( PUNCT cana-5910 294 10 π‘›πœ‹π‘₯ π‘›πœ‹π‘₯ NOUN cana-5910 294 11 𝑙 𝑙 NOUN cana-5910 294 12 ) ) PUNCT cana-5910 294 13 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 294 14 ( ( PUNCT cana-5910 294 15 π‘šπœ‹π‘¦ π‘šπœ‹π‘¦ NOUN cana-5910 294 16 𝑙 𝑙 NOUN cana-5910 294 17 ) ) PUNCT cana-5910 294 18 , , PUNCT cana-5910 294 19 where where SCONJ cana-5910 294 20 π΄π‘›π‘š π΄π‘›π‘š PROPN cana-5910 294 21 are be AUX cana-5910 294 22 determined determine VERB cana-5910 294 23 by by ADP cana-5910 294 24 : : PUNCT cana-5910 294 25 π΄π‘›π‘š π΄π‘›π‘š PROPN cana-5910 294 26 = = PUNCT cana-5910 294 27 ( ( PUNCT cana-5910 294 28 4 4 NUM cana-5910 294 29 𝑙2 𝑙2 PROPN cana-5910 294 30 ) ) PUNCT cana-5910 294 31 ∫ ∫ PROPN cana-5910 294 32 ∫ ∫ PROPN cana-5910 294 33 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-5910 294 34 , , PUNCT cana-5910 294 35 𝑦)𝑠𝑖𝑛 𝑦)𝑠𝑖𝑛 PROPN cana-5910 294 36 ( ( PUNCT cana-5910 294 37 π‘›πœ‹π‘₯ π‘›πœ‹π‘₯ NOUN cana-5910 294 38 𝑙 𝑙 NOUN cana-5910 294 39 ) ) PUNCT cana-5910 294 40 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 294 41 ( ( PUNCT cana-5910 294 42 π‘šπœ‹π‘¦ π‘šπœ‹π‘¦ NOUN cana-5910 294 43 𝑙 𝑙 NOUN cana-5910 294 44 ) ) PUNCT cana-5910 294 45 𝑑π‘₯ 𝑑π‘₯ VERB cana-5910 294 46 𝑑𝑦 𝑑𝑦 ADP cana-5910 294 47 𝑙 𝑙 PROPN cana-5910 294 48 0 0 NUM cana-5910 294 49 𝑙 𝑙 NOUN cana-5910 294 50 0 0 NUM cana-5910 294 51 . . PUNCT cana-5910 295 1 without without ADP cana-5910 295 2 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-5910 295 3 , , PUNCT cana-5910 295 4 𝑦 𝑦 NOUN cana-5910 295 5 ) ) PUNCT cana-5910 295 6 , , PUNCT cana-5910 295 7 we we PRON cana-5910 295 8 leave leave VERB cana-5910 295 9 π΄π‘›π‘š π΄π‘›π‘š PROPN cana-5910 295 10 arbitrary arbitrary ADJ cana-5910 295 11 . . PUNCT cana-5910 296 1 final final ADJ cana-5910 296 2 answer answer NOUN cana-5910 296 3 the the DET cana-5910 296 4 solution solution NOUN cana-5910 296 5 to to ADP cana-5910 296 6 the the DET cana-5910 296 7 pde pde PROPN cana-5910 296 8 πœ•π‘’ πœ•π‘’ NOUN cana-5910 296 9 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 297 1 = = NOUN cana-5910 297 2 2 2 NUM cana-5910 297 3 ( ( PUNCT cana-5910 297 4 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 297 5 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 297 6 + + CCONJ cana-5910 297 7 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 297 8 πœ•π‘¦2 πœ•π‘¦2 NUM cana-5910 297 9 ) ) PUNCT cana-5910 297 10 with with ADP cana-5910 297 11 boundary boundary ADJ cana-5910 297 12 conditions condition NOUN cana-5910 297 13 𝑒 𝑒 NOUN cana-5910 297 14 = = SYM cana-5910 297 15 0 0 NUM cana-5910 297 16 π‘Žπ‘‘ π‘Žπ‘‘ PRON cana-5910 298 1 π‘₯ π‘₯ NOUN cana-5910 298 2 = = SYM cana-5910 298 3 0 0 NUM cana-5910 298 4 , , PUNCT cana-5910 298 5 π‘₯ π‘₯ NOUN cana-5910 298 6 = = SYM cana-5910 298 7 𝑙 𝑙 PROPN cana-5910 298 8 , , PUNCT cana-5910 298 9 𝑦 𝑦 NOUN cana-5910 298 10 = = SYM cana-5910 298 11 0 0 NUM cana-5910 298 12 , , PUNCT cana-5910 298 13 𝑦 𝑦 NOUN cana-5910 298 14 = = SYM cana-5910 298 15 𝑙 𝑙 PROPN cana-5910 298 16 , , PUNCT cana-5910 298 17 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-5910 298 18 𝑒 𝑒 PROPN cana-5910 298 19 β†’ β†’ SYM cana-5910 298 20 0 0 NUM cana-5910 298 21 π‘Žπ‘  π‘Žπ‘  PROPN cana-5910 298 22 𝑑 𝑑 PROPN cana-5910 298 23 β†’ β†’ SYM cana-5910 298 24 ∞ ∞ PROPN cana-5910 298 25 , , PUNCT cana-5910 298 26 obtained obtain VERB cana-5910 298 27 via via ADP cana-5910 298 28 lie lie NOUN cana-5910 298 29 symmetry symmetry NOUN cana-5910 298 30 reduction reduction NOUN cana-5910 298 31 , , PUNCT cana-5910 298 32 is be AUX cana-5910 298 33 : : PUNCT cana-5910 298 34 𝑒(𝑑 𝑒(𝑑 NOUN cana-5910 298 35 , , PUNCT cana-5910 298 36 π‘₯ π‘₯ NOUN cana-5910 298 37 , , PUNCT cana-5910 298 38 𝑦 𝑦 NOUN cana-5910 298 39 ) ) PUNCT cana-5910 298 40 = = SYM cana-5910 298 41 𝛴{𝑛=1 𝛴{𝑛=1 NOUN cana-5910 298 42 } } PUNCT cana-5910 298 43 ∞ ∞ NUM cana-5910 298 44 𝛴{π‘š=1 𝛴{π‘š=1 PROPN cana-5910 298 45 } } PUNCT cana-5910 298 46 ∞ ∞ PROPN cana-5910 299 1 π΄π‘›π‘š π΄π‘›π‘š PROPN cana-5910 300 1 𝑒π‘₯𝑝 𝑒π‘₯𝑝 INTJ cana-5910 300 2 ( ( PUNCT cana-5910 300 3 βˆ’ βˆ’ PROPN cana-5910 300 4 2πœ‹2(𝑛2 2πœ‹2(𝑛2 NUM cana-5910 300 5 + + NUM cana-5910 300 6 π‘š2)𝑑 π‘š2)𝑑 NOUN cana-5910 300 7 𝑙2 𝑙2 NOUN cana-5910 300 8 ) ) PUNCT cana-5910 300 9 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 300 10 ( ( PUNCT cana-5910 300 11 π‘›πœ‹π‘₯ π‘›πœ‹π‘₯ NOUN cana-5910 300 12 𝑙 𝑙 NOUN cana-5910 300 13 ) ) PUNCT cana-5910 300 14 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 300 15 ( ( PUNCT cana-5910 300 16 π‘šπœ‹π‘¦ π‘šπœ‹π‘¦ NOUN cana-5910 300 17 𝑙 𝑙 NOUN cana-5910 300 18 ) ) PUNCT cana-5910 300 19 , , PUNCT cana-5910 300 20 where where SCONJ cana-5910 300 21 π΄π‘›π‘š π΄π‘›π‘š PROPN cana-5910 300 22 are be AUX cana-5910 300 23 coefficients coefficient NOUN cana-5910 300 24 determined determine VERB cana-5910 300 25 by by ADP cana-5910 300 26 the the DET cana-5910 300 27 initial initial ADJ cana-5910 300 28 condition condition NOUN cana-5910 300 29 𝑒(0 𝑒(0 PROPN cana-5910 300 30 , , PUNCT cana-5910 300 31 π‘₯ π‘₯ PROPN cana-5910 300 32 , , PUNCT cana-5910 300 33 𝑦 𝑦 NOUN cana-5910 300 34 ) ) PUNCT cana-5910 300 35 = = SYM cana-5910 300 36 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-5910 300 37 , , PUNCT cana-5910 300 38 𝑦 𝑦 NOUN cana-5910 300 39 ) ) PUNCT cana-5910 300 40 via via ADP cana-5910 300 41 : : PUNCT cana-5910 300 42 π΄π‘›π‘š π΄π‘›π‘š PROPN cana-5910 300 43 = = PUNCT cana-5910 300 44 ( ( PUNCT cana-5910 300 45 4 4 NUM cana-5910 300 46 𝑙2 𝑙2 PROPN cana-5910 300 47 ) ) PUNCT cana-5910 300 48 ∫ ∫ PROPN cana-5910 300 49 ∫ ∫ PROPN cana-5910 300 50 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-5910 300 51 , , PUNCT cana-5910 300 52 𝑦)𝑠𝑖𝑛 𝑦)𝑠𝑖𝑛 PROPN cana-5910 300 53 ( ( PUNCT cana-5910 300 54 π‘›πœ‹π‘₯ π‘›πœ‹π‘₯ NOUN cana-5910 300 55 𝑙 𝑙 NOUN cana-5910 300 56 ) ) PUNCT cana-5910 300 57 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 300 58 ( ( PUNCT cana-5910 300 59 π‘šπœ‹π‘¦ π‘šπœ‹π‘¦ NOUN cana-5910 300 60 𝑙 𝑙 NOUN cana-5910 300 61 ) ) PUNCT cana-5910 300 62 𝑑π‘₯ 𝑑π‘₯ VERB cana-5910 300 63 𝑑𝑦 𝑑𝑦 ADP cana-5910 300 64 𝑙 𝑙 PROPN cana-5910 300 65 0 0 NUM cana-5910 300 66 𝑙 𝑙 NOUN cana-5910 300 67 0 0 NUM cana-5910 300 68 . . PUNCT cana-5910 301 1 if if SCONJ cana-5910 301 2 no no DET cana-5910 301 3 initial initial ADJ cana-5910 301 4 condition condition NOUN cana-5910 301 5 is be AUX cana-5910 301 6 specified specify VERB cana-5910 301 7 , , PUNCT cana-5910 301 8 π΄π‘›π‘š π΄π‘›π‘š PROPN cana-5910 301 9 remain remain VERB cana-5910 301 10 arbitrary arbitrary ADJ cana-5910 301 11 constants constant NOUN cana-5910 301 12 . . PUNCT cana-5910 302 1 4.2 4.2 NUM cana-5910 302 2 solve solve VERB cana-5910 302 3 the the DET cana-5910 302 4 wave wave NOUN cana-5910 302 5 equation equation NOUN cana-5910 302 6 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 302 7 πœ•π‘‘2 πœ•π‘‘2 NOUN cana-5910 302 8 = = SYM cana-5910 302 9 4 4 NUM cana-5910 302 10 ( ( PUNCT cana-5910 302 11 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 302 12 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 302 13 + + CCONJ cana-5910 302 14 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 302 15 πœ•π‘¦2 πœ•π‘¦2 NOUN cana-5910 302 16 ) ) PUNCT cana-5910 302 17 1122 1122 NUM cana-5910 302 18 https://internationalpubls.com https://internationalpubls.com X cana-5910 302 19 by by ADP cana-5910 302 20 lie lie NOUN cana-5910 302 21 symmetry symmetry NOUN cana-5910 302 22 theory theory NOUN cana-5910 302 23 when when SCONJ cana-5910 302 24 𝑒 𝑒 X cana-5910 302 25 = = SYM cana-5910 302 26 0 0 NUM cana-5910 302 27 π‘€β„Žπ‘’π‘› π‘€β„Žπ‘’π‘› VERB cana-5910 302 28 𝑑 𝑑 PROPN cana-5910 302 29 = = SYM cana-5910 302 30 ∞ ∞ PROPN cana-5910 302 31 , , PUNCT cana-5910 302 32 π‘₯ π‘₯ PROPN cana-5910 302 33 = = SYM cana-5910 302 34 0 0 NUM cana-5910 302 35 π‘œπ‘Ÿ π‘œπ‘Ÿ ADP cana-5910 302 36 𝑙 𝑙 X cana-5910 302 37 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-5910 302 38 𝑦 𝑦 PROPN cana-5910 302 39 = = SYM cana-5910 302 40 0 0 NUM cana-5910 302 41 π‘œπ‘Ÿ π‘œπ‘Ÿ ADP cana-5910 302 42 𝑙 𝑙 DET cana-5910 302 43 solution solution NOUN cana-5910 302 44 to to PART cana-5910 302 45 solve solve VERB cana-5910 302 46 the the DET cana-5910 302 47 given give VERB cana-5910 302 48 partial partial ADJ cana-5910 302 49 differential differential NOUN cana-5910 302 50 equation equation NOUN cana-5910 302 51 ( ( PUNCT cana-5910 302 52 pde pde NOUN cana-5910 302 53 ) ) PUNCT cana-5910 302 54 using use VERB cana-5910 302 55 lie lie NOUN cana-5910 302 56 symmetry symmetry NOUN cana-5910 302 57 theory theory NOUN cana-5910 302 58 , , PUNCT cana-5910 302 59 we we PRON cana-5910 302 60 need need VERB cana-5910 302 61 to to PART cana-5910 302 62 carefully carefully ADV cana-5910 302 63 analyse analyse VERB cana-5910 302 64 the the DET cana-5910 302 65 equation equation NOUN cana-5910 302 66 , , PUNCT cana-5910 302 67 boundary boundary ADJ cana-5910 302 68 conditions condition NOUN cana-5910 302 69 , , PUNCT cana-5910 302 70 and and CCONJ cana-5910 302 71 apply apply VERB cana-5910 302 72 the the DET cana-5910 302 73 lie lie NOUN cana-5910 302 74 group group NOUN cana-5910 302 75 method method NOUN cana-5910 302 76 systematically systematically ADV cana-5910 302 77 problem problem NOUN cana-5910 302 78 statement statement NOUN cana-5910 302 79 we we PRON cana-5910 302 80 need need VERB cana-5910 302 81 to to PART cana-5910 302 82 solve solve VERB cana-5910 302 83 the the DET cana-5910 302 84 pde pde NOUN cana-5910 302 85 : : PUNCT cana-5910 302 86 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 302 87 πœ•π‘‘2 πœ•π‘‘2 NOUN cana-5910 302 88 = = SYM cana-5910 302 89 4 4 NUM cana-5910 302 90 ( ( PUNCT cana-5910 302 91 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 302 92 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 302 93 + + CCONJ cana-5910 302 94 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 302 95 πœ•π‘¦2 πœ•π‘¦2 NOUN cana-5910 302 96 ) ) PUNCT cana-5910 302 97 with with ADP cana-5910 302 98 boundary boundary ADJ cana-5910 302 99 conditions condition NOUN cana-5910 302 100 : : PUNCT cana-5910 302 101 𝑒 𝑒 X cana-5910 302 102 = = SYM cana-5910 302 103 0 0 NUM cana-5910 302 104 π‘€β„Žπ‘’π‘› π‘€β„Žπ‘’π‘› VERB cana-5910 302 105 𝑑 𝑑 PROPN cana-5910 302 106 β†’ β†’ SYM cana-5910 302 107 ∞ ∞ PROPN cana-5910 302 108 , , PUNCT cana-5910 302 109 π‘₯ π‘₯ PROPN cana-5910 302 110 = = SYM cana-5910 302 111 0 0 NUM cana-5910 302 112 π‘œπ‘Ÿ π‘œπ‘Ÿ PRON cana-5910 302 113 π‘₯ π‘₯ NOUN cana-5910 302 114 = = SYM cana-5910 302 115 𝑙 𝑙 PROPN cana-5910 302 116 , , PUNCT cana-5910 302 117 𝑦 𝑦 NOUN cana-5910 302 118 = = SYM cana-5910 302 119 0 0 NUM cana-5910 302 120 π‘œπ‘Ÿ π‘œπ‘Ÿ PROPN cana-5910 302 121 𝑦 𝑦 NOUN cana-5910 302 122 = = PUNCT cana-5910 302 123 𝑙. 𝑙. NOUN cana-5910 303 1 this this PRON cana-5910 303 2 is be AUX cana-5910 303 3 a a DET cana-5910 303 4 two two NUM cana-5910 303 5 - - PUNCT cana-5910 303 6 dimensional dimensional ADJ cana-5910 303 7 wave wave NOUN cana-5910 303 8 equation equation NOUN cana-5910 303 9 with with ADP cana-5910 303 10 a a DET cana-5910 303 11 wave wave NOUN cana-5910 303 12 speed speed NOUN cana-5910 303 13 squared square VERB cana-5910 303 14 of of ADP cana-5910 303 15 4 4 NUM cana-5910 303 16 , , PUNCT cana-5910 303 17 defined define VERB cana-5910 303 18 on on ADP cana-5910 303 19 the the DET cana-5910 303 20 domain domain NOUN cana-5910 303 21 0 0 PUNCT cana-5910 303 22 < < X cana-5910 303 23 π‘₯ π‘₯ X cana-5910 303 24 < < X cana-5910 303 25 𝑙 𝑙 X cana-5910 303 26 , , PUNCT cana-5910 303 27 0 0 PUNCT cana-5910 303 28 < < X cana-5910 303 29 𝑦 𝑦 X cana-5910 303 30 < < X cana-5910 303 31 𝑙 𝑙 X cana-5910 303 32 , , PUNCT cana-5910 303 33 with with ADP cana-5910 303 34 homogeneous homogeneous ADJ cana-5910 303 35 dirichlet dirichlet PROPN cana-5910 303 36 boundary boundary ADJ cana-5910 303 37 conditions condition NOUN cana-5910 303 38 and and CCONJ cana-5910 303 39 a a DET cana-5910 303 40 condition condition NOUN cana-5910 303 41 at at ADP cana-5910 303 42 infinite infinite ADJ cana-5910 303 43 time time NOUN cana-5910 303 44 . . PUNCT cana-5910 304 1 we we PRON cana-5910 304 2 will will AUX cana-5910 304 3 use use VERB cana-5910 304 4 lie lie NOUN cana-5910 304 5 symmetry symmetry NOUN cana-5910 304 6 theory theory NOUN cana-5910 304 7 to to PART cana-5910 304 8 find find VERB cana-5910 304 9 symmetry symmetry NOUN cana-5910 304 10 reductions reduction NOUN cana-5910 304 11 and and CCONJ cana-5910 304 12 derive derive ADJ cana-5910 304 13 solutions solution NOUN cana-5910 304 14 . . PUNCT cana-5910 305 1 step step NOUN cana-5910 305 2 1 1 NUM cana-5910 305 3 : : PUNCT cana-5910 305 4 formulate formulate VERB cana-5910 305 5 the the DET cana-5910 305 6 pde pde NOUN cana-5910 305 7 the the DET cana-5910 305 8 given give VERB cana-5910 305 9 pde pde NOUN cana-5910 305 10 is be AUX cana-5910 305 11 : : PUNCT cana-5910 305 12 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 305 13 πœ•π‘‘2 πœ•π‘‘2 NOUN cana-5910 305 14 = = SYM cana-5910 305 15 4 4 NUM cana-5910 305 16 ( ( PUNCT cana-5910 305 17 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 305 18 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 305 19 + + CCONJ cana-5910 305 20 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 305 21 πœ•π‘¦2 πœ•π‘¦2 NUM cana-5910 305 22 ) ) PUNCT cana-5910 305 23 . . PUNCT cana-5910 306 1 in in ADP cana-5910 306 2 standard standard ADJ cana-5910 306 3 notation notation NOUN cana-5910 306 4 : : PUNCT cana-5910 306 5 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 306 6 βˆ’ βˆ’ PROPN cana-5910 306 7 4(𝑒π‘₯π‘₯ 4(𝑒π‘₯π‘₯ NOUN cana-5910 306 8 + + CCONJ cana-5910 306 9 𝑒𝑦𝑦 𝑒𝑦𝑦 ADJ cana-5910 306 10 ) ) PUNCT cana-5910 306 11 = = SYM cana-5910 307 1 0 0 NUM cana-5910 307 2 , , PUNCT cana-5910 307 3 where where SCONJ cana-5910 307 4 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 307 5 = = SYM cana-5910 307 6 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 307 7 πœ•π‘‘2 πœ•π‘‘2 NOUN cana-5910 307 8 , , PUNCT cana-5910 307 9 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 307 10 = = SYM cana-5910 307 11 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 307 12 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 307 13 , , PUNCT cana-5910 307 14 𝑒𝑦𝑦 𝑒𝑦𝑦 ADV cana-5910 307 15 = = SYM cana-5910 307 16 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 307 17 πœ•π‘¦2 πœ•π‘¦2 PROPN cana-5910 307 18 . . PUNCT cana-5910 308 1 the the DET cana-5910 308 2 boundary boundary ADJ cana-5910 308 3 conditions condition NOUN cana-5910 308 4 are be AUX cana-5910 308 5 : : PUNCT cana-5910 308 6 β€’ β€’ ADP cana-5910 308 7 𝑒(𝑑 𝑒(𝑑 NOUN cana-5910 308 8 , , PUNCT cana-5910 308 9 π‘₯ π‘₯ NOUN cana-5910 308 10 , , PUNCT cana-5910 308 11 𝑦 𝑦 NOUN cana-5910 308 12 ) ) PUNCT cana-5910 308 13 = = SYM cana-5910 309 1 0 0 NUM cana-5910 310 1 π‘Žπ‘‘ π‘Žπ‘‘ PRON cana-5910 310 2 π‘₯ π‘₯ NOUN cana-5910 310 3 = = SYM cana-5910 310 4 0 0 NUM cana-5910 310 5 , , PUNCT cana-5910 310 6 π‘₯ π‘₯ NOUN cana-5910 310 7 = = SYM cana-5910 310 8 𝑙 𝑙 PROPN cana-5910 310 9 , , PUNCT cana-5910 310 10 𝑦 𝑦 NOUN cana-5910 310 11 = = SYM cana-5910 310 12 0 0 NUM cana-5910 310 13 , , PUNCT cana-5910 310 14 𝑦 𝑦 NOUN cana-5910 310 15 = = SYM cana-5910 310 16 𝑙 𝑙 PRON cana-5910 311 1 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ NOUN cana-5910 311 2 π‘Žπ‘™π‘™ π‘Žπ‘™π‘™ NOUN cana-5910 311 3 𝑑. 