id sid tid token lemma pos american_scientific_journal-1248 1 1 american american PROPN american_scientific_journal-1248 1 2 scientific scientific ADJ american_scientific_journal-1248 1 3 research research NOUN american_scientific_journal-1248 1 4 journal journal NOUN american_scientific_journal-1248 1 5 for for ADP american_scientific_journal-1248 1 6 engineering engineering NOUN american_scientific_journal-1248 1 7 , , PUNCT american_scientific_journal-1248 1 8 technology technology NOUN american_scientific_journal-1248 1 9 , , PUNCT american_scientific_journal-1248 1 10 and and CCONJ american_scientific_journal-1248 1 11 sciences science NOUN american_scientific_journal-1248 1 12 ( ( PUNCT american_scientific_journal-1248 1 13 asrjets asrjet NOUN american_scientific_journal-1248 1 14 ) ) PUNCT american_scientific_journal-1248 1 15 issn issn PROPN american_scientific_journal-1248 1 16 ( ( PUNCT american_scientific_journal-1248 1 17 print print NOUN american_scientific_journal-1248 1 18 ) ) PUNCT american_scientific_journal-1248 1 19 2313 2313 NUM american_scientific_journal-1248 1 20 - - SYM american_scientific_journal-1248 1 21 4410 4410 NUM american_scientific_journal-1248 1 22 , , PUNCT american_scientific_journal-1248 1 23 issn issn PROPN american_scientific_journal-1248 1 24 ( ( PUNCT american_scientific_journal-1248 1 25 online online ADJ american_scientific_journal-1248 1 26 ) ) PUNCT american_scientific_journal-1248 1 27 2313 2313 NUM american_scientific_journal-1248 1 28 - - SYM american_scientific_journal-1248 1 29 4402 4402 NUM american_scientific_journal-1248 1 30 Β© Β© PROPN american_scientific_journal-1248 1 31 global global ADJ american_scientific_journal-1248 1 32 society society NOUN american_scientific_journal-1248 1 33 of of ADP american_scientific_journal-1248 1 34 scientific scientific ADJ american_scientific_journal-1248 1 35 research research NOUN american_scientific_journal-1248 1 36 and and CCONJ american_scientific_journal-1248 1 37 researchers researcher NOUN american_scientific_journal-1248 1 38 http://asrjetsjournal.org/ http://asrjetsjournal.org/ ADJ american_scientific_journal-1248 1 39 solution solution NOUN american_scientific_journal-1248 1 40 of of ADP american_scientific_journal-1248 1 41 laplace`s laplace`s NOUN american_scientific_journal-1248 1 42 equation equation NOUN american_scientific_journal-1248 1 43 using use VERB american_scientific_journal-1248 1 44 numerical numerical ADJ american_scientific_journal-1248 1 45 methods method NOUN american_scientific_journal-1248 1 46 eltigani eltigani PROPN american_scientific_journal-1248 1 47 ismail ismail PROPN american_scientific_journal-1248 1 48 hassan hassan PROPN american_scientific_journal-1248 1 49 * * PUNCT american_scientific_journal-1248 1 50 department department PROPN american_scientific_journal-1248 1 51 of of ADP american_scientific_journal-1248 1 52 mathematics mathematic NOUN american_scientific_journal-1248 1 53 and and CCONJ american_scientific_journal-1248 1 54 statistics statistic NOUN american_scientific_journal-1248 1 55 , , PUNCT american_scientific_journal-1248 1 56 college college NOUN american_scientific_journal-1248 1 57 of of ADP american_scientific_journal-1248 1 58 science science NOUN american_scientific_journal-1248 1 59 , , PUNCT american_scientific_journal-1248 1 60 al al PROPN american_scientific_journal-1248 1 61 imam imam PROPN american_scientific_journal-1248 1 62 mohammad mohammad PROPN american_scientific_journal-1248 1 63 ibn ibn PROPN american_scientific_journal-1248 1 64 saud saud PROPN american_scientific_journal-1248 1 65 islamic islamic PROPN american_scientific_journal-1248 1 66 university university PROPN american_scientific_journal-1248 1 67 , , PUNCT american_scientific_journal-1248 1 68 riyadh riyadh PROPN american_scientific_journal-1248 1 69 11642 11642 NUM american_scientific_journal-1248 1 70 , , PUNCT american_scientific_journal-1248 2 1 ksa ksa PROPN american_scientific_journal-1248 2 2 . . PROPN american_scientific_journal-1248 2 3 department department PROPN american_scientific_journal-1248 2 4 of of ADP american_scientific_journal-1248 2 5 mathematics mathematics PROPN american_scientific_journal-1248 2 6 , , PUNCT american_scientific_journal-1248 2 7 college college NOUN american_scientific_journal-1248 2 8 of of ADP american_scientific_journal-1248 2 9 applied apply VERB american_scientific_journal-1248 2 10 science science NOUN american_scientific_journal-1248 2 11 and and CCONJ american_scientific_journal-1248 2 12 industrial industrial ADJ american_scientific_journal-1248 2 13 , , PUNCT american_scientific_journal-1248 2 14 university university NOUN american_scientific_journal-1248 2 15 of of ADP american_scientific_journal-1248 2 16 bahri bahri PROPN american_scientific_journal-1248 2 17 email email NOUN american_scientific_journal-1248 2 18 : : PUNCT american_scientific_journal-1248 2 19 ttfsr555@gmail.com ttfsr555@gmail.com X american_scientific_journal-1248 3 1 abstract abstract ADV american_scientific_journal-1248 3 2 this this DET american_scientific_journal-1248 3 3 paper paper NOUN american_scientific_journal-1248 3 4 is be AUX american_scientific_journal-1248 3 5 simple simple ADJ american_scientific_journal-1248 3 6 review review NOUN american_scientific_journal-1248 3 7 of of ADP american_scientific_journal-1248 3 8 the the DET american_scientific_journal-1248 3 9 solution solution NOUN american_scientific_journal-1248 3 10 of of ADP american_scientific_journal-1248 3 11 laplace`s laplace`s NOUN american_scientific_journal-1248 3 12 equation equation NOUN american_scientific_journal-1248 3 13 in in ADP american_scientific_journal-1248 3 14 rectangular rectangular ADJ american_scientific_journal-1248 3 15 coordinates coordinate NOUN american_scientific_journal-1248 3 16 system system NOUN american_scientific_journal-1248 3 17 , , PUNCT american_scientific_journal-1248 3 18 cylindrical cylindrical ADJ american_scientific_journal-1248 3 19 polar polar ADJ american_scientific_journal-1248 3 20 coordinates coordinate NOUN american_scientific_journal-1248 3 21 system system NOUN american_scientific_journal-1248 3 22 and and CCONJ american_scientific_journal-1248 3 23 spherical spherical ADJ american_scientific_journal-1248 3 24 polar polar ADJ american_scientific_journal-1248 3 25 coordinate coordinate NOUN american_scientific_journal-1248 3 26 system system NOUN american_scientific_journal-1248 3 27 . . PUNCT american_scientific_journal-1248 4 1 it it PRON american_scientific_journal-1248 4 2 also also ADV american_scientific_journal-1248 4 3 covers cover VERB american_scientific_journal-1248 4 4 numerical numerical ADJ american_scientific_journal-1248 4 5 method method NOUN american_scientific_journal-1248 4 6 in in ADP american_scientific_journal-1248 4 7 the the DET american_scientific_journal-1248 4 8 solution solution NOUN american_scientific_journal-1248 4 9 of of ADP american_scientific_journal-1248 4 10 laplace`s laplace`s PROPN american_scientific_journal-1248 4 11 equation equation NOUN american_scientific_journal-1248 4 12 . . PUNCT american_scientific_journal-1248 5 1 keywords keyword NOUN american_scientific_journal-1248 5 2 : : PUNCT american_scientific_journal-1248 5 3 laplace laplace NOUN american_scientific_journal-1248 5 4 equation equation NOUN american_scientific_journal-1248 5 5 ; ; PUNCT american_scientific_journal-1248 5 6 numerical numerical ADJ american_scientific_journal-1248 5 7 method method NOUN american_scientific_journal-1248 5 8 ; ; PUNCT american_scientific_journal-1248 5 9 laplacian laplacian ADJ american_scientific_journal-1248 5 10 difference difference NOUN american_scientific_journal-1248 5 11 ; ; PUNCT american_scientific_journal-1248 5 12 uniform uniform ADJ american_scientific_journal-1248 5 13 grid grid NOUN american_scientific_journal-1248 5 14 . . PUNCT american_scientific_journal-1248 6 1 1 1 X american_scientific_journal-1248 6 2 . . X american_scientific_journal-1248 6 3 introduction introduction NOUN american_scientific_journal-1248 6 4 during during ADP american_scientific_journal-1248 6 5 the the DET american_scientific_journal-1248 6 6 last last ADJ american_scientific_journal-1248 6 7 two two NUM american_scientific_journal-1248 6 8 centuries century NOUN american_scientific_journal-1248 6 9 several several ADJ american_scientific_journal-1248 6 10 methods method NOUN american_scientific_journal-1248 6 11 have have AUX american_scientific_journal-1248 6 12 been be AUX american_scientific_journal-1248 6 13 advanced advance VERB american_scientific_journal-1248 6 14 for for ADP american_scientific_journal-1248 6 15 solving solve VERB american_scientific_journal-1248 6 16 partial partial ADJ american_scientific_journal-1248 6 17 differential differential ADJ american_scientific_journal-1248 6 18 equations equation NOUN american_scientific_journal-1248 6 19 . . PUNCT american_scientific_journal-1248 7 1 among among ADP american_scientific_journal-1248 7 2 these these PRON american_scientific_journal-1248 7 3 we we PRON american_scientific_journal-1248 7 4 consider consider VERB american_scientific_journal-1248 7 5 only only ADV american_scientific_journal-1248 7 6 two two NUM american_scientific_journal-1248 7 7 techniques technique NOUN american_scientific_journal-1248 7 8 known know VERB american_scientific_journal-1248 7 9 as as ADP american_scientific_journal-1248 7 10 the the DET american_scientific_journal-1248 7 11 method method NOUN american_scientific_journal-1248 7 12 of of ADP american_scientific_journal-1248 7 13 separation separation NOUN american_scientific_journal-1248 7 14 of of ADP american_scientific_journal-1248 7 15 variables variable NOUN american_scientific_journal-1248 7 16 and and CCONJ american_scientific_journal-1248 7 17 laplace laplace NOUN american_scientific_journal-1248 7 18 transformation transformation NOUN american_scientific_journal-1248 7 19 , , PUNCT american_scientific_journal-1248 7 20 the the DET american_scientific_journal-1248 7 21 method method NOUN american_scientific_journal-1248 7 22 of of ADP american_scientific_journal-1248 7 23 separation separation NOUN american_scientific_journal-1248 7 24 of of ADP american_scientific_journal-1248 7 25 variables variable NOUN american_scientific_journal-1248 7 26 is be AUX american_scientific_journal-1248 7 27 perhaps perhaps ADV american_scientific_journal-1248 7 28 the the DET american_scientific_journal-1248 7 29 oldest old ADJ american_scientific_journal-1248 7 30 systematic systematic ADJ american_scientific_journal-1248 7 31 method method NOUN american_scientific_journal-1248 7 32 for for ADP american_scientific_journal-1248 7 33 solving solve VERB american_scientific_journal-1248 7 34 partial partial ADJ american_scientific_journal-1248 7 35 differential differential NOUN american_scientific_journal-1248 7 36 equations equation NOUN american_scientific_journal-1248 7 37 . . PUNCT american_scientific_journal-1248 8 1 it it PRON american_scientific_journal-1248 8 2 has have AUX american_scientific_journal-1248 8 3 been be AUX american_scientific_journal-1248 8 4 considerably considerably ADV american_scientific_journal-1248 8 5 refined refine VERB american_scientific_journal-1248 8 6 and and CCONJ american_scientific_journal-1248 8 7 generalized generalize VERB american_scientific_journal-1248 8 8 in in ADP american_scientific_journal-1248 8 9 meantime meantime NOUN american_scientific_journal-1248 8 10 and and CCONJ american_scientific_journal-1248 8 11 remains remain VERB american_scientific_journal-1248 8 12 a a DET american_scientific_journal-1248 8 13 method method NOUN american_scientific_journal-1248 8 14 of of ADP american_scientific_journal-1248 8 15 great great ADJ american_scientific_journal-1248 8 16 importance importance NOUN american_scientific_journal-1248 8 17 to to ADP american_scientific_journal-1248 8 18 day day NOUN american_scientific_journal-1248 8 19 . . PUNCT american_scientific_journal-1248 9 1 in in ADP american_scientific_journal-1248 9 2 this this DET american_scientific_journal-1248 9 3 study study NOUN american_scientific_journal-1248 9 4 we we PRON american_scientific_journal-1248 9 5 will will AUX american_scientific_journal-1248 9 6 review review VERB american_scientific_journal-1248 9 7 how how SCONJ american_scientific_journal-1248 9 8 the the DET american_scientific_journal-1248 9 9 method method NOUN american_scientific_journal-1248 9 10 of of ADP american_scientific_journal-1248 9 11 separation separation NOUN american_scientific_journal-1248 9 12 of of ADP american_scientific_journal-1248 9 13 variables variable NOUN american_scientific_journal-1248 9 14 . . PUNCT american_scientific_journal-1248 10 1 in in ADP american_scientific_journal-1248 10 2 solving solve VERB american_scientific_journal-1248 10 3 this this DET american_scientific_journal-1248 10 4 problem problem NOUN american_scientific_journal-1248 10 5 it it PRON american_scientific_journal-1248 10 6 is be AUX american_scientific_journal-1248 10 7 eventually eventually ADV american_scientific_journal-1248 10 8 necessary necessary ADJ american_scientific_journal-1248 10 9 to to PART american_scientific_journal-1248 10 10 consider consider VERB american_scientific_journal-1248 10 11 the the DET american_scientific_journal-1248 10 12 equations equation NOUN american_scientific_journal-1248 10 13 of of ADP american_scientific_journal-1248 10 14 whether whether SCONJ american_scientific_journal-1248 10 15 an an DET american_scientific_journal-1248 10 16 essentially essentially ADV american_scientific_journal-1248 10 17 arbitrary arbitrary ADJ american_scientific_journal-1248 10 18 function function NOUN american_scientific_journal-1248 10 19 can can AUX american_scientific_journal-1248 10 20 be be AUX american_scientific_journal-1248 10 21 expressed express VERB american_scientific_journal-1248 10 22 as as ADP american_scientific_journal-1248 10 23 an an DET american_scientific_journal-1248 10 24 infinite infinite ADJ american_scientific_journal-1248 10 25 series series NOUN american_scientific_journal-1248 10 26 of of ADP american_scientific_journal-1248 10 27 sine sine NOUN american_scientific_journal-1248 10 28 and and CCONJ american_scientific_journal-1248 10 29 cosine cosine NOUN american_scientific_journal-1248 10 30 functions function NOUN american_scientific_journal-1248 10 31 . . PUNCT american_scientific_journal-1248 11 1 series series NOUN american_scientific_journal-1248 11 2 of of ADP american_scientific_journal-1248 11 3 this this DET american_scientific_journal-1248 11 4 kind kind NOUN american_scientific_journal-1248 11 5 are be AUX american_scientific_journal-1248 11 6 called call VERB american_scientific_journal-1248 11 7 fourier fourier ADJ american_scientific_journal-1248 11 8 series series NOUN american_scientific_journal-1248 11 9 , , PUNCT american_scientific_journal-1248 11 10 then then ADV american_scientific_journal-1248 11 11 we we PRON american_scientific_journal-1248 11 12 will will AUX american_scientific_journal-1248 11 13 come come VERB american_scientific_journal-1248 11 14 across across ADP american_scientific_journal-1248 11 15 to to ADP american_scientific_journal-1248 11 16 in in ADP american_scientific_journal-1248 11 17 some some DET american_scientific_journal-1248 11 18 typical typical ADJ american_scientific_journal-1248 11 19 problems problem NOUN american_scientific_journal-1248 11 20 as as ADP american_scientific_journal-1248 11 21 the the DET american_scientific_journal-1248 11 22 laplace`s laplace`s NOUN american_scientific_journal-1248 11 23 equation equation NOUN american_scientific_journal-1248 11 24 solving solve VERB american_scientific_journal-1248 11 25 it it PRON american_scientific_journal-1248 11 26 by by ADP american_scientific_journal-1248 11 27 separation separation NOUN american_scientific_journal-1248 11 28 of of ADP american_scientific_journal-1248 11 29 variables variable NOUN american_scientific_journal-1248 11 30 and and CCONJ american_scientific_journal-1248 11 31 fourier fouri ADJ american_scientific_journal-1248 11 32 series series NOUN american_scientific_journal-1248 11 33 .by .by PROPN american_scientific_journal-1248 12 1 the the DET american_scientific_journal-1248 12 2 other other ADJ american_scientific_journal-1248 12 3 hand hand NOUN american_scientific_journal-1248 12 4 we we PRON american_scientific_journal-1248 12 5 will will AUX american_scientific_journal-1248 12 6 explain explain VERB american_scientific_journal-1248 12 7 in in ADP american_scientific_journal-1248 12 8 following follow VERB american_scientific_journal-1248 12 9 : : PUNCT american_scientific_journal-1248 12 10 ----------------------------------------------------------------------- ----------------------------------------------------------------------- PUNCT american_scientific_journal-1248 12 11 * * PUNCT american_scientific_journal-1248 12 12 corresponding correspond VERB american_scientific_journal-1248 12 13 author author NOUN american_scientific_journal-1248 12 14 . . PUNCT american_scientific_journal-1248 13 1 178 178 NUM american_scientific_journal-1248 13 2 http://asrjetsjournal.org/ http://asrjetsjournal.org/ PROPN american_scientific_journal-1248 13 3 american american ADJ american_scientific_journal-1248 13 4 scientific scientific ADJ american_scientific_journal-1248 13 5 research research NOUN american_scientific_journal-1248 13 6 journal journal NOUN american_scientific_journal-1248 13 7 for for ADP american_scientific_journal-1248 13 8 engineering engineering NOUN american_scientific_journal-1248 13 9 , , PUNCT american_scientific_journal-1248 13 10 technology technology NOUN american_scientific_journal-1248 13 11 , , PUNCT american_scientific_journal-1248 13 12 and and CCONJ american_scientific_journal-1248 13 13 sciences science NOUN american_scientific_journal-1248 13 14 ( ( PUNCT american_scientific_journal-1248 13 15 asrjets asrjet NOUN american_scientific_journal-1248 13 16 ) ) PUNCT american_scientific_journal-1248 13 17 ( ( PUNCT american_scientific_journal-1248 13 18 2016 2016 NUM american_scientific_journal-1248 13 19 ) ) PUNCT american_scientific_journal-1248 13 20 volume volume NOUN american_scientific_journal-1248 13 21 16 16 NUM american_scientific_journal-1248 13 22 , , PUNCT american_scientific_journal-1248 13 23 no no DET american_scientific_journal-1248 13 24 1 1 NUM american_scientific_journal-1248 13 25 , , PUNCT american_scientific_journal-1248 13 26 pp pp ADV american_scientific_journal-1248 13 27 178 178 NUM american_scientific_journal-1248 13 28 - - SYM american_scientific_journal-1248 13 29 192 192 NUM american_scientific_journal-1248 13 30 in in ADP american_scientific_journal-1248 13 31 first first ADV american_scientific_journal-1248 13 32 we we PRON american_scientific_journal-1248 13 33 define define VERB american_scientific_journal-1248 13 34 rectangular rectangular ADJ american_scientific_journal-1248 13 35 coordinate coordinate NOUN american_scientific_journal-1248 13 36 system system NOUN american_scientific_journal-1248 13 37 βˆ‡2π‘ˆ βˆ‡2π‘ˆ PROPN american_scientific_journal-1248 13 38 = = SYM american_scientific_journal-1248 13 39 πœ•2𝑒 πœ•2𝑒 NUM american_scientific_journal-1248 13 40 πœ•π‘₯2 πœ•π‘₯2 NOUN american_scientific_journal-1248 13 41 + + CCONJ american_scientific_journal-1248 13 42 πœ•2𝑒 πœ•2𝑒 PROPN american_scientific_journal-1248 13 43 πœ•π‘¦2 πœ•π‘¦2 ADJ american_scientific_journal-1248 13 44 = = SYM american_scientific_journal-1248 13 45 0 0 NUM american_scientific_journal-1248 13 46 in in ADP american_scientific_journal-1248 13 47 second second ADJ american_scientific_journal-1248 13 48 we we PRON american_scientific_journal-1248 13 49 transform transform VERB american_scientific_journal-1248 13 50 the the DET american_scientific_journal-1248 13 51 rectangular rectangular ADJ american_scientific_journal-1248 13 52 coordinates coordinate NOUN american_scientific_journal-1248 13 53 system system NOUN american_scientific_journal-1248 13 54 to to ADP american_scientific_journal-1248 13 55 cylindrical cylindrical ADJ american_scientific_journal-1248 13 56 coordinates coordinate NOUN american_scientific_journal-1248 13 57 system system NOUN american_scientific_journal-1248 13 58 is be AUX american_scientific_journal-1248 13 59 which which PRON american_scientific_journal-1248 13 60 comes come VERB american_scientific_journal-1248 13 61 as as ADP american_scientific_journal-1248 13 62 : : PUNCT american_scientific_journal-1248 13 63 π‘ˆ(π‘₯,𝑦. π‘ˆ(π‘₯,𝑦. NUM american_scientific_journal-1248 13 64 𝑧 𝑧 X american_scientific_journal-1248 13 65 ) ) PUNCT american_scientific_journal-1248 14 1 = = SYM american_scientific_journal-1248 14 2 ( ( PUNCT american_scientific_journal-1248 14 3 𝜌𝜌,πœƒπœƒ 𝜌𝜌,πœƒπœƒ NOUN american_scientific_journal-1248 14 4 , , PUNCT american_scientific_journal-1248 14 5 𝑧 𝑧 NOUN american_scientific_journal-1248 14 6 ) ) PUNCT american_scientific_journal-1248 14 7 and and CCONJ american_scientific_journal-1248 14 8 its its PRON american_scientific_journal-1248 14 9 solution solution NOUN american_scientific_journal-1248 14 10 given give VERB american_scientific_journal-1248 14 11 by by ADP american_scientific_journal-1248 14 12 : : PUNCT american_scientific_journal-1248 14 13 πœ•2𝑣 πœ•2𝑣 NOUN american_scientific_journal-1248 14 14 πœ•πœŒπœŒ2 πœ•πœŒπœŒ2 NOUN american_scientific_journal-1248 14 15 + + CCONJ american_scientific_journal-1248 14 16 1 1 NUM american_scientific_journal-1248 14 17 ρ ρ NUM american_scientific_journal-1248 14 18 πœ•π‘£ πœ•π‘£ NOUN american_scientific_journal-1248 14 19 βˆ‚Ο βˆ‚Ο PROPN american_scientific_journal-1248 15 1 + + CCONJ american_scientific_journal-1248 15 2 1 1 NUM american_scientific_journal-1248 15 3 ρ2 ρ2 VERB american_scientific_journal-1248 15 4 πœ•2𝑣 πœ•2𝑣 NOUN american_scientific_journal-1248 15 5 βˆ‚ΞΈ2 βˆ‚ΞΈ2 PRON american_scientific_journal-1248 15 6 + + NUM american_scientific_journal-1248 15 7 πœ•2𝑣 πœ•2𝑣 PROPN american_scientific_journal-1248 15 8 βˆ‚z2 βˆ‚z2 NOUN american_scientific_journal-1248 15 9 = = NOUN american_scientific_journal-1248 15 10 0 0 NUM american_scientific_journal-1248 15 11 in in ADP