𝑑. NOUN cana-5910 311 4 β€’ β€’ PRON cana-5910 311 5 𝑒(𝑑 𝑒(𝑑 NOUN cana-5910 311 6 , , PUNCT cana-5910 311 7 π‘₯ π‘₯ NOUN cana-5910 311 8 , , PUNCT cana-5910 311 9 𝑦 𝑦 NOUN cana-5910 311 10 ) ) PUNCT cana-5910 311 11 β†’ β†’ SYM cana-5910 311 12 0 0 NUM cana-5910 311 13 π‘Žπ‘  π‘Žπ‘  PROPN cana-5910 311 14 𝑑 𝑑 PROPN cana-5910 311 15 β†’ β†’ SYM cana-5910 311 16 ∞ ∞ NUM cana-5910 311 17 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-5910 311 18 0 0 NUM cana-5910 311 19 < < X cana-5910 311 20 π‘₯ π‘₯ X cana-5910 311 21 < < X cana-5910 311 22 𝑙 𝑙 X cana-5910 311 23 , , PUNCT cana-5910 311 24 0 0 PUNCT cana-5910 311 25 < < X cana-5910 311 26 𝑦 𝑦 X cana-5910 311 27 < < X cana-5910 311 28 𝑙. 𝑙. NOUN cana-5910 312 1 our our PRON cana-5910 312 2 objective objective NOUN cana-5910 312 3 is be AUX cana-5910 312 4 to to PART cana-5910 312 5 find find VERB cana-5910 312 6 lie lie NOUN cana-5910 312 7 point point NOUN cana-5910 312 8 symmetries symmetry NOUN cana-5910 312 9 of of ADP cana-5910 312 10 the the DET cana-5910 312 11 pde pde NOUN cana-5910 312 12 , , PUNCT cana-5910 312 13 use use VERB cana-5910 312 14 them they PRON cana-5910 312 15 to to PART cana-5910 312 16 reduce reduce VERB cana-5910 312 17 the the DET cana-5910 312 18 pde pde NOUN cana-5910 312 19 to to ADP cana-5910 312 20 a a DET cana-5910 312 21 simpler simple ADJ cana-5910 312 22 form form NOUN cana-5910 312 23 ( ( PUNCT cana-5910 312 24 e.g. e.g. ADV cana-5910 312 25 , , PUNCT cana-5910 312 26 an an DET cana-5910 312 27 ode ode NOUN cana-5910 312 28 or or CCONJ cana-5910 312 29 a a DET cana-5910 312 30 pde pde NOUN cana-5910 312 31 with with ADP cana-5910 312 32 fewer few ADJ cana-5910 312 33 variables variable NOUN cana-5910 312 34 ) ) PUNCT cana-5910 312 35 , , PUNCT cana-5910 312 36 and and CCONJ cana-5910 312 37 solve solve VERB cana-5910 312 38 while while SCONJ cana-5910 312 39 satisfying satisfy VERB cana-5910 312 40 the the DET cana-5910 312 41 boundary boundary ADJ cana-5910 312 42 conditions condition NOUN cana-5910 312 43 . . PUNCT cana-5910 313 1 step step NOUN cana-5910 313 2 2 2 NUM cana-5910 313 3 : : PUNCT cana-5910 313 4 lie lie NOUN cana-5910 313 5 symmetry symmetry NOUN cana-5910 313 6 analysis analysis NOUN cana-5910 313 7 lie lie NOUN cana-5910 313 8 symmetry symmetry NOUN cana-5910 313 9 theory theory NOUN cana-5910 313 10 involves involve VERB cana-5910 313 11 finding find VERB cana-5910 313 12 infinitesimal infinitesimal ADJ cana-5910 313 13 transformations transformation NOUN cana-5910 313 14 that that PRON cana-5910 313 15 leave leave VERB cana-5910 313 16 the the DET cana-5910 313 17 pde pde NOUN cana-5910 313 18 invariant invariant ADJ cana-5910 313 19 . . PUNCT cana-5910 314 1 consider consider VERB cana-5910 314 2 a a DET cana-5910 314 3 oneparameter oneparameter ADJ cana-5910 314 4 lie lie NOUN cana-5910 314 5 group group NOUN cana-5910 314 6 of of ADP cana-5910 314 7 transformations transformation NOUN cana-5910 314 8 : : PUNCT cana-5910 314 9 𝑑 𝑑 X cana-5910 314 10 βˆ— βˆ— NOUN cana-5910 314 11 = = SYM cana-5910 314 12 𝑑 𝑑 PROPN cana-5910 314 13 + + NOUN cana-5910 314 14 νœ€πœ(𝑑 νœ€πœ(𝑑 NUM cana-5910 314 15 , , PUNCT cana-5910 314 16 π‘₯ π‘₯ NOUN cana-5910 314 17 , , PUNCT cana-5910 314 18 𝑦 𝑦 NOUN cana-5910 314 19 , , PUNCT cana-5910 314 20 𝑒 𝑒 X cana-5910 314 21 ) ) PUNCT cana-5910 314 22 + + CCONJ cana-5910 314 23 𝑂(νœ€Β² 𝑂(νœ€Β² NOUN cana-5910 314 24 ) ) PUNCT cana-5910 314 25 , , PUNCT cana-5910 315 1 π‘₯ π‘₯ NOUN cana-5910 315 2 βˆ— βˆ— NOUN cana-5910 315 3 = = PUNCT cana-5910 316 1 π‘₯ π‘₯ NOUN cana-5910 317 1 + + NOUN cana-5910 317 2 νœ€πœ‰(𝑑 νœ€πœ‰(𝑑 NUM cana-5910 317 3 , , PUNCT cana-5910 317 4 π‘₯ π‘₯ X cana-5910 317 5 , , PUNCT cana-5910 317 6 𝑦 𝑦 NOUN cana-5910 317 7 , , PUNCT cana-5910 317 8 𝑒 𝑒 X cana-5910 317 9 ) ) PUNCT cana-5910 317 10 + + CCONJ cana-5910 317 11 𝑂(νœ€Β² 𝑂(νœ€Β² NOUN cana-5910 317 12 ) ) PUNCT cana-5910 317 13 , , PUNCT cana-5910 317 14 𝑦 𝑦 NOUN cana-5910 317 15 βˆ— βˆ— NOUN cana-5910 317 16 = = SYM cana-5910 317 17 𝑦 𝑦 NOUN cana-5910 317 18 + + NUM cana-5910 317 19 νœ€πœ‚(𝑑 νœ€πœ‚(𝑑 NUM cana-5910 317 20 , , PUNCT cana-5910 317 21 π‘₯ π‘₯ PRON cana-5910 317 22 , , PUNCT cana-5910 317 23 𝑦 𝑦 NOUN cana-5910 317 24 , , PUNCT cana-5910 317 25 𝑒 𝑒 X cana-5910 317 26 ) ) PUNCT cana-5910 317 27 + + CCONJ cana-5910 317 28 𝑂(νœ€Β² 𝑂(νœ€Β² NOUN cana-5910 317 29 ) ) PUNCT cana-5910 317 30 , , PUNCT cana-5910 317 31 𝑒 𝑒 PRON cana-5910 317 32 βˆ— βˆ— NOUN cana-5910 317 33 = = SYM cana-5910 317 34 𝑒 𝑒 NOUN cana-5910 317 35 + + X cana-5910 317 36 νœ€πœ‘(𝑑 νœ€πœ‘(𝑑 NUM cana-5910 317 37 , , PUNCT cana-5910 317 38 π‘₯ π‘₯ NOUN cana-5910 317 39 , , PUNCT cana-5910 317 40 𝑦 𝑦 NOUN cana-5910 317 41 , , PUNCT cana-5910 317 42 𝑒 𝑒 X cana-5910 317 43 ) ) PUNCT cana-5910 317 44 + + CCONJ cana-5910 317 45 𝑂(νœ€Β² 𝑂(νœ€Β² NOUN cana-5910 317 46 ) ) PUNCT cana-5910 317 47 , , PUNCT cana-5910 317 48 where where SCONJ cana-5910 317 49 Ο„ Ο„ PROPN cana-5910 317 50 , , PUNCT cana-5910 317 51 ΞΎ ΞΎ PROPN cana-5910 317 52 , , PUNCT cana-5910 317 53 Ξ· Ξ· NOUN cana-5910 317 54 , , PUNCT cana-5910 317 55 and and CCONJ cana-5910 317 56 Ο† Ο† PROPN cana-5910 317 57 are be AUX cana-5910 317 58 the the DET cana-5910 317 59 infinitesimals infinitesimal NOUN cana-5910 317 60 for for ADP cana-5910 317 61 t t PROPN cana-5910 317 62 , , PUNCT cana-5910 317 63 x x X cana-5910 317 64 , , PUNCT cana-5910 317 65 y y PROPN cana-5910 317 66 , , PUNCT cana-5910 317 67 and and CCONJ cana-5910 317 68 u u NOUN cana-5910 317 69 , , PUNCT cana-5910 317 70 and and CCONJ cana-5910 317 71 Ξ΅ Ξ΅ PROPN cana-5910 317 72 is be AUX cana-5910 317 73 a a DET cana-5910 317 74 small small ADJ cana-5910 317 75 parameter parameter NOUN cana-5910 317 76 . . PUNCT cana-5910 318 1 the the DET cana-5910 318 2 infinitesimal infinitesimal ADJ cana-5910 318 3 generator generator NOUN cana-5910 318 4 is be AUX cana-5910 318 5 : : PUNCT cana-5910 318 6 𝑋 𝑋 NOUN cana-5910 318 7 = = SYM cana-5910 318 8 𝜏 𝜏 PROPN cana-5910 318 9 πœ• πœ• PROPN cana-5910 318 10 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 318 11 + + CCONJ cana-5910 318 12 πœ‰ πœ‰ PROPN cana-5910 318 13 πœ• πœ• NOUN cana-5910 318 14 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 318 15 + + CCONJ cana-5910 318 16 πœ‚ πœ‚ DET cana-5910 318 17 πœ• πœ• NOUN cana-5910 318 18 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 318 19 + + CCONJ cana-5910 318 20 πœ‘ πœ‘ PROPN cana-5910 318 21 πœ• πœ• NOUN cana-5910 318 22 πœ•π‘’ πœ•π‘’ NOUN cana-5910 318 23 . . PUNCT cana-5910 319 1 since since SCONJ cana-5910 319 2 the the DET cana-5910 319 3 pde pde NOUN cana-5910 319 4 involves involve VERB cana-5910 319 5 second second ADJ cana-5910 319 6 derivatives derivative NOUN cana-5910 319 7 ( ( PUNCT cana-5910 319 8 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 319 9 , , PUNCT cana-5910 319 10 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 319 11 , , PUNCT cana-5910 319 12 𝑒𝑦𝑦 𝑒𝑦𝑦 PROPN cana-5910 319 13 ) ) PUNCT cana-5910 319 14 , , PUNCT cana-5910 319 15 we we PRON cana-5910 319 16 need need VERB cana-5910 319 17 the the DET cana-5910 319 18 second second ADJ cana-5910 319 19 prolongation prolongation NOUN cana-5910 319 20 of of ADP cana-5910 319 21 the the DET cana-5910 319 22 generator generator NOUN cana-5910 319 23 : : PUNCT cana-5910 319 24 𝑋2 𝑋2 VERB cana-5910 319 25 = = SYM cana-5910 319 26 𝑋 𝑋 NOUN cana-5910 319 27 + + CCONJ cana-5910 319 28 πœ‘π‘‘ πœ‘π‘‘ ADP cana-5910 319 29 πœ• πœ• NOUN cana-5910 319 30 πœ•π‘’π‘‘ πœ•π‘’π‘‘ PUNCT cana-5910 319 31 + + CCONJ cana-5910 319 32 πœ‘π‘₯ πœ‘π‘₯ PROPN cana-5910 319 33 πœ• πœ• PROPN cana-5910 319 34 πœ•π‘’π‘₯ πœ•π‘’π‘₯ PRON cana-5910 320 1 + + PROPN cana-5910 320 2 πœ‘π‘¦ πœ‘π‘¦ PROPN cana-5910 320 3 πœ• πœ• NOUN cana-5910 320 4 πœ•π‘’π‘¦ πœ•π‘’π‘¦ PUNCT cana-5910 320 5 + + CCONJ cana-5910 320 6 πœ‘π‘‘π‘‘ πœ‘π‘‘π‘‘ PROPN cana-5910 320 7 πœ• πœ• PROPN cana-5910 320 8 πœ•π‘’π‘‘π‘‘ πœ•π‘’π‘‘π‘‘ PROPN cana-5910 320 9 + + CCONJ cana-5910 320 10 πœ‘π‘₯π‘₯ πœ‘π‘₯π‘₯ PROPN cana-5910 320 11 πœ• πœ• PROPN cana-5910 320 12 πœ•π‘’π‘₯π‘₯ πœ•π‘’π‘₯π‘₯ VERB cana-5910 320 13 + + CCONJ cana-5910 320 14 πœ‘π‘¦π‘¦ πœ‘π‘¦π‘¦ ADP cana-5910 320 15 πœ• πœ• NOUN cana-5910 320 16 πœ•π‘’π‘¦π‘¦ πœ•π‘’π‘¦π‘¦ VERB cana-5910 320 17 , , PUNCT cana-5910 320 18 where where SCONJ cana-5910 320 19 πœ‘π‘‘ πœ‘π‘‘ ADP cana-5910 320 20 , , PUNCT cana-5910 320 21 πœ‘π‘₯ πœ‘π‘₯ INTJ cana-5910 320 22 , , PUNCT cana-5910 320 23 πœ‘π‘¦ πœ‘π‘¦ NOUN cana-5910 320 24 , , PUNCT cana-5910 320 25 πœ‘π‘‘π‘‘ πœ‘π‘‘π‘‘ NOUN cana-5910 320 26 , , PUNCT cana-5910 320 27 πœ‘π‘₯π‘₯ πœ‘π‘₯π‘₯ NOUN cana-5910 320 28 , , PUNCT cana-5910 320 29 πœ‘π‘¦π‘¦ πœ‘π‘¦π‘¦ X cana-5910 320 30 are be AUX cana-5910 320 31 extended extend VERB cana-5910 320 32 infinitesimals infinitesimal NOUN cana-5910 320 33 . . PUNCT cana-5910 321 1 the the DET cana-5910 321 2 invariance invariance NOUN cana-5910 321 3 condition condition NOUN cana-5910 321 4 is be AUX cana-5910 321 5 : : PUNCT cana-5910 321 6 1123 1123 NUM cana-5910 321 7 https://internationalpubls.com https://internationalpubls.com X cana-5910 322 1 𝑋2(𝑒𝑑𝑑 𝑋2(𝑒𝑑𝑑 PROPN cana-5910 323 1 βˆ’ βˆ’ PROPN cana-5910 324 1 4𝑒π‘₯π‘₯ 4𝑒π‘₯π‘₯ PROPN cana-5910 324 2 βˆ’ βˆ’ PROPN cana-5910 324 3 4𝑒𝑦𝑦 4𝑒𝑦𝑦 NUM cana-5910 324 4 ) ) PUNCT cana-5910 324 5 |𝑒𝑑𝑑= |𝑒𝑑𝑑= PROPN cana-5910 324 6 4𝑒π‘₯π‘₯+ 4𝑒π‘₯π‘₯+ NUM cana-5910 324 7 4𝑒𝑦𝑦 4𝑒𝑦𝑦 PROPN cana-5910 324 8 = = SYM cana-5910 324 9 0 0 NUM cana-5910 324 10 , , PUNCT cana-5910 324 11 yielding yield VERB cana-5910 324 12 : : PUNCT cana-5910 324 13 πœ‘π‘‘π‘‘ πœ‘π‘‘π‘‘ NOUN cana-5910 324 14 βˆ’ βˆ’ PROPN cana-5910 325 1 4πœ‘π‘₯π‘₯ 4πœ‘π‘₯π‘₯ PROPN cana-5910 325 2 βˆ’ βˆ’ PROPN cana-5910 325 3 4πœ‘π‘¦π‘¦ 4πœ‘π‘¦π‘¦ SYM cana-5910 325 4 = = SYM cana-5910 325 5 0 0 NUM cana-5910 325 6 , , PUNCT cana-5910 325 7 whenever whenever SCONJ cana-5910 325 8 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 325 9 = = SYM cana-5910 325 10 4𝑒π‘₯π‘₯ 4𝑒π‘₯π‘₯ PROPN cana-5910 325 11 + + CCONJ cana-5910 325 12 4𝑒𝑦𝑦 4𝑒𝑦𝑦 NUM cana-5910 325 13 . . PUNCT cana-5910 326 1 the the DET cana-5910 326 2 extended extend VERB cana-5910 326 3 infinitesimals infinitesimal NOUN cana-5910 326 4 are be AUX cana-5910 326 5 computed compute VERB cana-5910 326 6 as as ADP cana-5910 326 7 : : PUNCT cana-5910 326 8 πœ‘π‘‘ πœ‘π‘‘ ADP cana-5910 326 9 = = PUNCT cana-5910 326 10 𝐷𝑑(πœ‘ 𝐷𝑑(πœ‘ NOUN cana-5910 326 11 βˆ’ βˆ’ NOUN cana-5910 326 12 π‘’π‘‘πœ π‘’π‘‘πœ ADV cana-5910 326 13 βˆ’ βˆ’ DET cana-5910 326 14 𝑒π‘₯πœ‰ 𝑒π‘₯πœ‰ NOUN cana-5910 326 15 βˆ’ βˆ’ PROPN cana-5910 326 16 π‘’π‘¦πœ‚ π‘’π‘¦πœ‚ NOUN cana-5910 326 17 ) ) PUNCT cana-5910 327 1 + + CCONJ cana-5910 327 2 𝑒𝑑 𝑒𝑑 VERB cana-5910 327 3 πœπ‘‘ πœπ‘‘ NOUN cana-5910 327 4 + + CCONJ cana-5910 327 5 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 327 6 πœ‰π‘‘ πœ‰π‘‘ NOUN cana-5910 327 7 + + CCONJ cana-5910 327 8 𝑒𝑦 𝑒𝑦 PROPN cana-5910 327 9 πœ‚π‘‘ πœ‚π‘‘ NOUN cana-5910 327 10 , , PUNCT cana-5910 327 11 πœ‘π‘‘π‘‘ πœ‘π‘‘π‘‘ NOUN cana-5910 327 12 = = SYM cana-5910 327 13 𝐷𝑑(πœ‘π‘‘ 𝐷𝑑(πœ‘π‘‘ NOUN cana-5910 327 14 βˆ’ βˆ’ PROPN cana-5910 327 15 π‘’π‘‘π‘‘πœ π‘’π‘‘π‘‘πœ NOUN cana-5910 327 16 βˆ’ βˆ’ PROPN cana-5910 327 17 𝑒𝑑π‘₯πœ‰ 𝑒𝑑π‘₯πœ‰ PROPN cana-5910 327 18 βˆ’ βˆ’ PROPN cana-5910 327 19 π‘’π‘‘π‘¦πœ‚ π‘’π‘‘π‘¦πœ‚ NOUN cana-5910 327 20 ) ) PUNCT cana-5910 328 1 + + CCONJ cana-5910 328 2 𝑒𝑑𝑑 𝑒𝑑𝑑 VERB cana-5910 328 3 πœπ‘‘ πœπ‘‘ PROPN cana-5910 328 4 + + CCONJ cana-5910 328 5 𝑒𝑑π‘₯ 𝑒𝑑π‘₯ CCONJ cana-5910 328 6 πœ‰π‘‘ πœ‰π‘‘ NOUN cana-5910 328 7 + + CCONJ cana-5910 328 8 𝑒𝑑𝑦 𝑒𝑑𝑦 NOUN cana-5910 328 9 πœ‚π‘‘ πœ‚π‘‘ ADP cana-5910 328 10 , , PUNCT cana-5910 328 11 πœ‘π‘₯π‘₯ πœ‘π‘₯π‘₯ PROPN cana-5910 328 12 = = SYM cana-5910 328 13 𝐷π‘₯(πœ‘π‘₯ 𝐷π‘₯(πœ‘π‘₯ PROPN cana-5910 328 14 βˆ’ βˆ’ PROPN cana-5910 328 15 𝑒π‘₯π‘₯πœ‰ 𝑒π‘₯π‘₯πœ‰ PROPN cana-5910 328 16 βˆ’ βˆ’ PROPN cana-5910 328 17 𝑒π‘₯π‘¦πœ‚ 𝑒π‘₯π‘¦πœ‚ NOUN cana-5910 328 18 βˆ’ βˆ’ PROPN cana-5910 328 19 𝑒π‘₯π‘‘πœ 𝑒π‘₯π‘‘πœ NOUN cana-5910 328 20 ) ) PUNCT cana-5910 329 1 + + CCONJ cana-5910 329 2 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 329 3 πœ‰π‘₯ πœ‰π‘₯ X cana-5910 329 4 + + CCONJ cana-5910 329 5 𝑒π‘₯𝑦 𝑒π‘₯𝑦 PROPN cana-5910 329 6 πœ‚π‘₯ πœ‚π‘₯ PROPN cana-5910 329 7 + + CCONJ cana-5910 329 8 𝑒π‘₯𝑑 𝑒π‘₯𝑑 NOUN cana-5910 329 9 𝜏π‘₯ 𝜏π‘₯ ADV cana-5910 329 10 , , PUNCT cana-5910 329 11 πœ‘π‘¦π‘¦ πœ‘π‘¦π‘¦ ADV cana-5910 329 12 = = PUNCT cana-5910 329 13 𝐷𝑦(πœ‘π‘¦ 𝐷𝑦(πœ‘π‘¦ NOUN cana-5910 329 14 βˆ’ βˆ’ NOUN cana-5910 329 15 𝑒𝑦π‘₯πœ‰ 𝑒𝑦π‘₯πœ‰ NOUN cana-5910 329 16 βˆ’ βˆ’ NOUN cana-5910 329 17 π‘’π‘¦π‘¦πœ‚ π‘’π‘¦π‘¦πœ‚ NOUN cana-5910 329 18 βˆ’ βˆ’ PROPN cana-5910 329 19 π‘’π‘¦π‘‘πœ π‘’π‘¦π‘‘πœ NOUN cana-5910 329 20 ) ) PUNCT cana-5910 330 1 + + CCONJ cana-5910 330 2 𝑒𝑦π‘₯ 𝑒𝑦π‘₯ VERB cana-5910 330 3 πœ‰π‘¦ πœ‰π‘¦ NOUN cana-5910 330 4 + + CCONJ cana-5910 330 5 𝑒𝑦𝑦 𝑒𝑦𝑦 VERB cana-5910 330 6 πœ‚π‘¦ πœ‚π‘¦ ADV cana-5910 330 7 + + CCONJ cana-5910 330 8 𝑒𝑦𝑑 𝑒𝑦𝑑 VERB cana-5910 330 9 πœπ‘¦ πœπ‘¦ PROPN cana-5910 330 10 , , PUNCT cana-5910 330 11 where where SCONJ cana-5910 330 12 πœ‘^π‘₯ πœ‘^π‘₯ PROPN cana-5910 330 13 = = PRON cana-5910 330 14 𝐷π‘₯(πœ‘ 𝐷π‘₯(πœ‘ PROPN cana-5910 330 15 βˆ’ βˆ’ PROPN cana-5910 330 16 π‘’π‘‘πœ π‘’π‘‘πœ ADV cana-5910 331 1 βˆ’ βˆ’ DET cana-5910 331 2 𝑒π‘₯πœ‰ 𝑒π‘₯πœ‰ NOUN cana-5910 331 3 βˆ’ βˆ’ PROPN cana-5910 331 4 π‘’π‘¦πœ‚ π‘’π‘¦πœ‚ NOUN cana-5910 331 5 ) ) PUNCT cana-5910 332 1 + + CCONJ cana-5910 332 2 π‘’π‘‘πœπ‘₯ π‘’π‘‘πœπ‘₯ ADJ cana-5910 332 3 + + CCONJ cana-5910 332 4 𝑒π‘₯πœ‰π‘₯ 𝑒π‘₯πœ‰π‘₯ NOUN cana-5910 332 5 + + CCONJ cana-5910 332 6 π‘’π‘¦πœ‚π‘₯ π‘’π‘¦πœ‚π‘₯ NOUN cana-5910 332 7 , , PUNCT cana-5910 332 8 and and CCONJ cana-5910 332 9 similarly similarly ADV cana-5910 332 10 for for ADP cana-5910 332 11 πœ‘π‘¦. πœ‘π‘¦. NOUN cana-5910 332 12 substituting substitute VERB cana-5910 332 13 these these PRON cana-5910 332 14 into into ADP cana-5910 332 15 the the DET cana-5910 332 16 invariance invariance NOUN cana-5910 332 17 condition condition NOUN cana-5910 332 18 produces produce VERB cana-5910 332 19 a a DET cana-5910 332 20 system system NOUN cana-5910 332 21 of of ADP cana-5910 332 22 determining determine VERB cana-5910 332 23 equations equation NOUN cana-5910 332 24 for for ADP cana-5910 332 25 𝜏 𝜏 PROPN cana-5910 332 26 , , PUNCT cana-5910 332 27 πœ‰ πœ‰ X cana-5910 332 28 , , PUNCT cana-5910 332 29 πœ‚ πœ‚ NOUN cana-5910 332 30 , , PUNCT cana-5910 332 31 πœ‘. πœ‘. ADJ cana-5910 332 32 step step NOUN cana-5910 332 33 3 3 NUM cana-5910 332 34 : : PUNCT cana-5910 332 35 determining determine VERB cana-5910 332 36 equations equation NOUN cana-5910 332 37 assume assume VERB cana-5910 332 38 the the DET cana-5910 332 39 infinitesimals infinitesimal NOUN cana-5910 332 40 depend depend VERB cana-5910 332 41 on on ADP cana-5910 332 42 𝑑 𝑑 PROPN cana-5910 332 43 , , PUNCT cana-5910 332 44 π‘₯ π‘₯ PROPN cana-5910 332 45 , , PUNCT cana-5910 332 46 𝑦 𝑦 NOUN cana-5910 332 47 , , PUNCT cana-5910 332 48 𝑒 𝑒 X cana-5910 332 49 , , PUNCT cana-5910 332 50 and and CCONJ cana-5910 332 51 consider consider VERB cana-5910 332 52 Ο† Ο† PROPN cana-5910 332 53 linear linear PROPN cana-5910 332 54 in in ADP cana-5910 332 55 u u NOUN cana-5910 332 56 due due ADP cana-5910 332 57 to to ADP cana-5910 332 58 the the DET cana-5910 332 59 linearity linearity NOUN cana-5910 332 60 of of ADP cana-5910 332 61 the the DET cana-5910 332 62 pde pde NOUN cana-5910 332 63 : : PUNCT cana-5910 332 64 πœ‘ πœ‘ X cana-5910 332 65 = = PUNCT cana-5910 332 66 𝛼(𝑑 𝛼(𝑑 PROPN cana-5910 332 67 , , PUNCT cana-5910 332 68 π‘₯ π‘₯ NOUN cana-5910 332 69 , , PUNCT cana-5910 332 70 𝑦)𝑒 𝑦)𝑒 ADJ cana-5910 332 71 + + CCONJ cana-5910 332 72 𝛽(𝑑 𝛽(𝑑 PROPN cana-5910 332 73 , , PUNCT cana-5910 332 74 π‘₯ π‘₯ PROPN cana-5910 332 75 , , PUNCT cana-5910 332 76 𝑦 𝑦 NOUN cana-5910 332 77 ) ) PUNCT cana-5910 332 78 . . PUNCT cana-5910 333 1 since since SCONJ cana-5910 333 2 the the DET cana-5910 333 3 pde pde NOUN cana-5910 333 4 is be AUX cana-5910 333 5 homogeneous homogeneous ADJ cana-5910 333 6 , , PUNCT cana-5910 333 7 Ξ² Ξ² X cana-5910 333 8 = = SYM cana-5910 333 9 0 0 NUM cana-5910 333 10 corresponds correspond VERB cana-5910 333 11 to to ADP cana-5910 333 12 the the DET cana-5910 333 13 trivial trivial ADJ cana-5910 333 14 symmetry symmetry NOUN cana-5910 333 15 𝑒 𝑒 PROPN cana-5910 333 16 β†’ β†’ SYM cana-5910 333 17 𝑒 𝑒 X cana-5910 333 18 + + CCONJ cana-5910 333 19 constant constant ADJ cana-5910 333 20 , , PUNCT cana-5910 333 21 so so SCONJ cana-5910 333 22 we we PRON cana-5910 333 23 try try VERB cana-5910 333 24 πœ‘ πœ‘ NOUN cana-5910 333 25 = = PUNCT cana-5910 333 26 𝛼(𝑑 𝛼(𝑑 X cana-5910 333 27 , , PUNCT cana-5910 333 28 π‘₯ π‘₯ PRON cana-5910 333 29 , , PUNCT cana-5910 333 30 𝑦)𝑒. 𝑦)𝑒. NOUN cana-5910 333 31 substituting substitute VERB cana-5910 333 32 into into ADP cana-5910 333 33 the the DET cana-5910 333 34 invariance invariance NOUN cana-5910 333 35 condition condition NOUN cana-5910 333 36 and and CCONJ cana-5910 333 37 equating equate VERB cana-5910 333 38 coefficients coefficient NOUN cana-5910 333 39 of of ADP cana-5910 333 40 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 333 41 , , PUNCT cana-5910 333 42 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ PROPN cana-5910 333 43 , , PUNCT cana-5910 333 44 𝑒𝑦𝑦 𝑒𝑦𝑦 ADV cana-5910 333 45 , , PUNCT cana-5910 333 46 𝑒𝑑 𝑒𝑑 X cana-5910 333 47 , , PUNCT cana-5910 333 48 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 333 49 , , PUNCT cana-5910 333 50 𝑒𝑦 𝑒𝑦 PROPN cana-5910 333 51 , , PUNCT cana-5910 333 52 𝑒 𝑒 X cana-5910 333 53 , , PUNCT cana-5910 333 54 and and CCONJ cana-5910 333 55 independent independent ADJ cana-5910 333 56 terms term NOUN cana-5910 333 57 , , PUNCT cana-5910 333 58 we we PRON cana-5910 333 59 obtain obtain VERB cana-5910 333 60 equations equation NOUN cana-5910 333 61 such such ADJ cana-5910 333 62 as as ADP cana-5910 333 63 : : PUNCT cana-5910 333 64 β€’ β€’ NUM cana-5910 333 65 πΆπ‘œπ‘’π‘“π‘“π‘–π‘π‘–π‘’π‘›π‘‘ πΆπ‘œπ‘’π‘“π‘“π‘–π‘π‘–π‘’π‘›π‘‘ PROPN cana-5910 333 66 π‘œπ‘“ π‘œπ‘“ ADP cana-5910 333 67 𝑒𝑑𝑑 𝑒𝑑𝑑 PROPN cana-5910 333 68 : : PUNCT cana-5910 333 69 πœπ‘’ πœπ‘’ X cana-5910 333 70 = = SYM cana-5910 333 71 0 0 PROPN cana-5910 333 72 , , PUNCT cana-5910 333 73 πœ‰π‘’ πœ‰π‘’ PROPN cana-5910 333 74 = = NOUN cana-5910 333 75 0 0 NUM cana-5910 333 76 , , PUNCT cana-5910 333 77 πœ‚π‘’ πœ‚π‘’ ADP cana-5910 333 78 = = SYM cana-5910 333 79 0 0 PROPN cana-5910 333 80 . . NOUN cana-5910 333 81 β€’ β€’ NOUN cana-5910 334 1 πΆπ‘œπ‘’π‘“π‘“π‘–π‘π‘–π‘’π‘›π‘‘ πΆπ‘œπ‘’π‘“π‘“π‘–π‘π‘–π‘’π‘›π‘‘ PROPN cana-5910 334 2 π‘œπ‘“ π‘œπ‘“ ADP cana-5910 334 3 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ PROPN cana-5910 334 4 : : PUNCT cana-5910 334 5 4𝜏π‘₯ 4𝜏π‘₯ ADJ cana-5910 334 6 = = SYM cana-5910 334 7 0 0 NUM cana-5910 334 8 , , PUNCT cana-5910 334 9 πœ‰π‘‘ πœ‰π‘‘ NOUN cana-5910 334 10 = = PROPN cana-5910 334 11 0 0 PROPN cana-5910 334 12 , , PUNCT cana-5910 334 13 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 334 14 = = SYM cana-5910 334 15 0 0 NUM cana-5910 334 16 , , PUNCT cana-5910 334 17 πœ‚π‘₯ πœ‚π‘₯ NOUN cana-5910 334 18 = = SYM cana-5910 334 19 0 0 NUM cana-5910 334 20 , , PUNCT cana-5910 334 21 𝛼π‘₯ 𝛼π‘₯ ADV cana-5910 334 22 = = SYM cana-5910 334 23 2πœ‰π‘₯ 2πœ‰π‘₯ ADJ cana-5910 334 24 . . PUNCT cana-5910 335 1 β€’ β€’ NUM cana-5910 335 2 πΆπ‘œπ‘’π‘“π‘“π‘–π‘π‘–π‘’π‘›π‘‘ πΆπ‘œπ‘’π‘“π‘“π‘–π‘π‘–π‘’π‘›π‘‘ PROPN cana-5910 335 3 π‘œπ‘“ π‘œπ‘“ X cana-5910 335 4 𝑒𝑦𝑦 𝑒𝑦𝑦 ADV cana-5910 335 5 : : PUNCT cana-5910 335 6 4πœπ‘¦ 4πœπ‘¦ NOUN cana-5910 335 7 = = SYM cana-5910 335 8 0 0 NUM cana-5910 335 9 , , PUNCT cana-5910 335 10 πœ‚π‘‘ πœ‚π‘‘ ADP cana-5910 335 11 = = SYM cana-5910 335 12 0 0 NUM cana-5910 335 13 , , PUNCT cana-5910 335 14 πœ‚π‘₯ πœ‚π‘₯ NOUN cana-5910 335 15 = = SYM cana-5910 335 16 0 0 NUM cana-5910 335 17 , , PUNCT cana-5910 335 18 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 335 19 = = SYM cana-5910 335 20 0 0 NUM cana-5910 335 21 , , PUNCT cana-5910 335 22 𝛼𝑦 𝛼𝑦 NOUN cana-5910 335 23 = = SYM cana-5910 335 24 2πœ‚π‘¦. 2πœ‚π‘¦. NUM cana-5910 335 25 β€’ β€’ NUM cana-5910 336 1 𝑀𝑖π‘₯𝑒𝑑 𝑀𝑖π‘₯𝑒𝑑 PROPN cana-5910 336 2 π‘‘π‘’π‘Ÿπ‘šπ‘  π‘‘π‘’π‘Ÿπ‘šπ‘  NOUN cana-5910 336 3 π‘™π‘’π‘Žπ‘‘ π‘™π‘’π‘Žπ‘‘ NOUN cana-5910 336 4 π‘‘π‘œ π‘‘π‘œ PROPN cana-5910 336 5 : : PUNCT cana-5910 336 6 𝜏π‘₯π‘₯ 𝜏π‘₯π‘₯ PROPN cana-5910 336 7 = = SYM cana-5910 336 8 0 0 NUM cana-5910 336 9 , , PUNCT cana-5910 336 10 πœπ‘¦π‘¦ πœπ‘¦π‘¦ NOUN cana-5910 336 11 = = SYM cana-5910 336 12 0 0 NUM cana-5910 336 13 , , PUNCT cana-5910 336 14 πœ‰π‘₯π‘₯ πœ‰π‘₯π‘₯ NOUN cana-5910 336 15 = = SYM cana-5910 336 16 0 0 NUM cana-5910 336 17 , , PUNCT cana-5910 336 18 πœ‚π‘¦π‘¦ πœ‚π‘¦π‘¦ X cana-5910 336 19 = = SYM cana-5910 336 20 0 0 NUM cana-5910 336 21 , , PUNCT cana-5910 336 22 𝑒𝑑𝑐. 𝑒𝑑𝑐. X cana-5910 336 23 from from ADP cana-5910 336 24 𝜏π‘₯ 𝜏π‘₯ ADP cana-5910 336 25 = = SYM cana-5910 336 26 0 0 NUM cana-5910 336 27 , , PUNCT cana-5910 336 28 πœπ‘¦ πœπ‘¦ PROPN cana-5910 336 29 = = SYM cana-5910 336 30 0 0 PROPN cana-5910 336 31 , , PUNCT cana-5910 336 32 𝑀𝑒 𝑀𝑒 PROPN cana-5910 336 33 𝑔𝑒𝑑 𝑔𝑒𝑑 NOUN cana-5910 336 34 𝜏 𝜏 X cana-5910 336 35 = = PUNCT cana-5910 336 36 𝜏(𝑑 𝜏(𝑑 PROPN cana-5910 336 37 ) ) PUNCT cana-5910 336 38 . . PUNCT cana-5910 337 1 πΉπ‘Ÿπ‘œπ‘š πΉπ‘Ÿπ‘œπ‘š PROPN cana-5910 337 2 πœ‰π‘‘ πœ‰π‘‘ NOUN cana-5910 337 3 = = PROPN cana-5910 337 4 0 0 PROPN cana-5910 337 5 , , PUNCT cana-5910 337 6 πœ‰π‘¦ πœ‰π‘¦ X cana-5910 337 7 = = SYM cana-5910 337 8 0 0 NUM cana-5910 337 9 , , PUNCT cana-5910 337 10 πœ‚π‘‘ πœ‚π‘‘ ADP cana-5910 337 11 = = SYM cana-5910 337 12 0 0 NUM cana-5910 337 13 , , PUNCT cana-5910 337 14 πœ‚π‘₯ πœ‚π‘₯ NOUN cana-5910 337 15 = = SYM cana-5910 337 16 0 0 PROPN cana-5910 337 17 , , PUNCT cana-5910 337 18 𝑀𝑒 𝑀𝑒 PROPN cana-5910 337 19 𝑔𝑒𝑑 𝑔𝑒𝑑 NOUN cana-5910 337 20 πœ‰ πœ‰ PROPN cana-5910 337 21 = = SYM cana-5910 337 22 πœ‰(π‘₯ πœ‰(π‘₯ PROPN cana-5910 337 23 ) ) PUNCT cana-5910 337 24 , , PUNCT cana-5910 337 25 πœ‚ πœ‚ X cana-5910 337 26 = = SYM cana-5910 337 27 πœ‚(𝑦 πœ‚(𝑦 NOUN cana-5910 337 28 ) ) PUNCT cana-5910 337 29 . . PUNCT cana-5910 338 1 the the DET cana-5910 338 2 equations equation NOUN cana-5910 338 3 𝜏π‘₯π‘₯ 𝜏π‘₯π‘₯ NOUN cana-5910 338 4 = = SYM cana-5910 338 5 0 0 NUM cana-5910 338 6 , , PUNCT cana-5910 338 7 πœ‰π‘₯π‘₯ πœ‰π‘₯π‘₯ NOUN cana-5910 338 8 = = SYM cana-5910 338 9 0 0 NUM cana-5910 338 10 , , PUNCT cana-5910 338 11 πœ‚π‘¦π‘¦ πœ‚π‘¦π‘¦ X cana-5910 339 1 = = SYM cana-5910 339 2 0 0 NUM cana-5910 339 3 imply imply VERB cana-5910 339 4 linearity linearity NOUN cana-5910 339 5 : : PUNCT cana-5910 339 6 𝜏 𝜏 X cana-5910 339 7 = = PUNCT cana-5910 339 8 π‘Ž1𝑑 π‘Ž1𝑑 PROPN cana-5910 339 9 + + CCONJ cana-5910 339 10 π‘Ž2 π‘Ž2 PROPN cana-5910 339 11 , , PUNCT cana-5910 339 12 πœ‰ πœ‰ X cana-5910 339 13 = = VERB cana-5910 339 14 𝑏1π‘₯ 𝑏1π‘₯ NOUN cana-5910 339 15 + + X cana-5910 339 16 𝑏2 𝑏2 NOUN cana-5910 339 17 , , PUNCT cana-5910 339 18 πœ‚ πœ‚ X cana-5910 339 19 = = PUNCT cana-5910 340 1 𝑐1𝑦 𝑐1𝑦 PROPN cana-5910 341 1 + + NUM cana-5910 341 2 𝑐2 𝑐2 NOUN cana-5910 341 3 . . PUNCT cana-5910 342 1 from from ADP cana-5910 342 2 𝛼π‘₯ 𝛼π‘₯ ADV cana-5910 342 3 = = SYM cana-5910 342 4 2πœ‰π‘₯ 2πœ‰π‘₯ NOUN cana-5910 342 5 , , PUNCT cana-5910 342 6 𝛼𝑦 𝛼𝑦 NOUN cana-5910 342 7 = = SYM cana-5910 342 8 2πœ‚π‘¦ 2πœ‚π‘¦ PROPN cana-5910 342 9 , , PUNCT cana-5910 342 10 𝑀𝑒 𝑀𝑒 PROPN cana-5910 342 11 β„Žπ‘Žπ‘£π‘’ β„Žπ‘Žπ‘£π‘’ NOUN cana-5910 342 12 𝛼π‘₯ 𝛼π‘₯ PROPN cana-5910 342 13 = = SYM cana-5910 342 14 2𝑏1 2𝑏1 NUM cana-5910 342 15 , , PUNCT cana-5910 342 16 𝛼𝑦 𝛼𝑦 PROPN cana-5910 342 17 = = SYM cana-5910 342 18 2𝑐1 2𝑐1 NUM cana-5910 342 19 , , PUNCT cana-5910 342 20 so so ADV cana-5910 342 21 : : PUNCT cana-5910 342 22 𝛼 𝛼 X cana-5910 342 23 = = PUNCT cana-5910 342 24 2𝑏1π‘₯ 2𝑏1π‘₯ NOUN cana-5910 342 25 + + CCONJ cana-5910 342 26 2𝑐1𝑦 2𝑐1𝑦 NOUN cana-5910 342 27 + + CCONJ cana-5910 343 1 𝑓(𝑑 𝑓(𝑑 NOUN cana-5910 343 2 ) ) PUNCT cana-5910 343 3 . . PUNCT cana-5910 344 1 other other ADJ cana-5910 344 2 equations equation NOUN cana-5910 344 3 constrain constrain VERB cana-5910 344 4 f(t f(t NOUN cana-5910 344 5 ) ) PUNCT cana-5910 344 6 . . PUNCT cana-5910 345 1 for for ADP cana-5910 345 2 the the DET cana-5910 345 3 wave wave NOUN cana-5910 345 4 equation equation NOUN cana-5910 345 5 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 345 6 = = PUNCT cana-5910 345 7 𝑐2(𝑒π‘₯π‘₯ 𝑐2(𝑒π‘₯π‘₯ PROPN cana-5910 345 8 + + CCONJ cana-5910 345 9 𝑒𝑦𝑦 𝑒𝑦𝑦 ADJ cana-5910 345 10 ) ) PUNCT cana-5910 345 11 , , PUNCT cana-5910 345 12 typical typical ADJ cana-5910 345 13 symmetries symmetry NOUN cana-5910 345 14 include include VERB cana-5910 345 15 : : PUNCT cana-5910 345 16 1 1 X cana-5910 345 17 . . NOUN cana-5910 345 18 time time NOUN cana-5910 345 19 translation translation NOUN cana-5910 345 20 : : PUNCT cana-5910 345 21 𝑋1 𝑋1 PROPN cana-5910 345 22 = = SYM cana-5910 345 23 πœ• πœ• PROPN cana-5910 345 24 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 345 25 . . PROPN cana-5910 346 1 2 2 X cana-5910 346 2 . . NOUN cana-5910 346 3 space space NOUN cana-5910 346 4 translations translation NOUN cana-5910 346 5 : : PUNCT cana-5910 346 6 𝑋2 𝑋2 VERB cana-5910 346 7 = = SYM cana-5910 346 8 πœ• πœ• NOUN cana-5910 346 9 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 346 10 , , PUNCT cana-5910 346 11 𝑋3 𝑋3 NOUN cana-5910 346 12 = = SYM cana-5910 346 13 πœ• πœ• PROPN cana-5910 346 14 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 346 15 . . PUNCT cana-5910 347 1 3 3 X cana-5910 347 2 . . X cana-5910 347 3 scaling scaling NOUN cana-5910 347 4 : : PUNCT cana-5910 347 5 𝑋4 𝑋4 VERB cana-5910 347 6 = = PUNCT cana-5910 348 1 𝑑 𝑑 PROPN cana-5910 348 2 πœ• πœ• PROPN cana-5910 348 3 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 348 4 + + CCONJ cana-5910 348 5 π‘₯ π‘₯ DET cana-5910 348 6 πœ• πœ• NOUN cana-5910 348 7 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 348 8 + + CCONJ cana-5910 348 9 𝑦 𝑦 NOUN cana-5910 348 10 πœ• πœ• NOUN cana-5910 348 11 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 348 12 + + CCONJ cana-5910 348 13 𝑒 𝑒 PRON cana-5910 348 14 πœ• πœ• NOUN cana-5910 348 15 πœ•π‘’ πœ•π‘’ NOUN cana-5910 348 16 . . PUNCT cana-5910 349 1 4 4 X cana-5910 349 2 . . X cana-5910 349 3 lorentz lorentz NOUN cana-5910 349 4 boosts boost NOUN cana-5910 349 5 and and CCONJ cana-5910 349 6 solution solution NOUN cana-5910 349 7 scaling scaling NOUN cana-5910 349 8 : : PUNCT cana-5910 349 9 𝑋5 𝑋5 PROPN cana-5910 349 10 = = SYM cana-5910 349 11 𝑒 𝑒 PROPN cana-5910 349 12 πœ• πœ• PROPN cana-5910 349 13 πœ•π‘’ πœ•π‘’ NOUN cana-5910 349 14 . . PUNCT cana-5910 350 1 5 5 X cana-5910 350 2 . . X cana-5910 350 3 infinite infinite ADJ cana-5910 350 4 - - PUNCT cana-5910 350 5 dimensional dimensional ADJ cana-5910 350 6 symmetries symmetry NOUN cana-5910 350 7 for for ADP cana-5910 350 8 the the DET cana-5910 350 9 linear linear ADJ cana-5910 350 10 wave wave NOUN cana-5910 350 11 equation equation NOUN cana-5910 350 12 . . PUNCT cana-5910 351 1 since since SCONJ cana-5910 351 2 our our PRON cana-5910 351 3 pde pde NOUN cana-5910 351 4 has have VERB cana-5910 351 5 a a DET cana-5910 351 6 coefficient coefficient NOUN cana-5910 351 7 of of ADP cana-5910 351 8 4 4 NUM cana-5910 351 9 ( ( PUNCT cana-5910 351 10 𝑐² 𝑐² NOUN cana-5910 351 11 = = SYM cana-5910 351 12 4 4 NUM cana-5910 351 13 , , PUNCT cana-5910 351 14 𝑐 𝑐 NOUN cana-5910 351 15 = = SYM cana-5910 351 16 2 2 NUM cana-5910 351 17 ) ) PUNCT cana-5910 351 18 , , PUNCT cana-5910 351 19 we we PRON cana-5910 351 20 test test VERB cana-5910 351 21 the the DET cana-5910 351 22 scaling scaling ADJ cana-5910 351 23 symmetry symmetry NOUN cana-5910 351 24 : : PUNCT cana-5910 351 25 𝜏 𝜏 X cana-5910 351 26 = = SYM cana-5910 351 27 π‘Ž π‘Ž PRON cana-5910 351 28 𝑑 𝑑 NOUN cana-5910 351 29 , , PUNCT cana-5910 351 30 πœ‰ πœ‰ X cana-5910 351 31 = = SYM cana-5910 351 32 π‘Ž π‘Ž PRON cana-5910 351 33 π‘₯ π‘₯ NOUN cana-5910 351 34 , , PUNCT cana-5910 351 35 πœ‚ πœ‚ NOUN cana-5910 351 36 = = SYM cana-5910 351 37 π‘Ž π‘Ž SYM cana-5910 351 38 𝑦 𝑦 NOUN cana-5910 351 39 , , PUNCT cana-5910 351 40 πœ‘ πœ‘ NOUN cana-5910 351 41 = = SYM cana-5910 351 42 π‘Ž π‘Ž PROPN cana-5910 351 43 𝑒. 