american_scientific_journal-1248 15 12 third third ADJ american_scientific_journal-1248 15 13 we we PRON american_scientific_journal-1248 15 14 also also ADV american_scientific_journal-1248 15 15 transform transform VERB american_scientific_journal-1248 15 16 the the DET american_scientific_journal-1248 15 17 rectangular rectangular ADJ american_scientific_journal-1248 15 18 coordinates coordinate NOUN american_scientific_journal-1248 15 19 system system NOUN american_scientific_journal-1248 15 20 to to PART american_scientific_journal-1248 15 21 spherical spherical ADJ american_scientific_journal-1248 15 22 polar polar ADJ american_scientific_journal-1248 15 23 coordinates coordinate NOUN american_scientific_journal-1248 15 24 system system NOUN american_scientific_journal-1248 15 25 which which PRON american_scientific_journal-1248 15 26 comes come VERB american_scientific_journal-1248 15 27 as as ADP american_scientific_journal-1248 15 28 : : PUNCT american_scientific_journal-1248 15 29 π‘ˆ π‘ˆ PROPN american_scientific_journal-1248 15 30 = = SYM american_scientific_journal-1248 15 31 ( ( PUNCT american_scientific_journal-1248 15 32 π‘₯,𝑦 π‘₯,𝑦 INTJ american_scientific_journal-1248 15 33 , , PUNCT american_scientific_journal-1248 15 34 𝑧 𝑧 NOUN american_scientific_journal-1248 15 35 ) ) PUNCT american_scientific_journal-1248 15 36 = = SYM american_scientific_journal-1248 15 37 ( ( PUNCT american_scientific_journal-1248 15 38 π‘Ÿ,πœƒπœƒ,πœ™πœ™ π‘Ÿ,πœƒπœƒ,πœ™πœ™ NOUN american_scientific_journal-1248 15 39 ) ) PUNCT american_scientific_journal-1248 15 40 βˆ‡2π‘ˆ βˆ‡2π‘ˆ NOUN american_scientific_journal-1248 16 1 = = SYM american_scientific_journal-1248 16 2 πœ•2u πœ•2u NOUN american_scientific_journal-1248 16 3 βˆ‚r2 βˆ‚r2 NOUN american_scientific_journal-1248 16 4 + + CCONJ american_scientific_journal-1248 16 5 2 2 NUM american_scientific_journal-1248 16 6 r r NOUN american_scientific_journal-1248 16 7 πœ•u πœ•u NOUN american_scientific_journal-1248 17 1 βˆ‚r βˆ‚r X american_scientific_journal-1248 17 2 + + CCONJ american_scientific_journal-1248 17 3 1 1 NUM american_scientific_journal-1248 17 4 r2 r2 NOUN american_scientific_journal-1248 17 5 sin sin NOUN american_scientific_journal-1248 17 6 ΞΈ ΞΈ PROPN american_scientific_journal-1248 17 7 βˆ‚ βˆ‚ PUNCT american_scientific_journal-1248 18 1 βˆ‚ΞΈ βˆ‚ΞΈ PROPN american_scientific_journal-1248 18 2 ( ( PUNCT american_scientific_journal-1248 18 3 sinπœƒπœƒ sinπœƒπœƒ NOUN american_scientific_journal-1248 18 4 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-1248 18 5 πœ•πœƒπœƒ πœ•πœƒπœƒ PUNCT american_scientific_journal-1248 18 6 ) ) PUNCT american_scientific_journal-1248 19 1 + + CCONJ american_scientific_journal-1248 19 2 1 1 NUM american_scientific_journal-1248 19 3 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-1248 19 4 1 1 NUM american_scientific_journal-1248 19 5 sin2ΞΈ sin2ΞΈ NOUN american_scientific_journal-1248 19 6 βˆ‚2u βˆ‚2u X american_scientific_journal-1248 19 7 βˆ‚ΞΈ2 βˆ‚ΞΈ2 X american_scientific_journal-1248 19 8 = = SYM american_scientific_journal-1248 19 9 0 0 NUM american_scientific_journal-1248 19 10 in in ADP american_scientific_journal-1248 19 11 fourth fourth ADJ american_scientific_journal-1248 19 12 we we PRON american_scientific_journal-1248 19 13 explain explain VERB american_scientific_journal-1248 19 14 computerized computerized ADJ american_scientific_journal-1248 19 15 computation computation NOUN american_scientific_journal-1248 19 16 method method NOUN american_scientific_journal-1248 19 17 to to PART american_scientific_journal-1248 19 18 solve solve VERB american_scientific_journal-1248 19 19 general general ADJ american_scientific_journal-1248 19 20 solution solution NOUN american_scientific_journal-1248 19 21 of of ADP american_scientific_journal-1248 19 22 laplace`s laplace`s NOUN american_scientific_journal-1248 19 23 equation equation NOUN american_scientific_journal-1248 19 24 by by ADP american_scientific_journal-1248 19 25 finite finite ADJ american_scientific_journal-1248 19 26 different different ADJ american_scientific_journal-1248 19 27 equation equation NOUN american_scientific_journal-1248 19 28 . . PUNCT american_scientific_journal-1248 20 1 2 2 X american_scientific_journal-1248 20 2 . . X american_scientific_journal-1248 20 3 two two NUM american_scientific_journal-1248 20 4 dimensional dimensional ADJ american_scientific_journal-1248 20 5 heat heat NOUN american_scientific_journal-1248 20 6 flow flow NOUN american_scientific_journal-1248 20 7 consider consider VERB american_scientific_journal-1248 20 8 the the DET american_scientific_journal-1248 20 9 flow flow NOUN american_scientific_journal-1248 20 10 of of ADP american_scientific_journal-1248 20 11 heat heat NOUN american_scientific_journal-1248 20 12 in in ADP american_scientific_journal-1248 20 13 a a DET american_scientific_journal-1248 20 14 metal metal NOUN american_scientific_journal-1248 20 15 plate plate NOUN american_scientific_journal-1248 20 16 of of ADP american_scientific_journal-1248 20 17 uniform uniform ADJ american_scientific_journal-1248 20 18 thickness thickness PROPN american_scientific_journal-1248 20 19 𝛼𝛼(π‘π‘š 𝛼𝛼(π‘π‘š PROPN american_scientific_journal-1248 20 20 ) ) PUNCT american_scientific_journal-1248 20 21 , , PUNCT american_scientific_journal-1248 20 22 density density NOUN american_scientific_journal-1248 20 23 ( ( PUNCT american_scientific_journal-1248 20 24 𝜌𝜌 𝜌𝜌 X american_scientific_journal-1248 20 25 π‘”π‘Ÿβ„ π‘”π‘Ÿβ„ NOUN american_scientific_journal-1248 20 26 π‘π‘š3⁄ π‘π‘š3⁄ X american_scientific_journal-1248 20 27 ) ) PUNCT american_scientific_journal-1248 20 28 , , PUNCT american_scientific_journal-1248 20 29 specific specific ADJ american_scientific_journal-1248 20 30 heat heat NOUN american_scientific_journal-1248 20 31 𝑆(π‘π‘Žπ‘™. 𝑆(π‘π‘Žπ‘™. NOUN american_scientific_journal-1248 20 32 π‘”π‘Ÿβ„ π‘”π‘Ÿβ„ NOUN american_scientific_journal-1248 20 33 . . PUNCT american_scientific_journal-1248 21 1 deg)and deg)and ADJ american_scientific_journal-1248 21 2 thermal thermal ADJ american_scientific_journal-1248 21 3 conductivity conductivity NOUN american_scientific_journal-1248 21 4 𝐾(π‘π‘Žπ‘™ 𝐾(π‘π‘Žπ‘™ ADP american_scientific_journal-1248 21 5 π‘π‘šβ„ π‘π‘šβ„ PROPN american_scientific_journal-1248 21 6 . . PUNCT american_scientific_journal-1248 21 7 deg deg PROPN american_scientific_journal-1248 21 8 ) ) PUNCT american_scientific_journal-1248 21 9 . . PUNCT american_scientific_journal-1248 22 1 let let VERB american_scientific_journal-1248 22 2 𝑋 𝑋 PROPN american_scientific_journal-1248 22 3 ∘ ∘ PROPN american_scientific_journal-1248 22 4 π‘Œplane π‘Œplane PROPN american_scientific_journal-1248 22 5 be be AUX american_scientific_journal-1248 22 6 taken take VERB american_scientific_journal-1248 22 7 in in ADP american_scientific_journal-1248 22 8 face face NOUN american_scientific_journal-1248 22 9 the the DET american_scientific_journal-1248 22 10 point point NOUN american_scientific_journal-1248 22 11 [ [ X american_scientific_journal-1248 22 12 1 1 NUM american_scientific_journal-1248 22 13 ] ] PUNCT american_scientific_journal-1248 22 14 . . PUNCT american_scientific_journal-1248 23 1 if if SCONJ american_scientific_journal-1248 23 2 temperature temperature NOUN american_scientific_journal-1248 23 3 at at ADP american_scientific_journal-1248 23 4 any any DET american_scientific_journal-1248 23 5 point point NOUN american_scientific_journal-1248 23 6 is be AUX american_scientific_journal-1248 23 7 independent independent ADJ american_scientific_journal-1248 23 8 of of ADP american_scientific_journal-1248 23 9 the the DET american_scientific_journal-1248 23 10 𝑍coordinate 𝑍coordinate PROPN american_scientific_journal-1248 23 11 and and CCONJ american_scientific_journal-1248 23 12 depends depend VERB american_scientific_journal-1248 23 13 only only ADV american_scientific_journal-1248 23 14 on on ADP american_scientific_journal-1248 23 15 𝑋,π‘Œ 𝑋,π‘Œ NOUN american_scientific_journal-1248 23 16 and and CCONJ american_scientific_journal-1248 23 17 time time NOUN american_scientific_journal-1248 23 18 𝑑. 𝑑. VERB american_scientific_journal-1248 23 19 then then ADV american_scientific_journal-1248 23 20 the the DET american_scientific_journal-1248 23 21 flow flow NOUN american_scientific_journal-1248 23 22 is be AUX american_scientific_journal-1248 23 23 said say VERB american_scientific_journal-1248 23 24 to to PART american_scientific_journal-1248 23 25 be be AUX american_scientific_journal-1248 23 26 two two NUM american_scientific_journal-1248 23 27 dimensional dimensional ADJ american_scientific_journal-1248 23 28 .in .in PUNCT american_scientific_journal-1248 23 29 this this DET american_scientific_journal-1248 23 30 case case NOUN american_scientific_journal-1248 23 31 , , PUNCT american_scientific_journal-1248 23 32 the the DET american_scientific_journal-1248 23 33 heat heat NOUN american_scientific_journal-1248 23 34 flow flow NOUN american_scientific_journal-1248 23 35 is be AUX american_scientific_journal-1248 23 36 in in ADP american_scientific_journal-1248 23 37 the the DET american_scientific_journal-1248 23 38 π‘‹π‘Œ-plane π‘‹π‘Œ-plane PROPN american_scientific_journal-1248 23 39 only only ADV american_scientific_journal-1248 23 40 and and CCONJ american_scientific_journal-1248 23 41 is be AUX american_scientific_journal-1248 23 42 zero zero NUM american_scientific_journal-1248 23 43 along along ADP american_scientific_journal-1248 23 44 the the DET american_scientific_journal-1248 23 45 normal normal ADJ american_scientific_journal-1248 23 46 to to ADP american_scientific_journal-1248 23 47 the the DET american_scientific_journal-1248 23 48 π‘‹π‘Œ-plane π‘‹π‘Œ-plane PROPN american_scientific_journal-1248 23 49 . . PUNCT american_scientific_journal-1248 24 1 consider consider VERB american_scientific_journal-1248 24 2 a a DET american_scientific_journal-1248 24 3 rectangular rectangular ADJ american_scientific_journal-1248 24 4 element𝐴𝐡𝐢𝐷 element𝐴𝐡𝐢𝐷 NOUN american_scientific_journal-1248 24 5 of of ADP american_scientific_journal-1248 24 6 the the DET american_scientific_journal-1248 24 7 plane plane NOUN american_scientific_journal-1248 24 8 with with ADP american_scientific_journal-1248 24 9 sides side NOUN american_scientific_journal-1248 24 10 as as SCONJ american_scientific_journal-1248 24 11 shown show VERB american_scientific_journal-1248 24 12 in in ADP american_scientific_journal-1248 24 13 figure figure NOUN american_scientific_journal-1248 24 14 . . PUNCT american_scientific_journal-1248 25 1 by by ADP american_scientific_journal-1248 25 2 𝐴on 𝐴on PROPN american_scientific_journal-1248 25 3 the the DET american_scientific_journal-1248 25 4 amount amount NOUN american_scientific_journal-1248 25 5 of of ADP american_scientific_journal-1248 25 6 heat heat NOUN american_scientific_journal-1248 25 7 entering enter VERB american_scientific_journal-1248 25 8 the the DET american_scientific_journal-1248 25 9 element element NOUN american_scientific_journal-1248 25 10 , , PUNCT american_scientific_journal-1248 25 11 from from ADP american_scientific_journal-1248 25 12 the the DET american_scientific_journal-1248 25 13 side side NOUN american_scientific_journal-1248 25 14 ab ab PROPN american_scientific_journal-1248 25 15 = = PUNCT american_scientific_journal-1248 25 16 βˆ’kΞ±βˆ‚y βˆ’kΞ±βˆ‚y PROPN american_scientific_journal-1248 25 17 οΏ½ οΏ½ PROPN american_scientific_journal-1248 25 18 βˆ‚u βˆ‚u PROPN american_scientific_journal-1248 25 19 βˆ‚x βˆ‚x PROPN american_scientific_journal-1248 25 20 οΏ½ οΏ½ PROPN american_scientific_journal-1248 25 21 x x X american_scientific_journal-1248 25 22 . . PUNCT american_scientific_journal-1248 26 1 the the DET american_scientific_journal-1248 26 2 quantity quantity NOUN american_scientific_journal-1248 26 3 of of ADP american_scientific_journal-1248 26 4 heat heat NOUN american_scientific_journal-1248 26 5 flowing flow VERB american_scientific_journal-1248 26 6 out out ADP american_scientific_journal-1248 26 7 through through ADP american_scientific_journal-1248 26 8 the the DET american_scientific_journal-1248 26 9 side side NOUN american_scientific_journal-1248 26 10 𝐢𝐷 𝐢𝐷 NOUN american_scientific_journal-1248 26 11 per per ADP american_scientific_journal-1248 26 12 second second ADJ american_scientific_journal-1248 26 13 = = PUNCT american_scientific_journal-1248 26 14 βˆ’π‘˜π›Όπ›Όπœ•π‘₯ βˆ’π‘˜π›Όπ›Όπœ•π‘₯ PROPN american_scientific_journal-1248 26 15 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 16 πœ•π‘’ πœ•π‘’ PROPN american_scientific_journal-1248 26 17 πœ•π‘¦ πœ•π‘¦ NOUN american_scientific_journal-1248 26 18 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 19 𝑦+πœ•π‘¦ 𝑦+πœ•π‘¦ PROPN american_scientific_journal-1248 26 20 ( ( PUNCT american_scientific_journal-1248 26 21 1 1 NUM american_scientific_journal-1248 26 22 ) ) PUNCT american_scientific_journal-1248 26 23 and and CCONJ american_scientific_journal-1248 26 24 the the DET american_scientific_journal-1248 26 25 quantity quantity NOUN american_scientific_journal-1248 26 26 of of ADP american_scientific_journal-1248 26 27 heat heat NOUN american_scientific_journal-1248 26 28 flowing flow VERB american_scientific_journal-1248 26 29 out out ADP american_scientific_journal-1248 26 30 through through ADP american_scientific_journal-1248 26 31 the the DET american_scientific_journal-1248 26 32 side side NOUN american_scientific_journal-1248 26 33 𝐡𝐢 𝐡𝐢 PROPN american_scientific_journal-1248 26 34 per per ADP american_scientific_journal-1248 26 35 second second NOUN american_scientific_journal-1248 26 36 = = PUNCT american_scientific_journal-1248 26 37 βˆ’π‘˜π›Όπ›Όπœ•π‘¦ βˆ’π‘˜π›Όπ›Όπœ•π‘¦ NOUN american_scientific_journal-1248 26 38 οΏ½ οΏ½ NOUN american_scientific_journal-1248 26 39 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-1248 26 40 πœ•π‘₯ πœ•π‘₯ X american_scientific_journal-1248 26 41 οΏ½ οΏ½ X american_scientific_journal-1248 26 42 π‘₯+πœ•π‘₯ π‘₯+πœ•π‘₯ PROPN american_scientific_journal-1248 26 43 ( ( PUNCT american_scientific_journal-1248 26 44 2 2 NUM american_scientific_journal-1248 26 45 ) ) PUNCT american_scientific_journal-1248 26 46 hence hence ADV american_scientific_journal-1248 26 47 the the DET american_scientific_journal-1248 26 48 total total ADJ american_scientific_journal-1248 26 49 gain gain NOUN american_scientific_journal-1248 26 50 of of ADP american_scientific_journal-1248 26 51 heat heat NOUN american_scientific_journal-1248 26 52 by by ADP american_scientific_journal-1248 26 53 rectangular rectangular ADJ american_scientific_journal-1248 26 54 element element NOUN american_scientific_journal-1248 26 55 per per ADP american_scientific_journal-1248 26 56 second second ADJ american_scientific_journal-1248 26 57 179 179 NUM american_scientific_journal-1248 26 58 american american ADJ american_scientific_journal-1248 26 59 scientific scientific ADJ american_scientific_journal-1248 26 60 research research NOUN american_scientific_journal-1248 26 61 journal journal NOUN american_scientific_journal-1248 26 62 for for ADP american_scientific_journal-1248 26 63 engineering engineering NOUN american_scientific_journal-1248 26 64 , , PUNCT american_scientific_journal-1248 26 65 technology technology NOUN american_scientific_journal-1248 26 66 , , PUNCT american_scientific_journal-1248 26 67 and and CCONJ american_scientific_journal-1248 26 68 sciences science NOUN american_scientific_journal-1248 26 69 ( ( PUNCT american_scientific_journal-1248 26 70 asrjets asrjet NOUN american_scientific_journal-1248 26 71 ) ) PUNCT american_scientific_journal-1248 26 72 ( ( PUNCT american_scientific_journal-1248 26 73 2016 2016 NUM american_scientific_journal-1248 26 74 ) ) PUNCT american_scientific_journal-1248 26 75 volume volume NOUN american_scientific_journal-1248 26 76 16 16 NUM american_scientific_journal-1248 26 77 , , PUNCT american_scientific_journal-1248 26 78 no no DET american_scientific_journal-1248 26 79 1 1 NUM american_scientific_journal-1248 26 80 , , PUNCT american_scientific_journal-1248 26 81 pp pp ADV american_scientific_journal-1248 26 82 178 178 NUM american_scientific_journal-1248 26 83 - - SYM american_scientific_journal-1248 26 84 192 192 NUM american_scientific_journal-1248 26 85 = = PUNCT american_scientific_journal-1248 26 86 βˆ’π‘˜π›Όπ›Όπœ•π‘₯ βˆ’π‘˜π›Όπ›Όπœ•π‘₯ PROPN american_scientific_journal-1248 26 87 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 88 πœ•π‘’ πœ•π‘’ PROPN american_scientific_journal-1248 26 89 πœ•π‘¦ πœ•π‘¦ NOUN american_scientific_journal-1248 26 90 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 91 𝑦 𝑦 NOUN american_scientific_journal-1248 26 92 βˆ’ βˆ’ X american_scientific_journal-1248 26 93 π‘˜π›Όπ›Όπœ•π‘¦ π‘˜π›Όπ›Όπœ•π‘¦ PRON american_scientific_journal-1248 26 94 οΏ½ οΏ½ NOUN american_scientific_journal-1248 26 95 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-1248 26 96 πœ•π‘₯ πœ•π‘₯ X american_scientific_journal-1248 26 97 οΏ½ οΏ½ X american_scientific_journal-1248 26 98 π‘₯ π‘₯ PROPN american_scientific_journal-1248 26 99 + + CCONJ american_scientific_journal-1248 26 100 π‘˜π›Όπ›Όπœ•π‘₯ π‘˜π›Όπ›Όπœ•π‘₯ PROPN american_scientific_journal-1248 26 101 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 102 πœ•π‘’ πœ•π‘’ PROPN american_scientific_journal-1248 26 103 πœ•π‘¦ πœ•π‘¦ NOUN american_scientific_journal-1248 26 104 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 105 𝑦+πœ•π‘¦ 𝑦+πœ•π‘¦ PROPN american_scientific_journal-1248 26 106 + + CCONJ american_scientific_journal-1248 26 107 π‘˜π›Όπ›Όπœ•π‘¦ π‘˜π›Όπ›Όπœ•π‘¦ ADP american_scientific_journal-1248 26 108 οΏ½ οΏ½ NOUN american_scientific_journal-1248 26 109 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-1248 26 110 πœ•π‘₯ πœ•π‘₯ X american_scientific_journal-1248 26 111 οΏ½ οΏ½ NOUN american_scientific_journal-1248 26 112 π‘₯+πœ•π‘₯ π‘₯+πœ•π‘₯ NOUN american_scientific_journal-1248 26 113 = = SYM american_scientific_journal-1248 26 114 π‘˜π›Όπ›Όπœ•π‘₯ π‘˜π›Όπ›Όπœ•π‘₯ PROPN american_scientific_journal-1248 26 115 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 116 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 117 πœ•π‘’πœ•π‘₯ πœ•π‘’πœ•π‘₯ PROPN american_scientific_journal-1248 26 118 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 119 π‘₯+πœ•π‘₯ π‘₯+πœ•π‘₯ PROPN american_scientific_journal-1248 26 120 βˆ’ βˆ’ PROPN american_scientific_journal-1248 26 121 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 122 πœ•π‘’πœ•π‘₯ πœ•π‘’πœ•π‘₯ PROPN american_scientific_journal-1248 26 123 οΏ½ οΏ½ NOUN american_scientific_journal-1248 26 124 π‘₯ π‘₯ PRON american_scientific_journal-1248 26 125 πœ•π‘₯ πœ•π‘₯ X american_scientific_journal-1248 26 126 + + CCONJ american_scientific_journal-1248 26 127 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 128 πœ•π‘’πœ•π‘¦ πœ•π‘’πœ•π‘¦ PROPN american_scientific_journal-1248 26 129 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 130 𝑦+πœ•π‘¦ 𝑦+πœ•π‘¦ PROPN american_scientific_journal-1248 26 131 βˆ’ βˆ’ PROPN american_scientific_journal-1248 26 132 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 133 πœ•π‘’πœ•π‘¦ πœ•π‘’πœ•π‘¦ PROPN american_scientific_journal-1248 26 134 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 135 𝑦 𝑦 NOUN american_scientific_journal-1248 26 136 πœ•π‘¦ πœ•π‘¦ PROPN american_scientific_journal-1248 26 137 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 138 also also ADV american_scientific_journal-1248 26 139 the the DET american_scientific_journal-1248 26 140 rate rate NOUN american_scientific_journal-1248 26 141 of of ADP american_scientific_journal-1248 26 142 gain gain NOUN american_scientific_journal-1248 26 143 of of ADP american_scientific_journal-1248 26 144 heat heat NOUN american_scientific_journal-1248 26 145 by by ADP american_scientific_journal-1248 26 146 the the DET american_scientific_journal-1248 26 147 element element NOUN american_scientific_journal-1248 26 148 = = PUNCT american_scientific_journal-1248 26 149 πœŒπœŒπœ•π‘₯πœ•π‘¦π›Όπ›Όπ‘  πœŒπœŒπœ•π‘₯πœ•π‘¦π›Όπ›Όπ‘  ADJ american_scientific_journal-1248 26 150 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-1248 26 151 πœ•π‘‘ πœ•π‘‘ NOUN american_scientific_journal-1248 26 152 thus thus ADV american_scientific_journal-1248 26 153 equating equate VERB american_scientific_journal-1248 26 154 ( ( PUNCT american_scientific_journal-1248 26 155 1 1 NUM american_scientific_journal-1248 26 156 ) ) PUNCT american_scientific_journal-1248 26 157 and and CCONJ american_scientific_journal-1248 26 158 ( ( PUNCT american_scientific_journal-1248 26 159 2 2 X american_scientific_journal-1248 26 160 ) ) PUNCT american_scientific_journal-1248 26 161 = = SYM american_scientific_journal-1248 26 162 π‘˜π›Όπ›Όπœ•π‘₯πœ•π‘¦ π‘˜π›Όπ›Όπœ•π‘₯πœ•π‘¦ PROPN american_scientific_journal-1248 26 163 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 164 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 165 πœ•π‘’πœ•π‘₯ πœ•π‘’πœ•π‘₯ PROPN american_scientific_journal-1248 26 166 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 167 π‘₯+πœ•π‘₯ π‘₯+πœ•π‘₯ PROPN american_scientific_journal-1248 26 168 βˆ’ βˆ’ PROPN american_scientific_journal-1248 26 169 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 170 πœ•π‘’πœ•π‘¦ πœ•π‘’πœ•π‘¦ PROPN american_scientific_journal-1248 26 171 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 172 π‘₯ π‘₯ PRON american_scientific_journal-1248 26 173 πœ•π‘₯ πœ•π‘₯ X american_scientific_journal-1248 26 174 + + CCONJ american_scientific_journal-1248 26 175 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 176 πœ•π‘’πœ•π‘₯ πœ•π‘’πœ•π‘₯ PROPN american_scientific_journal-1248 26 177 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 178 𝑦+πœ•π‘¦ 𝑦+πœ•π‘¦ PROPN american_scientific_journal-1248 26 179 βˆ’ βˆ’ PROPN american_scientific_journal-1248 26 180 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 181 πœ•π‘’πœ•π‘¦ πœ•π‘’πœ•π‘¦ PROPN american_scientific_journal-1248 26 182 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 183 𝑦 𝑦 NOUN american_scientific_journal-1248 26 184 πœ•π‘¦ πœ•π‘¦ PROPN american_scientific_journal-1248 26 185 οΏ½ οΏ½ NOUN american_scientific_journal-1248 26 186 = = SYM