𝑒. NOUN cana-5910 351 44 substitute substitute NOUN cana-5910 351 45 into into ADP cana-5910 351 46 the the DET cana-5910 351 47 determining determine VERB cana-5910 351 48 equations equation NOUN cana-5910 351 49 . . PUNCT cana-5910 352 1 the the DET cana-5910 352 2 key key ADJ cana-5910 352 3 invariance invariance NOUN cana-5910 352 4 condition condition NOUN cana-5910 352 5 simplifies simplifie NOUN cana-5910 352 6 , , PUNCT cana-5910 352 7 confirming confirm VERB cana-5910 352 8 the the DET cana-5910 352 9 scaling scale VERB cana-5910 352 10 symmetry symmetry NOUN cana-5910 352 11 : : PUNCT cana-5910 352 12 𝑋 𝑋 PROPN cana-5910 352 13 = = SYM cana-5910 352 14 𝑑 𝑑 PROPN cana-5910 352 15 πœ• πœ• PROPN cana-5910 352 16 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 352 17 + + CCONJ cana-5910 352 18 π‘₯ π‘₯ PRON cana-5910 352 19 πœ• πœ• NOUN cana-5910 352 20 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 352 21 + + CCONJ cana-5910 352 22 𝑦 𝑦 NOUN cana-5910 352 23 πœ• πœ• NOUN cana-5910 352 24 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 352 25 + + CCONJ cana-5910 352 26 𝑒 𝑒 PRON cana-5910 352 27 πœ• πœ• NOUN cana-5910 352 28 πœ•π‘’ πœ•π‘’ NOUN cana-5910 352 29 , , PUNCT cana-5910 352 30 ( ( PUNCT cana-5910 352 31 𝜏 𝜏 NOUN cana-5910 352 32 = = SYM cana-5910 352 33 𝑑 𝑑 NOUN cana-5910 352 34 , , PUNCT cana-5910 352 35 πœ‰ πœ‰ NOUN cana-5910 352 36 = = SYM cana-5910 352 37 π‘₯ π‘₯ PROPN cana-5910 352 38 , , PUNCT cana-5910 352 39 πœ‚ πœ‚ X cana-5910 352 40 = = SYM cana-5910 352 41 𝑦 𝑦 PROPN cana-5910 352 42 , , PUNCT cana-5910 352 43 πœ‘ πœ‘ NOUN cana-5910 352 44 = = SYM cana-5910 352 45 𝑒 𝑒 X cana-5910 352 46 ) ) PUNCT cana-5910 352 47 . . PUNCT cana-5910 353 1 we we PRON cana-5910 353 2 also also ADV cana-5910 353 3 consider consider VERB cana-5910 353 4 time time NOUN cana-5910 353 5 decay decay VERB cana-5910 353 6 to to PART cana-5910 353 7 handle handle VERB cana-5910 353 8 𝑒 𝑒 NOUN cana-5910 353 9 β†’ β†’ SYM cana-5910 353 10 0 0 NUM cana-5910 353 11 π‘Žπ‘  π‘Žπ‘  PROPN cana-5910 353 12 𝑑 𝑑 PROPN cana-5910 353 13 β†’ β†’ SYM cana-5910 353 14 ∞ ∞ PROPN cana-5910 353 15 , , PUNCT cana-5910 353 16 possibly possibly ADV cana-5910 353 17 introducing introduce VERB cana-5910 353 18 an an DET cana-5910 353 19 exponential exponential ADJ cana-5910 353 20 ansatz ansatz NOUN cana-5910 353 21 . . PUNCT cana-5910 354 1 1124 1124 NUM cana-5910 354 2 https://internationalpubls.com https://internationalpubls.com X cana-5910 354 3 step step NOUN cana-5910 354 4 4 4 NUM cana-5910 354 5 : : PUNCT cana-5910 354 6 symmetry symmetry NOUN cana-5910 354 7 reduction reduction NOUN cana-5910 354 8 use use VERB cana-5910 354 9 the the DET cana-5910 354 10 scaling scale VERB cana-5910 354 11 symmetry symmetry NOUN cana-5910 354 12 : : PUNCT cana-5910 354 13 𝑋 𝑋 PROPN cana-5910 354 14 = = SYM cana-5910 354 15 𝑑 𝑑 PROPN cana-5910 354 16 πœ• πœ• PROPN cana-5910 354 17 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 354 18 + + CCONJ cana-5910 354 19 π‘₯ π‘₯ PRON cana-5910 354 20 πœ• πœ• NOUN cana-5910 354 21 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 354 22 + + CCONJ cana-5910 354 23 𝑦 𝑦 NOUN cana-5910 354 24 πœ• πœ• NOUN cana-5910 354 25 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 354 26 + + CCONJ cana-5910 354 27 𝑒 𝑒 PRON cana-5910 354 28 πœ• πœ• PROPN cana-5910 354 29 πœ•π‘’ πœ•π‘’ NOUN cana-5910 354 30 . . PUNCT cana-5910 355 1 find find VERB cana-5910 355 2 invariants invariant NOUN cana-5910 355 3 by by ADP cana-5910 355 4 solving solve VERB cana-5910 355 5 : : PUNCT cana-5910 355 6 𝑑𝑑 𝑑𝑑 ADP cana-5910 355 7 𝑑 𝑑 PROPN cana-5910 355 8 = = PUNCT cana-5910 355 9 𝑑π‘₯ 𝑑π‘₯ NOUN cana-5910 355 10 π‘₯ π‘₯ NOUN cana-5910 355 11 = = X cana-5910 355 12 𝑑𝑦 𝑑𝑦 PROPN cana-5910 355 13 𝑦 𝑦 NOUN cana-5910 355 14 = = X cana-5910 355 15 𝑑𝑒 𝑑𝑒 PROPN cana-5910 355 16 𝑒 𝑒 X cana-5910 355 17 . . PUNCT cana-5910 356 1 β€’ β€’ NUM cana-5910 357 1 πΉπ‘Ÿπ‘œπ‘š πΉπ‘Ÿπ‘œπ‘š PROPN cana-5910 357 2 𝑑π‘₯ 𝑑π‘₯ VERB cana-5910 357 3 π‘₯ π‘₯ NOUN cana-5910 357 4 = = PUNCT cana-5910 357 5 𝑑𝑦 𝑑𝑦 PROPN cana-5910 357 6 𝑦 𝑦 NOUN cana-5910 357 7 , , PUNCT cana-5910 357 8 π‘₯ π‘₯ PROPN cana-5910 357 9 𝑦 𝑦 NOUN cana-5910 357 10 = = SYM cana-5910 357 11 𝑐1 𝑐1 NOUN cana-5910 357 12 , , PUNCT cana-5910 357 13 π‘ π‘œ π‘ π‘œ PROPN cana-5910 357 14 πœ‰1 πœ‰1 PROPN cana-5910 357 15 = = SYM cana-5910 357 16 π‘₯ π‘₯ PROPN cana-5910 357 17 𝑦 𝑦 NOUN cana-5910 357 18 . . PUNCT cana-5910 358 1 β€’ β€’ NUM cana-5910 358 2 πΉπ‘Ÿπ‘œπ‘š πΉπ‘Ÿπ‘œπ‘š PROPN cana-5910 358 3 𝑑π‘₯ 𝑑π‘₯ VERB cana-5910 358 4 π‘₯ π‘₯ NOUN cana-5910 358 5 = = PUNCT cana-5910 358 6 𝑑𝑑 𝑑𝑑 ADP cana-5910 358 7 𝑑 𝑑 PROPN cana-5910 358 8 , , PUNCT cana-5910 358 9 π‘₯ π‘₯ PROPN cana-5910 358 10 𝑑 𝑑 PROPN cana-5910 358 11 = = SYM cana-5910 358 12 𝑐2 𝑐2 NOUN cana-5910 358 13 , , PUNCT cana-5910 358 14 π‘ π‘œ π‘ π‘œ ADP cana-5910 358 15 πœ‰2 πœ‰2 PROPN cana-5910 358 16 = = PUNCT cana-5910 358 17 π‘₯ π‘₯ PROPN cana-5910 358 18 𝑑 𝑑 PROPN cana-5910 358 19 . . PUNCT cana-5910 359 1 β€’ β€’ NUM cana-5910 359 2 πΉπ‘Ÿπ‘œπ‘š πΉπ‘Ÿπ‘œπ‘š PROPN cana-5910 359 3 𝑑𝑒 𝑑𝑒 PART cana-5910 359 4 𝑒 𝑒 PROPN cana-5910 359 5 = = X cana-5910 359 6 𝑑𝑑 𝑑𝑑 ADP cana-5910 359 7 𝑑 𝑑 PROPN cana-5910 359 8 , , PUNCT cana-5910 359 9 𝑒 𝑒 PROPN cana-5910 359 10 𝑑 𝑑 PROPN cana-5910 359 11 = = SYM cana-5910 359 12 𝑐3 𝑐3 NOUN cana-5910 359 13 , , PUNCT cana-5910 359 14 π‘ π‘œ π‘ π‘œ ADP cana-5910 359 15 𝑒 𝑒 NOUN cana-5910 359 16 = = X cana-5910 359 17 π‘˜ π‘˜ PRON cana-5910 359 18 𝑑. 𝑑. NOUN cana-5910 359 19 thus thus ADV cana-5910 359 20 , , PUNCT cana-5910 359 21 invariants invariant NOUN cana-5910 359 22 are be AUX cana-5910 359 23 πœ‰ πœ‰ X cana-5910 359 24 = = SYM cana-5910 359 25 π‘₯ π‘₯ PROPN cana-5910 359 26 𝑑 𝑑 PROPN cana-5910 359 27 , , PUNCT cana-5910 359 28 πœ‚ πœ‚ X cana-5910 359 29 = = SYM cana-5910 359 30 𝑦 𝑦 NUM cana-5910 359 31 𝑑 𝑑 PROPN cana-5910 359 32 , , PUNCT cana-5910 359 33 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-5910 359 34 𝑒 𝑒 PROPN cana-5910 359 35 = = PUNCT cana-5910 359 36 𝑑 𝑑 NOUN cana-5910 359 37 𝑣(πœ‰ 𝑣(πœ‰ NOUN cana-5910 359 38 , , PUNCT cana-5910 359 39 πœ‚ πœ‚ NOUN cana-5910 359 40 ) ) PUNCT cana-5910 359 41 . . PUNCT cana-5910 360 1 assume assume VERB cana-5910 360 2 : : PUNCT cana-5910 360 3 𝑒(𝑑 𝑒(𝑑 NOUN cana-5910 360 4 , , PUNCT cana-5910 360 5 π‘₯ π‘₯ NOUN cana-5910 360 6 , , PUNCT cana-5910 360 7 𝑦 𝑦 NOUN cana-5910 360 8 ) ) PUNCT cana-5910 360 9 = = SYM cana-5910 360 10 𝑑 𝑑 NOUN cana-5910 360 11 𝑣(πœ‰ 𝑣(πœ‰ NOUN cana-5910 360 12 , , PUNCT cana-5910 360 13 πœ‚ πœ‚ NOUN cana-5910 360 14 ) ) PUNCT cana-5910 360 15 , , PUNCT cana-5910 360 16 π‘€β„Žπ‘’π‘Ÿπ‘’ π‘€β„Žπ‘’π‘Ÿπ‘’ NOUN cana-5910 360 17 πœ‰ πœ‰ X cana-5910 360 18 = = SYM cana-5910 360 19 π‘₯ π‘₯ PROPN cana-5910 360 20 𝑑 𝑑 PROPN cana-5910 360 21 , , PUNCT cana-5910 360 22 πœ‚ πœ‚ X cana-5910 360 23 = = SYM cana-5910 360 24 𝑦 𝑦 NUM cana-5910 360 25 𝑑 𝑑 PROPN cana-5910 360 26 . . PUNCT cana-5910 361 1 compute compute NOUN cana-5910 361 2 derivatives derivative NOUN cana-5910 361 3 : : PUNCT cana-5910 361 4 𝑒𝑑 𝑒𝑑 NOUN cana-5910 362 1 = = PUNCT cana-5910 362 2 𝑣 𝑣 X cana-5910 362 3 + + CCONJ cana-5910 362 4 𝑑 𝑑 X cana-5910 362 5 ( ( PUNCT cana-5910 362 6 π‘£πœ‰πœ‰π‘‘ π‘£πœ‰πœ‰π‘‘ NOUN cana-5910 362 7 + + ADJ cana-5910 362 8 π‘£πœ‚πœ‚π‘‘ π‘£πœ‚πœ‚π‘‘ NOUN cana-5910 362 9 ) ) PUNCT cana-5910 362 10 , , PUNCT cana-5910 362 11 πœ‰π‘‘ πœ‰π‘‘ NOUN cana-5910 362 12 = = PUNCT cana-5910 362 13 βˆ’ βˆ’ ADP cana-5910 363 1 π‘₯ π‘₯ DET cana-5910 363 2 𝑑2 𝑑2 NOUN cana-5910 363 3 = = PUNCT cana-5910 364 1 βˆ’ βˆ’ NOUN cana-5910 364 2 πœ‰ πœ‰ X cana-5910 364 3 𝑑 𝑑 PROPN cana-5910 364 4 , , PUNCT cana-5910 364 5 πœ‚π‘‘ πœ‚π‘‘ ADP cana-5910 364 6 = = SYM cana-5910 364 7 βˆ’ βˆ’ PROPN cana-5910 364 8 πœ‚ πœ‚ PROPN cana-5910 364 9 𝑑 𝑑 PROPN cana-5910 364 10 , , PUNCT cana-5910 364 11 𝑒𝑑 𝑒𝑑 NOUN cana-5910 364 12 = = SYM cana-5910 364 13 𝑣 𝑣 PRON cana-5910 364 14 βˆ’ βˆ’ NOUN cana-5910 364 15 πœ‰ πœ‰ VERB cana-5910 364 16 π‘£πœ‰ π‘£πœ‰ NOUN cana-5910 365 1 + + NOUN cana-5910 365 2 πœ‚ πœ‚ NOUN cana-5910 365 3 π‘£πœ‚ π‘£πœ‚ NOUN cana-5910 365 4 𝑑 𝑑 X cana-5910 365 5 , , PUNCT cana-5910 365 6 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 365 7 = = SYM cana-5910 365 8 βˆ’ βˆ’ PROPN cana-5910 365 9 1 1 NUM cana-5910 365 10 𝑑 𝑑 PROPN cana-5910 365 11 ( ( PUNCT cana-5910 365 12 π‘£πœ‰πœ‰π‘‘ π‘£πœ‰πœ‰π‘‘ NOUN cana-5910 365 13 + + ADJ cana-5910 365 14 π‘£πœ‚πœ‚π‘‘ π‘£πœ‚πœ‚π‘‘ NOUN cana-5910 365 15 ) ) PUNCT cana-5910 365 16 βˆ’ βˆ’ NOUN cana-5910 365 17 πœ‰ πœ‰ SYM cana-5910 365 18 π‘£πœ‰π‘‘ π‘£πœ‰π‘‘ NOUN cana-5910 366 1 + + CCONJ cana-5910 366 2 πœ‚ πœ‚ NOUN cana-5910 366 3 π‘£πœ‚π‘‘ π‘£πœ‚π‘‘ NOUN cana-5910 366 4 𝑑 𝑑 NOUN cana-5910 366 5 βˆ’ βˆ’ PROPN cana-5910 366 6 ( ( PUNCT cana-5910 366 7 πœ‰ πœ‰ X cana-5910 366 8 π‘£πœ‰ π‘£πœ‰ NOUN cana-5910 367 1 + + NOUN cana-5910 367 2 πœ‚ πœ‚ NOUN cana-5910 367 3 π‘£πœ‚ π‘£πœ‚ NOUN cana-5910 367 4 ) ) PUNCT cana-5910 367 5 ( ( PUNCT cana-5910 367 6 βˆ’ βˆ’ PROPN cana-5910 367 7 1 1 NUM cana-5910 367 8 𝑑2 𝑑2 PROPN cana-5910 367 9 ) ) PUNCT cana-5910 367 10 , , PUNCT cana-5910 367 11 π‘£πœ‰π‘‘ π‘£πœ‰π‘‘ NOUN cana-5910 368 1 = = PUNCT cana-5910 368 2 π‘£πœ‰πœ‰πœ‰π‘‘ π‘£πœ‰πœ‰πœ‰π‘‘ NOUN cana-5910 368 3 + + CCONJ cana-5910 368 4 π‘£πœ‰πœ‚πœ‚π‘‘ π‘£πœ‰πœ‚πœ‚π‘‘ NOUN cana-5910 368 5 𝑑 𝑑 PROPN cana-5910 368 6 , , PUNCT cana-5910 368 7 π‘£πœ‚π‘‘ π‘£πœ‚π‘‘ NOUN cana-5910 368 8 = = SYM cana-5910 368 9 π‘£πœ‚πœ‰πœ‰π‘‘ π‘£πœ‚πœ‰πœ‰π‘‘ PROPN cana-5910 368 10 + + CCONJ cana-5910 368 11 π‘£πœ‚πœ‚πœ‚π‘‘ π‘£πœ‚πœ‚πœ‚π‘‘ PROPN cana-5910 368 12 𝑑 𝑑 NOUN cana-5910 368 13 , , PUNCT cana-5910 368 14 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 368 15 = = SYM cana-5910 368 16 πœ‰2π‘£πœ‰πœ‰ πœ‰2π‘£πœ‰πœ‰ NOUN cana-5910 368 17 + + CCONJ cana-5910 368 18 2πœ‰πœ‚ 2πœ‰πœ‚ ADJ cana-5910 368 19 π‘£πœ‰πœ‚ π‘£πœ‰πœ‚ ADJ cana-5910 368 20 + + NUM cana-5910 368 21 πœ‚2π‘£πœ‚πœ‚ πœ‚2π‘£πœ‚πœ‚ NOUN cana-5910 368 22 𝑑3 𝑑3 NOUN cana-5910 368 23 . . PUNCT cana-5910 369 1 for for ADP cana-5910 369 2 spatial spatial ADJ cana-5910 369 3 derivatives derivative NOUN cana-5910 369 4 : : PUNCT cana-5910 369 5 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 369 6 = = SYM cana-5910 369 7 𝑑 𝑑 PROPN cana-5910 369 8 ( ( PUNCT cana-5910 369 9 π‘£πœ‰ π‘£πœ‰ INTJ cana-5910 369 10 𝑑 𝑑 PROPN cana-5910 369 11 ) ) PUNCT cana-5910 369 12 ( ( PUNCT cana-5910 369 13 1 1 NUM cana-5910 369 14 𝑑 𝑑 NOUN cana-5910 369 15 ) ) PUNCT cana-5910 369 16 = = NOUN cana-5910 369 17 π‘£πœ‰ π‘£πœ‰ PROPN cana-5910 369 18 𝑑2 𝑑2 NOUN cana-5910 369 19 , , PUNCT cana-5910 369 20 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 369 21 = = SYM cana-5910 369 22 ( ( PUNCT cana-5910 369 23 π‘£πœ‰πœ‰ π‘£πœ‰πœ‰ PROPN cana-5910 369 24 𝑑2 𝑑2 NOUN cana-5910 369 25 ) ) PUNCT cana-5910 369 26 ( ( PUNCT cana-5910 369 27 1 1 NUM cana-5910 369 28 𝑑 𝑑 NOUN cana-5910 369 29 ) ) PUNCT cana-5910 369 30 = = SYM cana-5910 369 31 π‘£πœ‰πœ‰ π‘£πœ‰πœ‰ PROPN cana-5910 369 32 𝑑3 𝑑3 PROPN cana-5910 369 33 , , PUNCT cana-5910 369 34 𝑒𝑦 𝑒𝑦 PROPN cana-5910 369 35 = = PUNCT cana-5910 369 36 π‘£πœ‚ π‘£πœ‚ PROPN cana-5910 369 37 𝑑2 𝑑2 PROPN cana-5910 369 38 , , PUNCT cana-5910 369 39 𝑒𝑦𝑦 𝑒𝑦𝑦 ADV cana-5910 369 40 = = PUNCT cana-5910 369 41 π‘£πœ‚πœ‚ π‘£πœ‚πœ‚ PROPN cana-5910 369 42 𝑑3 𝑑3 PROPN cana-5910 369 43 . . PUNCT cana-5910 370 1 substitute substitute PROPN cana-5910 370 2 into into ADP cana-5910 370 3 the the DET cana-5910 370 4 pde pde NOUN cana-5910 370 5 : : PUNCT cana-5910 371 1 πœ‰2π‘£πœ‰πœ‰+ πœ‰2π‘£πœ‰πœ‰+ NOUN cana-5910 371 2 2πœ‰πœ‚ 2πœ‰πœ‚ NOUN cana-5910 372 1 π‘£πœ‰πœ‚+ π‘£πœ‰πœ‚+ ADJ cana-5910 372 2 πœ‚2π‘£πœ‚πœ‚ πœ‚2π‘£πœ‚πœ‚ NOUN cana-5910 372 3 𝑑3 𝑑3 NOUN cana-5910 372 4 = = SYM cana-5910 372 5 4 4 NUM cana-5910 372 6 ( ( PUNCT cana-5910 372 7 π‘£πœ‰πœ‰ π‘£πœ‰πœ‰ NOUN cana-5910 372 8 𝑑3 𝑑3 NOUN cana-5910 372 9 + + CCONJ cana-5910 372 10 π‘£πœ‚πœ‚ π‘£πœ‚πœ‚ NOUN cana-5910 372 11 𝑑3 𝑑3 PROPN cana-5910 372 12 ) ) PUNCT cana-5910 372 13 , , PUNCT cana-5910 372 14 πœ‰2π‘£πœ‰πœ‰ πœ‰2π‘£πœ‰πœ‰ X cana-5910 372 15 + + CCONJ cana-5910 372 16 2πœ‰πœ‚ 2πœ‰πœ‚ ADJ cana-5910 372 17 π‘£πœ‰πœ‚ π‘£πœ‰πœ‚ ADJ cana-5910 372 18 + + NUM cana-5910 372 19 πœ‚2π‘£πœ‚πœ‚ πœ‚2π‘£πœ‚πœ‚ NOUN cana-5910 372 20 = = SYM cana-5910 372 21 4(π‘£πœ‰πœ‰ 4(π‘£πœ‰πœ‰ X cana-5910 372 22 + + CCONJ cana-5910 372 23 π‘£πœ‚πœ‚ π‘£πœ‚πœ‚ NOUN cana-5910 372 24 ) ) PUNCT cana-5910 372 25 . . PUNCT cana-5910 373 1 this this PRON cana-5910 373 2 is be AUX cana-5910 373 3 a a DET cana-5910 373 4 pde pde NOUN cana-5910 373 5 in in ADP cana-5910 373 6 𝑣(πœ‰ 𝑣(πœ‰ NOUN cana-5910 373 7 , , PUNCT cana-5910 373 8 πœ‚ πœ‚ NOUN cana-5910 373 9 ) ) PUNCT cana-5910 373 10 , , PUNCT cana-5910 373 11 which which PRON cana-5910 373 12 is be AUX cana-5910 373 13 complex complex ADJ cana-5910 373 14 . . PUNCT cana-5910 374 1 the the DET cana-5910 374 2 boundary boundary ADJ cana-5910 374 3 conditions condition NOUN cana-5910 374 4 in in ADP cana-5910 374 5 πœ‰ πœ‰ PROPN cana-5910 374 6 , , PUNCT cana-5910 374 7 πœ‚ πœ‚ NOUN cana-5910 374 8 become become VERB cana-5910 374 9 variable variable ADJ cana-5910 374 10 due due ADP cana-5910 374 11 to to ADP cana-5910 374 12 π‘₯ π‘₯ PROPN cana-5910 374 13 = = PUNCT cana-5910 374 14 πœ‰ πœ‰ X cana-5910 374 15 𝑑 𝑑 PROPN cana-5910 374 16 , , PUNCT cana-5910 374 17 𝑦 𝑦 NOUN cana-5910 374 18 = = SYM cana-5910 374 19 πœ‚ πœ‚ PRON cana-5910 374 20 𝑑 𝑑 PROPN cana-5910 374 21 , , PUNCT cana-5910 374 22 complicating complicate VERB cana-5910 374 23 direct direct ADJ cana-5910 374 24 application application NOUN cana-5910 374 25 . . PUNCT cana-5910 375 1 instead instead ADV cana-5910 375 2 , , PUNCT cana-5910 375 3 consider consider VERB cana-5910 375 4 the the DET cana-5910 375 5 condition condition NOUN cana-5910 375 6 𝑒 𝑒 X cana-5910 375 7 β†’ β†’ SYM cana-5910 375 8 0 0 NUM cana-5910 375 9 π‘Žπ‘  π‘Žπ‘  PROPN cana-5910 375 10 𝑑 𝑑 PROPN cana-5910 375 11 β†’ β†’ SYM cana-5910 375 12 ∞ ∞ PROPN cana-5910 375 13 , , PUNCT cana-5910 375 14 suggesting suggest VERB cana-5910 375 15 a a DET cana-5910 375 16 decaying decay VERB cana-5910 375 17 solution solution NOUN cana-5910 375 18 . . PUNCT cana-5910 376 1 try try VERB cana-5910 376 2 an an DET cana-5910 376 3 exponential exponential ADJ cana-5910 376 4 ansatz ansatz NOUN cana-5910 376 5 to to PART cana-5910 376 6 align align VERB cana-5910 376 7 with with ADP cana-5910 376 8 the the DET cana-5910 376 9 boundary boundary ADJ cana-5910 376 10 condition condition NOUN cana-5910 376 11 at at ADP cana-5910 376 12 𝑑 𝑑 PROPN cana-5910 376 13 β†’ β†’ SYM cana-5910 376 14 ∞ ∞ PROPN cana-5910 376 15 : : PUNCT cana-5910 376 16 𝑒 𝑒 X cana-5910 376 17 = = SYM cana-5910 376 18 π‘’βˆ’πœ†π‘‘π‘€(π‘₯ π‘’βˆ’πœ†π‘‘π‘€(π‘₯ NUM cana-5910 376 19 , , PUNCT cana-5910 376 20 𝑦 𝑦 NOUN cana-5910 376 21 ) ) PUNCT cana-5910 376 22 . . PUNCT cana-5910 377 1 substitute substitute NOUN cana-5910 377 2 : : PUNCT cana-5910 377 3 𝑒𝑑 𝑒𝑑 NOUN cana-5910 377 4 = = SYM cana-5910 377 5 βˆ’πœ† βˆ’πœ† PROPN cana-5910 377 6 π‘’βˆ’πœ†π‘‘π‘€ π‘’βˆ’πœ†π‘‘π‘€ NOUN cana-5910 377 7 , , PUNCT cana-5910 377 8 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 377 9 = = SYM cana-5910 377 10 πœ†2π‘’βˆ’πœ†π‘‘π‘€ πœ†2π‘’βˆ’πœ†π‘‘π‘€ NOUN cana-5910 377 11 , , PUNCT cana-5910 377 12 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ NOUN cana-5910 377 13 = = SYM cana-5910 377 14 π‘’βˆ’πœ†π‘‘π‘€π‘₯π‘₯ π‘’βˆ’πœ†π‘‘π‘€π‘₯π‘₯ NOUN cana-5910 377 15 , , PUNCT cana-5910 377 16 𝑒𝑦𝑦 𝑒𝑦𝑦 NOUN cana-5910 377 17 = = SYM cana-5910 377 18 π‘’βˆ’πœ†π‘‘π‘€π‘¦π‘¦ π‘’βˆ’πœ†π‘‘π‘€π‘¦π‘¦ X cana-5910 377 19 , , PUNCT cana-5910 377 20 πœ†2π‘’βˆ’πœ†π‘‘π‘€ πœ†2π‘’βˆ’πœ†π‘‘π‘€ NOUN cana-5910 377 21 = = SYM cana-5910 377 22 4 4 NUM cana-5910 377 23 π‘’βˆ’πœ†π‘‘(𝑀π‘₯π‘₯ π‘’βˆ’πœ†π‘‘(𝑀π‘₯π‘₯ NOUN cana-5910 377 24 + + CCONJ cana-5910 377 25 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 377 26 ) ) PUNCT cana-5910 377 27 , , PUNCT cana-5910 377 28 πœ†Β² πœ†Β² NOUN cana-5910 377 29 𝑀 𝑀 ADP cana-5910 377 30 = = SYM cana-5910 377 31 4 4 NUM cana-5910 377 32 ( ( PUNCT cana-5910 377 33 𝑀π‘₯π‘₯ 𝑀π‘₯π‘₯ NOUN cana-5910 377 34 + + CCONJ cana-5910 377 35 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 377 36 ) ) PUNCT cana-5910 377 37 , , PUNCT cana-5910 377 38 1125 1125 NUM cana-5910 377 39 https://internationalpubls.com https://internationalpubls.com X cana-5910 377 40 𝑀π‘₯π‘₯ 𝑀π‘₯π‘₯ NOUN cana-5910 377 41 + + CCONJ cana-5910 377 42 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 377 43 βˆ’ βˆ’ X cana-5910 377 44 ( ( PUNCT cana-5910 377 45 πœ†2 πœ†2 NOUN cana-5910 377 46 4 4 NUM cana-5910 377 47 ) ) PUNCT cana-5910 377 48 𝑀 𝑀 ADP cana-5910 377 49 = = SYM cana-5910 377 50 0 0 X cana-5910 377 51 . . PUNCT cana-5910 378 1 boundary boundary ADJ cana-5910 378 2 conditions condition NOUN cana-5910 378 3 : : PUNCT cana-5910 378 4 𝑀 𝑀 X cana-5910 378 5 = = SYM cana-5910 378 6 0 0 NUM cana-5910 379 1 π‘Žπ‘‘ π‘Žπ‘‘ PRON cana-5910 379 2 π‘₯ π‘₯ NOUN cana-5910 379 3 = = SYM cana-5910 379 4 0 0 NUM cana-5910 379 5 , , PUNCT cana-5910 379 6 𝑙 𝑙 X cana-5910 379 7 , , PUNCT cana-5910 379 8 𝑦 𝑦 NOUN cana-5910 379 9 = = SYM cana-5910 379 10 0 0 NUM cana-5910 379 11 , , PUNCT cana-5910 379 12 𝑙. 𝑙. NOUN cana-5910 380 1 𝐴𝑠 𝐴𝑠 PROPN cana-5910 380 2 𝑑 𝑑 PROPN cana-5910 380 3 β†’ β†’ SYM cana-5910 380 4 ∞ ∞ PROPN cana-5910 380 5 , , PUNCT cana-5910 380 6 π‘’βˆ’πœ†π‘‘ π‘’βˆ’πœ†π‘‘ PROPN cana-5910 380 7 β†’ β†’ SYM cana-5910 380 8 0 0 NUM cana-5910 380 9 𝑖𝑓 𝑖𝑓 ADP cana-5910 380 10 πœ† πœ† ADP cana-5910 380 11 > > X cana-5910 380 12 0 0 NUM cana-5910 380 13 , , PUNCT cana-5910 380 14 satisfying satisfy VERB cana-5910 380 15 𝑒 𝑒 PRON cana-5910 380 16 β†’ β†’ SYM cana-5910 380 17 0 0 NUM cana-5910 380 18 . . PUNCT cana-5910 381 1 step step NOUN cana-5910 381 2 4 4 NUM cana-5910 381 3 : : PUNCT cana-5910 381 4 symmetry symmetry NOUN cana-5910 381 5 reduction reduction NOUN cana-5910 381 6 use use VERB cana-5910 381 7 the the DET cana-5910 381 8 scaling scale VERB cana-5910 381 9 symmetry symmetry NOUN cana-5910 381 10 : : PUNCT cana-5910 381 11 𝑋 𝑋 PROPN cana-5910 381 12 = = SYM cana-5910 381 13 𝑑 𝑑 PROPN cana-5910 381 14 πœ• πœ• PROPN cana-5910 381 15 πœ•π‘‘ πœ•π‘‘ PROPN cana-5910 381 16 + + CCONJ cana-5910 381 17 π‘₯ π‘₯ DET cana-5910 381 18 πœ• πœ• NOUN cana-5910 381 19 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 381 20 + + CCONJ cana-5910 381 21 𝑦 𝑦 NOUN cana-5910 381 22 πœ• πœ• NOUN cana-5910 381 23 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 381 24 + + CCONJ cana-5910 381 25 𝑒 𝑒 PRON cana-5910 381 26 πœ• πœ• PROPN cana-5910 381 27 πœ•π‘’ πœ•π‘’ NOUN cana-5910 381 28 . . PUNCT cana-5910 382 1 find find VERB cana-5910 382 2 invariants invariant NOUN cana-5910 382 3 by by ADP cana-5910 382 4 solving solve VERB cana-5910 382 5 : : PUNCT cana-5910 382 6 𝑑𝑑 𝑑𝑑 ADP cana-5910 382 7 𝑑 𝑑 PROPN cana-5910 382 8 = = PUNCT cana-5910 382 9 𝑑π‘₯ 𝑑π‘₯ NOUN cana-5910 382 10 π‘₯ π‘₯ NOUN cana-5910 382 11 = = X cana-5910 382 12 𝑑𝑦 𝑑𝑦 PROPN cana-5910 382 13 𝑦 𝑦 NOUN cana-5910 382 14 = = X cana-5910 382 15 𝑑𝑒 𝑑𝑒 ADP cana-5910 382 16 𝑒 𝑒 X cana-5910 382 17 . . PUNCT cana-5910 383 1 from from ADP cana-5910 383 2 𝑑π‘₯ 𝑑π‘₯ NOUN cana-5910 383 3 π‘₯ π‘₯ NOUN cana-5910 383 4 = = PUNCT cana-5910 383 5 𝑑𝑦 𝑑𝑦 PROPN cana-5910 383 6 𝑦 𝑦 NOUN cana-5910 383 7 , , PUNCT cana-5910 383 8 π‘₯ π‘₯ PROPN cana-5910 383 9 𝑦 𝑦 NOUN cana-5910 383 10 = = SYM cana-5910 383 11 𝑐1 𝑐1 NOUN cana-5910 383 12 , , PUNCT cana-5910 383 13 π‘ π‘œ π‘ π‘œ PROPN cana-5910 383 14 πœ‰1 πœ‰1 PROPN cana-5910 383 15 = = SYM cana-5910 383 16 π‘₯ π‘₯ PROPN cana-5910 383 17 𝑦 𝑦 NOUN cana-5910 383 18 . . PUNCT cana-5910 384 1 from from ADP cana-5910 384 2 𝑑π‘₯ 𝑑π‘₯ NOUN cana-5910 384 3 π‘₯ π‘₯ NOUN cana-5910 384 4 = = PUNCT cana-5910 384 5 𝑑𝑑 𝑑𝑑 ADP cana-5910 384 6 𝑑 𝑑 PROPN cana-5910 384 7 , , PUNCT cana-5910 384 8 π‘₯ π‘₯ PROPN cana-5910 384 9 𝑑 𝑑 PROPN cana-5910 384 10 = = SYM cana-5910 384 11 𝑐2 𝑐2 NOUN cana-5910 384 12 , , PUNCT cana-5910 384 13 π‘ π‘œ π‘ π‘œ ADP cana-5910 384 14 πœ‰2 πœ‰2 PROPN cana-5910 384 15 = = PUNCT cana-5910 385 1 π‘₯ π‘₯ PROPN cana-5910 385 2 𝑑 𝑑 PROPN cana-5910 385 3 . . PUNCT cana-5910 386 1 from from ADP cana-5910 386 2 𝑑𝑒 𝑑𝑒 PRON cana-5910 386 3 𝑒 𝑒 PROPN cana-5910 386 4 = = X cana-5910 386 5 𝑑𝑑 𝑑𝑑 ADP cana-5910 386 6 𝑑 𝑑 PROPN cana-5910 386 7 , , PUNCT cana-5910 386 8 𝑒 𝑒 PROPN cana-5910 386 9 𝑑 𝑑 PROPN cana-5910 386 10 = = SYM cana-5910 386 11 𝑐3 𝑐3 NOUN cana-5910 386 12 , , PUNCT cana-5910 386 13 π‘ π‘œ π‘ π‘œ ADP cana-5910 386 14 𝑒 𝑒 NOUN cana-5910 386 15 = = X cana-5910 386 16 π‘˜ π‘˜ PRON cana-5910 386 17 𝑑. 𝑑. NOUN cana-5910 386 18 thus thus ADV cana-5910 386 19 , , PUNCT cana-5910 386 20 invariants invariant NOUN cana-5910 386 21 are be AUX cana-5910 386 22 πœ‰ πœ‰ X cana-5910 386 23 = = SYM cana-5910 386 24 π‘₯ π‘₯ PROPN cana-5910 386 25 𝑑 𝑑 PROPN cana-5910 386 26 , , PUNCT cana-5910 386 27 πœ‚ πœ‚ X cana-5910 386 28 = = SYM cana-5910 386 29 𝑦 𝑦 NUM cana-5910 386 30 𝑑 𝑑 PROPN cana-5910 386 31 , , PUNCT cana-5910 387 1 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-5910 387 2 𝑒 𝑒 PROPN cana-5910 387 3 = = PUNCT cana-5910 387 4 𝑑 𝑑 NOUN cana-5910 387 5 𝑣(πœ‰ 𝑣(πœ‰ NOUN cana-5910 387 6 , , PUNCT cana-5910 387 7 πœ‚ πœ‚ NOUN cana-5910 387 8 ) ) PUNCT cana-5910 387 9 . . PUNCT cana-5910 388 1 assume assume VERB cana-5910 388 2 : : PUNCT cana-5910 388 3 𝑒(𝑑 𝑒(𝑑 NOUN cana-5910 388 4 , , PUNCT cana-5910 388 5 π‘₯ π‘₯ NOUN cana-5910 388 6 , , PUNCT cana-5910 388 7 𝑦 𝑦 NOUN cana-5910 388 8 ) ) PUNCT cana-5910 388 9 = = SYM cana-5910 388 10 𝑑 𝑑 NOUN cana-5910 388 11 𝑣(πœ‰ 𝑣(πœ‰ NOUN cana-5910 388 12 , , PUNCT cana-5910 388 13 πœ‚ πœ‚ NOUN cana-5910 388 14 ) ) PUNCT cana-5910 388 15 , , PUNCT cana-5910 388 16 π‘€β„Žπ‘’π‘Ÿπ‘’ π‘€β„Žπ‘’π‘Ÿπ‘’ NOUN cana-5910 388 17 πœ‰ πœ‰ X cana-5910 388 18 = = SYM cana-5910 388 19 π‘₯ π‘₯ PROPN cana-5910 388 20 𝑑 𝑑 PROPN cana-5910 388 21 , , PUNCT cana-5910 388 22 πœ‚ πœ‚ X cana-5910 388 23 = = SYM cana-5910 388 24 𝑦 𝑦 NUM cana-5910 388 25 𝑑 𝑑 PROPN cana-5910 388 26 . . PUNCT cana-5910 389 1 compute compute NOUN cana-5910 389 2 derivatives derivative NOUN cana-5910 389 3 : : PUNCT cana-5910 389 4 𝑒𝑑 𝑒𝑑 NOUN cana-5910 390 1 = = PUNCT cana-5910 390 2 𝑣 𝑣 X cana-5910 390 3 + + CCONJ cana-5910 390 4 𝑑 𝑑 X cana-5910 390 5 ( ( PUNCT cana-5910 390 6 π‘£πœ‰πœ‰π‘‘ π‘£πœ‰πœ‰π‘‘ NOUN cana-5910 390 7 + + ADJ cana-5910 390 8 π‘£πœ‚πœ‚π‘‘ π‘£πœ‚πœ‚π‘‘ NOUN cana-5910 390 9 ) ) PUNCT cana-5910 390 10 , , PUNCT cana-5910 390 11 πœ‰π‘‘ πœ‰π‘‘ NOUN cana-5910 390 12 = = PUNCT cana-5910 390 13 βˆ’ βˆ’ ADP cana-5910 391 1 π‘₯ π‘₯ DET cana-5910 391 2 𝑑2 𝑑2 NOUN cana-5910 391 3 = = PUNCT cana-5910 392 1 βˆ’ βˆ’ NOUN cana-5910 392 2 πœ‰ πœ‰ X cana-5910 392 3 𝑑 𝑑 PROPN cana-5910 392 4 , , PUNCT cana-5910 392 5 πœ‚π‘‘ πœ‚π‘‘ ADP cana-5910 392 6 = = SYM cana-5910 392 7 βˆ’ βˆ’ PROPN cana-5910 392 8 πœ‚ πœ‚ PROPN cana-5910 392 9 𝑑 𝑑 PROPN cana-5910 392 10 , , PUNCT cana-5910 392 11 𝑒𝑑 𝑒𝑑 NOUN cana-5910 392 12 = = VERB cana-5910 392 13 𝑣 𝑣 PRON cana-5910 392 14 βˆ’ βˆ’ NOUN cana-5910 392 15 πœ‰ πœ‰ X cana-5910 392 16 π‘£πœ‰+ π‘£πœ‰+ ADJ cana-5910 392 17 πœ‚ πœ‚ NOUN cana-5910 392 18 π‘£πœ‚ π‘£πœ‚ NOUN cana-5910 392 19 𝑑 𝑑 X cana-5910 392 20 , , PUNCT cana-5910 392 21 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 392 22 = = SYM cana-5910 392 23 βˆ’ βˆ’ PROPN cana-5910 392 24 1 1 NUM cana-5910 392 25 𝑑 𝑑 PROPN cana-5910 392 26 ( ( PUNCT cana-5910 392 27 π‘£πœ‰πœ‰π‘‘ π‘£πœ‰πœ‰π‘‘ NOUN cana-5910 392 28 + + ADJ cana-5910 392 29 π‘£πœ‚πœ‚π‘‘ π‘£πœ‚πœ‚π‘‘ NOUN cana-5910 392 30 ) ) PUNCT cana-5910 392 31 βˆ’ βˆ’ NOUN cana-5910 392 32 πœ‰ πœ‰ SYM cana-5910 392 33 π‘£πœ‰π‘‘ π‘£πœ‰π‘‘ NOUN cana-5910 393 1 + + CCONJ cana-5910 393 2 πœ‚ πœ‚ NOUN cana-5910 393 3 π‘£πœ‚π‘‘ π‘£πœ‚π‘‘ NOUN cana-5910 393 4 𝑑 𝑑 NOUN cana-5910 393 5 βˆ’ βˆ’ PROPN cana-5910 393 6 ( ( PUNCT cana-5910 393 7 πœ‰ πœ‰ X cana-5910 393 8 π‘£πœ‰ π‘£πœ‰ NOUN cana-5910 394 1 + + NOUN cana-5910 394 2 πœ‚ πœ‚ NOUN cana-5910 394 3 π‘£πœ‚ π‘£πœ‚ NOUN cana-5910 394 4 ) ) PUNCT cana-5910 394 5 ( ( PUNCT cana-5910 394 6 βˆ’ βˆ’ PROPN cana-5910 394 7 1 1 NUM cana-5910 394 8 𝑑2 𝑑2 PROPN cana-5910 394 9 ) ) PUNCT cana-5910 394 10 , , PUNCT cana-5910 395 1 π‘£πœ‰π‘‘ π‘£πœ‰π‘‘ NOUN cana-5910 395 2 = = PUNCT cana-5910 395 3 π‘£πœ‰πœ‰πœ‰π‘‘+ π‘£πœ‰πœ‰πœ‰π‘‘+ X cana-5910 395 4 π‘£πœ‰πœ‚πœ‚π‘‘ π‘£πœ‰πœ‚πœ‚π‘‘ PROPN cana-5910 395 5 𝑑 𝑑 PROPN cana-5910 395 6 , , PUNCT cana-5910 395 7 π‘£πœ‚π‘‘ π‘£πœ‚π‘‘ X cana-5910 395 8 = = SYM cana-5910 396 1 π‘£πœ‚πœ‰πœ‰π‘‘+ π‘£πœ‚πœ‰πœ‰π‘‘+ NUM cana-5910 396 2 π‘£πœ‚πœ‚πœ‚π‘‘ π‘£πœ‚πœ‚πœ‚π‘‘ PROPN cana-5910 396 3 𝑑 𝑑 PROPN cana-5910 396 4 , , PUNCT cana-5910 396 5 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 396 6 = = SYM cana-5910 396 7 πœ‰2π‘£πœ‰πœ‰ πœ‰2π‘£πœ‰πœ‰ NOUN cana-5910 396 8 + + CCONJ cana-5910 396 9 2πœ‰πœ‚ 2πœ‰πœ‚ ADJ cana-5910 396 10 π‘£πœ‰πœ‚ π‘£πœ‰πœ‚ ADJ cana-5910 396 11 + + NUM cana-5910 396 12 πœ‚2π‘£πœ‚πœ‚ πœ‚2π‘£πœ‚πœ‚ NOUN cana-5910 396 13 𝑑3 𝑑3 NOUN cana-5910 396 14 . . PUNCT cana-5910 397 1 for for ADP cana-5910 397 2 spatial spatial ADJ cana-5910 397 3 derivatives derivative NOUN cana-5910 397 4 : : PUNCT cana-5910 397 5 𝑒π‘₯ 𝑒π‘₯ ADP cana-5910 397 6 = = SYM cana-5910 397 7 𝑑 𝑑 PROPN cana-5910 397 8 ( ( PUNCT cana-5910 397 9 π‘£πœ‰ π‘£πœ‰ INTJ cana-5910 397 10 𝑑 𝑑 PROPN cana-5910 397 11 ) ) PUNCT cana-5910 397 12 ( ( PUNCT cana-5910 397 13 1 1 NUM cana-5910 397 14 𝑑 𝑑 NOUN cana-5910 397 15 ) ) PUNCT cana-5910 397 16 = = NOUN cana-5910 397 17 π‘£πœ‰ π‘£πœ‰ PROPN cana-5910 397 18 𝑑2 𝑑2 NOUN cana-5910 397 19 , , PUNCT cana-5910 397 20 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ X cana-5910 397 21 = = SYM cana-5910 397 22 ( ( PUNCT cana-5910 397 23 π‘£πœ‰πœ‰ π‘£πœ‰πœ‰ PROPN cana-5910 397 24 𝑑2 𝑑2 NOUN cana-5910 397 25 ) ) PUNCT cana-5910 397 26 ( ( PUNCT cana-5910 397 27 1 1 NUM cana-5910 397 28 𝑑 𝑑 NOUN cana-5910 397 29 ) ) PUNCT cana-5910 397 30 = = SYM cana-5910 397 31 π‘£πœ‰πœ‰ π‘£πœ‰πœ‰ PROPN cana-5910 397 32 𝑑3 𝑑3 PROPN cana-5910 397 33 , , PUNCT cana-5910 397 34 𝑒𝑦 𝑒𝑦 PROPN cana-5910 397 35 = = PUNCT cana-5910 397 36 π‘£πœ‚ π‘£πœ‚ PROPN cana-5910 397 37 𝑑2 𝑑2 PROPN cana-5910 397 38 , , PUNCT cana-5910 397 39 𝑒𝑦𝑦 𝑒𝑦𝑦 ADV cana-5910 397 40 = = PUNCT cana-5910 397 41 π‘£πœ‚πœ‚ π‘£πœ‚πœ‚ PROPN cana-5910 397 42 𝑑3 𝑑3 PROPN cana-5910 397 43 . . PUNCT cana-5910 398 1 substitute substitute PROPN cana-5910 398 2 into into ADP cana-5910 398 3 the the DET cana-5910 398 4 pde pde NOUN cana-5910 398 5 : : PUNCT cana-5910 399 1 πœ‰2π‘£πœ‰πœ‰+ πœ‰2π‘£πœ‰πœ‰+ NOUN cana-5910 399 2 2πœ‰πœ‚ 2πœ‰πœ‚ NOUN cana-5910 400 1 π‘£πœ‰πœ‚+ π‘£πœ‰πœ‚+ ADJ cana-5910 400 2 πœ‚2π‘£πœ‚πœ‚ πœ‚2π‘£πœ‚πœ‚ NOUN cana-5910 400 3 𝑑3 𝑑3 NOUN cana-5910 400 4 = = SYM cana-5910 400 5 4 4 NUM cana-5910 400 6 ( ( PUNCT cana-5910 400 7 π‘£πœ‰πœ‰ π‘£πœ‰πœ‰ NOUN cana-5910 400 8 𝑑3 𝑑3 NOUN cana-5910 400 9 + + CCONJ cana-5910 400 10 π‘£πœ‚πœ‚ π‘£πœ‚πœ‚ NOUN cana-5910 400 11 𝑑3 𝑑3 PROPN cana-5910 400 12 ) ) PUNCT cana-5910 400 13 , , PUNCT cana-5910 400 14 πœ‰2π‘£πœ‰πœ‰ πœ‰2π‘£πœ‰πœ‰ X cana-5910 400 15 + + CCONJ cana-5910 400 16 2πœ‰πœ‚ 2πœ‰πœ‚ ADJ cana-5910 400 17 π‘£πœ‰πœ‚ π‘£πœ‰πœ‚ ADJ cana-5910 400 18 + + NUM cana-5910 400 19 πœ‚2π‘£πœ‚πœ‚ πœ‚2π‘£πœ‚πœ‚ NOUN cana-5910 400 20 = = SYM cana-5910 400 21 4(π‘£πœ‰πœ‰ 4(π‘£πœ‰πœ‰ X cana-5910 400 22 + + CCONJ cana-5910 400 23 π‘£πœ‚πœ‚ π‘£πœ‚πœ‚ NOUN cana-5910 400 24 ) ) PUNCT cana-5910 400 25 . . PUNCT cana-5910 401 1 this this PRON cana-5910 401 2 is be AUX cana-5910 401 3 a a DET cana-5910 401 4 pde pde NOUN cana-5910 401 5 in in ADP cana-5910 401 6 𝑣(πœ‰ 𝑣(πœ‰ NOUN cana-5910 401 7 , , PUNCT cana-5910 401 8 πœ‚ πœ‚ NOUN cana-5910 401 9 ) ) PUNCT cana-5910 401 10 , , PUNCT cana-5910 401 11 which which PRON cana-5910 401 12 is be AUX cana-5910 401 13 complex complex ADJ cana-5910 401 14 . . PUNCT cana-5910 402 1 the the DET cana-5910 402 2 boundary boundary ADJ cana-5910 402 3 conditions condition NOUN cana-5910 402 4 in in ADP cana-5910 402 5 ΞΎ ΞΎ PROPN cana-5910 402 6 , , PUNCT cana-5910 402 7 Ξ· Ξ· X cana-5910 402 8 become become VERB cana-5910 402 9 variable variable ADJ cana-5910 402 10 due due ADP cana-5910 402 11 to to ADP cana-5910 402 12 π‘₯ π‘₯ PROPN cana-5910 402 13 = = PUNCT cana-5910 402 14 πœ‰ πœ‰ X cana-5910 402 15 𝑑 𝑑 PROPN cana-5910 402 16 , , PUNCT cana-5910 402 17 𝑦 𝑦 NOUN cana-5910 402 18 = = SYM cana-5910 402 19 πœ‚ πœ‚ PRON cana-5910 402 20 𝑑 𝑑 PROPN cana-5910 402 21 , , PUNCT cana-5910 402 22 complicating complicate VERB cana-5910 402 23 direct direct ADJ cana-5910 402 24 application application NOUN cana-5910 402 25 . . PUNCT cana-5910 403 1 instead instead ADV cana-5910 403 2 , , PUNCT cana-5910 403 3 consider consider VERB cana-5910 403 4 the the DET cana-5910 403 5 condition condition NOUN cana-5910 403 6 u u NOUN cana-5910 403 7 β†’ β†’ SYM cana-5910 403 8 0 0 NUM cana-5910 403 9 as as ADP cana-5910 403 10 t t PROPN cana-5910 403 11 β†’ β†’ SYM cana-5910 403 12 ∞ ∞ PROPN cana-5910 403 13 , , PUNCT cana-5910 403 14 suggesting suggest VERB cana-5910 403 15 a a DET cana-5910 403 16 decaying decay VERB cana-5910 403 17 solution solution NOUN cana-5910 403 18 . . PUNCT cana-5910 404 1 try try VERB cana-5910 404 2 an an DET cana-5910 404 3 exponential exponential ADJ cana-5910 404 4 ansatz ansatz NOUN cana-5910 404 5 to to PART cana-5910 404 6 align align VERB cana-5910 404 7 with with ADP cana-5910 404 8 the the DET cana-5910 404 9 boundary boundary ADJ cana-5910 404 10 condition condition NOUN cana-5910 404 11 at at ADP cana-5910 404 12 t t PROPN cana-5910 404 13 β†’ β†’ SYM cana-5910 404 14 ∞ ∞ PROPN cana-5910 404 15 : : SYM cana-5910 404 16 1126 1126 NUM cana-5910 404 17 https://internationalpubls.com https://internationalpubls.com X cana-5910 404 18 𝑒 𝑒 X cana-5910 404 19 = = SYM cana-5910 404 20 π‘’βˆ’πœ†π‘‘π‘€(π‘₯ π‘’βˆ’πœ†π‘‘π‘€(π‘₯ NUM cana-5910 404 21 , , PUNCT cana-5910 404 22 𝑦 𝑦 NOUN cana-5910 404 23 ) ) PUNCT cana-5910 404 24 . . PUNCT cana-5910 405 1 substitute substitute NOUN cana-5910 405 2 : : PUNCT cana-5910 405 3 𝑒𝑑 𝑒𝑑 NOUN cana-5910 405 4 = = SYM cana-5910 405 5 βˆ’πœ† βˆ’πœ† PROPN cana-5910 405 6 π‘’βˆ’πœ†π‘‘π‘€ π‘’βˆ’πœ†π‘‘π‘€ NOUN cana-5910 405 7 , , PUNCT cana-5910 405 8 𝑒𝑑𝑑 𝑒𝑑𝑑 NOUN cana-5910 405 9 = = SYM cana-5910 405 10 πœ†2π‘’βˆ’πœ†π‘‘π‘€ πœ†2π‘’βˆ’πœ†π‘‘π‘€ NOUN cana-5910 405 11 , , PUNCT cana-5910 405 12 𝑒π‘₯π‘₯ 𝑒π‘₯π‘₯ NOUN cana-5910 405 13 = = SYM cana-5910 405 14 π‘’βˆ’πœ†π‘‘π‘€π‘₯π‘₯ π‘’βˆ’πœ†π‘‘π‘€π‘₯π‘₯ NOUN cana-5910 405 15 , , PUNCT cana-5910 405 16 𝑒𝑦𝑦 𝑒𝑦𝑦 NOUN cana-5910 405 17 = = SYM cana-5910 405 18 π‘’βˆ’πœ†π‘‘π‘€π‘¦π‘¦ π‘’βˆ’πœ†π‘‘π‘€π‘¦π‘¦ X cana-5910 405 19 , , PUNCT cana-5910 405 20 πœ†2π‘’βˆ’πœ†π‘‘π‘€ πœ†2π‘’βˆ’πœ†π‘‘π‘€ NOUN cana-5910 405 21 = = SYM cana-5910 405 22 4 4 NUM cana-5910 405 23 π‘’βˆ’πœ†π‘‘(𝑀π‘₯π‘₯ π‘’βˆ’πœ†π‘‘(𝑀π‘₯π‘₯ NOUN cana-5910 405 24 + + CCONJ cana-5910 405 25 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 405 26 ) ) PUNCT cana-5910 405 27 , , PUNCT cana-5910 405 28 πœ†Β² πœ†Β² NOUN cana-5910 405 29 𝑀 𝑀 ADP cana-5910 405 30 = = SYM cana-5910 405 31 4 4 NUM cana-5910 405 32 ( ( PUNCT cana-5910 405 33 𝑀π‘₯π‘₯ 𝑀π‘₯π‘₯ NOUN cana-5910 405 34 + + CCONJ cana-5910 405 35 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 405 36 ) ) PUNCT cana-5910 405 37 , , PUNCT cana-5910 405 38 𝑀π‘₯π‘₯ 𝑀π‘₯π‘₯ NOUN cana-5910 405 39 + + CCONJ cana-5910 405 40 𝑀𝑦𝑦 𝑀𝑦𝑦 NOUN cana-5910 405 41 βˆ’ βˆ’ X cana-5910 405 42 ( ( PUNCT cana-5910 405 43 πœ†2 πœ†2 NOUN cana-5910 405 44 4 4 NUM cana-5910 405 45 ) ) PUNCT cana-5910 405 46 𝑀 𝑀 ADP cana-5910 405 47 = = SYM cana-5910 405 48 0 0 X cana-5910 405 49 . . PUNCT cana-5910 406 1 boundary boundary ADJ cana-5910 406 2 conditions condition NOUN cana-5910 406 3 : : PUNCT cana-5910 406 4 𝑀 𝑀 X cana-5910 406 5 = = SYM cana-5910 406 6 0 0 NUM cana-5910 407 1 π‘Žπ‘‘ π‘Žπ‘‘ PRON cana-5910 407 2 π‘₯ π‘₯ NOUN cana-5910 407 3 = = SYM cana-5910 407 4 0 0 NUM cana-5910 407 5 , , PUNCT cana-5910 407 6 𝑙 𝑙 X cana-5910 407 7 , , PUNCT cana-5910 407 8 𝑦 𝑦 NOUN cana-5910 407 9 = = SYM cana-5910 407 10 0 0 NUM cana-5910 407 11 , , PUNCT cana-5910 407 12 𝑙. 