american_scientific_journal-1248 26 187 πœŒπœŒπœ•π‘₯πœ•π‘¦π›Όπ›Όπ‘  πœŒπœŒπœ•π‘₯πœ•π‘¦π›Όπ›Όπ‘  ADJ american_scientific_journal-1248 26 188 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-1248 26 189 πœ•π‘‘ πœ•π‘‘ NOUN american_scientific_journal-1248 26 190 ( ( PUNCT american_scientific_journal-1248 26 191 3 3 X american_scientific_journal-1248 26 192 ) ) PUNCT american_scientific_journal-1248 26 193 dividing divide VERB american_scientific_journal-1248 26 194 both both DET american_scientific_journal-1248 26 195 sides side NOUN american_scientific_journal-1248 26 196 by by ADP american_scientific_journal-1248 26 197 π›Όπ›Όπœ•π‘₯πœ•π‘¦ π›Όπ›Όπœ•π‘₯πœ•π‘¦ PROPN american_scientific_journal-1248 26 198 and and CCONJ american_scientific_journal-1248 26 199 taking take VERB american_scientific_journal-1248 26 200 limit limit NOUN american_scientific_journal-1248 26 201 as as ADP american_scientific_journal-1248 26 202 β†’ β†’ X american_scientific_journal-1248 26 203 0 0 NUM american_scientific_journal-1248 26 204 , , PUNCT american_scientific_journal-1248 26 205 we we PRON american_scientific_journal-1248 26 206 get get VERB american_scientific_journal-1248 26 207 : : PUNCT american_scientific_journal-1248 26 208 π‘˜ π‘˜ DET american_scientific_journal-1248 26 209 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 210 πœ•2𝑒 πœ•2𝑒 PROPN american_scientific_journal-1248 26 211 πœ•π‘₯2 πœ•π‘₯2 NOUN american_scientific_journal-1248 26 212 + + CCONJ american_scientific_journal-1248 26 213 πœ•2𝑒 πœ•2𝑒 PROPN american_scientific_journal-1248 26 214 πœ•π‘¦2 πœ•π‘¦2 PROPN american_scientific_journal-1248 26 215 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 216 = = SYM american_scientific_journal-1248 26 217 πœŒπœŒπ‘  πœŒπœŒπ‘  NOUN american_scientific_journal-1248 26 218 πœ•π‘¦ πœ•π‘¦ PROPN american_scientific_journal-1248 26 219 πœ•π‘‘ πœ•π‘‘ ADJ american_scientific_journal-1248 26 220 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-1248 26 221 πœ•π‘‘ πœ•π‘‘ NOUN american_scientific_journal-1248 26 222 = = PROPN american_scientific_journal-1248 26 223 𝐢2 𝐢2 PROPN american_scientific_journal-1248 26 224 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 225 πœ• πœ• NOUN american_scientific_journal-1248 26 226 2𝑒 2𝑒 NUM american_scientific_journal-1248 26 227 πœ•π‘₯2 πœ•π‘₯2 NOUN american_scientific_journal-1248 26 228 + + CCONJ american_scientific_journal-1248 26 229 πœ•2𝑒 πœ•2𝑒 PROPN american_scientific_journal-1248 26 230 πœ•π‘¦2 πœ•π‘¦2 PROPN american_scientific_journal-1248 26 231 οΏ½ οΏ½ PROPN american_scientific_journal-1248 26 232 ( ( PUNCT american_scientific_journal-1248 26 233 4 4 NUM american_scientific_journal-1248 26 234 ) ) PUNCT american_scientific_journal-1248 26 235 where where SCONJ american_scientific_journal-1248 26 236 𝐢2 𝐢2 SCONJ american_scientific_journal-1248 26 237 = = X american_scientific_journal-1248 26 238 π‘˜/πœŒπœŒπ‘  π‘˜/πœŒπœŒπ‘  PROPN american_scientific_journal-1248 26 239 is be AUX american_scientific_journal-1248 26 240 the the DET american_scientific_journal-1248 26 241 diffusivity diffusivity NOUN american_scientific_journal-1248 26 242 . . PUNCT american_scientific_journal-1248 27 1 hence hence ADV american_scientific_journal-1248 27 2 the the DET american_scientific_journal-1248 27 3 equation equation NOUN american_scientific_journal-1248 27 4 ( ( PUNCT american_scientific_journal-1248 27 5 4 4 X american_scientific_journal-1248 27 6 ) ) PUNCT american_scientific_journal-1248 27 7 gives give VERB american_scientific_journal-1248 27 8 the the DET american_scientific_journal-1248 27 9 temperature temperature NOUN american_scientific_journal-1248 27 10 distribution distribution NOUN american_scientific_journal-1248 27 11 of of ADP american_scientific_journal-1248 27 12 plane plane NOUN american_scientific_journal-1248 27 13 in in ADP american_scientific_journal-1248 27 14 the the DET american_scientific_journal-1248 27 15 transit transit NOUN american_scientific_journal-1248 27 16 state state NOUN american_scientific_journal-1248 28 1 [ [ X american_scientific_journal-1248 28 2 1 1 NUM american_scientific_journal-1248 28 3 ] ] PUNCT american_scientific_journal-1248 28 4 . . PUNCT american_scientific_journal-1248 29 1 3 3 X american_scientific_journal-1248 29 2 . . X american_scientific_journal-1248 29 3 the the DET american_scientific_journal-1248 29 4 diffusion diffusion NOUN american_scientific_journal-1248 29 5 equation equation NOUN american_scientific_journal-1248 29 6 in in ADP american_scientific_journal-1248 29 7 two two NUM american_scientific_journal-1248 29 8 - - PUNCT american_scientific_journal-1248 29 9 dimensions dimension NOUN american_scientific_journal-1248 29 10 when when SCONJ american_scientific_journal-1248 29 11 cylindrical cylindrical ADJ american_scientific_journal-1248 29 12 co co NOUN american_scientific_journal-1248 29 13 - - NOUN american_scientific_journal-1248 29 14 ordinates ordinate NOUN american_scientific_journal-1248 29 15 𝜌𝜌,πœƒπœƒ 𝜌𝜌,πœƒπœƒ NOUN american_scientific_journal-1248 29 16 , , PUNCT american_scientific_journal-1248 29 17 𝑧 𝑧 PROPN american_scientific_journal-1248 29 18 are be AUX american_scientific_journal-1248 29 19 used use VERB american_scientific_journal-1248 29 20 see see NOUN american_scientific_journal-1248 30 1 [ [ X american_scientific_journal-1248 30 2 2],we 2],we PROPN american_scientific_journal-1248 30 3 have have VERB american_scientific_journal-1248 30 4 π‘₯ π‘₯ NOUN american_scientific_journal-1248 30 5 = = SYM american_scientific_journal-1248 30 6 𝜌𝜌 𝜌𝜌 PROPN american_scientific_journal-1248 30 7 cos cos PROPN american_scientific_journal-1248 30 8 πœƒπœƒ πœƒπœƒ PROPN american_scientific_journal-1248 30 9 , , PUNCT american_scientific_journal-1248 30 10 y y PROPN american_scientific_journal-1248 30 11 = = SYM american_scientific_journal-1248 30 12 ρ ρ PROPN american_scientific_journal-1248 30 13 sinπœƒπœƒ sinπœƒπœƒ NOUN american_scientific_journal-1248 30 14 , , PUNCT american_scientific_journal-1248 30 15 z z PROPN american_scientific_journal-1248 30 16 = = SYM american_scientific_journal-1248 30 17 zand zand PROPN american_scientific_journal-1248 30 18 the the DET american_scientific_journal-1248 30 19 equation equation NOUN american_scientific_journal-1248 30 20 of of ADP american_scientific_journal-1248 30 21 the the DET american_scientific_journal-1248 30 22 conduction conduction NOUN american_scientific_journal-1248 30 23 of of ADP american_scientific_journal-1248 30 24 heat heat NOUN american_scientific_journal-1248 30 25 comes come VERB american_scientific_journal-1248 30 26 πœ• πœ• NOUN american_scientific_journal-1248 30 27 2𝑣 2𝑣 NUM american_scientific_journal-1248 30 28 πœ•πœŒ2 πœ•πœŒ2 NOUN american_scientific_journal-1248 30 29 + + CCONJ american_scientific_journal-1248 30 30 1 1 NUM american_scientific_journal-1248 30 31 ρ ρ NUM american_scientific_journal-1248 30 32 πœ•π‘£ πœ•π‘£ NOUN american_scientific_journal-1248 30 33 βˆ‚Ο βˆ‚Ο PROPN american_scientific_journal-1248 31 1 + + CCONJ american_scientific_journal-1248 31 2 1 1 NUM american_scientific_journal-1248 31 3 ρ2 ρ2 VERB american_scientific_journal-1248 31 4 πœ•2𝑣 πœ•2𝑣 NOUN american_scientific_journal-1248 31 5 βˆ‚ΞΈ2 βˆ‚ΞΈ2 DET american_scientific_journal-1248 31 6 + + NUM american_scientific_journal-1248 31 7 πœ•2𝑣 πœ•2𝑣 PROPN american_scientific_journal-1248 31 8 βˆ‚z2 βˆ‚z2 NOUN american_scientific_journal-1248 31 9 = = SYM american_scientific_journal-1248 31 10 0 0 PUNCT american_scientific_journal-1248 31 11 ( ( PUNCT american_scientific_journal-1248 31 12 5 5 X american_scientific_journal-1248 31 13 ) ) PUNCT american_scientific_journal-1248 31 14 let let VERB american_scientific_journal-1248 31 15 us we PRON american_scientific_journal-1248 31 16 first first ADV american_scientific_journal-1248 31 17 consider consider VERB american_scientific_journal-1248 31 18 solutions solution NOUN american_scientific_journal-1248 31 19 which which PRON american_scientific_journal-1248 31 20 are be AUX american_scientific_journal-1248 31 21 independent independent ADJ american_scientific_journal-1248 31 22 of of ADP american_scientific_journal-1248 31 23 𝑧 𝑧 PRON american_scientific_journal-1248 31 24 𝐢(π‘₯ 𝐢(π‘₯ X american_scientific_journal-1248 31 25 + + NUM american_scientific_journal-1248 31 26 πœ•π‘₯,𝑦 πœ•π‘₯,𝑦 ADJ american_scientific_journal-1248 31 27 ) ) PUNCT american_scientific_journal-1248 31 28 + + NUM american_scientific_journal-1248 31 29 πœ•π‘¦ πœ•π‘¦ NOUN american_scientific_journal-1248 31 30 ) ) PUNCT american_scientific_journal-1248 31 31 𝐡(π‘₯ 𝐡(π‘₯ PROPN american_scientific_journal-1248 32 1 + + PUNCT american_scientific_journal-1248 32 2 πœ•π‘₯,𝑦 πœ•π‘₯,𝑦 PROPN american_scientific_journal-1248 32 3 ) ) PUNCT american_scientific_journal-1248 32 4 𝐷(π‘₯ 𝐷(π‘₯ NUM american_scientific_journal-1248 32 5 , , PUNCT american_scientific_journal-1248 32 6 𝑦 𝑦 NOUN american_scientific_journal-1248 32 7 ) ) PUNCT american_scientific_journal-1248 32 8 𝐴(π‘₯,𝑦 𝐴(π‘₯,𝑦 NOUN american_scientific_journal-1248 32 9 ) ) PUNCT american_scientific_journal-1248 32 10 figure1 figure1 PROPN american_scientific_journal-1248 33 1 180 180 NUM american_scientific_journal-1248 33 2 american american ADJ american_scientific_journal-1248 33 3 scientific scientific ADJ american_scientific_journal-1248 33 4 research research NOUN american_scientific_journal-1248 33 5 journal journal NOUN american_scientific_journal-1248 33 6 for for ADP american_scientific_journal-1248 33 7 engineering engineering NOUN american_scientific_journal-1248 33 8 , , PUNCT american_scientific_journal-1248 33 9 technology technology NOUN american_scientific_journal-1248 33 10 , , PUNCT american_scientific_journal-1248 33 11 and and CCONJ american_scientific_journal-1248 33 12 sciences science NOUN american_scientific_journal-1248 33 13 ( ( PUNCT american_scientific_journal-1248 33 14 asrjets asrjet NOUN american_scientific_journal-1248 33 15 ) ) PUNCT american_scientific_journal-1248 33 16 ( ( PUNCT american_scientific_journal-1248 33 17 2016 2016 NUM american_scientific_journal-1248 33 18 ) ) PUNCT american_scientific_journal-1248 33 19 volume volume NOUN american_scientific_journal-1248 33 20 16 16 NUM american_scientific_journal-1248 33 21 , , PUNCT american_scientific_journal-1248 33 22 no no DET american_scientific_journal-1248 33 23 1 1 NUM american_scientific_journal-1248 33 24 , , PUNCT american_scientific_journal-1248 33 25 pp pp ADV american_scientific_journal-1248 33 26 178 178 NUM american_scientific_journal-1248 33 27 - - SYM american_scientific_journal-1248 33 28 192 192 NUM american_scientific_journal-1248 33 29 πœ•2𝑣 πœ•2𝑣 NOUN american_scientific_journal-1248 33 30 πœ•πœŒπœŒ2 πœ•πœŒπœŒ2 NOUN american_scientific_journal-1248 33 31 + + CCONJ american_scientific_journal-1248 33 32 1 1 NUM american_scientific_journal-1248 33 33 ρ ρ NUM american_scientific_journal-1248 33 34 πœ•π‘£ πœ•π‘£ NOUN american_scientific_journal-1248 33 35 βˆ‚Ο βˆ‚Ο PROPN american_scientific_journal-1248 34 1 + + CCONJ american_scientific_journal-1248 34 2 1 1 NUM american_scientific_journal-1248 34 3 ρ2 ρ2 VERB american_scientific_journal-1248 34 4 πœ•2𝑣 πœ•2𝑣 NOUN american_scientific_journal-1248 34 5 βˆ‚ΞΈ2 βˆ‚ΞΈ2 X american_scientific_journal-1248 34 6 = = SYM american_scientific_journal-1248 34 7 0 0 NUM american_scientific_journal-1248 35 1 ( ( PUNCT american_scientific_journal-1248 35 2 6 6 NUM american_scientific_journal-1248 35 3 ) ) PUNCT american_scientific_journal-1248 35 4 the the DET american_scientific_journal-1248 35 5 equation equation NOUN american_scientific_journal-1248 35 6 for for ADP american_scientific_journal-1248 35 7 𝑅 𝑅 PROPN american_scientific_journal-1248 35 8 is be AUX american_scientific_journal-1248 35 9 πœ• πœ• NOUN american_scientific_journal-1248 35 10 2𝑅 2𝑅 PROPN american_scientific_journal-1248 35 11 πœ•πœŒ2 πœ•πœŒ2 NOUN american_scientific_journal-1248 35 12 + + CCONJ american_scientific_journal-1248 35 13 1 1 NUM american_scientific_journal-1248 35 14 ρ ρ NUM american_scientific_journal-1248 35 15 πœ•π‘… πœ•π‘… PROPN american_scientific_journal-1248 36 1 βˆ‚Ο βˆ‚Ο PROPN american_scientific_journal-1248 36 2 + + CCONJ american_scientific_journal-1248 36 3 1 1 NUM american_scientific_journal-1248 36 4 ρ2 ρ2 NOUN american_scientific_journal-1248 36 5 πœ•2𝑅 πœ•2𝑅 ADV american_scientific_journal-1248 36 6 βˆ‚ΞΈ2 βˆ‚ΞΈ2 PROPN american_scientific_journal-1248 36 7 = = SYM american_scientific_journal-1248 36 8 0 0 NUM american_scientific_journal-1248 36 9 ( ( PUNCT american_scientific_journal-1248 36 10 7 7 NUM american_scientific_journal-1248 36 11 ) ) PUNCT american_scientific_journal-1248 36 12 by by ADP american_scientific_journal-1248 36 13 separation separation NOUN american_scientific_journal-1248 36 14 of of ADP american_scientific_journal-1248 36 15 variables variable NOUN american_scientific_journal-1248 36 16 π‘…βˆ–βˆ–πœ“πœ“ π‘…βˆ–βˆ–πœ“πœ“ PROPN american_scientific_journal-1248 37 1 + + CCONJ american_scientific_journal-1248 37 2 1 1 NUM american_scientific_journal-1248 37 3 𝜌𝜌 𝜌𝜌 ADP american_scientific_journal-1248 37 4 π‘…βˆ–πœ“πœ“ π‘…βˆ–πœ“πœ“ NOUN american_scientific_journal-1248 37 5 + + CCONJ american_scientific_journal-1248 37 6 1 1 NUM american_scientific_journal-1248 37 7 𝜌𝜌2 𝜌𝜌2 NOUN american_scientific_journal-1248 37 8 π‘…πœ“πœ“βˆ–βˆ– π‘…πœ“πœ“βˆ–βˆ– NOUN american_scientific_journal-1248 37 9 = = SYM american_scientific_journal-1248 37 10 0 0 NUM american_scientific_journal-1248 37 11 ( ( PUNCT american_scientific_journal-1248 37 12 8) 8) NUM american_scientific_journal-1248 37 13 π‘…πœ“πœ“ π‘…πœ“πœ“ PROPN american_scientific_journal-1248 37 14 π‘…βˆ–βˆ– π‘…βˆ–βˆ– X american_scientific_journal-1248 37 15 𝑅 𝑅 PROPN american_scientific_journal-1248 37 16 + + ADP american_scientific_journal-1248 37 17 1 1 NUM american_scientific_journal-1248 37 18 𝜌𝜌 𝜌𝜌 ADP american_scientific_journal-1248 37 19 π‘…βˆ– π‘…βˆ– PROPN american_scientific_journal-1248 37 20 𝑅 𝑅 PROPN american_scientific_journal-1248 37 21 + + NOUN american_scientific_journal-1248 37 22 πœ“πœ“βˆ–βˆ– πœ“πœ“βˆ–βˆ– NOUN american_scientific_journal-1248 37 23 πœ“πœ“ πœ“πœ“ ADJ american_scientific_journal-1248 37 24 = = PROPN american_scientific_journal-1248 37 25 βˆ’πœ† βˆ’πœ† PROPN american_scientific_journal-1248 37 26 ( ( PUNCT american_scientific_journal-1248 37 27 9 9 NUM american_scientific_journal-1248 37 28 ) ) PUNCT american_scientific_journal-1248 37 29 𝜌𝜌2 𝜌𝜌2 VERB american_scientific_journal-1248 37 30 π‘…βˆ–βˆ– π‘…βˆ–βˆ– PROPN american_scientific_journal-1248 37 31 𝑅 𝑅 PROPN american_scientific_journal-1248 37 32 + + PROPN american_scientific_journal-1248 37 33 ρ ρ PROPN american_scientific_journal-1248 37 34 π‘…βˆ– π‘…βˆ– PROPN american_scientific_journal-1248 37 35 𝑅 𝑅 PROPN american_scientific_journal-1248 37 36 + + PROPN american_scientific_journal-1248 37 37 Ξ»2ρ2 Ξ»2ρ2 PROPN american_scientific_journal-1248 37 38 βˆ’ βˆ’ PROPN american_scientific_journal-1248 37 39 πœ“πœ“βˆ–βˆ– πœ“πœ“βˆ–βˆ– NOUN american_scientific_journal-1248 37 40 πœ“πœ“ πœ“πœ“ ADJ american_scientific_journal-1248 37 41 = = PUNCT american_scientific_journal-1248 37 42 πœ‡πœ‡2 πœ‡πœ‡2 NOUN american_scientific_journal-1248 37 43 ( ( PUNCT american_scientific_journal-1248 37 44 10 10 NUM american_scientific_journal-1248 37 45 ) ) PUNCT american_scientific_journal-1248 37 46 from from ADP american_scientific_journal-1248 37 47 ( ( PUNCT american_scientific_journal-1248 37 48 10),we 10),we NUM american_scientific_journal-1248 37 49 πœ‡πœ‡2πœ“πœ“ πœ‡πœ‡2πœ“πœ“ NUM american_scientific_journal-1248 37 50 = = SYM american_scientific_journal-1248 37 51 0 0 NUM american_scientific_journal-1248 38 1 ( ( PUNCT american_scientific_journal-1248 38 2 11 11 NUM american_scientific_journal-1248 38 3 ) ) PUNCT american_scientific_journal-1248 38 4 𝜌𝜌2π‘…βˆ–βˆ– 𝜌𝜌2π‘…βˆ–βˆ– NUM american_scientific_journal-1248 38 5 + + CCONJ american_scientific_journal-1248 38 6 πœŒπœŒπ‘…βˆ– πœŒπœŒπ‘…βˆ– NOUN american_scientific_journal-1248 39 1 + + CCONJ american_scientific_journal-1248 39 2 ( ( PUNCT american_scientific_journal-1248 39 3 Ξ»2ρ2 Ξ»2ρ2 ADP american_scientific_journal-1248 39 4 βˆ’ βˆ’ PROPN american_scientific_journal-1248 39 5 πœ‡πœ‡2)𝑅 πœ‡πœ‡2)𝑅 PROPN american_scientific_journal-1248 39 6 = = SYM american_scientific_journal-1248 39 7 0 0 NUM american_scientific_journal-1248 40 1 ( ( PUNCT american_scientific_journal-1248 40 2 12 12 NUM american_scientific_journal-1248 40 3 ) ) PUNCT american_scientific_journal-1248 40 4 the the DET american_scientific_journal-1248 40 5 solution solution NOUN american_scientific_journal-1248 40 6 of of ADP american_scientific_journal-1248 40 7 ( ( PUNCT american_scientific_journal-1248 40 8 11 11 NUM american_scientific_journal-1248 40 9 ) ) PUNCT american_scientific_journal-1248 40 10 and and CCONJ american_scientific_journal-1248 40 11 ( ( PUNCT american_scientific_journal-1248 40 12 12 12 NUM american_scientific_journal-1248 40 13 ) ) PUNCT american_scientific_journal-1248 40 14 given give VERB american_scientific_journal-1248 40 15 respectively respectively ADV american_scientific_journal-1248 40 16 by by ADP american_scientific_journal-1248 40 17 : : PUNCT american_scientific_journal-1248 40 18 πœ“πœ“(πœƒπœƒ πœ“πœ“(πœƒπœƒ PROPN american_scientific_journal-1248 40 19 ) ) PUNCT american_scientific_journal-1248 41 1 = = SYM american_scientific_journal-1248 41 2 𝐴1 𝐴1 PROPN american_scientific_journal-1248 41 3 cosπœ‡πœ‡πœƒπœƒ cosπœ‡πœ‡πœƒπœƒ VERB american_scientific_journal-1248 41 4 + + CCONJ american_scientific_journal-1248 41 5 𝐴2 𝐴2 PROPN american_scientific_journal-1248 41 6 sin sin NOUN american_scientific_journal-1248 41 7 πœ‡πœ‡πœƒπœƒ πœ‡πœ‡πœƒπœƒ NOUN american_scientific_journal-1248 41 8 ( ( PUNCT american_scientific_journal-1248 41 9 13 13 NUM american_scientific_journal-1248 41 10 ) ) PUNCT american_scientific_journal-1248 41 11 𝑅(𝜌𝜌 𝑅(𝜌𝜌 ADJ american_scientific_journal-1248 41 12 ) ) PUNCT american_scientific_journal-1248 41 13 = = SYM american_scientific_journal-1248 41 14 𝐡1π½πœ‡(πœ†πœŒπœŒ 𝐡1π½πœ‡(πœ†πœŒπœŒ NOUN american_scientific_journal-1248 41 15 ) ) PUNCT american_scientific_journal-1248 41 16 + + CCONJ american_scientific_journal-1248 42 1 𝐡2π‘Œπœ‡(πœ†πœŒπœŒ 𝐡2π‘Œπœ‡(πœ†πœŒπœŒ NOUN american_scientific_journal-1248 42 2 ) ) PUNCT american_scientific_journal-1248 42 3 ( ( PUNCT american_scientific_journal-1248 42 4 14 14 NUM american_scientific_journal-1248 42 5 ) ) PUNCT american_scientific_journal-1248 42 6 the the DET american_scientific_journal-1248 42 7 general general ADJ american_scientific_journal-1248 42 8 solution solution NOUN american_scientific_journal-1248 42 9 is be AUX american_scientific_journal-1248 42 10 𝑉(𝜌𝜌,πœƒπœƒ 𝑉(𝜌𝜌,πœƒπœƒ NOUN american_scientific_journal-1248 42 11 ) ) PUNCT american_scientific_journal-1248 42 12 = = PUNCT american_scientific_journal-1248 43 1 [ [ X american_scientific_journal-1248 43 2 𝐴1 𝐴1 X american_scientific_journal-1248 43 3 cos cos PROPN american_scientific_journal-1248 43 4 πœ‡πœ‡πœƒπœƒ πœ‡πœ‡πœƒπœƒ PROPN american_scientific_journal-1248 43 5 + + CCONJ american_scientific_journal-1248 43 6 𝐴2 𝐴2 PROPN american_scientific_journal-1248 43 7 sin sin PROPN american_scientific_journal-1248 43 8 πœ‡πœ‡πœƒπœƒ] πœ‡πœ‡πœƒπœƒ] NOUN american_scientific_journal-1248 43 9 οΏ½ οΏ½ NOUN american_scientific_journal-1248 43 10 𝐡1π½πœ‡(πœ†πœŒπœŒ 𝐡1π½πœ‡(πœ†πœŒπœŒ NOUN american_scientific_journal-1248 43 11 ) ) PUNCT american_scientific_journal-1248 43 12 + + CCONJ american_scientific_journal-1248 43 13 𝐡2π‘Œπœ‡(πœ†πœŒπœŒ 𝐡2π‘Œπœ‡(πœ†πœŒπœŒ PROPN american_scientific_journal-1248 43 14 ) ) PUNCT american_scientific_journal-1248 43 15 οΏ½ οΏ½ PROPN american_scientific_journal-1248 43 16 ( ( PUNCT american_scientific_journal-1248 43 17 15 15 NUM american_scientific_journal-1248 43 18 ) ) PUNCT american_scientific_journal-1248 43 19 since since SCONJ american_scientific_journal-1248 43 20 𝑉 𝑉 PROPN american_scientific_journal-1248 43 21 is be AUX american_scientific_journal-1248 43 22 bounded bound VERB american_scientific_journal-1248 43 23 𝜌𝜌 𝜌𝜌 ADP american_scientific_journal-1248 44 1 = = SYM american_scientific_journal-1248 44 2 0 0 PROPN american_scientific_journal-1248 44 3 then then ADV american_scientific_journal-1248 44 4 𝐡2 𝐡2 NOUN american_scientific_journal-1248 44 5 = = SYM american_scientific_journal-1248 44 6 0 0 PROPN american_scientific_journal-1248 44 7 gives give VERB american_scientific_journal-1248 44 8 𝑉(𝜌𝜌,πœƒπœƒ 𝑉(𝜌𝜌,πœƒπœƒ NOUN american_scientific_journal-1248 44 9 ) ) PUNCT american_scientific_journal-1248 44 10 = = PUNCT american_scientific_journal-1248 45 1 [ [ X american_scientific_journal-1248 45 2 𝐴1 𝐴1 X american_scientific_journal-1248 45 3 cos cos PROPN american_scientific_journal-1248 45 4 πœ‡πœ‡πœƒπœƒ πœ‡πœ‡πœƒπœƒ PROPN american_scientific_journal-1248 45 5 + + CCONJ american_scientific_journal-1248 45 6 𝐴2 𝐴2 PROPN american_scientific_journal-1248 45 7 sin sin PROPN american_scientific_journal-1248 45 8 πœ‡πœ‡πœƒπœƒ] πœ‡πœ‡πœƒπœƒ] NOUN american_scientific_journal-1248 45 9 οΏ½ οΏ½ NOUN american_scientific_journal-1248 45 10 𝐡1π½πœ‡(πœ†πœŒπœŒ 𝐡1π½πœ‡(πœ†πœŒπœŒ NOUN american_scientific_journal-1248 45 11 ) ) PUNCT american_scientific_journal-1248 45 12 οΏ½ οΏ½ PROPN american_scientific_journal-1248 45 13 ( ( PUNCT american_scientific_journal-1248 45 14 16 16 NUM american_scientific_journal-1248 45 15 ) ) PUNCT american_scientific_journal-1248 45 16 an an DET american_scientific_journal-1248 45 17 initial initial ADJ american_scientific_journal-1248 45 18 distribution distribution NOUN american_scientific_journal-1248 45 19 of of ADP american_scientific_journal-1248 45 20 concentration concentration NOUN american_scientific_journal-1248 45 21 , , PUNCT american_scientific_journal-1248 45 22 let let VERB american_scientific_journal-1248 45 23 𝑉 𝑉 PROPN american_scientific_journal-1248 45 24 = = SYM american_scientific_journal-1248 45 25 𝑓(𝜌𝜌,πœƒπœƒ 𝑓(𝜌𝜌,πœƒπœƒ PROPN american_scientific_journal-1248 45 26 ) ) PUNCT american_scientific_journal-1248 45 27 when when SCONJ american_scientific_journal-1248 45 28 𝑑 𝑑 X american_scientific_journal-1248 45 29 = = VERB american_scientific_journal-1248 45 30 0 0 NUM american_scientific_journal-1248 45 31 , , PUNCT american_scientific_journal-1248 45 32 we we PRON american_scientific_journal-1248 45 33 may may AUX american_scientific_journal-1248 45 34 try try VERB american_scientific_journal-1248 45 35 to to PART american_scientific_journal-1248 45 36 satisfy satisfy VERB american_scientific_journal-1248 45 37 the the DET american_scientific_journal-1248 45 38 condition condition NOUN american_scientific_journal-1248 45 39 , , PUNCT american_scientific_journal-1248 45 40 when when SCONJ american_scientific_journal-1248 45 41 πœ‡πœ‡ πœ‡πœ‡ ADP american_scientific_journal-1248 45 42 = = PUNCT american_scientific_journal-1248 45 43 𝑛 𝑛 PROPN american_scientific_journal-1248 45 44 and and CCONJ american_scientific_journal-1248 45 45 v v NOUN american_scientific_journal-1248 45 46 must must AUX american_scientific_journal-1248 45 47 have have VERB american_scientific_journal-1248 45 48 period period NOUN american_scientific_journal-1248 45 49 2πœ‹πœ‹ 2πœ‹πœ‹ VERB american_scientific_journal-1248 45 50 in in ADP american_scientific_journal-1248 45 51 the the DET american_scientific_journal-1248 45 52 variable variable ADJ american_scientific_journal-1248 45 53 πœƒπœƒ. πœƒπœƒ. NOUN american_scientific_journal-1248 45 54 ( ( PUNCT american_scientific_journal-1248 45 55 17 17 NUM american_scientific_journal-1248 45 56 ) ) PUNCT american_scientific_journal-1248 45 57 𝑉(𝜌𝜌,πœƒπœƒ 𝑉(𝜌𝜌,πœƒπœƒ NOUN american_scientific_journal-1248 45 58 ) ) PUNCT american_scientific_journal-1248 45 59 = = PUNCT american_scientific_journal-1248 46 1 [ [ X american_scientific_journal-1248 46 2 𝐴1 𝐴1 X american_scientific_journal-1248 46 3 cos cos PROPN american_scientific_journal-1248 46 4 πœ‡πœ‡πœƒπœƒ πœ‡πœ‡πœƒπœƒ PROPN american_scientific_journal-1248 46 5 + + CCONJ american_scientific_journal-1248 46 6 𝐴2 𝐴2 PROPN american_scientific_journal-1248 46 7 sin sin PROPN american_scientific_journal-1248 46 8 πœ‡πœ‡πœƒπœƒ] πœ‡πœ‡πœƒπœƒ] NOUN american_scientific_journal-1248 46 9 οΏ½ οΏ½ NOUN american_scientific_journal-1248 46 10 𝐡1π½πœ‡(πœ†πœŒπœŒ 𝐡1π½πœ‡(πœ†πœŒπœŒ NOUN american_scientific_journal-1248 46 11 ) ) PUNCT american_scientific_journal-1248 46 12 οΏ½ οΏ½ PROPN american_scientific_journal-1248 46 13 now now ADV american_scientific_journal-1248 46 14 by by ADP american_scientific_journal-1248 46 15 substituting substitute VERB american_scientific_journal-1248 46 16 boundary boundary ADJ american_scientific_journal-1248 46 17 condition condition NOUN american_scientific_journal-1248 46 18 𝑉(1 𝑉(1 NUM american_scientific_journal-1248 46 19 , , PUNCT american_scientific_journal-1248 46 20 πœƒπœƒ πœƒπœƒ PRON american_scientific_journal-1248 46 21 ) ) PUNCT american_scientific_journal-1248 46 22 = = NOUN american_scientific_journal-1248 46 23 0 0 SYM american_scientific_journal-1248 46 24 181 181 NUM american_scientific_journal-1248 46 25 american american ADJ american_scientific_journal-1248 46 26 scientific scientific ADJ american_scientific_journal-1248 46 27 research research NOUN american_scientific_journal-1248 46 28 journal journal NOUN american_scientific_journal-1248 46 29 for for ADP american_scientific_journal-1248 46 30 engineering engineering NOUN american_scientific_journal-1248 46 31 , , PUNCT american_scientific_journal-1248 46 32 technology technology NOUN american_scientific_journal-1248 46 33 , , PUNCT american_scientific_journal-1248 46 34 and and CCONJ american_scientific_journal-1248 46 35 sciences science NOUN american_scientific_journal-1248 46 36 ( ( PUNCT american_scientific_journal-1248 46 37 asrjets asrjet NOUN american_scientific_journal-1248 46 38 ) ) PUNCT american_scientific_journal-1248 46 39 ( ( PUNCT american_scientific_journal-1248 46 40 2016 2016 NUM american_scientific_journal-1248 46 41 ) ) PUNCT american_scientific_journal-1248 46 42 volume volume NOUN american_scientific_journal-1248 46 43 16 16 NUM american_scientific_journal-1248 46 44 , , PUNCT american_scientific_journal-1248 46 45 no no DET american_scientific_journal-1248 46 46 1 1 NUM american_scientific_journal-1248 46 47 , , PUNCT american_scientific_journal-1248 46 48 pp pp ADV american_scientific_journal-1248 46 49 178 178 NUM american_scientific_journal-1248 46 50 - - SYM american_scientific_journal-1248 46 51 192 192 NUM american_scientific_journal-1248 46 52 𝑉(𝜌𝜌,πœƒπœƒ 𝑉(𝜌𝜌,πœƒπœƒ NOUN american_scientific_journal-1248 46 53 ) ) PUNCT american_scientific_journal-1248 46 54 = = PUNCT american_scientific_journal-1248 47 1 [ [ X american_scientific_journal-1248 47 2 𝐴1 𝐴1 PROPN american_scientific_journal-1248 47 3 cosπ‘›πœƒπœƒ cosπ‘›πœƒπœƒ NOUN american_scientific_journal-1248 47 4 + + CCONJ american_scientific_journal-1248 47 5 𝐴2 𝐴2 PROPN american_scientific_journal-1248 47 6 sinπ‘›πœƒπœƒ][𝐡𝐽𝑛(πœ†πœŒπœŒ sinπ‘›πœƒπœƒ][𝐡𝐽𝑛(πœ†πœŒπœŒ NOUN american_scientific_journal-1248 47 7 ) ) PUNCT american_scientific_journal-1248 47 8 ] ] PUNCT american_scientific_journal-1248 47 9 ( ( PUNCT american_scientific_journal-1248 47 10 18 18 NUM american_scientific_journal-1248 47 11 ) ) PUNCT american_scientific_journal-1248 47 12 now now ADV american_scientific_journal-1248 47 13 equation equation NOUN american_scientific_journal-1248 47 14 ( ( PUNCT american_scientific_journal-1248 47 15 17 17 NUM american_scientific_journal-1248 47 16 ) ) PUNCT american_scientific_journal-1248 47 17 can can AUX american_scientific_journal-1248 47 18 be be AUX american_scientific_journal-1248 47 19 expressed express VERB american_scientific_journal-1248 47 20 us we PRON american_scientific_journal-1248 47 21 𝑉(𝜌𝜌,πœƒπœƒ 𝑉(𝜌𝜌,πœƒπœƒ NOUN american_scientific_journal-1248 47 22 ) ) PUNCT american_scientific_journal-1248 47 23 = = PUNCT american_scientific_journal-1248 48 1 [ [ X american_scientific_journal-1248 48 2 𝐴 𝐴 PROPN american_scientific_journal-1248 48 3 cosπ‘›πœƒπœƒ cosπ‘›πœƒπœƒ NOUN american_scientific_journal-1248 48 4 + + X american_scientific_journal-1248 48 5 𝐡 𝐡 NOUN american_scientific_journal-1248 48 6 sinπ‘›πœƒπœƒ][𝐽𝑛(πœ†π‘šπ‘›πœŒπœŒ sinπ‘›πœƒπœƒ][𝐽𝑛(πœ†π‘šπ‘›πœŒπœŒ PROPN american_scientific_journal-1248 48 7 ) ) PUNCT american_scientific_journal-1248 48 8 ] ] PUNCT american_scientific_journal-1248 48 9 ( ( PUNCT american_scientific_journal-1248 48 10 19 19 NUM american_scientific_journal-1248 48 11 ) ) PUNCT american_scientific_journal-1248 48 12 by by ADP american_scientific_journal-1248 48 13 using use VERB american_scientific_journal-1248 48 14 super super ADJ american_scientific_journal-1248 48 15 position position NOUN american_scientific_journal-1248 48 16 𝑉(𝜌𝜌,πœƒπœƒ 𝑉(𝜌𝜌,πœƒπœƒ NOUN american_scientific_journal-1248 48 17 ) ) PUNCT american_scientific_journal-1248 48 18 = = PUNCT american_scientific_journal-1248 48 19 βˆ‘ βˆ‘ PUNCT american_scientific_journal-1248 48 20 βˆ‘ βˆ‘ PUNCT american_scientific_journal-1248 48 21 [ [ X american_scientific_journal-1248 48 22 amncosπ‘›πœƒπœƒ amncosπ‘›πœƒπœƒ NOUN american_scientific_journal-1248 48 23 + + CCONJ american_scientific_journal-1248 48 24 bmn bmn NOUN american_scientific_journal-1248 48 25 sinπ‘›πœƒπœƒ]𝐽𝑛(πœ†π‘šπ‘›πœŒπœŒ)∞ sinπ‘›πœƒπœƒ]𝐽𝑛(πœ†π‘šπ‘›πœŒπœŒ)∞ NOUN american_scientific_journal-1248 48 26 m=0 m=0 PROPN american_scientific_journal-1248 48 27 ∞ ∞ NUM american_scientific_journal-1248 48 28 𝑛=0 𝑛=0 NOUN american_scientific_journal-1248 48 29 ( ( PUNCT american_scientific_journal-1248 48 30 20 20 NUM american_scientific_journal-1248 48 31 ) ) PUNCT american_scientific_journal-1248 48 32 we we PRON american_scientific_journal-1248 48 33 let let VERB american_scientific_journal-1248 48 34 𝑉 𝑉 PROPN american_scientific_journal-1248 48 35 = = SYM american_scientific_journal-1248 48 36 𝑓(𝜌𝜌,πœƒπœƒ 𝑓(𝜌𝜌,πœƒπœƒ PROPN american_scientific_journal-1248 48 37 ) ) PUNCT american_scientific_journal-1248 48 38 gives give VERB american_scientific_journal-1248 48 39 𝑓(𝜌𝜌,πœƒπœƒ 𝑓(𝜌𝜌,πœƒπœƒ NOUN american_scientific_journal-1248 48 40 ) ) PUNCT american_scientific_journal-1248 49 1 = = SYM american_scientific_journal-1248 49 2 βˆ‘ βˆ‘ PUNCT american_scientific_journal-1248 49 3 π·π‘›βˆž π·π‘›βˆž PROPN american_scientific_journal-1248 49 4 n=0 n=0 NUM american_scientific_journal-1248 49 5 cosπ‘›πœƒπœƒ cosπ‘›πœƒπœƒ NOUN american_scientific_journal-1248 50 1 + + CCONJ american_scientific_journal-1248 50 2 𝐸𝑛 𝐸𝑛 ADP american_scientific_journal-1248 50 3 sinπ‘›πœƒπœƒ(21)formula sinπ‘›πœƒπœƒ(21)formula PROPN american_scientific_journal-1248 50 4 ( ( PUNCT american_scientific_journal-1248 50 5 21 21 NUM american_scientific_journal-1248 50 6 ) ) PUNCT american_scientific_journal-1248 50 7 represent represent VERB american_scientific_journal-1248 50 8 fourier fourier ADJ american_scientific_journal-1248 50 9 series series PROPN american_scientific_journal-1248 50 10 of𝑓(𝜌𝜌,πœƒπœƒ of𝑓(𝜌𝜌,πœƒπœƒ NOUN american_scientific_journal-1248 50 11 ) ) PUNCT american_scientific_journal-1248 50 12 with with ADP american_scientific_journal-1248 50 13 period period NOUN american_scientific_journal-1248 50 14 2πœ‹πœ‹ 2πœ‹πœ‹ VERB american_scientific_journal-1248 51 1 𝐷𝑛 𝐷𝑛 NOUN american_scientific_journal-1248 51 2 = = NOUN american_scientific_journal-1248 51 3 1 1 NUM american_scientific_journal-1248 51 4 πœ‹ πœ‹ NOUN american_scientific_journal-1248 51 5 ∫ ∫ PROPN american_scientific_journal-1248 51 6 𝑓(𝜌𝜌,πœƒπœƒ 𝑓(𝜌𝜌,πœƒπœƒ PROPN american_scientific_journal-1248 51 7 ) ) PUNCT american_scientific_journal-1248 51 8 cosπ‘›πœƒπœƒ cosπ‘›πœƒπœƒ ADJ american_scientific_journal-1248 51 9 π‘‘πœƒπœƒ2πœ‹ π‘‘πœƒπœƒ2πœ‹ PROPN american_scientific_journal-1248 51 10 0 0 SYM american_scientific_journal-1248 51 11 and𝐸𝑛 and𝐸𝑛 PROPN american_scientific_journal-1248 51 12 = = SYM american_scientific_journal-1248 51 13 1 1 NUM american_scientific_journal-1248 51 14 πœ‹ πœ‹ NOUN american_scientific_journal-1248 51 15 ∫ ∫ PROPN american_scientific_journal-1248 51 16 𝑓(𝜌𝜌,πœƒπœƒ 𝑓(𝜌𝜌,πœƒπœƒ PROPN american_scientific_journal-1248 51 17 ) ) PUNCT american_scientific_journal-1248 51 18 sinπ‘›πœƒπœƒ sinπ‘›πœƒπœƒ VERB american_scientific_journal-1248 51 19 π‘‘πœƒπœƒ2πœ‹ π‘‘πœƒπœƒ2πœ‹ PROPN american_scientific_journal-1248 51 20 0 0 NUM american_scientific_journal-1248 51 21 for for ADP american_scientific_journal-1248 51 22 𝑛 𝑛 PRON american_scientific_journal-1248 51 23 = = SYM american_scientific_journal-1248 51 24 1,2 1,2 NUM american_scientific_journal-1248 51 25 , , PUNCT american_scientific_journal-1248 51 26 … … PUNCT american_scientific_journal-1248 51 27 we we PRON american_scientific_journal-1248 51 28 have have VERB american_scientific_journal-1248 51 29 π΄π‘šπ‘› π΄π‘šπ‘› NOUN american_scientific_journal-1248 51 30 = = SYM american_scientific_journal-1248 51 31 2 2 NUM american_scientific_journal-1248 51 32 𝐽𝑛+1 𝐽𝑛+1 X american_scientific_journal-1248 51 33 2 2 NUM american_scientific_journal-1248 51 34 ( ( PUNCT american_scientific_journal-1248 51 35 πœ†π‘šπ‘›)∫ πœ†π‘šπ‘›)∫ NOUN american_scientific_journal-1248 51 36 πœŒπœŒπ½π‘›(πœ†π‘šπ‘›πœŒπœŒ)π·π‘›π‘‘πœŒπœŒ πœŒπœŒπ½π‘›(πœ†π‘šπ‘›πœŒπœŒ)π·π‘›π‘‘πœŒπœŒ NUM american_scientific_journal-1248 51 37 1 1 NUM american_scientific_journal-1248 51 38 0 0 NUM american_scientific_journal-1248 51 39 and and CCONJ american_scientific_journal-1248 51 40 π΅π‘šπ‘› π΅π‘šπ‘› PROPN american_scientific_journal-1248 51 41 = = SYM american_scientific_journal-1248 51 42 2 2 NUM american_scientific_journal-1248 51 43 𝐽𝑛+1 𝐽𝑛+1 X american_scientific_journal-1248 51 44 2 2 NUM american_scientific_journal-1248 51 45 ( ( PUNCT american_scientific_journal-1248 51 46 πœ†π‘šπ‘›)∫ πœ†π‘šπ‘›)∫ NOUN american_scientific_journal-1248 51 47 πœŒπœŒπ½π‘›(πœ†π‘šπ‘›πœŒπœŒ)πΈπ‘›π‘‘πœŒπœŒ πœŒπœŒπ½π‘›(πœ†π‘šπ‘›πœŒπœŒ)πΈπ‘›π‘‘πœŒπœŒ X american_scientific_journal-1248 51 48 1 1 NUM american_scientific_journal-1248 51 49 0 0 NUM american_scientific_journal-1248 51 50 4 4 NUM american_scientific_journal-1248 51 51 . . PUNCT american_scientific_journal-1248 52 1 the the DET american_scientific_journal-1248 52 2 elementary elementary ADJ american_scientific_journal-1248 52 3 solution solution NOUN american_scientific_journal-1248 52 4 if if SCONJ american_scientific_journal-1248 52 5 we we PRON american_scientific_journal-1248 52 6 solve solve VERB american_scientific_journal-1248 52 7 βˆ‡2π‘ˆ βˆ‡2π‘ˆ PROPN american_scientific_journal-1248 52 8 = = SYM american_scientific_journal-1248 52 9 0 0 NUM american_scientific_journal-1248 52 10 in in ADP american_scientific_journal-1248 52 11 spherical spherical ADJ american_scientific_journal-1248 52 12 coordinates coordinate NOUN american_scientific_journal-1248 52 13 ( ( PUNCT american_scientific_journal-1248 52 14 π‘Ÿ,πœƒπœƒ,πœ™πœ™ π‘Ÿ,πœƒπœƒ,πœ™πœ™ NOUN american_scientific_journal-1248 52 15 ) ) PUNCT american_scientific_journal-1248 52 16 βˆ‡2π‘ˆ βˆ‡2π‘ˆ NOUN american_scientific_journal-1248 52 17 = = SYM american_scientific_journal-1248 52 18 1 1 NUM american_scientific_journal-1248 52 19 β„Ž1β„Ž2β„Ž3 β„Ž1β„Ž2β„Ž3 PROPN american_scientific_journal-1248 52 20 [ [ PUNCT american_scientific_journal-1248 52 21 πœ• πœ• NOUN american_scientific_journal-1248 52 22 πœ•π‘ž1 πœ•π‘ž1 NOUN american_scientific_journal-1248 52 23 [ [ PUNCT american_scientific_journal-1248 52 24 β„Ž2β„Ž3 β„Ž2β„Ž3 PROPN american_scientific_journal-1248 52 25 β„Ž1 β„Ž1 PROPN american_scientific_journal-1248 52 26 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-1248 52 27 πœ•π‘ž1 πœ•π‘ž1 NOUN american_scientific_journal-1248 52 28 ] ] X american_scientific_journal-1248 53 1 + + CCONJ american_scientific_journal-1248 53 2 πœ• πœ• PROPN american_scientific_journal-1248 53 3 πœ•π‘ž2 πœ•π‘ž2 PROPN american_scientific_journal-1248 53 4 [ [ PUNCT american_scientific_journal-1248 53 5 β„Ž1β„Ž3 β„Ž1β„Ž3 VERB american_scientific_journal-1248 53 6 β„Ž2 β„Ž2 ADJ american_scientific_journal-1248 53 7 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-1248 53 8 πœ•π‘ž2 πœ•π‘ž2 NOUN american_scientific_journal-1248 53 9 ] ] PUNCT american_scientific_journal-1248 54 1 + + CCONJ american_scientific_journal-1248 54 2 πœ• πœ• NOUN american_scientific_journal-1248 54 3 πœ•π‘ž3 πœ•π‘ž3 NOUN american_scientific_journal-1248 54 4 [ [ PUNCT american_scientific_journal-1248 54 5 β„Ž2β„Ž1 β„Ž2β„Ž1 ADV american_scientific_journal-1248 54 6 β„Ž3 β„Ž3 ADJ american_scientific_journal-1248 54 7 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-1248 54 8 πœ•π‘ž3 πœ•π‘ž3 NOUN american_scientific_journal-1248 54 9 ] ] PUNCT american_scientific_journal-1248 54 10 whereβ„Ž1 whereβ„Ž1 NOUN american_scientific_journal-1248 54 11 = = SYM american_scientific_journal-1248 54 12 1 1 NUM american_scientific_journal-1248 54 13 , , PUNCT american_scientific_journal-1248 54 14 β„Ž2 β„Ž2 NOUN american_scientific_journal-1248 54 15 = = PUNCT american_scientific_journal-1248 54 16 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-1248 54 17 and and CCONJ american_scientific_journal-1248 54 18 β„Ž3 β„Ž3 NOUN american_scientific_journal-1248 54 19 = = SYM american_scientific_journal-1248 54 20 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-1248 54 21 sinπœƒπœƒ sinπœƒπœƒ NOUN american_scientific_journal-1248 54 22 and and CCONJ american_scientific_journal-1248 54 23 π‘ž1 π‘ž1 PROPN american_scientific_journal-1248 54 24 = = SYM american_scientific_journal-1248 54 25 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-1248 54 26 , , PUNCT american_scientific_journal-1248 54 27 π‘ž2 π‘ž2 PROPN american_scientific_journal-1248 54 28 = = SYM american_scientific_journal-1248 54 29 πœƒπœƒ πœƒπœƒ INTJ american_scientific_journal-1248 54 30 , , PUNCT american_scientific_journal-1248 54 31 and and CCONJ american_scientific_journal-1248 54 32 π‘ž3 π‘ž3 PROPN american_scientific_journal-1248 54 33 = = PROPN american_scientific_journal-1248 54 34 πœ™πœ™.if πœ™πœ™.if PROPN american_scientific_journal-1248 54 35 π‘ˆ π‘ˆ PROPN american_scientific_journal-1248 54 36 is be AUX american_scientific_journal-1248 54 37 independent independent ADJ american_scientific_journal-1248 54 38 ofπœ™πœ™ ofπœ™πœ™ NOUN american_scientific_journal-1248 54 39 with with ADP american_scientific_journal-1248 54 40 boundary boundary ADJ american_scientific_journal-1248 54 41 condition condition NOUN american_scientific_journal-1248 54 42 π‘ˆ(1,πœƒπœƒ π‘ˆ(1,πœƒπœƒ NOUN american_scientific_journal-1248 54 43 ) ) PUNCT american_scientific_journal-1248 54 44 = = SYM american_scientific_journal-1248 54 45 𝑓(πœƒπœƒ 𝑓(πœƒπœƒ NOUN american_scientific_journal-1248 54 46 ) ) PUNCT american_scientific_journal-1248 54 47 the the DET american_scientific_journal-1248 54 48 solution solution NOUN american_scientific_journal-1248 54 49 of of ADP american_scientific_journal-1248 54 50 βˆ‡2π‘ˆ βˆ‡2π‘ˆ PROPN american_scientific_journal-1248 54 51 = = SYM american_scientific_journal-1248 54 52 0 0 NUM american_scientific_journal-1248 54 53 in in ADP american_scientific_journal-1248 54 54 spherical spherical ADJ american_scientific_journal-1248 54 55 coordinates coordinate NOUN american_scientific_journal-1248 54 56 which which PRON american_scientific_journal-1248 54 57 is be AUX american_scientific_journal-1248 54 58 independent independent ADJ american_scientific_journal-1248 54 59 of of ADP american_scientific_journal-1248 54 60 πœ™πœ™ πœ™πœ™ NOUN american_scientific_journal-1248 54 61 expressed express VERB american_scientific_journal-1248 54 62 as as ADP american_scientific_journal-1248 54 63 : : PUNCT american_scientific_journal-1248 54 64 π‘Ÿ2 π‘Ÿ2 ADJ american_scientific_journal-1248 54 65 πœ•2𝑒 πœ•2𝑒 ADJ american_scientific_journal-1248 54 66 πœ•π‘Ÿ2 πœ•π‘Ÿ2 NOUN american_scientific_journal-1248 54 67 + + CCONJ american_scientific_journal-1248 54 68 2π‘Ÿ 2π‘Ÿ NUM american_scientific_journal-1248 54 69 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-1248 54 70 πœ•π‘Ÿ πœ•π‘Ÿ NOUN american_scientific_journal-1248 55 1 + + CCONJ american_scientific_journal-1248 55 2 1 1 NUM american_scientific_journal-1248 55 3 sin sin NOUN american_scientific_journal-1248 55 4 πœƒπœƒ πœƒπœƒ PROPN american_scientific_journal-1248 55 5 πœ• πœ• PROPN american_scientific_journal-1248 55 6 πœ•πœƒπœƒ πœ•πœƒπœƒ X american_scientific_journal-1248 55 7 οΏ½ οΏ½ PROPN american_scientific_journal-1248 55 8 sinπœƒπœƒ sinπœƒπœƒ AUX american_scientific_journal-1248 55 9 πœ•π‘’ πœ•π‘’ NOUN american_scientific_journal-1248 55 10 πœ•πœƒπœƒ πœ•πœƒπœƒ X american_scientific_journal-1248 55 11 οΏ½ οΏ½ PROPN american_scientific_journal-1248 55 12 = = SYM american_scientific_journal-1248 55 13 0 0 NUM american_scientific_journal-1248 55 14 ( ( PUNCT american_scientific_journal-1248 55 15 22 22 NUM american_scientific_journal-1248 55 16 ) ) PUNCT american_scientific_journal-1248 55 17 take take VERB american_scientific_journal-1248 55 18 𝑒(π‘Ÿ,πœƒπœƒ 𝑒(π‘Ÿ,πœƒπœƒ NOUN american_scientific_journal-1248 55 19 ) ) PUNCT american_scientific_journal-1248 55 20 = = SYM american_scientific_journal-1248 55 21 𝑅(π‘Ÿ)πœ“πœ“(πœƒπœƒ)(23 𝑅(π‘Ÿ)πœ“πœ“(πœƒπœƒ)(23 NOUN american_scientific_journal-1248 55 22 ) ) PUNCT american_scientific_journal-1248 55 23 we we PRON american_scientific_journal-1248 55 24 get get VERB american_scientific_journal-1248 55 25 π‘Ÿ2𝑅\\πœ“πœ“ π‘Ÿ2𝑅\\πœ“πœ“ PRON american_scientific_journal-1248 55 26 + + ADJ american_scientific_journal-1248 55 27 2π‘Ÿπ‘…\πœ“πœ“ 2π‘Ÿπ‘…\πœ“πœ“ NUM american_scientific_journal-1248 55 28 + + CCONJ american_scientific_journal-1248 55 29 𝑅 𝑅 PROPN american_scientific_journal-1248 55 30 sinπœƒπœƒ sinπœƒπœƒ NOUN american_scientific_journal-1248 55 31 πœ• πœ• PROPN american_scientific_journal-1248 55 32 πœ•πœƒπœƒ πœ•πœƒπœƒ X american_scientific_journal-1248 55 33 οΏ½ οΏ½ PROPN american_scientific_journal-1248 55 34 sin sin NOUN american_scientific_journal-1248 55 35 πœƒπœƒ πœƒπœƒ PROPN american_scientific_journal-1248 55 36 πœ“πœ“\ πœ“πœ“\ NOUN american_scientific_journal-1248 55 37 οΏ½ οΏ½ NOUN american_scientific_journal-1248 55 38 = = SYM american_scientific_journal-1248 55 39 0 0 NUM american_scientific_journal-1248 55 40 182 182 NUM american_scientific_journal-1248 55 41 american american ADJ american_scientific_journal-1248 55 42 scientific scientific ADJ american_scientific_journal-1248 55 43 research research NOUN american_scientific_journal-1248 55 44 journal journal NOUN american_scientific_journal-1248 55 45 for for ADP american_scientific_journal-1248 55 46 engineering engineering NOUN american_scientific_journal-1248 55 47 , , PUNCT american_scientific_journal-1248 55 48 technology technology NOUN american_scientific_journal-1248 55 49 , , PUNCT american_scientific_journal-1248 55 50 and and CCONJ american_scientific_journal-1248 55 51 sciences science NOUN american_scientific_journal-1248 55 52 ( ( PUNCT american_scientific_journal-1248 55 53 asrjets asrjet NOUN american_scientific_journal-1248 55 54 ) ) PUNCT american_scientific_journal-1248 55 55 ( ( PUNCT american_scientific_journal-1248 55 56 2016 2016 NUM american_scientific_journal-1248 55 57 ) ) PUNCT american_scientific_journal-1248 55 58 volume volume NOUN american_scientific_journal-1248 55 59 16 16 NUM american_scientific_journal-1248 55 60 , , PUNCT american_scientific_journal-1248 55 61 no no DET american_scientific_journal-1248 55 62 1 1 NUM american_scientific_journal-1248 55 63 , , PUNCT american_scientific_journal-1248 55 64 pp pp ADV american_scientific_journal-1248 55 65 178 178 NUM american_scientific_journal-1248 55 66 - - SYM american_scientific_journal-1248 55 67 192 192 NUM american_scientific_journal-1248 55 68 dividing divide VERB american_scientific_journal-1248 55 69 the the DET american_scientific_journal-1248 55 70 above above ADJ american_scientific_journal-1248 55 71 equation equation NOUN american_scientific_journal-1248 55 72 by by ADP american_scientific_journal-1248 55 73 π‘…πœ“πœ“ π‘…πœ“πœ“ PROPN american_scientific_journal-1248 55 74 , , PUNCT american_scientific_journal-1248 55 75 then then ADV american_scientific_journal-1248 55 76 we we PRON american_scientific_journal-1248 55 77 get get VERB american_scientific_journal-1248 55 78 π‘Ÿ2 π‘Ÿ2 NOUN american_scientific_journal-1248 55 79 𝑅\\ 𝑅\\ PROPN american_scientific_journal-1248 55 80 𝑅 𝑅 PROPN american_scientific_journal-1248 55 81 + + PROPN american_scientific_journal-1248 55 82 2π‘Ÿ 2π‘Ÿ PROPN american_scientific_journal-1248 55 83 𝑅\ 𝑅\ NOUN american_scientific_journal-1248 55 84 𝑅 𝑅 NOUN american_scientific_journal-1248 