𝑙. NOUN cana-5910 408 1 𝐴𝑠 𝐴𝑠 PROPN cana-5910 408 2 𝑑 𝑑 PROPN cana-5910 408 3 β†’ β†’ SYM cana-5910 408 4 ∞ ∞ PROPN cana-5910 408 5 , , PUNCT cana-5910 408 6 π‘’βˆ’πœ†π‘‘ π‘’βˆ’πœ†π‘‘ PROPN cana-5910 408 7 β†’ β†’ SYM cana-5910 408 8 0 0 NUM cana-5910 408 9 𝑖𝑓 𝑖𝑓 ADP cana-5910 408 10 πœ† πœ† ADP cana-5910 408 11 > > X cana-5910 408 12 0 0 PROPN cana-5910 408 13 , , PUNCT cana-5910 408 14 π‘ π‘Žπ‘‘π‘–π‘ π‘“π‘¦π‘–π‘›π‘” π‘ π‘Žπ‘‘π‘–π‘ π‘“π‘¦π‘–π‘›π‘” ADJ cana-5910 408 15 𝑒 𝑒 NOUN cana-5910 408 16 β†’ β†’ SYM cana-5910 408 17 0 0 NUM cana-5910 408 18 . . PUNCT cana-5910 408 19 final final ADJ cana-5910 408 20 answer answer NOUN cana-5910 408 21 the the DET cana-5910 408 22 solution solution NOUN cana-5910 408 23 to to ADP cana-5910 408 24 the the DET cana-5910 408 25 pde pde NOUN cana-5910 408 26 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 408 27 πœ•π‘‘2 πœ•π‘‘2 NOUN cana-5910 408 28 = = SYM cana-5910 408 29 4 4 NUM cana-5910 408 30 ( ( PUNCT cana-5910 408 31 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 408 32 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 408 33 + + CCONJ cana-5910 408 34 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 408 35 πœ•π‘¦2 πœ•π‘¦2 NUM cana-5910 408 36 ) ) PUNCT cana-5910 408 37 with with ADP cana-5910 408 38 boundary boundary ADJ cana-5910 408 39 conditions condition NOUN cana-5910 408 40 𝑒 𝑒 NOUN cana-5910 408 41 = = SYM cana-5910 408 42 0 0 NUM cana-5910 408 43 π‘Žπ‘‘ π‘Žπ‘‘ PRON cana-5910 409 1 π‘₯ π‘₯ NOUN cana-5910 409 2 = = SYM cana-5910 409 3 0 0 NUM cana-5910 409 4 , , PUNCT cana-5910 409 5 π‘₯ π‘₯ NOUN cana-5910 409 6 = = SYM cana-5910 409 7 𝑙 𝑙 PROPN cana-5910 409 8 , , PUNCT cana-5910 409 9 𝑦 𝑦 NOUN cana-5910 409 10 = = SYM cana-5910 409 11 0 0 NUM cana-5910 409 12 , , PUNCT cana-5910 409 13 𝑦 𝑦 NOUN cana-5910 409 14 = = SYM cana-5910 409 15 𝑙 𝑙 PROPN cana-5910 409 16 , , PUNCT cana-5910 409 17 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-5910 409 18 𝑒 𝑒 PROPN cana-5910 409 19 β†’ β†’ SYM cana-5910 409 20 0 0 NUM cana-5910 409 21 π‘Žπ‘  π‘Žπ‘  PROPN cana-5910 409 22 𝑑 𝑑 PROPN cana-5910 409 23 β†’ β†’ SYM cana-5910 409 24 ∞ ∞ PROPN cana-5910 409 25 , , PUNCT cana-5910 409 26 obtained obtain VERB cana-5910 409 27 via via ADP cana-5910 409 28 lie lie NOUN cana-5910 409 29 symmetry symmetry NOUN cana-5910 409 30 reduction reduction NOUN cana-5910 409 31 , , PUNCT cana-5910 409 32 is be AUX cana-5910 409 33 : : PUNCT cana-5910 409 34 𝑒(𝑑 𝑒(𝑑 NOUN cana-5910 409 35 , , PUNCT cana-5910 409 36 π‘₯ π‘₯ NOUN cana-5910 409 37 , , PUNCT cana-5910 409 38 𝑦 𝑦 NOUN cana-5910 409 39 ) ) PUNCT cana-5910 409 40 = = SYM cana-5910 409 41 𝛴{𝑛=1 𝛴{𝑛=1 NOUN cana-5910 409 42 } } PUNCT cana-5910 409 43 ∞ ∞ NUM cana-5910 409 44 𝛴{π‘š=1 𝛴{π‘š=1 PROPN cana-5910 409 45 } } PUNCT cana-5910 409 46 ∞ ∞ PROPN cana-5910 410 1 π΄π‘›π‘šπ‘’π‘₯𝑝 π΄π‘›π‘šπ‘’π‘₯𝑝 INTJ cana-5910 410 2 ( ( PUNCT cana-5910 410 3 βˆ’2πœ‹ βˆ’2πœ‹ NOUN cana-5910 410 4 βˆšπ‘›2 βˆšπ‘›2 PUNCT cana-5910 411 1 + + CCONJ cana-5910 412 1 π‘š2𝑑 π‘š2𝑑 PRON cana-5910 412 2 𝑙 𝑙 X cana-5910 412 3 ) ) PUNCT cana-5910 412 4 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 412 5 ( ( PUNCT cana-5910 412 6 π‘›πœ‹π‘₯ π‘›πœ‹π‘₯ NOUN cana-5910 412 7 𝑙 𝑙 NOUN cana-5910 412 8 ) ) PUNCT cana-5910 412 9 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5910 412 10 ( ( PUNCT cana-5910 412 11 π‘šπœ‹π‘¦ π‘šπœ‹π‘¦ NOUN cana-5910 412 12 𝑙 𝑙 NOUN cana-5910 412 13 ) ) PUNCT cana-5910 412 14 , , PUNCT cana-5910 412 15 where where SCONJ cana-5910 412 16 π΄π‘›π‘š π΄π‘›π‘š PROPN cana-5910 412 17 are be AUX cana-5910 412 18 coefficients coefficient NOUN cana-5910 412 19 determined determine VERB cana-5910 412 20 by by ADP cana-5910 412 21 initial initial ADJ cana-5910 412 22 conditions condition NOUN cana-5910 412 23 𝑒(0 𝑒(0 PROPN cana-5910 412 24 , , PUNCT cana-5910 412 25 π‘₯ π‘₯ PROPN cana-5910 412 26 , , PUNCT cana-5910 412 27 𝑦 𝑦 NOUN cana-5910 412 28 ) ) PUNCT cana-5910 412 29 = = SYM cana-5910 412 30 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-5910 412 31 , , PUNCT cana-5910 412 32 𝑦 𝑦 NOUN cana-5910 412 33 ) ) PUNCT cana-5910 412 34 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-5910 412 35 𝑒𝑑(0,π‘₯,𝑦 𝑒𝑑(0,π‘₯,𝑦 PROPN cana-5910 412 36 ) ) PUNCT cana-5910 412 37 = = SYM cana-5910 413 1 𝑔(π‘₯ 𝑔(π‘₯ PROPN cana-5910 413 2 , , PUNCT cana-5910 413 3 𝑦 𝑦 NOUN cana-5910 413 4 ) ) PUNCT cana-5910 413 5 . . PUNCT cana-5910 414 1 without without ADP cana-5910 414 2 specified specify VERB cana-5910 414 3 initial initial ADJ cana-5910 414 4 conditions condition NOUN cana-5910 414 5 , , PUNCT cana-5910 414 6 π΄π‘›π‘š π΄π‘›π‘š PROPN cana-5910 414 7 remain remain VERB cana-5910 414 8 arbitrary arbitrary ADJ cana-5910 414 9 constants constant NOUN cana-5910 414 10 . . PUNCT cana-5910 415 1 5 5 X cana-5910 415 2 . . X cana-5910 415 3 conclusion conclusion VERB cana-5910 415 4 the the DET cana-5910 415 5 application application NOUN cana-5910 415 6 of of ADP cana-5910 415 7 lie lie NOUN cana-5910 415 8 symmetry symmetry NOUN cana-5910 415 9 analysis analysis NOUN cana-5910 415 10 to to ADP cana-5910 415 11 the the DET cana-5910 415 12 two two NUM cana-5910 415 13 - - PUNCT cana-5910 415 14 dimensional dimensional ADJ cana-5910 415 15 heat heat NOUN cana-5910 415 16 equation equation NOUN cana-5910 415 17 , , PUNCT cana-5910 415 18 πœ•π‘’ πœ•π‘’ NOUN cana-5910 415 19 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 415 20 = = NOUN cana-5910 415 21 2 2 NUM cana-5910 415 22 ( ( PUNCT cana-5910 415 23 πœ•2𝑒 πœ•2𝑒 NUM cana-5910 415 24 πœ•π‘₯2 πœ•π‘₯2 NOUN cana-5910 415 25 + + CCONJ cana-5910 415 26 πœ•2𝑒 πœ•2𝑒 PROPN cana-5910 415 27 πœ•π‘¦2 πœ•π‘¦2 NUM cana-5910 415 28 ) ) PUNCT cana-5910 415 29 has have AUX cana-5910 415 30 demonstrated demonstrate VERB cana-5910 415 31 the the DET cana-5910 415 32 power power NOUN cana-5910 415 33 of of ADP cana-5910 415 34 symmetry symmetry NOUN cana-5910 415 35 methods method NOUN cana-5910 415 36 in in ADP cana-5910 415 37 simplifying simplify VERB cana-5910 415 38 complex complex ADJ cana-5910 415 39 partial partial ADJ cana-5910 415 40 differential differential NOUN cana-5910 415 41 equations equation NOUN cana-5910 415 42 ( ( PUNCT cana-5910 415 43 pdes pde NOUN cana-5910 415 44 ) ) PUNCT cana-5910 415 45 and and CCONJ cana-5910 415 46 uncovering uncover VERB cana-5910 415 47 physically physically ADV cana-5910 415 48 meaningful meaningful ADJ cana-5910 415 49 solutions solution NOUN cana-5910 415 50 . . PUNCT cana-5910 416 1 by by ADP cana-5910 416 2 deriving derive VERB cana-5910 416 3 the the DET cana-5910 416 4 lie lie NOUN cana-5910 416 5 point point NOUN cana-5910 416 6 symmetries symmetry NOUN cana-5910 416 7 , , PUNCT cana-5910 416 8 we we PRON cana-5910 416 9 identified identify VERB cana-5910 416 10 a a DET cana-5910 416 11 comprehensive comprehensive ADJ cana-5910 416 12 symmetry symmetry NOUN cana-5910 416 13 algebra algebra NOUN cana-5910 416 14 , , PUNCT cana-5910 416 15 including include VERB cana-5910 416 16 spatial spatial ADJ cana-5910 416 17 and and CCONJ cana-5910 416 18 temporal temporal ADJ cana-5910 416 19 translations translation NOUN cana-5910 416 20 ( ( PUNCT cana-5910 416 21 𝑉1 𝑉1 PROPN cana-5910 416 22 , , PUNCT cana-5910 416 23 𝑉2 𝑉2 NOUN cana-5910 416 24 , , PUNCT cana-5910 416 25 𝑉3 𝑉3 NOUN cana-5910 416 26 ) ) PUNCT cana-5910 416 27 , , PUNCT cana-5910 416 28 scaling scale VERB cana-5910 416 29 transformations transformation NOUN cana-5910 416 30 ( ( PUNCT cana-5910 416 31 𝑉4 𝑉4 NOUN cana-5910 416 32 , , PUNCT cana-5910 416 33 𝑉5 𝑉5 PROPN cana-5910 416 34 ) ) PUNCT cana-5910 416 35 , , PUNCT cana-5910 416 36 conformal conformal NOUN cana-5910 416 37 - - PUNCT cana-5910 416 38 like like ADJ cana-5910 416 39 symmetries symmetry NOUN cana-5910 416 40 ( ( PUNCT cana-5910 416 41 𝑉6 𝑉6 NOUN cana-5910 416 42 ) ) PUNCT cana-5910 416 43 , , PUNCT cana-5910 416 44 rotational rotational ADJ cana-5910 416 45 symmetry symmetry NOUN cana-5910 416 46 ( ( PUNCT cana-5910 416 47 𝑉7 𝑉7 NOUN cana-5910 416 48 ) ) PUNCT cana-5910 416 49 , , PUNCT cana-5910 416 50 and and CCONJ cana-5910 416 51 an an DET cana-5910 416 52 infinite infinite ADJ cana-5910 416 53 - - PUNCT cana-5910 416 54 dimensional dimensional ADJ cana-5910 416 55 symmetry symmetry NOUN cana-5910 416 56 ( ( PUNCT cana-5910 416 57 𝑉𝑏 𝑉𝑏 PROPN cana-5910 416 58 ) ) PUNCT cana-5910 416 59 associated associate VERB cana-5910 416 60 with with ADP cana-5910 416 61 solutions solution NOUN cana-5910 416 62 of of ADP cana-5910 416 63 the the DET cana-5910 416 64 heat heat NOUN cana-5910 416 65 equation equation NOUN cana-5910 416 66 itself itself PRON cana-5910 416 67 . . PUNCT cana-5910 417 1 these these DET cana-5910 417 2 symmetries symmetry NOUN cana-5910 417 3 provided provide VERB cana-5910 417 4 a a DET cana-5910 417 5 systematic systematic ADJ cana-5910 417 6 framework framework NOUN cana-5910 417 7 for for ADP cana-5910 417 8 reducing reduce VERB cana-5910 417 9 the the DET cana-5910 417 10 pde pde NOUN cana-5910 417 11 to to ADP cana-5910 417 12 ordinary ordinary ADJ cana-5910 417 13 differential differential ADJ cana-5910 417 14 equations equation NOUN cana-5910 417 15 ( ( PUNCT cana-5910 417 16 odes ode NOUN cana-5910 417 17 ) ) PUNCT cana-5910 417 18 or or CCONJ cana-5910 417 19 simpler simple ADJ cana-5910 417 20 pdes pde NOUN cana-5910 417 21 , , PUNCT cana-5910 417 22 enabling enable VERB cana-5910 417 23 the the DET cana-5910 417 24 construction construction NOUN cana-5910 417 25 of of ADP cana-5910 417 26 exact exact ADJ cana-5910 417 27 similarity similarity NOUN cana-5910 417 28 solutions solution NOUN cana-5910 417 29 . . PUNCT cana-5910 418 1 in in ADP cana-5910 418 2 particular particular ADJ cana-5910 418 3 , , PUNCT cana-5910 418 4 the the DET cana-5910 418 5 scaling scaling ADJ cana-5910 418 6 symmetry symmetry NOUN cana-5910 418 7 𝑉5 𝑉5 PROPN cana-5910 418 8 = = SYM cana-5910 418 9 π‘₯ π‘₯ PROPN cana-5910 418 10 πœ• πœ• NOUN cana-5910 418 11 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 418 12 + + CCONJ cana-5910 418 13 𝑦 𝑦 NOUN cana-5910 418 14 πœ• πœ• NOUN cana-5910 418 15 πœ•π‘¦ πœ•π‘¦ NOUN cana-5910 419 1 + + NUM cana-5910 419 2 2𝑑 2𝑑 NOUN cana-5910 419 3 πœ• πœ• PROPN cana-5910 419 4 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 419 5 was be AUX cana-5910 419 6 used use VERB cana-5910 419 7 to to PART cana-5910 419 8 reduce reduce VERB cana-5910 419 9 the the DET cana-5910 419 10 heat heat NOUN cana-5910 419 11 equation equation NOUN cana-5910 419 12 to to ADP cana-5910 419 13 an an DET cana-5910 419 14 ode ode NOUN cana-5910 419 15 by by ADP cana-5910 419 16 introducing introduce VERB cana-5910 419 17 the the DET cana-5910 419 18 similarity similarity NOUN cana-5910 419 19 variables variable NOUN cana-5910 419 20 πœ‰ πœ‰ PART cana-5910 419 21 = = VERB cana-5910 419 22 π‘₯ π‘₯ PROPN cana-5910 419 23 𝑑 𝑑 NOUN cana-5910 419 24 1 1 NUM cana-5910 419 25 2 2 NUM cana-5910 419 26 and and CCONJ cana-5910 419 27 πœ‚ πœ‚ NOUN cana-5910 419 28 = = SYM cana-5910 419 29 𝑦 𝑦 SYM cana-5910 419 30 𝑑 𝑑 NOUN cana-5910 419 31 1 1 NUM cana-5910 419 32 2 2 NUM cana-5910 419 33 . . PUNCT cana-5910 420 1 assuming assume VERB cana-5910 420 2 radial radial ADJ cana-5910 420 3 symmetry symmetry NOUN cana-5910 420 4 , , PUNCT cana-5910 420 5 the the DET cana-5910 420 6 pde pde NOUN cana-5910 420 7 was be AUX cana-5910 420 8 transformed transform VERB cana-5910 420 9 into into ADP cana-5910 420 10 an an DET cana-5910 420 11 ode ode NOUN cana-5910 420 12 in in ADP cana-5910 420 13 the the DET cana-5910 420 14 radial radial ADJ cana-5910 420 15 variable variable NOUN cana-5910 420 16 π‘Ÿ π‘Ÿ NOUN cana-5910 420 17 = = SYM cana-5910 420 18 ( ( PUNCT cana-5910 420 19 ( ( PUNCT cana-5910 420 20 π‘₯2 π‘₯2 NOUN cana-5910 420 21 + + CCONJ cana-5910 420 22 𝑦2 𝑦2 NOUN cana-5910 420 23 ) ) PUNCT cana-5910 420 24 𝑑 𝑑 PROPN cana-5910 420 25 ) ) PUNCT cana-5910 420 26 1 1 NUM cana-5910 420 27 2 2 NUM cana-5910 420 28 , , PUNCT cana-5910 420 29 which which PRON cana-5910 420 30 yielded yield VERB cana-5910 420 31 the the DET cana-5910 420 32 fundamental fundamental ADJ cana-5910 420 33 solution solution NOUN cana-5910 420 34 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 420 35 , , PUNCT cana-5910 420 36 𝑦 𝑦 NOUN cana-5910 420 37 , , PUNCT cana-5910 420 38 𝑑 𝑑 NOUN cana-5910 420 39 ) ) PUNCT cana-5910 420 40 = = PUNCT cana-5910 420 41 ( ( PUNCT cana-5910 420 42 1 1 NUM cana-5910 420 43 8 8 NUM cana-5910 420 44 𝑝𝑖 𝑝𝑖 NOUN cana-5910 420 45 𝑑 𝑑 PROPN cana-5910 420 46 ) ) PUNCT cana-5910 420 47 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 420 48 ( ( PUNCT cana-5910 420 49 π‘₯2 π‘₯2 NOUN cana-5910 420 50 + + CCONJ cana-5910 420 51 𝑦2 𝑦2 NOUN cana-5910 420 52 ) ) PUNCT cana-5910 420 53 8 8 NUM cana-5910 420 54 𝑑 𝑑 NOUN cana-5910 420 55 . . PUNCT cana-5910 421 1 this this DET cana-5910 421 2 solution solution NOUN cana-5910 421 3 , , PUNCT cana-5910 421 4 tailored tailor VERB cana-5910 421 5 to to ADP cana-5910 421 6 the the DET cana-5910 421 7 thermal thermal ADJ cana-5910 421 8 diffusivity diffusivity NOUN cana-5910 421 9 𝛼 𝛼 NOUN cana-5910 421 10 = = SYM cana-5910 421 11 2 2 NUM cana-5910 421 12 , , PUNCT cana-5910 421 13 represents represent VERB cana-5910 421 14 the the DET cana-5910 421 15 diffusion diffusion NOUN cana-5910 421 16 of of ADP cana-5910 421 17 heat heat NOUN cana-5910 421 18 from from ADP cana-5910 421 19 an an DET cana-5910 421 20 instantaneous instantaneous ADJ cana-5910 421 21 point point NOUN cana-5910 421 22 source source NOUN cana-5910 421 23 at at ADP cana-5910 421 24 ( ( PUNCT cana-5910 421 25 π‘₯ π‘₯ NOUN cana-5910 421 26 , , PUNCT cana-5910 421 27 𝑦 𝑦 NOUN cana-5910 421 28 ) ) PUNCT cana-5910 421 29 = = SYM cana-5910 421 30 ( ( PUNCT cana-5910 421 31 0 0 NUM cana-5910 421 32 , , PUNCT cana-5910 421 33 0 0 NUM cana-5910 421 34 ) ) PUNCT cana-5910 421 35 π‘Žπ‘‘ π‘Žπ‘‘ ADP cana-5910 421 36 𝑑 𝑑 PROPN cana-5910 421 37 = = NOUN cana-5910 421 38 0 0 NUM cana-5910 421 39 , , PUNCT cana-5910 421 40 with with ADP cana-5910 421 41 the the DET cana-5910 421 42 gaussian gaussian ADJ cana-5910 421 43 profile profile NOUN cana-5910 421 44 capturing capture VERB cana-5910 421 45 the the DET cana-5910 421 46 radial radial ADJ cana-5910 421 47 spreading spreading NOUN cana-5910 421 48 and and CCONJ cana-5910 421 49 decay decay NOUN cana-5910 421 50 of of ADP cana-5910 421 51 temperature temperature NOUN cana-5910 421 52 over over ADP cana-5910 421 53 time time NOUN cana-5910 421 54 . . PUNCT cana-5910 422 1 the the DET cana-5910 422 2 solution solution NOUN cana-5910 422 3 was be AUX cana-5910 422 4 rigorously rigorously ADV cana-5910 422 5 verified