55 85 = = SYM american_scientific_journal-1248 55 86 βˆ’ βˆ’ PROPN american_scientific_journal-1248 55 87 1 1 NUM american_scientific_journal-1248 55 88 πœ“πœ“ πœ“πœ“ ADP american_scientific_journal-1248 55 89 sinπœƒπœƒ sinπœƒπœƒ PROPN american_scientific_journal-1248 55 90 πœ• πœ• PROPN american_scientific_journal-1248 55 91 πœ•πœƒπœƒ πœ•πœƒπœƒ X american_scientific_journal-1248 55 92 οΏ½ οΏ½ PROPN american_scientific_journal-1248 55 93 sin sin NOUN american_scientific_journal-1248 55 94 πœƒπœƒ πœƒπœƒ PROPN american_scientific_journal-1248 55 95 πœ“πœ“\ πœ“πœ“\ NOUN american_scientific_journal-1248 55 96 οΏ½ οΏ½ NOUN american_scientific_journal-1248 55 97 = = SYM american_scientific_journal-1248 55 98 πœ†2 πœ†2 NOUN american_scientific_journal-1248 55 99 π‘Ÿ2𝑅\\ π‘Ÿ2𝑅\\ PROPN american_scientific_journal-1248 55 100 + + CCONJ american_scientific_journal-1248 55 101 2π‘Ÿπ‘…\ 2π‘Ÿπ‘…\ ADJ american_scientific_journal-1248 55 102 + + CCONJ american_scientific_journal-1248 55 103 πœ†2𝑅 πœ†2𝑅 CCONJ american_scientific_journal-1248 55 104 = = SYM american_scientific_journal-1248 55 105 0 0 NUM american_scientific_journal-1248 55 106 ( ( PUNCT american_scientific_journal-1248 55 107 24 24 NUM american_scientific_journal-1248 55 108 ) ) PUNCT american_scientific_journal-1248 55 109 and and CCONJ american_scientific_journal-1248 55 110 πœ• πœ• PROPN american_scientific_journal-1248 55 111 πœ•πœƒπœƒ πœ•πœƒπœƒ SYM american_scientific_journal-1248 55 112 οΏ½ οΏ½ PROPN american_scientific_journal-1248 56 1 sinπœƒπœƒ sinπœƒπœƒ PROPN american_scientific_journal-1248 56 2 πœ“πœ“\ πœ“πœ“\ VERB american_scientific_journal-1248 56 3 οΏ½ οΏ½ PROPN american_scientific_journal-1248 56 4 βˆ’ βˆ’ PROPN american_scientific_journal-1248 56 5 πœ†2ψ πœ†2ψ X american_scientific_journal-1248 56 6 sin sin NOUN american_scientific_journal-1248 56 7 ΞΈ ΞΈ NOUN american_scientific_journal-1248 56 8 = = SYM american_scientific_journal-1248 56 9 0 0 NUM american_scientific_journal-1248 56 10 sinπœƒπœƒ sinπœƒπœƒ PROPN american_scientific_journal-1248 56 11 πœ•2πœ“πœ“ πœ•2πœ“πœ“ PROPN american_scientific_journal-1248 56 12 πœ•πœƒπœƒ2 πœ•πœƒπœƒ2 X american_scientific_journal-1248 56 13 + + X american_scientific_journal-1248 56 14 cos cos PROPN american_scientific_journal-1248 56 15 πœƒπœƒ πœƒπœƒ PROPN american_scientific_journal-1248 56 16 πœ•πœ“πœ“ πœ•πœ“πœ“ PUNCT american_scientific_journal-1248 56 17 πœ•πœƒπœƒ πœ•πœƒπœƒ PUNCT american_scientific_journal-1248 56 18 βˆ’ βˆ’ NOUN american_scientific_journal-1248 56 19 πœ†2πœ“πœ“ πœ†2πœ“πœ“ SYM american_scientific_journal-1248 56 20 sinπœƒπœƒ sinπœƒπœƒ NOUN american_scientific_journal-1248 56 21 = = SYM american_scientific_journal-1248 56 22 0 0 NUM american_scientific_journal-1248 56 23 ( ( PUNCT american_scientific_journal-1248 56 24 25 25 NUM american_scientific_journal-1248 56 25 ) ) PUNCT american_scientific_journal-1248 56 26 the the DET american_scientific_journal-1248 56 27 solution solution NOUN american_scientific_journal-1248 56 28 of of ADP american_scientific_journal-1248 56 29 ( ( PUNCT american_scientific_journal-1248 56 30 24 24 NUM american_scientific_journal-1248 56 31 ) ) PUNCT american_scientific_journal-1248 56 32 is be AUX american_scientific_journal-1248 56 33 given give VERB american_scientific_journal-1248 56 34 by by ADP american_scientific_journal-1248 56 35 𝑅(π‘Ÿ 𝑅(π‘Ÿ NOUN american_scientific_journal-1248 56 36 ) ) PUNCT american_scientific_journal-1248 56 37 = = PUNCT american_scientific_journal-1248 57 1 π΄π‘Ÿπ‘› π΄π‘Ÿπ‘› NOUN american_scientific_journal-1248 57 2 + + CCONJ american_scientific_journal-1248 57 3 𝐡 𝐡 NOUN american_scientific_journal-1248 57 4 π‘Ÿπ‘›+1 π‘Ÿπ‘›+1 NUM american_scientific_journal-1248 57 5 ( ( PUNCT american_scientific_journal-1248 57 6 26 26 NUM american_scientific_journal-1248 57 7 ) ) PUNCT american_scientific_journal-1248 57 8 whereπœ†2 whereπœ†2 NOUN american_scientific_journal-1248 58 1 = = PUNCT american_scientific_journal-1248 58 2 βˆ’π‘›(𝑛 βˆ’π‘›(𝑛 X american_scientific_journal-1248 58 3 + + ADP american_scientific_journal-1248 58 4 1 1 NUM american_scientific_journal-1248 58 5 ) ) PUNCT american_scientific_journal-1248 58 6 ( ( PUNCT american_scientific_journal-1248 58 7 27 27 NUM american_scientific_journal-1248 58 8 ) ) PUNCT american_scientific_journal-1248 58 9 from from ADP american_scientific_journal-1248 58 10 ( ( PUNCT american_scientific_journal-1248 58 11 27 27 NUM american_scientific_journal-1248 58 12 ) ) PUNCT american_scientific_journal-1248 58 13 and and CCONJ american_scientific_journal-1248 58 14 equation(25 equation(25 NOUN american_scientific_journal-1248 58 15 ) ) PUNCT american_scientific_journal-1248 58 16 sinπœƒπœƒ sinπœƒπœƒ VERB american_scientific_journal-1248 58 17 πœ•2πœ“πœ“ πœ•2πœ“πœ“ PROPN american_scientific_journal-1248 58 18 πœ•πœƒπœƒ2 πœ•πœƒπœƒ2 X american_scientific_journal-1248 58 19 + + X american_scientific_journal-1248 58 20 cos cos PROPN american_scientific_journal-1248 58 21 πœƒπœƒ πœƒπœƒ PROPN american_scientific_journal-1248 58 22 πœ•πœ“πœ“ πœ•πœ“πœ“ NUM american_scientific_journal-1248 58 23 πœ•πœƒπœƒ πœ•πœƒπœƒ PUNCT american_scientific_journal-1248 59 1 + + CCONJ american_scientific_journal-1248 59 2 𝑛(𝑛 𝑛(𝑛 PROPN american_scientific_journal-1248 59 3 + + CCONJ american_scientific_journal-1248 59 4 1 1 X american_scientific_journal-1248 59 5 ) ) PUNCT american_scientific_journal-1248 59 6 sin sin NOUN american_scientific_journal-1248 59 7 πœƒπœƒ πœƒπœƒ INTJ american_scientific_journal-1248 59 8 πœ“πœ“ πœ“πœ“ NOUN american_scientific_journal-1248 59 9 = = SYM american_scientific_journal-1248 59 10 0 0 NUM american_scientific_journal-1248 59 11 ( ( PUNCT american_scientific_journal-1248 59 12 28 28 NUM american_scientific_journal-1248 59 13 ) ) PUNCT american_scientific_journal-1248 59 14 equation equation NOUN american_scientific_journal-1248 59 15 ( ( PUNCT american_scientific_journal-1248 59 16 28 28 NUM american_scientific_journal-1248 59 17 ) ) PUNCT american_scientific_journal-1248 59 18 is be AUX american_scientific_journal-1248 59 19 a a DET american_scientific_journal-1248 59 20 legendre legendre NOUN american_scientific_journal-1248 59 21 ode ode PROPN american_scientific_journal-1248 59 22 , , PUNCT american_scientific_journal-1248 59 23 with with ADP american_scientific_journal-1248 59 24 general general ADJ american_scientific_journal-1248 59 25 solution solution NOUN american_scientific_journal-1248 59 26 πœ“πœ“(πœƒπœƒ πœ“πœ“(πœƒπœƒ PROPN american_scientific_journal-1248 59 27 ) ) PUNCT american_scientific_journal-1248 60 1 = = PRON american_scientific_journal-1248 60 2 𝐢𝑃𝑛(cos 𝐢𝑃𝑛(co NOUN american_scientific_journal-1248 60 3 πœƒπœƒ πœƒπœƒ PRON american_scientific_journal-1248 60 4 ) ) PUNCT american_scientific_journal-1248 60 5 + + NUM american_scientific_journal-1248 60 6 𝐷𝑄𝑛(cos 𝐷𝑄𝑛(cos X american_scientific_journal-1248 60 7 πœƒπœƒ πœƒπœƒ ADJ american_scientific_journal-1248 60 8 ) ) PUNCT american_scientific_journal-1248 60 9 ( ( PUNCT american_scientific_journal-1248 60 10 29 29 NUM american_scientific_journal-1248 60 11 ) ) PUNCT american_scientific_journal-1248 60 12 𝑒(π‘Ÿ,πœƒπœƒ 𝑒(π‘Ÿ,πœƒπœƒ NOUN american_scientific_journal-1248 60 13 ) ) PUNCT american_scientific_journal-1248 60 14 = = SYM american_scientific_journal-1248 60 15 οΏ½ οΏ½ PROPN american_scientific_journal-1248 61 1 π΄π‘Ÿπ‘› π΄π‘Ÿπ‘› NOUN american_scientific_journal-1248 61 2 + + CCONJ american_scientific_journal-1248 61 3 𝐡 𝐡 PROPN american_scientific_journal-1248 61 4 π‘Ÿπ‘›+1 π‘Ÿπ‘›+1 NUM american_scientific_journal-1248 61 5 οΏ½ οΏ½ PROPN american_scientific_journal-1248 62 1 [ [ X american_scientific_journal-1248 62 2 𝐢𝑃𝑛(cos 𝐢𝑃𝑛(co NOUN american_scientific_journal-1248 62 3 πœƒπœƒ πœƒπœƒ PRON american_scientific_journal-1248 62 4 ) ) PUNCT american_scientific_journal-1248 62 5 + + NUM american_scientific_journal-1248 62 6 𝐷𝑄𝑛(cos 𝐷𝑄𝑛(cos X american_scientific_journal-1248 62 7 πœƒπœƒ πœƒπœƒ NOUN american_scientific_journal-1248 62 8 ) ) PUNCT american_scientific_journal-1248 62 9 ] ] PUNCT american_scientific_journal-1248 62 10 ( ( PUNCT american_scientific_journal-1248 62 11 30 30 NUM american_scientific_journal-1248 62 12 ) ) PUNCT american_scientific_journal-1248 62 13 if if SCONJ american_scientific_journal-1248 62 14 𝑒(π‘Ÿ,πœƒπœƒ 𝑒(π‘Ÿ,πœƒπœƒ NOUN american_scientific_journal-1248 62 15 ) ) PUNCT american_scientific_journal-1248 62 16 represent represent VERB american_scientific_journal-1248 62 17 temperature temperature NOUN american_scientific_journal-1248 62 18 on on ADP american_scientific_journal-1248 62 19 sphere sphere NOUN american_scientific_journal-1248 62 20 radius radius NOUN american_scientific_journal-1248 62 21 with with ADP american_scientific_journal-1248 62 22 center center NOUN american_scientific_journal-1248 62 23 at at ADP american_scientific_journal-1248 62 24 origin origin NOUN american_scientific_journal-1248 62 25 , , PUNCT american_scientific_journal-1248 62 26 there there ADV american_scientific_journal-1248 62 27 the the DET american_scientific_journal-1248 62 28 heat heat NOUN american_scientific_journal-1248 62 29 must must AUX american_scientific_journal-1248 62 30 be be AUX american_scientific_journal-1248 62 31 bounded bound VERB american_scientific_journal-1248 62 32 . . PUNCT american_scientific_journal-1248 63 1 when when SCONJ american_scientific_journal-1248 63 2 πœƒπœƒ πœƒπœƒ PROPN american_scientific_journal-1248 63 3 = = SYM american_scientific_journal-1248 63 4 0 0 NUM american_scientific_journal-1248 63 5 orπœ‹πœ‹ orπœ‹πœ‹ ADJ american_scientific_journal-1248 63 6 ( ( PUNCT american_scientific_journal-1248 63 7 along along ADP american_scientific_journal-1248 63 8 𝑍 𝑍 PROPN american_scientific_journal-1248 63 9 axis axis NOUN american_scientific_journal-1248 63 10 in in ADP american_scientific_journal-1248 63 11 spherical spherical ADJ american_scientific_journal-1248 63 12 coordinates coordinate NOUN american_scientific_journal-1248 63 13 ) ) PUNCT american_scientific_journal-1248 63 14 when when SCONJ american_scientific_journal-1248 63 15 𝑄𝑛(1 𝑄𝑛(1 NOUN american_scientific_journal-1248 63 16 ) ) PUNCT american_scientific_journal-1248 63 17 β†’ β†’ SYM american_scientific_journal-1248 63 18 ∞ ∞ NOUN american_scientific_journal-1248 63 19 this this PRON american_scientific_journal-1248 63 20 provided provide VERB american_scientific_journal-1248 63 21 that that DET american_scientific_journal-1248 63 22 𝐷 𝐷 NOUN american_scientific_journal-1248 63 23 = = SYM american_scientific_journal-1248 63 24 0 0 NUM american_scientific_journal-1248 63 25 ( ( PUNCT american_scientific_journal-1248 63 26 31 31 NUM american_scientific_journal-1248 63 27 ) ) PUNCT american_scientific_journal-1248 64 1 to to PART american_scientific_journal-1248 64 2 avoid avoid VERB american_scientific_journal-1248 64 3 infinitely infinitely ADV american_scientific_journal-1248 64 4 temperature temperature NOUN american_scientific_journal-1248 64 5 at at ADP american_scientific_journal-1248 64 6 center center NOUN american_scientific_journal-1248 64 7 of of ADP american_scientific_journal-1248 64 8 sphere sphere NOUN american_scientific_journal-1248 64 9 π‘Ÿ π‘Ÿ NOUN american_scientific_journal-1248 64 10 = = SYM american_scientific_journal-1248 64 11 0 0 NUM american_scientific_journal-1248 64 12 we we PRON american_scientific_journal-1248 64 13 take take VERB american_scientific_journal-1248 64 14 𝐡 𝐡 NOUN american_scientific_journal-1248 64 15 = = SYM american_scientific_journal-1248 64 16 0 0 NUM american_scientific_journal-1248 64 17 ( ( PUNCT american_scientific_journal-1248 64 18 32 32 NUM american_scientific_journal-1248 64 19 ) ) PUNCT american_scientific_journal-1248 64 20 now now ADV american_scientific_journal-1248 64 21 from from ADP american_scientific_journal-1248 64 22 ( ( PUNCT american_scientific_journal-1248 64 23 31 31 NUM american_scientific_journal-1248 64 24 ) ) PUNCT american_scientific_journal-1248 64 25 and and CCONJ american_scientific_journal-1248 64 26 ( ( PUNCT american_scientific_journal-1248 64 27 32 32 NUM american_scientific_journal-1248 64 28 ) ) PUNCT american_scientific_journal-1248 64 29 the the DET american_scientific_journal-1248 64 30 solution solution NOUN american_scientific_journal-1248 64 31 is be AUX american_scientific_journal-1248 64 32 given give VERB american_scientific_journal-1248 64 33 by by ADP american_scientific_journal-1248 64 34 ( ( PUNCT american_scientific_journal-1248 64 35 30 30 NUM american_scientific_journal-1248 64 36 ) ) PUNCT american_scientific_journal-1248 64 37 𝑒(π‘Ÿ,πœƒπœƒ 𝑒(π‘Ÿ,πœƒπœƒ NOUN american_scientific_journal-1248 64 38 ) ) PUNCT american_scientific_journal-1248 65 1 = = SYM american_scientific_journal-1248 65 2 πΈπ‘Ÿπ‘›π‘ƒπ‘›(cos πΈπ‘Ÿπ‘›π‘ƒπ‘›(cos X american_scientific_journal-1248 65 3 πœƒπœƒ πœƒπœƒ PROPN american_scientific_journal-1248 65 4 ) ) PUNCT american_scientific_journal-1248 65 5 ( ( PUNCT american_scientific_journal-1248 65 6 33 33 NUM american_scientific_journal-1248 65 7 ) ) PUNCT american_scientific_journal-1248 65 8 183 183 NUM american_scientific_journal-1248 65 9 american american ADJ american_scientific_journal-1248 65 10 scientific scientific ADJ american_scientific_journal-1248 65 11 research research NOUN american_scientific_journal-1248 65 12 journal journal NOUN american_scientific_journal-1248 65 13 for for ADP american_scientific_journal-1248 65 14 engineering engineering NOUN american_scientific_journal-1248 65 15 , , PUNCT american_scientific_journal-1248 65 16 technology technology NOUN american_scientific_journal-1248 65 17 , , PUNCT american_scientific_journal-1248 65 18 and and CCONJ american_scientific_journal-1248 65 19 sciences science NOUN american_scientific_journal-1248 65 20 ( ( PUNCT american_scientific_journal-1248 65 21 asrjets asrjet NOUN american_scientific_journal-1248 65 22 ) ) PUNCT american_scientific_journal-1248 65 23 ( ( PUNCT american_scientific_journal-1248 65 24 2016 2016 NUM american_scientific_journal-1248 65 25 ) ) PUNCT american_scientific_journal-1248 65 26 volume volume NOUN american_scientific_journal-1248 65 27 16 16 NUM american_scientific_journal-1248 65 28 , , PUNCT american_scientific_journal-1248 65 29 no no DET american_scientific_journal-1248 65 30 1 1 NUM american_scientific_journal-1248 65 31 , , PUNCT american_scientific_journal-1248 65 32 pp pp ADV american_scientific_journal-1248 65 33 178 178 NUM american_scientific_journal-1248 65 34 - - SYM american_scientific_journal-1248 65 35 192 192 NUM american_scientific_journal-1248 65 36 by by ADP american_scientific_journal-1248 65 37 the the DET american_scientific_journal-1248 65 38 supper supper NOUN american_scientific_journal-1248 65 39 position position NOUN american_scientific_journal-1248 65 40 ( ( PUNCT american_scientific_journal-1248 65 41 33 33 NUM american_scientific_journal-1248 65 42 ) ) PUNCT american_scientific_journal-1248 65 43 can can AUX american_scientific_journal-1248 65 44 be be AUX american_scientific_journal-1248 65 45 written write VERB american_scientific_journal-1248 65 46 as as ADP american_scientific_journal-1248 65 47 𝑒(π‘Ÿ,πœƒπœƒ 𝑒(π‘Ÿ,πœƒπœƒ NOUN american_scientific_journal-1248 65 48 ) ) PUNCT american_scientific_journal-1248 65 49 = = SYM american_scientific_journal-1248 65 50 οΏ½ οΏ½ NOUN american_scientific_journal-1248 65 51 πΈπ‘›π‘Ÿπ‘›π‘ƒπ‘›(cosπœƒπœƒ πΈπ‘›π‘Ÿπ‘›π‘ƒπ‘›(cosπœƒπœƒ NOUN american_scientific_journal-1248 65 52 ) ) PUNCT american_scientific_journal-1248 65 53 ( ( PUNCT american_scientific_journal-1248 65 54 34 34 NUM american_scientific_journal-1248 65 55 ) ) PUNCT american_scientific_journal-1248 65 56 ∞ ∞ NUM american_scientific_journal-1248 65 57 𝑛=0 𝑛=0 PROPN american_scientific_journal-1248 65 58 by by ADP american_scientific_journal-1248 65 59 applying apply VERB american_scientific_journal-1248 65 60 the the DET american_scientific_journal-1248 65 61 boundary boundary ADJ american_scientific_journal-1248 65 62 condition condition NOUN american_scientific_journal-1248 65 63 𝑒(1,πœƒπœƒ 𝑒(1,πœƒπœƒ NOUN american_scientific_journal-1248 65 64 ) ) PUNCT american_scientific_journal-1248 66 1 = = SYM american_scientific_journal-1248 66 2 𝑓(πœƒπœƒ 𝑓(πœƒπœƒ NOUN american_scientific_journal-1248 66 3 ) ) PUNCT american_scientific_journal-1248 66 4 into into ADP american_scientific_journal-1248 66 5 ( ( PUNCT american_scientific_journal-1248 66 6 34 34 NUM american_scientific_journal-1248 66 7 ) ) PUNCT american_scientific_journal-1248 66 8 . . PUNCT american_scientific_journal-1248 67 1 𝑓(πœƒπœƒ 𝑓(πœƒπœƒ PROPN american_scientific_journal-1248 67 2 ) ) PUNCT american_scientific_journal-1248 68 1 = = PUNCT american_scientific_journal-1248 68 2 οΏ½ οΏ½ PROPN american_scientific_journal-1248 68 3 enpn(cos enpn(cos PROPN american_scientific_journal-1248 68 4 ΞΈ ΞΈ PROPN american_scientific_journal-1248 68 5 ) ) PUNCT american_scientific_journal-1248 68 6 ( ( PUNCT american_scientific_journal-1248 68 7 35 35 NUM american_scientific_journal-1248 68 8 ) ) PUNCT american_scientific_journal-1248 68 9 ∞ ∞ NUM american_scientific_journal-1248 69 1 𝑛=0 𝑛=0 PROPN american_scientific_journal-1248 69 2 let let VERB american_scientific_journal-1248 69 3 πœ‡πœ‡ πœ‡πœ‡ VERB american_scientific_journal-1248 69 4 = = VERB american_scientific_journal-1248 69 5 cos cos X american_scientific_journal-1248 69 6 πœƒπœƒ,thenπœƒπœƒ πœƒπœƒ,thenπœƒπœƒ PROPN american_scientific_journal-1248 69 7 = = SYM american_scientific_journal-1248 69 8 π‘π‘œπ‘ βˆ’1πœ‡πœ‡ π‘π‘œπ‘ βˆ’1πœ‡πœ‡ PROPN american_scientific_journal-1248 69 9 𝑓(π‘π‘œπ‘ βˆ’1πœ‡πœ‡ 𝑓(π‘π‘œπ‘ βˆ’1πœ‡πœ‡ PROPN american_scientific_journal-1248 69 10 ) ) PUNCT american_scientific_journal-1248 69 11 = = SYM american_scientific_journal-1248 69 12 οΏ½ οΏ½ PROPN american_scientific_journal-1248 69 13 enpn(Β΅ enpn(Β΅ PROPN american_scientific_journal-1248 69 14 ) ) PUNCT american_scientific_journal-1248 69 15 ( ( PUNCT american_scientific_journal-1248 69 16 36 36 NUM american_scientific_journal-1248 69 17 ) ) PUNCT american_scientific_journal-1248 69 18 ∞ ∞ NUM american_scientific_journal-1248 69 19 𝑛=0 𝑛=0 PROPN american_scientific_journal-1248 69 20 now now ADV american_scientific_journal-1248 69 21 𝐸𝑛 𝐸𝑛 ADP american_scientific_journal-1248 69 22 = = SYM american_scientific_journal-1248 69 23 ( ( PUNCT american_scientific_journal-1248 69 24 2𝑛 2𝑛 NOUN american_scientific_journal-1248 69 25 + + CCONJ american_scientific_journal-1248 69 26 1 1 X american_scientific_journal-1248 69 27 ) ) SYM american_scientific_journal-1248 69 28 2 2 NUM american_scientific_journal-1248 69 29 οΏ½ οΏ½ NOUN american_scientific_journal-1248 69 30 𝑓(π‘π‘œπ‘ βˆ’1πœ‡πœ‡)𝑃𝑛(πœ‡πœ‡)π‘‘πœ‡πœ‡ 𝑓(π‘π‘œπ‘ βˆ’1πœ‡πœ‡)𝑃𝑛(πœ‡πœ‡)π‘‘πœ‡πœ‡ PROPN american_scientific_journal-1248 69 31 ( ( PUNCT american_scientific_journal-1248 69 32 37 37 NUM american_scientific_journal-1248 69 33 ) ) PUNCT american_scientific_journal-1248 69 34 1 1 NUM american_scientific_journal-1248 69 35 βˆ’1 βˆ’1 NOUN american_scientific_journal-1248 69 36 from from ADP american_scientific_journal-1248 69 37 ( ( PUNCT american_scientific_journal-1248 69 38 34 34 NUM american_scientific_journal-1248 69 39 ) ) PUNCT american_scientific_journal-1248 69 40 , , PUNCT american_scientific_journal-1248 69 41 we we PRON american_scientific_journal-1248 69 42 have have VERB american_scientific_journal-1248 69 43 𝑒(π‘Ÿ,πœƒπœƒ 𝑒(π‘Ÿ,πœƒπœƒ NOUN american_scientific_journal-1248 69 44 ) ) PUNCT american_scientific_journal-1248 69 45 = = SYM american_scientific_journal-1248 69 46 οΏ½ οΏ½ PROPN american_scientific_journal-1248 69 47 οΏ½ οΏ½ PROPN american_scientific_journal-1248 69 48 ( ( PUNCT american_scientific_journal-1248 69 49 2𝑛 2𝑛 PROPN american_scientific_journal-1248 69 50 + + CCONJ american_scientific_journal-1248 69 51 1 1 X american_scientific_journal-1248 69 52 ) ) SYM american_scientific_journal-1248 69 53 2 2 NUM american_scientific_journal-1248 69 54 οΏ½ οΏ½ PROPN american_scientific_journal-1248 69 55 𝑓(πœƒπœƒ)𝑃𝑛(cos 𝑓(πœƒπœƒ)𝑃𝑛(cos PROPN american_scientific_journal-1248 69 56 πœƒπœƒ πœƒπœƒ NOUN american_scientific_journal-1248 69 57 ) ) PUNCT american_scientific_journal-1248 69 58 sinπœƒπœƒπ‘‘πœƒπœƒ sinπœƒπœƒπ‘‘πœƒπœƒ NOUN american_scientific_journal-1248 69 59 πœ‹ πœ‹ X american_scientific_journal-1248 69 60 0 0 NUM american_scientific_journal-1248 69 61 οΏ½ οΏ½ PROPN american_scientific_journal-1248 70 1 [ [ X american_scientific_journal-1248 70 2 π‘Ÿπ‘›π‘ƒπ‘›(cos π‘Ÿπ‘›π‘ƒπ‘›(cos X american_scientific_journal-1248 70 3 πœƒπœƒ πœƒπœƒ NOUN american_scientific_journal-1248 70 4 ) ) PUNCT american_scientific_journal-1248 70 5 ] ] PUNCT american_scientific_journal-1248 71 1 ∞ ∞ NUM american_scientific_journal-1248 71 2 𝑛=0 𝑛=0 PROPN american_scientific_journal-1248 71 3 . . PUNCT american_scientific_journal-1248 72 1 5 5 X american_scientific_journal-1248 72 2 . . X american_scientific_journal-1248 72 3 the the DET american_scientific_journal-1248 72 4 laplacian laplacian ADJ american_scientific_journal-1248 72 5 difference difference NOUN american_scientific_journal-1248 72 6 equation equation NOUN american_scientific_journal-1248 72 7 central central ADJ american_scientific_journal-1248 72 8 differences difference NOUN american_scientific_journal-1248 72 9 based base VERB american_scientific_journal-1248 72 10 on on ADP american_scientific_journal-1248 72 11 the the DET american_scientific_journal-1248 72 12 grid grid NOUN american_scientific_journal-1248 72 13 and and CCONJ american_scientific_journal-1248 72 14 scheme scheme NOUN american_scientific_journal-1248 72 15 used use VERB american_scientific_journal-1248 72 16 for for ADP american_scientific_journal-1248 72 17 the the DET american_scientific_journal-1248 72 18 finite finite ADJ american_scientific_journal-1248 72 19 difference difference NOUN american_scientific_journal-1248 72 20 solution solution NOUN american_scientific_journal-1248 72 21 in in ADP american_scientific_journal-1248 72 22 two two NUM american_scientific_journal-1248 72 23 independent independent ADJ american_scientific_journal-1248 72 24 variables variable NOUN american_scientific_journal-1248 72 25 such such ADJ american_scientific_journal-1248 72 26 as as ADP american_scientific_journal-1248 72 27 the the DET american_scientific_journal-1248 72 28 laplace laplace NOUN american_scientific_journal-1248 72 29 equation equation NOUN american_scientific_journal-1248 72 30 [ [ X american_scientific_journal-1248 72 31 3 3 NUM american_scientific_journal-1248 72 32 ] ] PUNCT american_scientific_journal-1248 72 33 . . PUNCT american_scientific_journal-1248 73 1 πœ•2𝑇 πœ•2𝑇 PROPN american_scientific_journal-1248 73 2 πœ•π‘₯2 πœ•π‘₯2 PROPN american_scientific_journal-1248 74 1 = = PRON american_scientific_journal-1248 74 2 𝑇𝑖+1,𝑗 𝑇𝑖+1,𝑗 PROPN american_scientific_journal-1248 74 3 βˆ’ βˆ’ PROPN american_scientific_journal-1248 74 4 2𝑇𝑖,𝑗 2𝑇𝑖,𝑗 NUM american_scientific_journal-1248 74 5 + + CCONJ american_scientific_journal-1248 74 6 π‘‡π‘–βˆ’1,𝑗 π‘‡π‘–βˆ’1,𝑗 PRON american_scientific_journal-1248 74 7 βˆ‡π‘₯2 βˆ‡π‘₯2 PROPN american_scientific_journal-1248 74 8 ( ( PUNCT american_scientific_journal-1248 74 9 38 38 NUM american_scientific_journal-1248 74 10 ) ) PUNCT american_scientific_journal-1248 74 11 and and CCONJ american_scientific_journal-1248 74 12 πœ•2𝑇 πœ•2𝑇 PROPN american_scientific_journal-1248 74 13 πœ•π‘¦2 πœ•π‘¦2 PROPN american_scientific_journal-1248 74 14 = = SYM american_scientific_journal-1248 75 1 𝑇𝑖,𝑗+1 𝑇𝑖,𝑗+1 PRON american_scientific_journal-1248 76 1 βˆ’ βˆ’ NOUN american_scientific_journal-1248 76 2 2𝑇𝑖,𝑗 2𝑇𝑖,𝑗 NUM american_scientific_journal-1248 76 3 + + CCONJ american_scientific_journal-1248 76 4 𝑇𝑖,π‘—βˆ’1 𝑇𝑖,π‘—βˆ’1 PROPN american_scientific_journal-1248 76 5 βˆ‡π‘¦2 βˆ‡π‘¦2 NUM american_scientific_journal-1248 76 6 ( ( PUNCT american_scientific_journal-1248 76 7 39 39 NUM american_scientific_journal-1248 76 8 ) ) PUNCT american_scientific_journal-1248 76 9 respectively respectively ADV american_scientific_journal-1248 76 10 which which PRON american_scientific_journal-1248 76 11 have have VERB american_scientific_journal-1248 76 12 errors error NOUN american_scientific_journal-1248 76 13 of of ADP american_scientific_journal-1248 76 14 o[βˆ‡(π‘₯)2 o[βˆ‡(π‘₯)2 NOUN american_scientific_journal-1248 76 15 ] ] X american_scientific_journal-1248 76 16 and and CCONJ american_scientific_journal-1248 76 17 o[βˆ‡(𝑦)2 o[βˆ‡(𝑦)2 NOUN american_scientific_journal-1248 76 18 ] ] PUNCT american_scientific_journal-1248 76 19 substituting substitute VERB american_scientific_journal-1248 76 20 these these DET american_scientific_journal-1248 76 21 expressions expression NOUN american_scientific_journal-1248 76 22 into into ADP american_scientific_journal-1248 76 23 equation equation NOUN american_scientific_journal-1248 76 24 into into ADP american_scientific_journal-1248 76 25 equation equation NOUN american_scientific_journal-1248 76 26 184 184 NUM american_scientific_journal-1248 76 27 american american PROPN american_scientific_journal-1248 76 28 scientific scientific ADJ american_scientific_journal-1248 76 29 research research NOUN american_scientific_journal-1248 76 30 journal journal NOUN american_scientific_journal-1248 76 31 for for ADP american_scientific_journal-1248 76 32 engineering engineering NOUN american_scientific_journal-1248 76 33 , , PUNCT american_scientific_journal-1248 76 34 technology technology NOUN american_scientific_journal-1248 76 35 , , PUNCT american_scientific_journal-1248 76 36 and and CCONJ american_scientific_journal-1248 76 37 sciences science NOUN american_scientific_journal-1248 76 38 ( ( PUNCT american_scientific_journal-1248 76 39 asrjets asrjet NOUN american_scientific_journal-1248 76 40 ) ) PUNCT american_scientific_journal-1248 76 41 ( ( PUNCT american_scientific_journal-1248 76 42 2016 2016 NUM american_scientific_journal-1248 76 43 ) ) PUNCT american_scientific_journal-1248 76 44 volume volume NOUN american_scientific_journal-1248 76 45 16 16 NUM american_scientific_journal-1248 76 46 , , PUNCT american_scientific_journal-1248 76 47 no no DET american_scientific_journal-1248 76 48 1 1 NUM american_scientific_journal-1248 76 49 , , PUNCT american_scientific_journal-1248 76 50 pp pp ADV american_scientific_journal-1248 76 51 178 178 NUM american_scientific_journal-1248 76 52 - - SYM american_scientific_journal-1248 76 53 192 192 NUM american_scientific_journal-1248 76 54 πœ•2𝑇 πœ•2𝑇 NOUN american_scientific_journal-1248 76 55 πœ•π‘₯2 πœ•π‘₯2 NOUN american_scientific_journal-1248 76 56 + + CCONJ american_scientific_journal-1248 76 57 πœ•2𝑇 πœ•2𝑇 PROPN american_scientific_journal-1248 76 58 πœ•π‘¦2 πœ•π‘¦2 PROPN american_scientific_journal-1248 76 59 = = SYM american_scientific_journal-1248 76 60 0 0 PROPN american_scientific_journal-1248 76 61 gives give VERB american_scientific_journal-1248 76 62 𝑇𝑖+1,𝑗 𝑇𝑖+1,𝑗 PROPN american_scientific_journal-1248 76 63 βˆ’ βˆ’ PROPN american_scientific_journal-1248 76 64 2𝑇𝑖,𝑗 2𝑇𝑖,𝑗 NUM american_scientific_journal-1248 76 65 + + CCONJ american_scientific_journal-1248 76 66 π‘‡π‘–βˆ’1,𝑗 π‘‡π‘–βˆ’1,𝑗 PRON american_scientific_journal-1248 76 67 βˆ‡π‘₯2 βˆ‡π‘₯2 NOUN american_scientific_journal-1248 76 68 + + CCONJ american_scientific_journal-1248 76 69 𝑇𝑖,𝑗+1 𝑇𝑖,𝑗+1 PROPN american_scientific_journal-1248 76 70 βˆ’ βˆ’ NOUN american_scientific_journal-1248 76 71 2𝑇𝑖,𝑗 2𝑇𝑖,𝑗 NUM american_scientific_journal-1248 76 72 + + CCONJ american_scientific_journal-1248 76 73 𝑇𝑖,π‘—βˆ’1 𝑇𝑖,π‘—βˆ’1 PROPN american_scientific_journal-1248 76 74 βˆ‡π‘¦2 βˆ‡π‘¦2 NUM american_scientific_journal-1248 76 75 = = SYM american_scientific_journal-1248 76 76 0 0 PUNCT american_scientific_journal-1248 76 77 ( ( PUNCT american_scientific_journal-1248 76 78 40 40 NUM american_scientific_journal-1248 76 79 ) ) PUNCT american_scientific_journal-1248 76 80 for for ADP american_scientific_journal-1248 76 81 the the DET american_scientific_journal-1248 76 82 square square ADJ american_scientific_journal-1248 76 83 grid grid NOUN american_scientific_journal-1248 76 84 , , PUNCT american_scientific_journal-1248 77 1 βˆ‡π‘₯ βˆ‡π‘₯ PROPN american_scientific_journal-1248 77 2 = = PUNCT american_scientific_journal-1248 77 3 βˆ‡π‘¦ βˆ‡π‘¦ PROPN american_scientific_journal-1248 77 4 and and CCONJ american_scientific_journal-1248 77 5 by by ADP american_scientific_journal-1248 77 6 collection collection NOUN american_scientific_journal-1248 77 7 of of ADP american_scientific_journal-1248 77 8 terms term NOUN american_scientific_journal-1248 77 9 , , PUNCT american_scientific_journal-1248 77 10 the the DET american_scientific_journal-1248 77 11 equation equation NOUN american_scientific_journal-1248 77 12 becomes become VERB american_scientific_journal-1248 77 13 𝑇𝑖+1,𝑗 𝑇𝑖+1,𝑗 PROPN american_scientific_journal-1248 77 14 + + CCONJ american_scientific_journal-1248 77 15 tiβˆ’1,j tiβˆ’1,j NOUN american_scientific_journal-1248 77 16 + + CCONJ american_scientific_journal-1248 77 17 ti ti NOUN american_scientific_journal-1248 77 18 , , PUNCT american_scientific_journal-1248 77 19 j+1 j+1 PROPN american_scientific_journal-1248 77 20 + + PUNCT american_scientific_journal-1248 77 21 ti ti NOUN american_scientific_journal-1248 77 22 , , PUNCT american_scientific_journal-1248 77 23 jβˆ’1 jβˆ’1 PROPN american_scientific_journal-1248 77 24 βˆ’ βˆ’ PROPN american_scientific_journal-1248 77 25 4ti 4ti NOUN american_scientific_journal-1248 77 26 , , PUNCT american_scientific_journal-1248 77 27 j j PROPN american_scientific_journal-1248 77 28 = = SYM american_scientific_journal-1248 77 29 0 0 PROPN american_scientific_journal-1248 77 30 ( ( PUNCT american_scientific_journal-1248 77 31 41 41 NUM american_scientific_journal-1248 77 32 ) ) PUNCT american_scientific_journal-1248 77 33 this this DET american_scientific_journal-1248 77 34 relationship relationship NOUN american_scientific_journal-1248 77 35 , , PUNCT american_scientific_journal-1248 77 36 which which PRON american_scientific_journal-1248 77 37 holds hold VERB american_scientific_journal-1248 77 38 for for ADP american_scientific_journal-1248 77 39 all all DET american_scientific_journal-1248 77 40 interior interior ADJ american_scientific_journal-1248 77 41 points point NOUN american_scientific_journal-1248 77 42 on on ADP american_scientific_journal-1248 77 43 the the DET american_scientific_journal-1248 77 44 plate plate NOUN american_scientific_journal-1248 77 45 , , PUNCT american_scientific_journal-1248 77 46 is be AUX american_scientific_journal-1248 77 47 referred refer VERB american_scientific_journal-1248 77 48 to to ADP american_scientific_journal-1248 77 49 as as ADP american_scientific_journal-1248 77 50 the the DET american_scientific_journal-1248 77 51 laplacian laplacian ADJ american_scientific_journal-1248 77 52 different different ADJ american_scientific_journal-1248 77 53 equation equation NOUN american_scientific_journal-1248 77 54 . . PUNCT american_scientific_journal-1248 78 1 a a DET american_scientific_journal-1248 78 2 heated heated ADJ american_scientific_journal-1248 78 3 plate plate NOUN american_scientific_journal-1248 78 4 where where SCONJ american_scientific_journal-1248 78 5 boundary boundary ADJ american_scientific_journal-1248 78 6 temperature temperature NOUN american_scientific_journal-1248 78 7 are be AUX american_scientific_journal-1248 78 8 hold hold VERB american_scientific_journal-1248 78 9 at at ADP american_scientific_journal-1248 78 10 const const NOUN american_scientific_journal-1248 78 11 balance balance NOUN american_scientific_journal-1248 78 12 ant ant ADJ american_scientific_journal-1248 78 13 levels level NOUN american_scientific_journal-1248 78 14 . . PUNCT american_scientific_journal-1248 79 1 this this PRON american_scientific_journal-1248 79 2 called call VERB american_scientific_journal-1248 79 3 adirichlet adirichlet PROPN american_scientific_journal-1248 79 4 boundary boundary ADJ american_scientific_journal-1248 79 5 condition condition NOUN american_scientific_journal-1248 79 6 . . PUNCT american_scientific_journal-1248 80 1 where where SCONJ american_scientific_journal-1248 80 2 the the DET american_scientific_journal-1248 80 3 edges edge NOUN american_scientific_journal-1248 80 4 are be AUX american_scientific_journal-1248 80 5 hold hold VERB american_scientific_journal-1248 80 6 at at ADP american_scientific_journal-1248 80 7 constant constant ADJ american_scientific_journal-1248 80 8 temperature temperature NOUN american_scientific_journal-1248 80 9 for for ADP american_scientific_journal-1248 80 10 the the DET american_scientific_journal-1248 80 11 case case NOUN american_scientific_journal-1248 80 12 illustrated illustrate VERB american_scientific_journal-1248 80 13 in in ADP american_scientific_journal-1248 80 14 figure figure NOUN american_scientific_journal-1248 80 15 2 2 NUM american_scientific_journal-1248 80 16 balance balance NOUN american_scientific_journal-1248 80 17 figurec1 figurec1 NOUN american_scientific_journal-1248 80 18 is be AUX american_scientific_journal-1248 80 19 , , PUNCT american_scientific_journal-1248 80 20 according accord VERB american_scientific_journal-1248 80 21 to to ADP american_scientific_journal-1248 80 22 equation equation NOUN american_scientific_journal-1248 80 23 ( ( PUNCT american_scientific_journal-1248 80 24 41 41 NUM american_scientific_journal-1248 80 25 ) ) PUNCT american_scientific_journal-1248 80 26 𝑇21 𝑇21 PROPN american_scientific_journal-1248 80 27 + + CCONJ american_scientific_journal-1248 80 28 t01 t01 PROPN american_scientific_journal-1248 80 29 + + CCONJ american_scientific_journal-1248 80 30 t12 t12 PROPN american_scientific_journal-1248 80 31 + + CCONJ american_scientific_journal-1248 80 32 t10 t10 NOUN american_scientific_journal-1248 80 33 βˆ’ βˆ’ PROPN american_scientific_journal-1248 80 34 4t11 4t11 NOUN american_scientific_journal-1248 80 35 = = SYM american_scientific_journal-1248 80 36 0 0 PUNCT american_scientific_journal-1248 81 1 ( ( PUNCT american_scientific_journal-1248 81 2 42 42 NUM american_scientific_journal-1248 81 3 ) ) PUNCT american_scientific_journal-1248 81 4 and and CCONJ american_scientific_journal-1248 81 5 t10 t10 NUM american_scientific_journal-1248 81 6 = = SYM american_scientific_journal-1248 81 7 0however 0however PROPN american_scientific_journal-1248 81 8 , , PUNCT american_scientific_journal-1248 81 9 t01 t01 PROPN american_scientific_journal-1248 81 10 = = SYM american_scientific_journal-1248 81 11 75 75 NUM american_scientific_journal-1248 81 12 therefore therefore ADV american_scientific_journal-1248 81 13 equation equation NOUN american_scientific_journal-1248 81 14 πœ• πœ• PROPN american_scientific_journal-1248 81 15 2𝑇 2𝑇 PROPN american_scientific_journal-1248 81 16 πœ•π‘₯2 πœ•π‘₯2 PROPN american_scientific_journal-1248 81 17 + + CCONJ american_scientific_journal-1248 81 18 πœ•2𝑇 πœ•2𝑇 PROPN american_scientific_journal-1248 81 19 πœ•π‘¦2 πœ•π‘¦2 PROPN american_scientific_journal-1248 81 20 = = SYM american_scientific_journal-1248 81 21 0 0 NUM american_scientific_journal-1248 81 22 , , PUNCT american_scientific_journal-1248 81 23 can can AUX american_scientific_journal-1248 81 24 be be AUX american_scientific_journal-1248 81 25 expressed express VERB american_scientific_journal-1248 81 26 as as ADP american_scientific_journal-1248 81 27 βˆ’4𝑇11 βˆ’4𝑇11 PROPN american_scientific_journal-1248 81 28 + + CCONJ american_scientific_journal-1248 81 29 𝑇12 𝑇12 PROPN american_scientific_journal-1248 82 1 + + PROPN american_scientific_journal-1248 82 2 𝑇21 𝑇21 PROPN american_scientific_journal-1248 82 3 = = SYM american_scientific_journal-1248 82 4 βˆ’75 βˆ’75 PROPN american_scientific_journal-1248 82 5 similar similar ADJ american_scientific_journal-1248 82 6 equation equation NOUN american_scientific_journal-1248 82 7 can can AUX american_scientific_journal-1248 82 8 be be AUX american_scientific_journal-1248 82 9 developed develop VERB american_scientific_journal-1248 82 10 for for ADP american_scientific_journal-1248 82 11 the the DET american_scientific_journal-1248 82 12 other other ADJ american_scientific_journal-1248 82 13 interior interior ADJ american_scientific_journal-1248 82 14 points point NOUN american_scientific_journal-1248 82 15 . . PUNCT american_scientific_journal-1248 83 1 the the DET american_scientific_journal-1248 83 2 result result NOUN american_scientific_journal-1248 83 3 is be AUX american_scientific_journal-1248 83 4 the the DET american_scientific_journal-1248 83 5 following follow VERB american_scientific_journal-1248 83 6 set set NOUN american_scientific_journal-1248 83 7 of of ADP american_scientific_journal-1248 83 8 nine nine NUM american_scientific_journal-1248 83 9 simultaneous simultaneous ADJ american_scientific_journal-1248 83 10 equations equation NOUN american_scientific_journal-1248 83 11 with with ADP american_scientific_journal-1248 83 12 nine nine NUM american_scientific_journal-1248 83 13 unknowns unknown NOUN american_scientific_journal-1248 83 14 . . PUNCT american_scientific_journal-1248 84 1 4𝑇11 4𝑇11 NOUN american_scientific_journal-1248 85 1 βˆ’ βˆ’ PROPN american_scientific_journal-1248 86 1 𝑇21 𝑇21 PROPN american_scientific_journal-1248 86 2 βˆ’ βˆ’ NUM american_scientific_journal-1248 86 3 𝑇12 𝑇12 PROPN american_scientific_journal-1248 86 4 = = PRON american_scientific_journal-1248 87 1 75 75 NUM american_scientific_journal-1248 87 2 βˆ’π‘‡114𝑇21βˆ’π‘‡11βˆ’π‘‡22 βˆ’π‘‡114𝑇21βˆ’π‘‡11βˆ’π‘‡22 NUM american_scientific_journal-1248 87 3 = = SYM american_scientific_journal-1248 87 4 0 0 PUNCT american_scientific_journal-1248 87 5 βˆ’π‘‡214𝑇31βˆ’π‘‡32 βˆ’π‘‡214𝑇31βˆ’π‘‡32 X american_scientific_journal-1248 88 1 = = SYM american_scientific_journal-1248 88 2 50 50 NUM american_scientific_journal-1248 88 3 ( ( PUNCT american_scientific_journal-1248 88 4 1,3 1,3 NUM american_scientific_journal-1248 88 5 ) ) PUNCT american_scientific_journal-1248 88 6 ( ( PUNCT american_scientific_journal-1248 88 7 2,3 2,3 NUM american_scientific_journal-1248 88 8 ) ) PUNCT american_scientific_journal-1248 88 9 ( ( PUNCT american_scientific_journal-1248 88 10 1,2 1,2 NUM american_scientific_journal-1248 88 11 ) ) PUNCT american_scientific_journal-1248 88 12 ( ( PUNCT american_scientific_journal-1248 88 13 2,2 2,2 NUM american_scientific_journal-1248 88 14 ) ) PUNCT american_scientific_journal-1248 88 15 ( ( PUNCT american_scientific_journal-1248 88 16 3,2 3,2 NUM american_scientific_journal-1248 88 17 ) ) PUNCT american_scientific_journal-1248 88 18 ( ( PUNCT american_scientific_journal-1248 88 19 1,1 1,1 NUM american_scientific_journal-1248 88 20 ) ) PUNCT american_scientific_journal-1248 88 21 ( ( PUNCT american_scientific_journal-1248 88 22 2,1 2,1 NUM american_scientific_journal-1248 88 23 ) ) PUNCT american_scientific_journal-1248 88 24 ( ( PUNCT american_scientific_journal-1248 88 25 3,1 3,1 NUM american_scientific_journal-1248 88 26 ) ) PUNCT american_scientific_journal-1248 88 27 100 100 NUM american_scientific_journal-1248 88 28 75c 75c NOUN american_scientific_journal-1248 88 29 50c 50c NOUN american_scientific_journal-1248 88 30 Ω’ Ω’ NOUN american_scientific_journal-1248 88 31 β—Œ β—Œ PROPN american_scientific_journal-1248 88 32 0c 0c NOUN american_scientific_journal-1248 88 33 Ω’ Ω’ NOUN american_scientific_journal-1248 88 34 β—Œ β—Œ ADV american_scientific_journal-1248 88 35 figure figure VERB american_scientific_journal-1248 88 36 2 2 NUM american_scientific_journal-1248 88 37 185 185 NUM american_scientific_journal-1248 88 38 american american ADJ american_scientific_journal-1248 88 39 scientific scientific ADJ american_scientific_journal-1248 88 40 research research NOUN american_scientific_journal-1248 88 41 journal journal NOUN american_scientific_journal-1248 88 42 for for ADP american_scientific_journal-1248 88 43 engineering engineering NOUN american_scientific_journal-1248 88 44 , , PUNCT american_scientific_journal-1248 88 45 technology technology NOUN american_scientific_journal-1248 88 46 , , PUNCT american_scientific_journal-1248 88 47 and and CCONJ american_scientific_journal-1248 88 48 sciences science NOUN american_scientific_journal-1248 88 49 ( ( PUNCT american_scientific_journal-1248 88 50 asrjets asrjet NOUN american_scientific_journal-1248 88 51 ) ) PUNCT american_scientific_journal-1248 88 52 ( ( PUNCT american_scientific_journal-1248 88 53 2016 2016 NUM american_scientific_journal-1248 88 54 ) ) PUNCT american_scientific_journal-1248 88 55 volume volume NOUN american_scientific_journal-1248 88 56 16 16 NUM american_scientific_journal-1248 88 57 , , PUNCT american_scientific_journal-1248 88 58 no no DET american_scientific_journal-1248 88 59 1 1 NUM american_scientific_journal-1248 88 60 , , PUNCT american_scientific_journal-1248 88 61 pp pp ADV american_scientific_journal-1248 88 62 178 178 NUM american_scientific_journal-1248 88 63 - - SYM american_scientific_journal-1248 88 64 192 192 NUM american_scientific_journal-1248 88 65 βˆ’π‘‡11 βˆ’π‘‡11 PROPN american_scientific_journal-1248 88 66 4𝑇12βˆ’π‘‡22βˆ’π‘‡13 4𝑇12βˆ’π‘‡22βˆ’π‘‡13 NUM american_scientific_journal-1248 88 67 = = SYM american_scientific_journal-1248 88 68 75 75 NUM american_scientific_journal-1248 88 69 βˆ’π‘‡21βˆ’π‘‡124𝑇22βˆ’π‘‡32βˆ’π‘‡23 βˆ’π‘‡21βˆ’π‘‡124𝑇22βˆ’π‘‡32βˆ’π‘‡23 NOUN american_scientific_journal-1248 88 70 = = SYM american_scientific_journal-1248 88 71 0 0 PUNCT american_scientific_journal-1248 89 1 βˆ’π‘‡31βˆ’π‘‡224𝑇32𝑇33 βˆ’π‘‡31βˆ’π‘‡224𝑇32𝑇33 NOUN american_scientific_journal-1248 89 2 = = NOUN american_scientific_journal-1248 89 3 50 50 NUM american_scientific_journal-1248 89 4 βˆ’π‘‡124𝑇13βˆ’π‘‡23 βˆ’π‘‡124𝑇13βˆ’π‘‡23 NOUN american_scientific_journal-1248 89 5 = = SYM american_scientific_journal-1248 89 6 175 175 NUM american_scientific_journal-1248 89 7 βˆ’t22βˆ’t134𝑇23βˆ’t33 βˆ’t22βˆ’t134𝑇23βˆ’t33 NOUN american_scientific_journal-1248 89 8 = = NOUN american_scientific_journal-1248 89 9 100 100 NUM american_scientific_journal-1248 89 10 βˆ’t32βˆ’π‘‡234𝑇33 βˆ’t32βˆ’π‘‡234𝑇33 NOUN american_scientific_journal-1248 89 11 = = NUM american_scientific_journal-1248 89 12 150 150 NUM american_scientific_journal-1248 89 13 5.1 5.1 NUM american_scientific_journal-1248 89 14 example example NOUN american_scientific_journal-1248 89 15 temperature temperature NOUN american_scientific_journal-1248 89 16 of of ADP american_scientific_journal-1248 89 17 heated heated ADJ american_scientific_journal-1248 89 18 plate plate NOUN american_scientific_journal-1248 89 19 with with ADP american_scientific_journal-1248 89 20 fixed fix VERB american_scientific_journal-1248 89 21 boundary boundary ADJ american_scientific_journal-1248 89 22 conditions condition NOUN american_scientific_journal-1248 89 23 problem problem NOUN american_scientific_journal-1248 89 24 statement statement NOUN american_scientific_journal-1248 89 25 [ [ X american_scientific_journal-1248 89 26 4 4 NUM american_scientific_journal-1248 89 27 ] ] PUNCT american_scientific_journal-1248 89 28 . . PUNCT american_scientific_journal-1248 90 1 use use VERB american_scientific_journal-1248 90 2 lobmann lobmann PROPN american_scientific_journal-1248 90 3 's 's PART american_scientific_journal-1248 90 4 method method NOUN american_scientific_journal-1248 90 5 ( ( PUNCT american_scientific_journal-1248 90 6 gauss gauss ADJ american_scientific_journal-1248 90 7 - - PUNCT american_scientific_journal-1248 90 8 seidel seidel NOUN american_scientific_journal-1248 90 9 ) ) PUNCT american_scientific_journal-1248 90 10 to to PART american_scientific_journal-1248 90 11 solve solve VERB american_scientific_journal-1248 90 12 for for ADP american_scientific_journal-1248 90 13 temperature temperature NOUN american_scientific_journal-1248 90 14 of of ADP american_scientific_journal-1248 90 15 the the DET american_scientific_journal-1248 90 16 heated heated ADJ american_scientific_journal-1248 90 17 in in ADP american_scientific_journal-1248 90 18 figure figure NOUN american_scientific_journal-1248 90 19 ( ( PUNCT american_scientific_journal-1248 90 20 4.2).employ 4.2).employ NOUN american_scientific_journal-1248 90 21 over over ADP american_scientific_journal-1248 90 22 relaxation relaxation NOUN american_scientific_journal-1248 90 23 with with ADP american_scientific_journal-1248 90 24 a a DET american_scientific_journal-1248 90 25 value value NOUN american_scientific_journal-1248 90 26 of of ADP american_scientific_journal-1248 90 27 1.5 1.5 NUM american_scientific_journal-1248 