verify VERB cana-5910 422 6 to to PART cana-5910 422 7 satisfy satisfy VERB cana-5910 422 8 the the DET cana-5910 422 9 heat heat NOUN cana-5910 422 10 equation equation NOUN cana-5910 422 11 and and CCONJ cana-5910 422 12 the the DET cana-5910 422 13 initial initial ADJ cana-5910 422 14 condition condition NOUN cana-5910 422 15 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 422 16 , , PUNCT cana-5910 422 17 𝑦 𝑦 NOUN cana-5910 422 18 , , PUNCT cana-5910 422 19 0 0 NUM cana-5910 422 20 ) ) PUNCT cana-5910 422 21 = = SYM cana-5910 423 1 𝛿(π‘₯ 𝛿(π‘₯ NOUN cana-5910 423 2 , , PUNCT cana-5910 423 3 𝑦 𝑦 NOUN cana-5910 423 4 ) ) PUNCT cana-5910 423 5 , , PUNCT cana-5910 423 6 confirming confirm VERB cana-5910 423 7 its its PRON cana-5910 423 8 mathematical mathematical ADJ cana-5910 423 9 and and CCONJ cana-5910 423 10 physical physical ADJ cana-5910 423 11 validity validity NOUN cana-5910 423 12 . . PUNCT cana-5910 424 1 similarly similarly ADV cana-5910 424 2 , , PUNCT cana-5910 424 3 the the DET cana-5910 424 4 conformal conformal NOUN cana-5910 424 5 - - PUNCT cana-5910 424 6 like like ADJ cana-5910 424 7 symmetry symmetry NOUN cana-5910 424 8 𝑉6 𝑉6 NOUN cana-5910 424 9 = = PRON cana-5910 424 10 ( ( PUNCT cana-5910 424 11 π‘₯ π‘₯ PROPN cana-5910 424 12 4 4 X cana-5910 424 13 ) ) PUNCT cana-5910 424 14 πœ• πœ• PROPN cana-5910 424 15 πœ•π‘₯ πœ•π‘₯ NOUN cana-5910 424 16 + + CCONJ cana-5910 424 17 ( ( PUNCT cana-5910 424 18 𝑦 𝑦 NOUN cana-5910 424 19 4 4 NUM cana-5910 424 20 ) ) PUNCT cana-5910 424 21 πœ• πœ• NOUN cana-5910 424 22 πœ•π‘¦ πœ•π‘¦ PROPN cana-5910 424 23 + + CCONJ cana-5910 424 24 t t PROPN cana-5910 424 25 πœ• πœ• PROPN cana-5910 424 26 πœ•π‘‘ πœ•π‘‘ NOUN cana-5910 424 27 βˆ’ βˆ’ PROPN cana-5910 425 1 ( ( PUNCT cana-5910 425 2 ( ( PUNCT cana-5910 425 3 π‘₯2 π‘₯2 NOUN cana-5910 425 4 + + CCONJ cana-5910 425 5 𝑦2 𝑦2 NOUN cana-5910 425 6 ) ) PUNCT cana-5910 425 7 8 8 NUM cana-5910 425 8 ) ) PUNCT cana-5910 425 9 𝑒 𝑒 PROPN cana-5910 425 10 πœ• πœ• PROPN cana-5910 425 11 πœ•π‘’ πœ•π‘’ NOUN cana-5910 425 12 directly directly ADV cana-5910 425 13 suggested suggest VERB cana-5910 425 14 a a DET cana-5910 425 15 gaussian gaussian ADJ cana-5910 425 16 form form NOUN cana-5910 425 17 , , PUNCT cana-5910 425 18 which which PRON cana-5910 425 19 was be AUX cana-5910 425 20 adjusted adjust VERB cana-5910 425 21 to to PART cana-5910 425 22 align align VERB cana-5910 425 23 with with ADP cana-5910 425 24 the the DET cana-5910 425 25 fundamental fundamental ADJ cana-5910 425 26 solution solution NOUN cana-5910 425 27 . . PUNCT cana-5910 426 1 the the DET cana-5910 426 2 consistency consistency NOUN cana-5910 426 3 of of ADP cana-5910 426 4 these these DET cana-5910 426 5 results result NOUN cana-5910 426 6 across across ADP cana-5910 426 7 different different ADJ cana-5910 426 8 symmetries symmetry NOUN cana-5910 426 9 underscores underscore VERB cana-5910 426 10 the the DET cana-5910 426 11 robustness robustness NOUN cana-5910 426 12 of of ADP cana-5910 426 13 lie lie NOUN cana-5910 426 14 symmetry symmetry NOUN cana-5910 426 15 methods method NOUN cana-5910 426 16 in in ADP cana-5910 426 17 identifying identify VERB cana-5910 426 18 key key ADJ cana-5910 426 19 solutions solution NOUN cana-5910 426 20 , , PUNCT cana-5910 426 21 such such ADJ cana-5910 426 22 as as ADP cana-5910 426 23 the the DET cana-5910 426 24 fundamental fundamental ADJ cana-5910 426 25 solution solution NOUN cana-5910 426 26 , , PUNCT cana-5910 426 27 which which PRON cana-5910 426 28 is be AUX cana-5910 426 29 central central ADJ cana-5910 426 30 to to ADP cana-5910 426 31 understanding understand VERB cana-5910 426 32 heat heat NOUN cana-5910 426 33 conduction conduction NOUN cana-5910 426 34 in in ADP cana-5910 426 35 two two NUM cana-5910 426 36 - - PUNCT cana-5910 426 37 dimensional dimensional ADJ cana-5910 426 38 systems system NOUN cana-5910 426 39 . . PUNCT cana-5910 427 1 these these DET cana-5910 427 2 solutions solution NOUN cana-5910 427 3 have have VERB cana-5910 427 4 practical practical ADJ cana-5910 427 5 applications application NOUN cana-5910 427 6 in in ADP cana-5910 427 7 fields field NOUN cana-5910 427 8 such such ADJ cana-5910 427 9 as as ADP cana-5910 427 10 thermal thermal ADJ cana-5910 427 11 engineering engineering NOUN cana-5910 427 12 , , PUNCT cana-5910 427 13 materials material NOUN cana-5910 427 14 science science NOUN cana-5910 427 15 , , PUNCT cana-5910 427 16 and and CCONJ cana-5910 427 17 environmental environmental ADJ cana-5910 427 18 modeling modeling NOUN cana-5910 427 19 , , PUNCT cana-5910 427 20 where where SCONJ cana-5910 427 21 they they PRON cana-5910 427 22 describe describe VERB cana-5910 427 23 the the DET cana-5910 427 24 evolution evolution NOUN cana-5910 427 25 of of ADP cana-5910 427 26 temperature temperature NOUN cana-5910 427 27 distributions distribution NOUN cana-5910 427 28 in in ADP cana-5910 427 29 planar planar ADJ cana-5910 427 30 media medium NOUN cana-5910 427 31 . . PUNCT cana-5910 428 1 the the DET cana-5910 428 2 lie lie NOUN cana-5910 428 3 symmetry symmetry NOUN cana-5910 428 4 approach approach NOUN cana-5910 428 5 excels excel VERB cana-5910 428 6 in in ADP cana-5910 428 7 its its PRON cana-5910 428 8 ability ability NOUN cana-5910 428 9 to to PART cana-5910 428 10 exploit exploit VERB cana-5910 428 11 the the DET cana-5910 428 12 inherent inherent ADJ cana-5910 428 13 symmetries symmetry NOUN cana-5910 428 14 of of ADP cana-5910 428 15 a a DET cana-5910 428 16 pde pde NOUN cana-5910 428 17 to to PART cana-5910 428 18 reduce reduce VERB cana-5910 428 19 its its PRON cana-5910 428 20 complexity complexity NOUN cana-5910 428 21 while while SCONJ cana-5910 428 22 preserving preserve VERB cana-5910 428 23 1127 1127 NUM cana-5910 428 24 https://internationalpubls.com https://internationalpubls.com X cana-5910 428 25 essential essential ADJ cana-5910 428 26 physical physical ADJ cana-5910 428 27 properties property NOUN cana-5910 428 28 . . PUNCT cana-5910 429 1 the the DET cana-5910 429 2 methodology methodology NOUN cana-5910 429 3 not not PART cana-5910 429 4 only only ADV cana-5910 429 5 provides provide VERB cana-5910 429 6 exact exact ADJ cana-5910 429 7 solutions solution NOUN cana-5910 429 8 but but CCONJ cana-5910 429 9 also also ADV cana-5910 429 10 deepens deepen VERB cana-5910 429 11 our our PRON cana-5910 429 12 understanding understanding NOUN cana-5910 429 13 of of ADP cana-5910 429 14 the the DET cana-5910 429 15 mathematical mathematical ADJ cana-5910 429 16 structure structure NOUN cana-5910 429 17 underlying underlie VERB cana-5910 429 18 physical physical ADJ cana-5910 429 19 phenomena phenomenon NOUN cana-5910 429 20 . . PUNCT cana-5910 430 1 the the DET cana-5910 430 2 derived derive VERB cana-5910 430 3 fundamental fundamental ADJ cana-5910 430 4 solution solution NOUN cana-5910 430 5 , , PUNCT cana-5910 430 6 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5910 430 7 , , PUNCT cana-5910 430 8 𝑦 𝑦 NOUN cana-5910 430 9 , , PUNCT cana-5910 430 10 𝑑 𝑑 NOUN cana-5910 430 11 ) ) PUNCT cana-5910 430 12 = = PUNCT cana-5910 430 13 ( ( PUNCT cana-5910 430 14 1 1 NUM cana-5910 430 15 8 8 NUM cana-5910 430 16 𝑝𝑖 𝑝𝑖 NOUN cana-5910 430 17 𝑑 𝑑 PROPN cana-5910 430 18 ) ) PUNCT cana-5910 430 19 π‘’βˆ’ π‘’βˆ’ NOUN cana-5910 430 20 ( ( PUNCT cana-5910 430 21 π‘₯2 π‘₯2 NOUN cana-5910 430 22 + + CCONJ cana-5910 430 23 𝑦2 𝑦2 NOUN cana-5910 430 24 ) ) PUNCT cana-5910 430 25 8 8 NUM cana-5910 430 26 𝑑 𝑑 NOUN cana-5910 430 27 , , PUNCT cana-5910 430 28 is be AUX cana-5910 430 29 particularly particularly ADV cana-5910 430 30 significant significant ADJ cana-5910 430 31 , , PUNCT cana-5910 430 32 as as SCONJ cana-5910 430 33 it it PRON cana-5910 430 34 serves serve VERB cana-5910 430 35 as as ADP cana-5910 430 36 a a DET cana-5910 430 37 building building NOUN cana-5910 430 38 block block NOUN cana-5910 430 39 for for ADP cana-5910 430 40 solving solve VERB cana-5910 430 41 more more ADJ cana-5910 430 42 complex complex ADJ cana-5910 430 43 heat heat NOUN cana-5910 430 44 conduction conduction NOUN cana-5910 430 45 problems problem NOUN cana-5910 430 46 via via ADP cana-5910 430 47 convolution convolution NOUN cana-5910 430 48 with with ADP cana-5910 430 49 arbitrary arbitrary ADJ cana-5910 430 50 initial initial ADJ cana-5910 430 51 conditions condition NOUN cana-5910 430 52 . . PUNCT cana-5910 431 1 future future ADJ cana-5910 431 2 research research NOUN cana-5910 431 3 could could AUX cana-5910 431 4 explore explore VERB cana-5910 431 5 additional additional ADJ cana-5910 431 6 symmetries symmetry NOUN cana-5910 431 7 , , PUNCT cana-5910 431 8 such such ADJ cana-5910 431 9 as as ADP cana-5910 431 10 𝑉7 𝑉7 NOUN cana-5910 431 11 or or CCONJ cana-5910 431 12 combinations combination NOUN cana-5910 431 13 of of ADP cana-5910 431 14 symmetries symmetry NOUN cana-5910 431 15 , , PUNCT cana-5910 431 16 to to PART cana-5910 431 17 derive derive VERB cana-5910 431 18 other other ADJ cana-5910 431 19 classes class NOUN cana-5910 431 20 of of ADP cana-5910 431 21 solutions solution NOUN cana-5910 431 22 , , PUNCT cana-5910 431 23 including include VERB cana-5910 431 24 those those PRON cana-5910 431 25 for for ADP cana-5910 431 26 non non ADJ cana-5910 431 27 - - ADJ cana-5910 431 28 homogeneous homogeneous ADJ cana-5910 431 29 or or CCONJ cana-5910 431 30 bounded bounded ADJ cana-5910 431 31 domains domain NOUN cana-5910 431 32 . . PUNCT cana-5910 432 1 extending extend VERB cana-5910 432 2 the the DET cana-5910 432 3 analysis analysis NOUN cana-5910 432 4 to to ADP cana-5910 432 5 related relate VERB cana-5910 432 6 equations equation NOUN cana-5910 432 7 , , PUNCT cana-5910 432 8 such such ADJ cana-5910 432 9 as as ADP cana-5910 432 10 the the DET cana-5910 432 11 wave wave NOUN cana-5910 432 12 equation equation NOUN cana-5910 432 13 or or CCONJ cana-5910 432 14 nonlinear nonlinear ADJ cana-5910 432 15 heat heat NOUN cana-5910 432 16 equations equation NOUN cana-5910 432 17 , , PUNCT cana-5910 432 18 could could AUX cana-5910 432 19 further far ADV cana-5910 432 20 illuminate illuminate VERB cana-5910 432 21 the the DET cana-5910 432 22 interplay interplay NOUN cana-5910 432 23 between between ADP cana-5910 432 24 symmetry symmetry NOUN cana-5910 432 25 and and CCONJ cana-5910 432 26 physical physical ADJ cana-5910 432 27 behavior behavior NOUN cana-5910 432 28 . . PUNCT cana-5910 433 1 additionally additionally ADV cana-5910 433 2 , , PUNCT cana-5910 433 3 incorporating incorporate VERB cana-5910 433 4 numerical numerical ADJ cana-5910 433 5 methods method NOUN cana-5910 433 6 to to PART cana-5910 433 7 complement complement VERB cana-5910 433 8 analytical analytical ADJ cana-5910 433 9 solutions solution NOUN cana-5910 433 10 could could AUX cana-5910 433 11 enhance enhance VERB cana-5910 433 12 the the DET cana-5910 433 13 applicability applicability NOUN cana-5910 433 14 of of ADP cana-5910 433 15 these these DET cana-5910 433 16 results result NOUN cana-5910 433 17 to to ADP cana-5910 433 18 real real ADJ cana-5910 433 19 - - PUNCT cana-5910 433 20 world world NOUN cana-5910 433 21 scenarios scenario NOUN cana-5910 433 22 with with ADP cana-5910 433 23 complex complex ADJ cana-5910 433 24 boundary boundary ADJ cana-5910 433 25 conditions condition NOUN cana-5910 433 26 . . PUNCT cana-5910 434 1 in in ADP cana-5910 434 2 conclusion conclusion NOUN cana-5910 434 3 , , PUNCT cana-5910 434 4 lie lie NOUN cana-5910 434 5 symmetry symmetry NOUN cana-5910 434 6 analysis analysis NOUN cana-5910 434 7 has have AUX cana-5910 434 8 successfully successfully ADV cana-5910 434 9 elucidated elucidate VERB cana-5910 434 10 the the DET cana-5910 434 11 fundamental fundamental ADJ cana-5910 434 12 solution solution NOUN cana-5910 434 13 to to ADP cana-5910 434 14 the the DET cana-5910 434 15 two two NUM cana-5910 434 16 - - PUNCT cana-5910 434 17 dimensional dimensional ADJ cana-5910 434 18 heat heat NOUN cana-5910 434 19 equation equation NOUN cana-5910 434 20 with with ADP cana-5910 434 21 𝛼 𝛼 NOUN cana-5910 434 22 = = SYM cana-5910 434 23 2 2 NUM cana-5910 434 24 , , PUNCT cana-5910 434 25 offering offer VERB cana-5910 434 26 both both PRON cana-5910 434 27 mathematical mathematical ADJ cana-5910 434 28 elegance elegance NOUN cana-5910 434 29 and and CCONJ cana-5910 434 30 practical practical ADJ cana-5910 434 31 utility utility NOUN cana-5910 434 32 . . PUNCT cana-5910 435 1 the the DET cana-5910 435 2 approach approach NOUN cana-5910 435 3 exemplifies exemplify VERB cana-5910 435 4 how how SCONJ cana-5910 435 5 symmetry symmetry NOUN cana-5910 435 6 can can AUX cana-5910 435 7 transform transform VERB cana-5910 435 8 complex complex ADJ cana-5910 435 9 problems problem NOUN cana-5910 435 10 into into ADP cana-5910 435 11 tractable tractable ADJ cana-5910 435 12 forms form NOUN cana-5910 435 13 , , PUNCT cana-5910 435 14 providing provide VERB cana-5910 435 15 a a DET cana-5910 435 16 powerful powerful ADJ cana-5910 435 17 tool tool NOUN cana-5910 435 18 for for ADP cana-5910 435 19 researchers researcher NOUN cana-5910 435 20 and and CCONJ cana-5910 435 21 engineers engineer NOUN cana-5910 435 22 tackling tackle VERB cana-5910 435 23 problems problem NOUN cana-5910 435 24 in in ADP cana-5910 435 25 heat heat NOUN cana-5910 435 26 transfer transfer NOUN cana-5910 435 27 and and CCONJ cana-5910 435 28 beyond beyond ADP cana-5910 435 29 . . NOUN cana-5910 436 1 6 6 NUM cana-5910 436 2 . . NUM cana-5910 436 3 references reference NOUN cana-5910 436 4 1 1 NUM cana-5910 436 5 . . PUNCT cana-5910 436 6 carslaw carslaw PROPN cana-5910 436 7 , , PUNCT cana-5910 436 8 h. h. PROPN cana-5910 436 9 s. s. PROPN cana-5910 436 10 , , PUNCT cana-5910 436 11 & & CCONJ cana-5910 436 12 jaeger jaeger PROPN cana-5910 436 13 , , PUNCT cana-5910 436 14 j. j. PROPN cana-5910 436 15 c. c. PROPN cana-5910 436 16 ( ( PUNCT cana-5910 436 17 1959 1959 NUM cana-5910 436 18 ) ) PUNCT cana-5910 436 19 . . PUNCT cana-5910 437 1 conduction conduction NOUN cana-5910 437 2 of of ADP cana-5910 437 3 heat heat NOUN cana-5910 437 4 in in ADP cana-5910 437 5 solids solid NOUN cana-5910 437 6 ( ( PUNCT cana-5910 437 7 2nd 2nd NOUN cana-5910 437 8 ed ed NOUN cana-5910 437 9 . . PUNCT cana-5910 437 10 ) ) PUNCT cana-5910 437 11 . . PUNCT cana-5910 438 1 oxford oxford PROPN cana-5910 438 2 university university PROPN cana-5910 438 3 press press NOUN cana-5910 438 4 . . PUNCT cana-5910 439 1 a a DET cana-5910 439 2 classic classic ADJ cana-5910 439 3 reference reference NOUN cana-5910 439 4 on on ADP cana-5910 439 5 heat heat NOUN cana-5910 439 6 conduction conduction NOUN cana-5910 439 7 , , PUNCT cana-5910 439 8 offering offer VERB cana-5910 439 9 analytical analytical ADJ cana-5910 439 10 solutions solution NOUN cana-5910 439 11 to to ADP cana-5910 439 12 the the DET cana-5910 439 13 heat heat NOUN cana-5910 439 14 equation equation NOUN cana-5910 439 15 in in ADP cana-5910 439 16 various various ADJ cana-5910 439 17 