90 28 for for ADP american_scientific_journal-1248 90 29 the the DET american_scientific_journal-1248 90 30 weighting weighting NOUN american_scientific_journal-1248 90 31 factor factor NOUN american_scientific_journal-1248 90 32 and and CCONJ american_scientific_journal-1248 90 33 iterate iterate VERB american_scientific_journal-1248 90 34 πœ€πœ€π‘Ž πœ€πœ€π‘Ž NOUN american_scientific_journal-1248 90 35 = = SYM american_scientific_journal-1248 90 36 1 1 NUM american_scientific_journal-1248 90 37 % % NOUN american_scientific_journal-1248 90 38 . . PUNCT american_scientific_journal-1248 91 1 solution solution NOUN american_scientific_journal-1248 91 2 from from ADP american_scientific_journal-1248 91 3 equation equation NOUN american_scientific_journal-1248 91 4 𝑇𝑖,𝑗 𝑇𝑖,𝑗 NOUN american_scientific_journal-1248 91 5 = = SYM american_scientific_journal-1248 91 6 𝑇𝑖+1,𝑗+π‘‡π‘–βˆ’1,𝑗+𝑇𝑖,𝑗+1+𝑇𝑖,π‘—βˆ’1 𝑇𝑖+1,𝑗+π‘‡π‘–βˆ’1,𝑗+𝑇𝑖,𝑗+1+𝑇𝑖,π‘—βˆ’1 PROPN american_scientific_journal-1248 91 7 4 4 NUM american_scientific_journal-1248 91 8 at𝑖 at𝑖 NOUN american_scientific_journal-1248 91 9 = = SYM american_scientific_journal-1248 91 10 1 1 NUM american_scientific_journal-1248 91 11 , , PUNCT american_scientific_journal-1248 91 12 𝑗 𝑗 NOUN american_scientific_journal-1248 91 13 = = SYM american_scientific_journal-1248 91 14 1 1 NUM american_scientific_journal-1248 91 15 is𝑇11 is𝑇11 VERB american_scientific_journal-1248 91 16 = = SYM american_scientific_journal-1248 91 17 0 0 NUM american_scientific_journal-1248 91 18 + + NOUN american_scientific_journal-1248 91 19 75 75 NUM american_scientific_journal-1248 91 20 + + SYM american_scientific_journal-1248 91 21 0 0 NUM american_scientific_journal-1248 91 22 + + NOUN american_scientific_journal-1248 91 23 0 0 NUM american_scientific_journal-1248 91 24 4 4 NUM american_scientific_journal-1248 91 25 = = SYM american_scientific_journal-1248 91 26 18.75 18.75 NUM american_scientific_journal-1248 91 27 and and CCONJ american_scientific_journal-1248 91 28 applying apply VERB american_scientific_journal-1248 91 29 relaxation relaxation NOUN american_scientific_journal-1248 91 30 yield yield NOUN american_scientific_journal-1248 91 31 𝑇11 𝑇11 PROPN american_scientific_journal-1248 91 32 = = SYM american_scientific_journal-1248 91 33 1.5(18.75 1.5(18.75 NUM american_scientific_journal-1248 91 34 ) ) PUNCT american_scientific_journal-1248 92 1 + + CCONJ american_scientific_journal-1248 92 2 ( ( PUNCT american_scientific_journal-1248 92 3 1 1 NUM american_scientific_journal-1248 92 4 βˆ’ βˆ’ PROPN american_scientific_journal-1248 92 5 1.5)0 1.5)0 PROPN american_scientific_journal-1248 92 6 = = SYM american_scientific_journal-1248 92 7 28.125 28.125 NUM american_scientific_journal-1248 92 8 𝑇21 𝑇21 PROPN american_scientific_journal-1248 92 9 = = SYM american_scientific_journal-1248 92 10 0 0 PUNCT american_scientific_journal-1248 93 1 + + NUM american_scientific_journal-1248 93 2 28.125 28.125 NUM american_scientific_journal-1248 93 3 + + CCONJ american_scientific_journal-1248 93 4 0 0 NUM american_scientific_journal-1248 94 1 + + CCONJ american_scientific_journal-1248 94 2 0 0 NUM american_scientific_journal-1248 94 3 4 4 NUM american_scientific_journal-1248 94 4 = = SYM american_scientific_journal-1248 94 5 7.03125 7.03125 NUM american_scientific_journal-1248 94 6 𝑇21 𝑇21 PROPN american_scientific_journal-1248 94 7 = = SYM american_scientific_journal-1248 94 8 1.5(7.03125 1.5(7.03125 NUM american_scientific_journal-1248 94 9 ) ) PUNCT american_scientific_journal-1248 95 1 + + CCONJ american_scientific_journal-1248 95 2 ( ( PUNCT american_scientific_journal-1248 95 3 1 1 NUM american_scientific_journal-1248 95 4 βˆ’ βˆ’ PROPN american_scientific_journal-1248 95 5 1.5)0 1.5)0 NUM american_scientific_journal-1248 95 6 = = SYM american_scientific_journal-1248 95 7 10.54688 10.54688 NUM american_scientific_journal-1248 95 8 𝑇31 𝑇31 NOUN american_scientific_journal-1248 95 9 = = SYM american_scientific_journal-1248 95 10 50 50 NUM american_scientific_journal-1248 95 11 + + NUM american_scientific_journal-1248 95 12 10.54688 10.54688 NUM american_scientific_journal-1248 95 13 + + SYM american_scientific_journal-1248 95 14 0 0 NUM american_scientific_journal-1248 96 1 + + CCONJ american_scientific_journal-1248 96 2 0 0 NUM american_scientific_journal-1248 96 3 4 4 NUM american_scientific_journal-1248 96 4 = = SYM american_scientific_journal-1248 96 5 15.13672 15.13672 NUM american_scientific_journal-1248 96 6 𝑇31 𝑇31 NOUN american_scientific_journal-1248 96 7 = = SYM american_scientific_journal-1248 96 8 1.5(15.13672 1.5(15.13672 NOUN american_scientific_journal-1248 96 9 ) ) PUNCT american_scientific_journal-1248 97 1 + + CCONJ american_scientific_journal-1248 97 2 ( ( PUNCT american_scientific_journal-1248 97 3 1 1 NUM american_scientific_journal-1248 97 4 βˆ’ βˆ’ PROPN american_scientific_journal-1248 97 5 1.5)0 1.5)0 NUM american_scientific_journal-1248 97 6 = = SYM american_scientific_journal-1248 97 7 22.70508 22.70508 NUM american_scientific_journal-1248 97 8 the the DET american_scientific_journal-1248 97 9 computation computation NOUN american_scientific_journal-1248 97 10 is be AUX american_scientific_journal-1248 97 11 repeated repeat VERB american_scientific_journal-1248 97 12 for for ADP american_scientific_journal-1248 97 13 other other ADJ american_scientific_journal-1248 97 14 rows row NOUN american_scientific_journal-1248 97 15 to to PART american_scientific_journal-1248 97 16 give give VERB american_scientific_journal-1248 97 17 𝑇12 𝑇12 PROPN american_scientific_journal-1248 97 18 = = SYM american_scientific_journal-1248 97 19 38.67188 38.67188 NUM american_scientific_journal-1248 97 20 𝑇22 𝑇22 PROPN american_scientific_journal-1248 97 21 = = SYM american_scientific_journal-1248 97 22 18.45703 18.45703 NUM american_scientific_journal-1248 97 23 𝑇32 𝑇32 PROPN american_scientific_journal-1248 97 24 = = SYM american_scientific_journal-1248 97 25 34.18579 34.18579 NUM american_scientific_journal-1248 97 26 𝑇13 𝑇13 NOUN american_scientific_journal-1248 97 27 = = SYM american_scientific_journal-1248 97 28 80.12696 80.12696 NUM american_scientific_journal-1248 97 29 𝑇23 𝑇23 PROPN american_scientific_journal-1248 97 30 = = SYM american_scientific_journal-1248 97 31 74.46900 74.46900 NUM american_scientific_journal-1248 97 32 𝑇33 𝑇33 PROPN american_scientific_journal-1248 97 33 = = SYM american_scientific_journal-1248 97 34 96.99554 96.99554 NUM american_scientific_journal-1248 97 35 186 186 NUM american_scientific_journal-1248 97 36 american american ADJ american_scientific_journal-1248 97 37 scientific scientific ADJ american_scientific_journal-1248 97 38 research research NOUN american_scientific_journal-1248 97 39 journal journal NOUN american_scientific_journal-1248 97 40 for for ADP american_scientific_journal-1248 97 41 engineering engineering NOUN american_scientific_journal-1248 97 42 , , PUNCT american_scientific_journal-1248 97 43 technology technology NOUN american_scientific_journal-1248 97 44 , , PUNCT american_scientific_journal-1248 97 45 and and CCONJ american_scientific_journal-1248 97 46 sciences science NOUN american_scientific_journal-1248 97 47 ( ( PUNCT american_scientific_journal-1248 97 48 asrjets asrjet NOUN american_scientific_journal-1248 97 49 ) ) PUNCT american_scientific_journal-1248 97 50 ( ( PUNCT american_scientific_journal-1248 97 51 2016 2016 NUM american_scientific_journal-1248 97 52 ) ) PUNCT american_scientific_journal-1248 97 53 volume volume NOUN american_scientific_journal-1248 97 54 16 16 NUM american_scientific_journal-1248 97 55 , , PUNCT american_scientific_journal-1248 97 56 no no DET american_scientific_journal-1248 97 57 1 1 NUM american_scientific_journal-1248 97 58 , , PUNCT american_scientific_journal-1248 97 59 pp pp ADV american_scientific_journal-1248 97 60 178 178 NUM american_scientific_journal-1248 97 61 - - SYM american_scientific_journal-1248 97 62 192 192 NUM american_scientific_journal-1248 97 63 because because SCONJ american_scientific_journal-1248 97 64 all all DET american_scientific_journal-1248 97 65 the the DET american_scientific_journal-1248 97 66 𝑇𝑖,𝑗`𝑠 𝑇𝑖,𝑗`𝑠 PROPN american_scientific_journal-1248 97 67 are be AUX american_scientific_journal-1248 97 68 initially initially ADV american_scientific_journal-1248 97 69 zero zero NUM american_scientific_journal-1248 97 70 , , PUNCT american_scientific_journal-1248 97 71 all all DET american_scientific_journal-1248 97 72 πœ€πœ€π‘Ž πœ€πœ€π‘Ž NOUN american_scientific_journal-1248 97 73 for for ADP american_scientific_journal-1248 97 74 the the DET american_scientific_journal-1248 97 75 first first ADJ american_scientific_journal-1248 97 76 iteration iteration NOUN american_scientific_journal-1248 97 77 will will AUX american_scientific_journal-1248 97 78 be be AUX american_scientific_journal-1248 97 79 100 100 NUM american_scientific_journal-1248 97 80 % % NOUN american_scientific_journal-1248 97 81 . . PUNCT american_scientific_journal-1248 98 1 for for ADP american_scientific_journal-1248 98 2 the the DET american_scientific_journal-1248 98 3 second second ADJ american_scientific_journal-1248 98 4 iteration iteration NOUN american_scientific_journal-1248 98 5 the the DET american_scientific_journal-1248 98 6 results result NOUN american_scientific_journal-1248 98 7 are be AUX american_scientific_journal-1248 98 8 : : PUNCT american_scientific_journal-1248 99 1 𝑇11 𝑇11 PROPN american_scientific_journal-1248 99 2 = = PUNCT american_scientific_journal-1248 99 3 32.51953 32.51953 NUM american_scientific_journal-1248 99 4 𝑇21 𝑇21 PROPN american_scientific_journal-1248 99 5 = = PUNCT american_scientific_journal-1248 99 6 22.35718 22.35718 NUM american_scientific_journal-1248 100 1 𝑇31 𝑇31 NOUN american_scientific_journal-1248 100 2 = = SYM american_scientific_journal-1248 100 3 28.60108 28.60108 NUM american_scientific_journal-1248 100 4 𝑇12 𝑇12 PROPN american_scientific_journal-1248 100 5 = = SYM american_scientific_journal-1248 100 6 57.95288 57.95288 NUM american_scientific_journal-1248 100 7 𝑇22 𝑇22 PROPN american_scientific_journal-1248 100 8 = = SYM american_scientific_journal-1248 100 9 61.63333 61.63333 NUM american_scientific_journal-1248 100 10 𝑇32 𝑇32 PROPN american_scientific_journal-1248 100 11 = = PUNCT american_scientific_journal-1248 100 12 71.86833 71.86833 NUM american_scientific_journal-1248 101 1 𝑇13 𝑇13 NOUN american_scientific_journal-1248 101 2 = = SYM american_scientific_journal-1248 101 3 75.21973 75.21973 NUM american_scientific_journal-1248 101 4 𝑇23 𝑇23 PROPN american_scientific_journal-1248 101 5 = = SYM american_scientific_journal-1248 101 6 87.95872 87.95872 NUM american_scientific_journal-1248 101 7 𝑇33 𝑇33 PROPN american_scientific_journal-1248 101 8 = = SYM american_scientific_journal-1248 101 9 67.68736 67.68736 NUM american_scientific_journal-1248 101 10 the the DET american_scientific_journal-1248 101 11 error error NOUN american_scientific_journal-1248 101 12 for𝑇11 for𝑇11 NOUN american_scientific_journal-1248 101 13 can can AUX american_scientific_journal-1248 101 14 be be AUX american_scientific_journal-1248 101 15 estimated estimate VERB american_scientific_journal-1248 101 16 as as ADP american_scientific_journal-1248 101 17 equation equation NOUN american_scientific_journal-1248 101 18 βˆ’πœ•π‘ž βˆ’πœ•π‘ž NOUN american_scientific_journal-1248 101 19 πœ•π‘₯ πœ•π‘₯ NOUN american_scientific_journal-1248 101 20 βˆ’ βˆ’ ADP american_scientific_journal-1248 101 21 πœ•π‘ž πœ•π‘ž VERB american_scientific_journal-1248 101 22 πœ•π‘ž πœ•π‘ž ADJ american_scientific_journal-1248 101 23 = = SYM american_scientific_journal-1248 101 24 0 0 NUM american_scientific_journal-1248 101 25 οΏ½ οΏ½ PROPN american_scientific_journal-1248 101 26 ( ( PUNCT american_scientific_journal-1248 101 27 πœ€πœ€π‘Ž)𝑖,𝑗 πœ€πœ€π‘Ž)𝑖,𝑗 PROPN american_scientific_journal-1248 101 28 οΏ½ οΏ½ PROPN american_scientific_journal-1248 101 29 = = SYM american_scientific_journal-1248 101 30 οΏ½ οΏ½ PROPN american_scientific_journal-1248 101 31 32 32 NUM american_scientific_journal-1248 101 32 βˆ’ βˆ’ NOUN american_scientific_journal-1248 101 33 1953 1953 NUM american_scientific_journal-1248 101 34 βˆ’ βˆ’ PROPN american_scientific_journal-1248 101 35 28 28 NUM american_scientific_journal-1248 101 36 βˆ’ βˆ’ PROPN american_scientific_journal-1248 101 37 12500 12500 NUM american_scientific_journal-1248 101 38 32 32 NUM american_scientific_journal-1248 101 39 βˆ’ βˆ’ NUM american_scientific_journal-1248 101 40 51953 51953 NUM american_scientific_journal-1248 101 41 οΏ½ οΏ½ PROPN american_scientific_journal-1248 101 42 100 100 NUM american_scientific_journal-1248 101 43 % % NOUN american_scientific_journal-1248 101 44 = = NOUN american_scientific_journal-1248 101 45 13.5 13.5 NUM american_scientific_journal-1248 101 46 % % NOUN american_scientific_journal-1248 101 47 because because SCONJ american_scientific_journal-1248 101 48 this this DET american_scientific_journal-1248 101 49 value value NOUN american_scientific_journal-1248 101 50 is be AUX american_scientific_journal-1248 101 51 the the DET american_scientific_journal-1248 101 52 stopping stopping NOUN american_scientific_journal-1248 101 53 criterion criterion NOUN american_scientific_journal-1248 101 54 of of ADP american_scientific_journal-1248 101 55 1 1 NUM american_scientific_journal-1248 101 56 % % NOUN american_scientific_journal-1248 101 57 the the DET american_scientific_journal-1248 101 58 computation computation NOUN american_scientific_journal-1248 101 59 is be AUX american_scientific_journal-1248 101 60 continued continue VERB american_scientific_journal-1248 101 61 . . PUNCT american_scientific_journal-1248 102 1 the the DET american_scientific_journal-1248 102 2 ninth ninth ADJ american_scientific_journal-1248 102 3 iteration iteration NOUN american_scientific_journal-1248 102 4 gives give VERB american_scientific_journal-1248 102 5 the the DET american_scientific_journal-1248 102 6 result result NOUN american_scientific_journal-1248 102 7 : : PUNCT american_scientific_journal-1248 103 1 𝑇11 𝑇11 PROPN american_scientific_journal-1248 103 2 = = PUNCT american_scientific_journal-1248 103 3 43.0061 43.0061 NUM american_scientific_journal-1248 104 1 𝑇21 𝑇21 PROPN american_scientific_journal-1248 104 2 = = SYM american_scientific_journal-1248 104 3 33.29755 33.29755 NUM american_scientific_journal-1248 104 4 𝑇31 𝑇31 PROPN american_scientific_journal-1248 104 5 = = SYM american_scientific_journal-1248 104 6 33.88506 33.88506 NUM american_scientific_journal-1248 104 7 𝑇12 𝑇12 PROPN american_scientific_journal-1248 104 8 = = SYM american_scientific_journal-1248 104 9 63.21152 63.21152 NUM american_scientific_journal-1248 104 10 𝑇22 𝑇22 PROPN american_scientific_journal-1248 104 11 = = SYM american_scientific_journal-1248 104 12 56.11238 56.11238 NUM american_scientific_journal-1248 104 13 𝑇32 𝑇32 NOUN american_scientific_journal-1248 104 14 = = SYM american_scientific_journal-1248 104 15 52.33999 52.33999 NUM american_scientific_journal-1248 104 16 𝑇13 𝑇13 NOUN american_scientific_journal-1248 104 17 = = SYM american_scientific_journal-1248 104 18 78.58718 78.58718 NUM american_scientific_journal-1248 104 19 𝑇23 𝑇23 PROPN american_scientific_journal-1248 104 20 = = PROPN american_scientific_journal-1248 104 21 76.06402 76.06402 NUM american_scientific_journal-1248 104 22 𝑇33 𝑇33 PROPN american_scientific_journal-1248 104 23 = = PUNCT american_scientific_journal-1248 104 24 69.71050 69.71050 NUM american_scientific_journal-1248 104 25 where where SCONJ american_scientific_journal-1248 104 26 the the DET american_scientific_journal-1248 104 27 maximum maximum ADJ american_scientific_journal-1248 104 28 error error NOUN american_scientific_journal-1248 104 29 is be AUX american_scientific_journal-1248 104 30 0.71 0.71 NUM american_scientific_journal-1248 104 31 % % NOUN american_scientific_journal-1248 104 32 . . PUNCT american_scientific_journal-1248 105 1 6 6 X american_scientific_journal-1248 105 2 . . X american_scientific_journal-1248 105 3 explicit explicit ADJ american_scientific_journal-1248 105 4 solution solution NOUN american_scientific_journal-1248 105 5 for for ADP american_scientific_journal-1248 105 6 uniform uniform ADJ american_scientific_journal-1248 105 7 grid grid NOUN american_scientific_journal-1248 105 8 increments increment NOUN american_scientific_journal-1248 105 9 consider consider VERB american_scientific_journal-1248 105 10 a a DET american_scientific_journal-1248 105 11 rectangular rectangular ADJ american_scientific_journal-1248 105 12 domain domain NOUN american_scientific_journal-1248 105 13 where where SCONJ american_scientific_journal-1248 105 14 the the DET american_scientific_journal-1248 105 15 increments increment NOUN american_scientific_journal-1248 105 16 in in ADP american_scientific_journal-1248 105 17 both both CCONJ american_scientific_journal-1248 105 18 π‘₯ π‘₯ NOUN american_scientific_journal-1248 105 19 and and CCONJ american_scientific_journal-1248 105 20 𝑦 𝑦 NOUN american_scientific_journal-1248 105 21 are be AUX american_scientific_journal-1248 105 22 uniform uniform ADJ american_scientific_journal-1248 105 23 [ [ X american_scientific_journal-1248 105 24 5 5 NUM american_scientific_journal-1248 105 25 ] ] PUNCT american_scientific_journal-1248 105 26 . . PUNCT american_scientific_journal-1248 106 1 the the DET american_scientific_journal-1248 106 2 appropriate appropriate ADJ american_scientific_journal-1248 106 3 equation equation NOUN american_scientific_journal-1248 106 4 to to PART american_scientific_journal-1248 106 5 use use VERB american_scientific_journal-1248 106 6 for for ADP american_scientific_journal-1248 106 7 an an DET american_scientific_journal-1248 106 8 explicit explicit ADJ american_scientific_journal-1248 106 9 solution solution NOUN american_scientific_journal-1248 106 10 to to ADP american_scientific_journal-1248 106 11 the the DET american_scientific_journal-1248 106 12 laplace laplace NOUN american_scientific_journal-1248 106 13 equation equation NOUN american_scientific_journal-1248 106 14 is be AUX american_scientific_journal-1248 106 15 𝑇𝑖𝑗 𝑇𝑖𝑗 PROPN american_scientific_journal-1248 106 16 = = SYM american_scientific_journal-1248 106 17 π‘‡π‘–βˆ’1,𝑗+𝑇𝑖+1,𝑗+𝛽2 π‘‡π‘–βˆ’1,𝑗+𝑇𝑖+1,𝑗+𝛽2 PROPN american_scientific_journal-1248 106 18 οΏ½ οΏ½ NOUN american_scientific_journal-1248 106 19 𝑇𝑖,𝑗+1+𝑇𝑖,𝑗+1 𝑇𝑖,𝑗+1+𝑇𝑖,𝑗+1 PROPN american_scientific_journal-1248 106 20 οΏ½ οΏ½ PROPN american_scientific_journal-1248 106 21 2(1+𝛽2 2(1+𝛽2 NOUN american_scientific_journal-1248 106 22 ) ) PUNCT american_scientific_journal-1248 106 23 ( ( PUNCT american_scientific_journal-1248 106 24 43 43 NUM american_scientific_journal-1248 106 25 ) ) PUNCT american_scientific_journal-1248 106 26 where where SCONJ american_scientific_journal-1248 106 27 𝛽𝛽 𝛽𝛽 VERB american_scientific_journal-1248 106 28 = = NOUN american_scientific_journal-1248 106 29 Ξ΄π‘₯ Ξ΄π‘₯ ADV american_scientific_journal-1248 106 30 δ𝑦 δ𝑦 ADP american_scientific_journal-1248 106 31 the the DET american_scientific_journal-1248 106 32 solution solution NOUN american_scientific_journal-1248 106 33 will will AUX american_scientific_journal-1248 106 34 start start VERB american_scientific_journal-1248 106 35 by by ADP american_scientific_journal-1248 106 36 loading load VERB american_scientific_journal-1248 106 37 the the DET american_scientific_journal-1248 106 38 boundary boundary ADJ american_scientific_journal-1248 106 39 conditions condition NOUN american_scientific_journal-1248 106 40 , , PUNCT american_scientific_journal-1248 106 41 and and CCONJ american_scientific_journal-1248 106 42 then then ADV american_scientific_journal-1248 106 43 calculating calculate VERB american_scientific_journal-1248 106 44 the the DET american_scientific_journal-1248 106 45 values value NOUN american_scientific_journal-1248 106 46 of of ADP american_scientific_journal-1248 106 47 𝑇𝑖𝑗 𝑇𝑖𝑗 PROPN american_scientific_journal-1248 106 48 in in ADP american_scientific_journal-1248 106 49 the the DET american_scientific_journal-1248 106 50 interior interior ADJ american_scientific_journal-1248 106 51 points point NOUN american_scientific_journal-1248 106 52 of of ADP american_scientific_journal-1248 106 53 domain domain NOUN american_scientific_journal-1248 106 54 . . PUNCT american_scientific_journal-1248 107 1 while while SCONJ american_scientific_journal-1248 107 2 we we PRON american_scientific_journal-1248 107 3 are be AUX american_scientific_journal-1248 107 4 initially initially ADV american_scientific_journal-1248 107 5 temped tempe VERB american_scientific_journal-1248 107 6 to to ADP american_scientific_journal-1248 107 7 calculate𝑇𝑖𝑗 calculate𝑇𝑖𝑗 X american_scientific_journal-1248 107 8 only only ADV american_scientific_journal-1248 107 9 once once ADV american_scientific_journal-1248 107 10 with with ADP american_scientific_journal-1248 107 11 equation equation NOUN american_scientific_journal-1248 107 12 ( ( PUNCT american_scientific_journal-1248 107 13 43 43 NUM american_scientific_journal-1248 107 14 ) ) PUNCT american_scientific_journal-1248 107 15 , , PUNCT american_scientific_journal-1248 107 16 it it PRON american_scientific_journal-1248 107 17 should should AUX american_scientific_journal-1248 107 18 be be AUX american_scientific_journal-1248 107 19 mentioned mention VERB american_scientific_journal-1248 107 20 that that SCONJ american_scientific_journal-1248 107 21 these these DET american_scientific_journal-1248 107 22 values value NOUN american_scientific_journal-1248 107 23 are be AUX american_scientific_journal-1248 107 24 only only ADV american_scientific_journal-1248 107 25 a a DET american_scientific_journal-1248 107 26 fist fist ADJ american_scientific_journal-1248 107 27 approximation approximation NOUN american_scientific_journal-1248 107 28 to to ADP american_scientific_journal-1248 107 29 the the DET american_scientific_journal-1248 107 30 solution solution NOUN american_scientific_journal-1248 107 31 .we .we PUNCT american_scientific_journal-1248 108 1 should should AUX american_scientific_journal-1248 108 2 , , PUNCT american_scientific_journal-1248 108 3 therefore