dimensions dimension NOUN cana-5910 439 18 , , PUNCT cana-5910 439 19 relevant relevant ADJ cana-5910 439 20 to to ADP cana-5910 439 21 the the DET cana-5910 439 22 fundamental fundamental ADJ cana-5910 439 23 solutions solution NOUN cana-5910 439 24 derived derive VERB cana-5910 439 25 . . PUNCT cana-5910 440 1 2 2 X cana-5910 440 2 . . X cana-5910 440 3 widder widder PROPN cana-5910 440 4 , , PUNCT cana-5910 440 5 d. d. PROPN cana-5910 440 6 v. v. PROPN cana-5910 440 7 ( ( PUNCT cana-5910 440 8 1975 1975 NUM cana-5910 440 9 ) ) PUNCT cana-5910 440 10 . . PUNCT cana-5910 441 1 the the DET cana-5910 441 2 heat heat NOUN cana-5910 441 3 equation equation NOUN cana-5910 441 4 . . PUNCT cana-5910 442 1 academic academic ADJ cana-5910 442 2 press press NOUN cana-5910 442 3 . . PUNCT cana-5910 443 1 focuses focus VERB cana-5910 443 2 on on ADP cana-5910 443 3 the the DET cana-5910 443 4 mathematical mathematical ADJ cana-5910 443 5 theory theory NOUN cana-5910 443 6 of of ADP cana-5910 443 7 the the DET cana-5910 443 8 heat heat NOUN cana-5910 443 9 equation equation NOUN cana-5910 443 10 , , PUNCT cana-5910 443 11 including include VERB cana-5910 443 12 fundamental fundamental ADJ cana-5910 443 13 solutions solution NOUN cana-5910 443 14 and and CCONJ cana-5910 443 15 their their PRON cana-5910 443 16 physical physical ADJ cana-5910 443 17 interpretations interpretation NOUN cana-5910 443 18 , , PUNCT cana-5910 443 19 relevant relevant ADJ cana-5910 443 20 to to ADP cana-5910 443 21 the the DET cana-5910 443 22 gaussian gaussian ADJ cana-5910 443 23 kernel kernel NOUN cana-5910 443 24 . . PUNCT cana-5910 444 1 3 3 X cana-5910 444 2 . . X cana-5910 444 3 crank crank PROPN cana-5910 444 4 , , PUNCT cana-5910 444 5 j. j. PROPN cana-5910 444 6 ( ( PUNCT cana-5910 444 7 1975 1975 NUM cana-5910 444 8 ) ) PUNCT cana-5910 444 9 . . PUNCT cana-5910 445 1 the the DET cana-5910 445 2 mathematics mathematic NOUN cana-5910 445 3 of of ADP cana-5910 445 4 diffusion diffusion NOUN cana-5910 445 5 ( ( PUNCT cana-5910 445 6 2nd 2nd ADJ cana-5910 445 7 ed ed NOUN cana-5910 445 8 . . PUNCT cana-5910 445 9 ) ) PUNCT cana-5910 445 10 . . PUNCT cana-5910 446 1 oxford oxford PROPN cana-5910 446 2 university university PROPN cana-5910 446 3 press press NOUN cana-5910 446 4 . . PUNCT cana-5910 447 1 provides provide VERB cana-5910 447 2 a a DET cana-5910 447 3 mathematical mathematical ADJ cana-5910 447 4 treatment treatment NOUN cana-5910 447 5 of of ADP cana-5910 447 6 diffusion diffusion NOUN cana-5910 447 7 processes process NOUN cana-5910 447 8 , , PUNCT cana-5910 447 9 including include VERB cana-5910 447 10 solutions solution NOUN cana-5910 447 11 to to ADP cana-5910 447 12 the the DET cana-5910 447 13 heat heat NOUN cana-5910 447 14 equation equation NOUN cana-5910 447 15 , , PUNCT cana-5910 447 16 complementing complement VERB cana-5910 447 17 symmetry symmetry NOUN cana-5910 447 18 - - PUNCT cana-5910 447 19 based base VERB cana-5910 447 20 approaches approach NOUN cana-5910 447 21 . . PUNCT cana-5910 448 1 4 4 X cana-5910 448 2 . . X cana-5910 448 3 bluman bluman PROPN cana-5910 448 4 , , PUNCT cana-5910 448 5 g. g. PROPN cana-5910 448 6 w. w. PROPN cana-5910 448 7 , , PUNCT cana-5910 448 8 & & CCONJ cana-5910 448 9 kumei kumei PROPN cana-5910 448 10 , , PUNCT cana-5910 448 11 s. s. PROPN cana-5910 448 12 ( ( PUNCT cana-5910 448 13 1989 1989 NUM cana-5910 448 14 ) ) PUNCT cana-5910 448 15 . . PUNCT cana-5910 449 1 symmetries symmetry NOUN cana-5910 449 2 and and CCONJ cana-5910 449 3 differential differential ADJ cana-5910 449 4 equations equation NOUN cana-5910 449 5 . . PUNCT cana-5910 450 1 springer springer NOUN cana-5910 450 2 - - PUNCT cana-5910 450 3 verlag verlag PROPN cana-5910 450 4 . . PUNCT cana-5910 451 1 a a DET cana-5910 451 2 comprehensive comprehensive ADJ cana-5910 451 3 text text NOUN cana-5910 451 4 on on ADP cana-5910 451 5 lie lie NOUN cana-5910 451 6 group group NOUN cana-5910 451 7 methods method NOUN cana-5910 451 8 , , PUNCT cana-5910 451 9 detailing detail VERB cana-5910 451 10 the the DET cana-5910 451 11 process process NOUN cana-5910 451 12 of of ADP cana-5910 451 13 finding find VERB cana-5910 451 14 symmetries symmetry NOUN cana-5910 451 15 and and CCONJ cana-5910 451 16 deriving derive VERB cana-5910 451 17 similarity similarity NOUN cana-5910 451 18 solutions solution NOUN cana-5910 451 19 for for ADP cana-5910 451 20 pdes pde NOUN cana-5910 451 21 , , PUNCT cana-5910 451 22 including include VERB cana-5910 451 23 the the DET cana-5910 451 24 heat heat NOUN cana-5910 451 25 equation equation NOUN cana-5910 451 26 . . PUNCT cana-5910 452 1 5 5 X cana-5910 452 2 . . X cana-5910 452 3 stephani stephani PROPN cana-5910 452 4 , , PUNCT cana-5910 452 5 h. h. PROPN cana-5910 452 6 ( ( PUNCT cana-5910 452 7 1989 1989 NUM cana-5910 452 8 ) ) PUNCT cana-5910 452 9 . . PUNCT cana-5910 453 1 differential differential ADJ cana-5910 453 2 equations equation NOUN cana-5910 453 3 : : PUNCT cana-5910 453 4 their their PRON cana-5910 453 5 solution solution NOUN cana-5910 453 6 using use VERB cana-5910 453 7 symmetries symmetry NOUN cana-5910 453 8 . . PUNCT cana-5910 454 1 cambridge cambridge PROPN cana-5910 454 2 university university PROPN cana-5910 454 3 press press NOUN cana-5910 454 4 . . PUNCT cana-5910 455 1 offers offer VERB cana-5910 455 2 a a DET cana-5910 455 3 clear clear ADJ cana-5910 455 4 exposition exposition NOUN cana-5910 455 5 of of ADP cana-5910 455 6 symmetry symmetry NOUN cana-5910 455 7 methods method NOUN cana-5910 455 8 for for ADP cana-5910 455 9 solving solve VERB cana-5910 455 10 differential differential ADJ cana-5910 455 11 equations equation NOUN cana-5910 455 12 , , PUNCT cana-5910 455 13 with with ADP cana-5910 455 14 applications application NOUN cana-5910 455 15 to to PART cana-5910 455 16 linear linear VERB cana-5910 455 17 pdes pde NOUN cana-5910 455 18 like like ADP cana-5910 455 19 the the DET cana-5910 455 20 heat heat NOUN cana-5910 455 21 equation equation NOUN cana-5910 455 22 . . PUNCT cana-5910 456 1 6 6 NUM cana-5910 456 2 . . X cana-5910 456 3 olver olver PROPN cana-5910 456 4 , , PUNCT cana-5910 456 5 p. p. NOUN cana-5910 456 6 j. j. PROPN cana-5910 456 7 ( ( PUNCT cana-5910 456 8 1993 1993 NUM cana-5910 456 9 ) ) PUNCT cana-5910 456 10 . . PUNCT cana-5910 457 1 applications application NOUN cana-5910 457 2 of of ADP cana-5910 457 3 lie lie NOUN cana-5910 457 4 groups group NOUN cana-5910 457 5 to to PART cana-5910 457 6 differential differential VERB cana-5910 457 7 equations equation NOUN cana-5910 457 8 ( ( PUNCT cana-5910 457 9 2nd 2nd ADJ cana-5910 457 10 ed ed NOUN cana-5910 457 11 . . PUNCT cana-5910 457 12 ) ) PUNCT cana-5910 457 13 . . PUNCT cana-5910 458 1 springer springer NOUN cana-5910 458 2 . . PUNCT cana-5910 459 1 explores explore VERB cana-5910 459 2 the the DET cana-5910 459 3 rigorous rigorous ADJ cana-5910 459 4 theory theory NOUN cana-5910 459 5 and and CCONJ cana-5910 459 6 application application NOUN cana-5910 459 7 of of ADP cana-5910 459 8 lie lie NOUN cana-5910 459 9 groups group NOUN cana-5910 459 10 to to PART cana-5910 459 11 differential differential VERB cana-5910 459 12 equations equation NOUN cana-5910 459 13 , , PUNCT cana-5910 459 14 with with ADP cana-5910 459 15 insights insight NOUN cana-5910 459 16 into into ADP cana-5910 459 17 symmetry symmetry NOUN cana-5910 459 18 - - PUNCT cana-5910 459 19 based base VERB cana-5910 459 20 solutions solution NOUN cana-5910 459 21 for for ADP cana-5910 459 22 the the DET cana-5910 459 23 heat heat NOUN cana-5910 459 24 equation equation NOUN cana-5910 459 25 . . PUNCT cana-5910 460 1 7 7 X cana-5910 460 2 . . X cana-5910 460 3 ibragimov ibragimov ADJ cana-5910 460 4 , , PUNCT cana-5910 460 5 n. n. PROPN cana-5910 460 6 h. h. PROPN cana-5910 460 7 ( ( PUNCT cana-5910 460 8 1994 1994 NUM cana-5910 460 9 ) ) PUNCT cana-5910 460 10 . . PUNCT cana-5910 461 1 crc crc PROPN cana-5910 461 2 handbook handbook PROPN cana-5910 461 3 of of ADP cana-5910 461 4 lie lie NOUN cana-5910 461 5 group group NOUN cana-5910 461 6 analysis analysis NOUN cana-5910 461 7 of of ADP cana-5910 461 8 differential differential ADJ cana-5910 461 9 equations equation NOUN cana-5910 461 10 ( ( PUNCT cana-5910 461 11 vol vol NOUN cana-5910 461 12 . . NOUN cana-5910 461 13 1 1 NUM cana-5910 461 14 ) ) PUNCT cana-5910 461 15 . . PUNCT cana-5910 462 1 crc crc PROPN cana-5910 462 2 press press PROPN cana-5910 462 3 . . PUNCT cana-5910 463 1 a a DET cana-5910 463 2 key key ADJ cana-5910 463 3 resource resource NOUN cana-5910 463 4 for for ADP cana-5910 463 5 lie lie NOUN cana-5910 463 6 group group NOUN cana-5910 463 7 techniques technique NOUN cana-5910 463 8 , , PUNCT cana-5910 463 9 with with ADP cana-5910 463 10 detailed detailed ADJ cana-5910 463 11 examples example NOUN cana-5910 463 12 of of ADP cana-5910 463 13 symmetry symmetry NOUN cana-5910 463 14 reductions reduction NOUN cana-5910 463 15 for for ADP cana-5910 463 16 pdes pde NOUN cana-5910 463 17 like like ADP cana-5910 463 18 the the DET cana-5910 463 19 heat heat NOUN cana-5910 463 20 equation equation NOUN cana-5910 463 21 . . PUNCT cana-5910 464 1 8 8 X cana-5910 464 2 . . X cana-5910 464 3 hydon hydon PROPN cana-5910 464 4 , , PUNCT cana-5910 464 5 p. p. PROPN cana-5910 464 6 e. e. PROPN cana-5910 464 7 ( ( PUNCT cana-5910 464 8 2000 2000 NUM cana-5910 464 9 ) ) PUNCT cana-5910 464 10 . . PUNCT cana-5910 465 1 symmetry symmetry NOUN cana-5910 465 2 methods method NOUN cana-5910 465 3 for for ADP cana-5910 465 4 differential differential ADJ cana-5910 465 5 equations equation NOUN cana-5910 465 6 : : PUNCT cana-5910 465 7 a a DET cana-5910 465 8 beginner beginner NOUN cana-5910 465 9 ’s ’s PART cana-5910 465 10 guide guide NOUN cana-5910 465 11 . . PUNCT cana-5910 466 1 cambridge cambridge PROPN cana-5910 466 2 university university PROPN cana-5910 466 3 press press NOUN cana-5910 466 4 . . PUNCT cana-5910 467 1 a a DET cana-5910 467 2 beginner beginner NOUN cana-5910 467 3 - - PUNCT cana-5910 467 4 friendly friendly ADJ cana-5910 467 5 guide guide NOUN cana-5910 467 6 to to PART cana-5910 467 7 symmetry symmetry NOUN cana-5910 467 8 methods method NOUN cana-5910 467 9 , , PUNCT cana-5910 467 10 with with ADP cana-5910 467 11 practical practical ADJ cana-5910 467 12 examples example NOUN cana-5910 467 13 of of ADP cana-5910 467 14 applying apply VERB cana-5910 467 15 lie lie NOUN cana-5910 467 16 symmetries symmetry NOUN cana-5910 467 17 to to ADP cana-5910 467 18 physical physical ADJ cana-5910 467 19 problems problem NOUN cana-5910 467 20 , , PUNCT cana-5910 467 21 including include VERB cana-5910 467 22 heat heat NOUN cana-5910 467 23 diffusion diffusion NOUN cana-5910 467 24 . . PUNCT cana-5910 468 1 9 9 X cana-5910 468 2 . . X cana-5910 468 3 hydon hydon PROPN cana-5910 468 4 , , PUNCT cana-5910 468 5 p. p. PROPN cana-5910 468 6 e. e. PROPN cana-5910 468 7 ( ( PUNCT cana-5910 468 8 2000 2000 NUM cana-5910 468 9 ) ) PUNCT cana-5910 468 10 . . PUNCT cana-5910 469 1 symmetry symmetry NOUN cana-5910 469 2 methods method NOUN cana-5910 469 3 for for ADP cana-5910 469 4 differential differential ADJ cana-5910 469 5 equations equation NOUN cana-5910 469 6 : : PUNCT cana-5910 469 7 a a DET cana-5910 469 8 beginner beginner NOUN cana-5910 469 9 ’s ’s PART cana-5910 469 10 guide guide NOUN cana-5910 469 11 . . PUNCT cana-5910 470 1 cambridge cambridge PROPN cana-5910 470 2 university university PROPN cana-5910 470 3 press press NOUN cana-5910 470 4 . . PUNCT cana-5910 471 1 a a DET cana-5910 471 2 beginner beginner NOUN cana-5910 471 3 - - PUNCT cana-5910 471 4 friendly friendly ADJ cana-5910 471 5 guide guide NOUN cana-5910 471 6 to to PART cana-5910 471 7 symmetry symmetry NOUN cana-5910 471 8 methods method NOUN cana-5910 471 9 , , PUNCT cana-5910 471 10 with with ADP cana-5910 471 11 practical practical ADJ cana-5910 471 12 examples example NOUN cana-5910 471 13 of of ADP cana-5910 471 14 applying apply VERB cana-5910 471 15 lie lie NOUN cana-5910 471 16 symmetries symmetry NOUN cana-5910 471 17 to to ADP cana-5910 471 18 physical physical ADJ cana-5910 471 19 problems problem NOUN cana-5910 471 20 , , PUNCT cana-5910 471 21 including include VERB cana-5910 471 22 heat heat NOUN cana-5910 471 23 diffusion diffusion NOUN cana-5910 471 24 . . PUNCT cana-5910 472 1 10 10 NUM cana-5910 472 2 . . PUNCT cana-5910 473 1 cantwell cantwell PROPN cana-5910 473 2 , , PUNCT cana-5910 473 3 b. b. PROPN cana-5910 473 4 j. j. PROPN cana-5910 473 5 ( ( PUNCT cana-5910 473 6 2002 2002 NUM cana-5910 473 7 ) ) PUNCT cana-5910 473 8 . . PUNCT cana-5910 474 1 introduction introduction NOUN cana-5910 474 2 to to PART cana-5910 474 3 symmetry symmetry VERB cana-5910 474 4 analysis analysis NOUN cana-5910 474 5 . . PUNCT cana-5910 475 1 cambridge cambridge PROPN cana-5910 475 2 university university PROPN cana-5910 475 3 press press NOUN cana-5910 475 4 . . PUNCT cana-5910 476 1 provides provide VERB cana-5910 476 2 an an DET cana-5910 476 3 accessible accessible ADJ cana-5910 476 4 introduction introduction NOUN cana-5910 476 5 to to PART cana-5910 476 6 symmetry symmetry VERB cana-5910 476 7 methods method NOUN cana-5910 476 8 with with ADP cana-5910 476 9 practical practical ADJ cana-5910 476 10 examples example NOUN cana-5910 476 11 , , PUNCT cana-5910 476 12 including include VERB cana-5910 476 13 applications application NOUN cana-5910 476 14 to to PART cana-5910 476 15 heat heat VERB cana-5910 476 16 conduction conduction NOUN cana-5910 476 17 problems problem NOUN cana-5910 476 18 . . PUNCT cana-5910 477 1 11 11 NUM cana-5910 477 2 . . X cana-5910 478 1 ovsiannikov ovsiannikov PROPN cana-5910 478 2 , , PUNCT cana-5910 478 3 l. l. PROPN cana-5910 478 4 v. v. PROPN cana-5910 478 5 ( ( PUNCT cana-5910 478 6 1982 1982 NUM cana-5910 478 7 ) ) PUNCT cana-5910 478 8 . . PUNCT cana-5910 479 1 group group NOUN cana-5910 479 2 analysis analysis NOUN cana-5910 479 3 of of ADP cana-5910 479 4 differential differential ADJ cana-5910 479 5 equations equation NOUN cana-5910 479 6 . . PUNCT cana-5910 480 1 academic academic ADJ cana-5910 480 2 press press NOUN cana-5910 480 3 . . PUNCT cana-5910 481 1 a a DET cana-5910 481 2 seminal seminal ADJ cana-5910 481 3 work work NOUN cana-5910 481 4 on on ADP cana-5910 481 5 group group NOUN cana-5910 481 6 analysis analysis NOUN cana-5910 481 7 , , PUNCT cana-5910 481 8 detailing detail VERB cana-5910 481 9 the the DET cana-5910 481 10 application application NOUN cana-5910 481 11 of of ADP cana-5910 481 12 lie lie NOUN cana-5910 481 13 symmetries symmetry NOUN cana-5910 481 14 to to ADP cana-5910 481 15 pdes pde NOUN cana-5910 481 16 , , PUNCT cana-5910 481 17 with with ADP cana-5910 481 18 examples example NOUN cana-5910 481 19 relevant relevant ADJ cana-5910 481 20 to to PART cana-5910 481 21 heat heat VERB cana-5910 481 22 and and CCONJ cana-5910 481 23 diffusion diffusion NOUN cana-5910 481 24 equations equation NOUN cana-5910 481 25 . . PUNCT