therefore ADV american_scientific_journal-1248 108 4 , , PUNCT american_scientific_journal-1248 108 5 add add VERB american_scientific_journal-1248 108 6 additional additional ADJ american_scientific_journal-1248 108 7 index index NOUN american_scientific_journal-1248 108 8 , , PUNCT american_scientific_journal-1248 108 9 π‘˜ π‘˜ NOUN american_scientific_journal-1248 108 10 , , PUNCT american_scientific_journal-1248 108 11 representing represent VERB american_scientific_journal-1248 108 12 the the DET american_scientific_journal-1248 108 13 current current ADJ american_scientific_journal-1248 108 14 iteration iteration NOUN american_scientific_journal-1248 108 15 , , PUNCT american_scientific_journal-1248 108 16 to to ADP american_scientific_journal-1248 108 17 each each DET american_scientific_journal-1248 108 18 solution solution NOUN american_scientific_journal-1248 108 19 value value NOUN american_scientific_journal-1248 108 20 .the .the PUNCT american_scientific_journal-1248 108 21 solution solution NOUN american_scientific_journal-1248 108 22 values value NOUN american_scientific_journal-1248 108 23 will will AUX american_scientific_journal-1248 108 24 now now ADV american_scientific_journal-1248 108 25 be be AUX american_scientific_journal-1248 108 26 referred refer VERB american_scientific_journal-1248 108 27 to to ADP american_scientific_journal-1248 108 28 as as ADP american_scientific_journal-1248 108 29 π‘‡π‘–π‘—π‘˜ π‘‡π‘–π‘—π‘˜ PROPN american_scientific_journal-1248 108 30 , , PUNCT american_scientific_journal-1248 108 31 and and CCONJ american_scientific_journal-1248 108 32 equation equation NOUN american_scientific_journal-1248 108 33 ( ( PUNCT american_scientific_journal-1248 108 34 43 43 NUM american_scientific_journal-1248 108 35 ) ) PUNCT american_scientific_journal-1248 108 36 will will AUX american_scientific_journal-1248 108 37 be be AUX american_scientific_journal-1248 108 38 modified modify VERB american_scientific_journal-1248 108 39 to to PART american_scientific_journal-1248 108 40 read read VERB american_scientific_journal-1248 108 41 : : PUNCT american_scientific_journal-1248 108 42 187 187 NUM american_scientific_journal-1248 108 43 american american ADJ american_scientific_journal-1248 108 44 scientific scientific ADJ american_scientific_journal-1248 108 45 research research NOUN american_scientific_journal-1248 108 46 journal journal NOUN american_scientific_journal-1248 108 47 for for ADP american_scientific_journal-1248 108 48 engineering engineering NOUN american_scientific_journal-1248 108 49 , , PUNCT american_scientific_journal-1248 108 50 technology technology NOUN american_scientific_journal-1248 108 51 , , PUNCT american_scientific_journal-1248 108 52 and and CCONJ american_scientific_journal-1248 108 53 sciences science NOUN american_scientific_journal-1248 108 54 ( ( PUNCT american_scientific_journal-1248 108 55 asrjets asrjet NOUN american_scientific_journal-1248 108 56 ) ) PUNCT american_scientific_journal-1248 108 57 ( ( PUNCT american_scientific_journal-1248 108 58 2016 2016 NUM american_scientific_journal-1248 108 59 ) ) PUNCT american_scientific_journal-1248 108 60 volume volume NOUN american_scientific_journal-1248 108 61 16 16 NUM american_scientific_journal-1248 108 62 , , PUNCT american_scientific_journal-1248 108 63 no no DET american_scientific_journal-1248 108 64 1 1 NUM american_scientific_journal-1248 108 65 , , PUNCT american_scientific_journal-1248 108 66 pp pp ADV american_scientific_journal-1248 108 67 178 178 NUM american_scientific_journal-1248 108 68 - - SYM american_scientific_journal-1248 108 69 192 192 NUM american_scientific_journal-1248 108 70 𝑇𝑖,π‘—π‘˜+1 𝑇𝑖,π‘—π‘˜+1 NOUN american_scientific_journal-1248 108 71 = = SYM american_scientific_journal-1248 108 72 π‘‡π‘˜π‘–βˆ’1,𝑗+π‘‡π‘˜π‘–+1,𝑗+𝛽2 π‘‡π‘˜π‘–βˆ’1,𝑗+π‘‡π‘˜π‘–+1,𝑗+𝛽2 NOUN american_scientific_journal-1248 108 73 οΏ½ οΏ½ NOUN american_scientific_journal-1248 108 74 π‘‡π‘˜π‘–,π‘—βˆ’1+π‘‡π‘˜π‘–,𝑗+1 π‘‡π‘˜π‘–,π‘—βˆ’1+π‘‡π‘˜π‘–,𝑗+1 PROPN american_scientific_journal-1248 108 75 οΏ½ οΏ½ PROPN american_scientific_journal-1248 108 76 2(1+𝛽2 2(1+𝛽2 NOUN american_scientific_journal-1248 108 77 ) ) PUNCT american_scientific_journal-1248 108 78 ( ( PUNCT american_scientific_journal-1248 108 79 44 44 NUM american_scientific_journal-1248 108 80 ) ) PUNCT american_scientific_journal-1248 108 81 the the DET american_scientific_journal-1248 108 82 iterative iterative NOUN american_scientific_journal-1248 108 83 process process NOUN american_scientific_journal-1248 108 84 should should AUX american_scientific_journal-1248 108 85 be be AUX american_scientific_journal-1248 108 86 repeated repeat VERB american_scientific_journal-1248 108 87 until until SCONJ american_scientific_journal-1248 108 88 convergence convergence NOUN american_scientific_journal-1248 108 89 is be AUX american_scientific_journal-1248 108 90 achieved achieve VERB american_scientific_journal-1248 108 91 in in ADP american_scientific_journal-1248 108 92 every every DET american_scientific_journal-1248 108 93 interior interior ADJ american_scientific_journal-1248 108 94 point point NOUN american_scientific_journal-1248 108 95 of of ADP american_scientific_journal-1248 108 96 the the DET american_scientific_journal-1248 108 97 domain domain NOUN american_scientific_journal-1248 108 98 , , PUNCT american_scientific_journal-1248 108 99 or or CCONJ american_scientific_journal-1248 108 100 until until ADP american_scientific_journal-1248 108 101 a a DET american_scientific_journal-1248 108 102 maximum maximum ADJ american_scientific_journal-1248 108 103 number number NOUN american_scientific_journal-1248 108 104 of of ADP american_scientific_journal-1248 108 105 iterations iteration NOUN american_scientific_journal-1248 108 106 , , PUNCT american_scientific_journal-1248 108 107 say say VERB american_scientific_journal-1248 108 108 100 100 NUM american_scientific_journal-1248 108 109 , , PUNCT american_scientific_journal-1248 108 110 have have AUX american_scientific_journal-1248 108 111 been be AUX american_scientific_journal-1248 108 112 performed perform VERB american_scientific_journal-1248 108 113 . . PUNCT american_scientific_journal-1248 109 1 convergence convergence NOUN american_scientific_journal-1248 109 2 can can AUX american_scientific_journal-1248 109 3 be be AUX american_scientific_journal-1248 109 4 achieved achieve VERB american_scientific_journal-1248 109 5 , , PUNCT american_scientific_journal-1248 109 6 for for ADP american_scientific_journal-1248 109 7 example example NOUN american_scientific_journal-1248 109 8 , , PUNCT american_scientific_journal-1248 109 9 if if SCONJ american_scientific_journal-1248 109 10 , , PUNCT american_scientific_journal-1248 109 11 given give VERB american_scientific_journal-1248 109 12 a a DET american_scientific_journal-1248 109 13 tolerance tolerance NOUN american_scientific_journal-1248 109 14 value value NOUN american_scientific_journal-1248 109 15 πœ€πœ€ πœ€πœ€ DET american_scientific_journal-1248 109 16 maximum maximum ADJ american_scientific_journal-1248 109 17 difference difference NOUN american_scientific_journal-1248 109 18 two two NUM american_scientific_journal-1248 109 19 consecutive consecutive ADJ american_scientific_journal-1248 109 20 iterations iteration NOUN american_scientific_journal-1248 109 21 is be AUX american_scientific_journal-1248 109 22 less less ADJ american_scientific_journal-1248 109 23 than than ADP american_scientific_journal-1248 109 24 the the DET american_scientific_journal-1248 109 25 tolerance tolerance NOUN american_scientific_journal-1248 109 26 , , PUNCT american_scientific_journal-1248 109 27 , , PUNCT american_scientific_journal-1248 109 28 i.e. i.e. X american_scientific_journal-1248 109 29 , , PUNCT american_scientific_journal-1248 109 30 if if SCONJ american_scientific_journal-1248 109 31 max max PROPN american_scientific_journal-1248 109 32 𝑖,𝑗 𝑖,𝑗 PROPN american_scientific_journal-1248 109 33 οΏ½ οΏ½ PROPN american_scientific_journal-1248 109 34 𝑇𝑖,π‘—π‘˜+1 𝑇𝑖,π‘—π‘˜+1 NUM american_scientific_journal-1248 109 35 βˆ’ βˆ’ NOUN american_scientific_journal-1248 109 36 π‘‡π‘˜π‘–,𝑗 π‘‡π‘˜π‘–,𝑗 NUM american_scientific_journal-1248 109 37 οΏ½ οΏ½ PROPN american_scientific_journal-1248 109 38 ≀ ≀ PROPN american_scientific_journal-1248 109 39 πœ€πœ€. πœ€πœ€. PRON american_scientific_journal-1248 109 40 consider consider VERB american_scientific_journal-1248 109 41 , , PUNCT american_scientific_journal-1248 109 42 as as ADP american_scientific_journal-1248 109 43 an an DET american_scientific_journal-1248 109 44 example example NOUN american_scientific_journal-1248 109 45 , , PUNCT american_scientific_journal-1248 109 46 a a DET american_scientific_journal-1248 109 47 rectangular rectangular ADJ american_scientific_journal-1248 109 48 domain domain NOUN american_scientific_journal-1248 109 49 of of ADP american_scientific_journal-1248 109 50 length length NOUN american_scientific_journal-1248 109 51 𝐿 𝐿 PROPN american_scientific_journal-1248 109 52 = = PROPN american_scientific_journal-1248 109 53 5π‘π‘š 5π‘π‘š NOUN american_scientific_journal-1248 109 54 , , PUNCT american_scientific_journal-1248 109 55 and and CCONJ american_scientific_journal-1248 109 56 height height NOUN american_scientific_journal-1248 109 57 𝐻 𝐻 PROPN american_scientific_journal-1248 109 58 = = SYM american_scientific_journal-1248 109 59 3.5π‘π‘š 3.5π‘π‘š PROPN american_scientific_journal-1248 109 60 , , PUNCT american_scientific_journal-1248 109 61 with with ADP american_scientific_journal-1248 109 62 increments increment NOUN american_scientific_journal-1248 110 1 βˆ†π‘₯ βˆ†π‘₯ PROPN american_scientific_journal-1248 110 2 = = SYM american_scientific_journal-1248 110 3 1π‘π‘š 1π‘π‘š PROPN american_scientific_journal-1248 110 4 , , PUNCT american_scientific_journal-1248 110 5 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN american_scientific_journal-1248 110 6 βˆ†π‘¦ βˆ†π‘¦ PROPN american_scientific_journal-1248 110 7 = = SYM american_scientific_journal-1248 110 8 0.5π‘π‘š 0.5π‘π‘š PROPN american_scientific_journal-1248 110 9 , , PUNCT american_scientific_journal-1248 110 10 as as SCONJ american_scientific_journal-1248 110 11 illustrated illustrate VERB american_scientific_journal-1248 110 12 in in ADP american_scientific_journal-1248 110 13 the the DET american_scientific_journal-1248 110 14 figure figure NOUN american_scientific_journal-1248 110 15 blow blow NOUN american_scientific_journal-1248 110 16 . . PUNCT american_scientific_journal-1248 111 1 there there PRON american_scientific_journal-1248 111 2 will will AUX american_scientific_journal-1248 111 3 be be AUX american_scientific_journal-1248 111 4 𝑛 𝑛 PROPN american_scientific_journal-1248 111 5 = = PUNCT american_scientific_journal-1248 111 6 𝐿 𝐿 PROPN american_scientific_journal-1248 111 7 βˆ†π‘₯ βˆ†π‘₯ PROPN american_scientific_journal-1248 111 8 οΏ½ οΏ½ PROPN american_scientific_journal-1248 111 9 sub sub NOUN american_scientific_journal-1248 111 10 - - NOUN american_scientific_journal-1248 111 11 intervals interval NOUN american_scientific_journal-1248 111 12 in in ADP american_scientific_journal-1248 111 13 π‘₯ π‘₯ PRON american_scientific_journal-1248 111 14 , , PUNCT american_scientific_journal-1248 111 15 andπ‘š andπ‘š ADJ american_scientific_journal-1248 111 16 = = SYM american_scientific_journal-1248 111 17 𝐻 𝐻 PROPN american_scientific_journal-1248 111 18 βˆ†π‘¦ βˆ†π‘¦ PROPN american_scientific_journal-1248 111 19 οΏ½ οΏ½ PROPN american_scientific_journal-1248 111 20 sub sub NOUN american_scientific_journal-1248 111 21 - - NOUN american_scientific_journal-1248 111 22 intervals interval NOUN american_scientific_journal-1248 111 23 in in ADP american_scientific_journal-1248 111 24 , , PUNCT american_scientific_journal-1248 111 25 with with ADP american_scientific_journal-1248 111 26 π‘₯𝑖 π‘₯𝑖 NUM american_scientific_journal-1248 111 27 = = PUNCT american_scientific_journal-1248 111 28 ( ( PUNCT american_scientific_journal-1248 111 29 𝑖 𝑖 NOUN american_scientific_journal-1248 111 30 βˆ’ βˆ’ PROPN american_scientific_journal-1248 111 31 1)βˆ†π‘₯ 1)βˆ†π‘₯ NUM american_scientific_journal-1248 111 32 , , PUNCT american_scientific_journal-1248 111 33 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ NOUN american_scientific_journal-1248 111 34 𝑖 𝑖 NOUN american_scientific_journal-1248 111 35 = = SYM american_scientific_journal-1248 111 36 1,2 1,2 NUM american_scientific_journal-1248 111 37 , , PUNCT american_scientific_journal-1248 111 38 … … PUNCT american_scientific_journal-1248 111 39 , , PUNCT american_scientific_journal-1248 111 40 𝑛 𝑛 PRON american_scientific_journal-1248 111 41 + + NOUN american_scientific_journal-1248 111 42 1 1 NUM american_scientific_journal-1248 111 43 , , PUNCT american_scientific_journal-1248 111 44 and and CCONJ american_scientific_journal-1248 111 45 𝑦𝑗 𝑦𝑗 NOUN american_scientific_journal-1248 111 46 = = PUNCT american_scientific_journal-1248 111 47 ( ( PUNCT american_scientific_journal-1248 111 48 𝑗 𝑗 INTJ american_scientific_journal-1248 111 49 βˆ’ βˆ’ PROPN american_scientific_journal-1248 111 50 1)βˆ†π‘¦ 1)βˆ†π‘¦ NUM american_scientific_journal-1248 111 51 , , PUNCT american_scientific_journal-1248 111 52 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ NOUN american_scientific_journal-1248 111 53 𝑗 𝑗 NOUN american_scientific_journal-1248 111 54 = = SYM american_scientific_journal-1248 111 55 1,2 1,2 NUM american_scientific_journal-1248 111 56 , , PUNCT american_scientific_journal-1248 111 57 … … PUNCT american_scientific_journal-1248 111 58 , , PUNCT american_scientific_journal-1248 111 59 π‘š π‘š X american_scientific_journal-1248 111 60 + + NOUN american_scientific_journal-1248 111 61 1 1 NUM american_scientific_journal-1248 111 62 the the DET american_scientific_journal-1248 111 63 boundary boundary ADJ american_scientific_journal-1248 111 64 conditions condition NOUN american_scientific_journal-1248 111 65 are be AUX american_scientific_journal-1248 111 66 given give VERB american_scientific_journal-1248 111 67 as as SCONJ american_scientific_journal-1248 111 68 follows follow VERB american_scientific_journal-1248 111 69 : : PUNCT american_scientific_journal-1248 111 70 𝑇𝑖𝑗 𝑇𝑖𝑗 NOUN american_scientific_journal-1248 111 71 = = SYM american_scientific_journal-1248 111 72 5 5 NUM american_scientific_journal-1248 111 73 along along ADP american_scientific_journal-1248 111 74 the the DET american_scientific_journal-1248 111 75 left left ADJ american_scientific_journal-1248 111 76 and and CCONJ american_scientific_journal-1248 111 77 right right ADJ american_scientific_journal-1248 111 78 sides side NOUN american_scientific_journal-1248 111 79 of of ADP american_scientific_journal-1248 111 80 the the DET american_scientific_journal-1248 111 81 domain domain NOUN american_scientific_journal-1248 111 82 , , PUNCT american_scientific_journal-1248 111 83 while while SCONJ american_scientific_journal-1248 111 84 the the DET american_scientific_journal-1248 111 85 temperature temperature NOUN american_scientific_journal-1248 111 86 are be AUX american_scientific_journal-1248 111 87 given give VERB american_scientific_journal-1248 111 88 by by ADP american_scientific_journal-1248 111 89 the the DET american_scientific_journal-1248 111 90 function function NOUN american_scientific_journal-1248 111 91 𝑇𝑏(π‘₯ 𝑇𝑏(π‘₯ NOUN american_scientific_journal-1248 111 92 ) ) PUNCT american_scientific_journal-1248 111 93 = = SYM american_scientific_journal-1248 111 94 5 5 X american_scientific_journal-1248 111 95 . . X american_scientific_journal-1248 112 1 π‘₯. π‘₯. NOUN american_scientific_journal-1248 112 2 ( ( PUNCT american_scientific_journal-1248 112 3 1 1 NUM american_scientific_journal-1248 112 4 βˆ’ βˆ’ PROPN american_scientific_journal-1248 112 5 π‘₯ π‘₯ NOUN american_scientific_journal-1248 112 6 ) ) PUNCT american_scientific_journal-1248 112 7 for for ADP american_scientific_journal-1248 112 8 the the DET american_scientific_journal-1248 112 9 top top ADJ american_scientific_journal-1248 112 10 and and CCONJ american_scientific_journal-1248 112 11 bottom bottom ADJ american_scientific_journal-1248 112 12 sides side NOUN american_scientific_journal-1248 112 13 of of ADP american_scientific_journal-1248 112 14 the the DET american_scientific_journal-1248 112 15 domain domain NOUN american_scientific_journal-1248 112 16 , , PUNCT american_scientific_journal-1248 112 17 respectively respectively ADV american_scientific_journal-1248 112 18 [ [ X american_scientific_journal-1248 112 19 5 5 NUM american_scientific_journal-1248 112 20 ] ] PUNCT american_scientific_journal-1248 112 21 . . PUNCT american_scientific_journal-1248 113 1 solution solution NOUN american_scientific_journal-1248 113 2 is be AUX american_scientific_journal-1248 113 3 achieved achieve VERB american_scientific_journal-1248 113 4 by by ADP american_scientific_journal-1248 113 5 using use VERB american_scientific_journal-1248 113 6 function function NOUN american_scientific_journal-1248 113 7 laplaceexplicit.min laplaceexplicit.min X american_scientific_journal-1248 113 8 matlab matlab PROPN american_scientific_journal-1248 113 9 : : PUNCT american_scientific_journal-1248 113 10 188 188 NUM american_scientific_journal-1248 113 11 american american ADJ american_scientific_journal-1248 113 12 scientific scientific ADJ american_scientific_journal-1248 113 13 research research NOUN american_scientific_journal-1248 113 14 journal journal NOUN american_scientific_journal-1248 113 15 for for ADP american_scientific_journal-1248 113 16 engineering engineering NOUN american_scientific_journal-1248 113 17 , , PUNCT american_scientific_journal-1248 113 18 technology technology NOUN american_scientific_journal-1248 113 19 , , PUNCT american_scientific_journal-1248 113 20 and and CCONJ american_scientific_journal-1248 113 21 sciences science NOUN american_scientific_journal-1248 113 22 ( ( PUNCT american_scientific_journal-1248 113 23 asrjets asrjet NOUN american_scientific_journal-1248 113 24 ) ) PUNCT american_scientific_journal-1248 113 25 ( ( PUNCT american_scientific_journal-1248 113 26 2016 2016 NUM american_scientific_journal-1248 113 27 ) ) PUNCT american_scientific_journal-1248 113 28 volume volume NOUN american_scientific_journal-1248 113 29 16 16 NUM american_scientific_journal-1248 113 30 , , PUNCT american_scientific_journal-1248 113 31 no no DET american_scientific_journal-1248 113 32 1 1 NUM american_scientific_journal-1248 113 33 , , PUNCT american_scientific_journal-1248 113 34 pp pp ADV american_scientific_journal-1248 113 35 178 178 NUM american_scientific_journal-1248 113 36 - - SYM american_scientific_journal-1248 113 37 192 192 NUM american_scientific_journal-1248 113 38 function function NOUN american_scientific_journal-1248 114 1 [ [ X american_scientific_journal-1248 114 2 x x X american_scientific_journal-1248 114 3 , , PUNCT american_scientific_journal-1248 114 4 y y PROPN american_scientific_journal-1248 114 5 , , PUNCT american_scientific_journal-1248 114 6 t]= t]= PROPN american_scientific_journal-1248 114 7 laplaceexplicit(n laplaceexplicit(n PROPN american_scientific_journal-1248 114 8 , , PUNCT american_scientific_journal-1248 114 9 m m PROPN american_scientific_journal-1248 114 10 , , PUNCT american_scientific_journal-1248 114 11 dx dx PROPN american_scientific_journal-1248 114 12 , , PUNCT american_scientific_journal-1248 114 13 dy dy X american_scientific_journal-1248 114 14 ) ) PUNCT american_scientific_journal-1248 114 15 echo echo VERB american_scientific_journal-1248 114 16 off off ADP american_scientific_journal-1248 114 17 ; ; PUNCT american_scientific_journal-1248 114 18 numgrid(n numgrid(n PROPN american_scientific_journal-1248 114 19 , , PUNCT american_scientific_journal-1248 114 20 m m PROPN american_scientific_journal-1248 114 21 ) ) PUNCT american_scientific_journal-1248 114 22 ; ; PUNCT american_scientific_journal-1248 115 1 r r NOUN american_scientific_journal-1248 115 2 = = SYM american_scientific_journal-1248 115 3 5.0 5.0 NUM american_scientific_journal-1248 115 4 ; ; PUNCT american_scientific_journal-1248 115 5 t t PROPN american_scientific_journal-1248 115 6 = = SYM american_scientific_journal-1248 115 7 r*ones(n+1,m+1 r*ones(n+1,m+1 PROPN american_scientific_journal-1248 115 8 ) ) PUNCT american_scientific_journal-1248 115 9 ; ; PUNCT american_scientific_journal-1248 115 10 % % NOUN american_scientific_journal-1248 115 11 all all DET american_scientific_journal-1248 115 12 t(i t(i PROPN american_scientific_journal-1248 115 13 , , PUNCT american_scientific_journal-1248 115 14 j j NOUN american_scientific_journal-1248 115 15 ) ) PUNCT american_scientific_journal-1248 115 16 = = SYM american_scientific_journal-1248 115 17 1 1 NUM american_scientific_journal-1248 115 18 includes include VERB american_scientific_journal-1248 115 19 all all DET american_scientific_journal-1248 115 20 boundary boundary ADJ american_scientific_journal-1248 115 21 conditions condition NOUN american_scientific_journal-1248 115 22 x x PUNCT american_scientific_journal-1248 115 23 = = PUNCT american_scientific_journal-1248 116 1 [ [ X american_scientific_journal-1248 116 2 0 0 NUM american_scientific_journal-1248 116 3 : : PUNCT american_scientific_journal-1248 116 4 dx dx PROPN american_scientific_journal-1248 116 5 : : PUNCT american_scientific_journal-1248 116 6 n*dx];y=[0 n*dx];y=[0 NUM american_scientific_journal-1248 116 7 : : PUNCT american_scientific_journal-1248 116 8 dy dy NOUN american_scientific_journal-1248 116 9 : : PUNCT american_scientific_journal-1248 116 10 m*dy m*dy NOUN american_scientific_journal-1248 116 11 ] ] PUNCT american_scientific_journal-1248 116 12 ; ; PUNCT american_scientific_journal-1248 116 13 % % NOUN american_scientific_journal-1248 116 14 x x PUNCT american_scientific_journal-1248 116 15 and and CCONJ american_scientific_journal-1248 116 16 y y PROPN american_scientific_journal-1248 116 17 vectors vector NOUN american_scientific_journal-1248 116 18 for for ADP american_scientific_journal-1248 116 19 i i PRON american_scientific_journal-1248 116 20 = = NOUN american_scientific_journal-1248 116 21 1 1 NUM american_scientific_journal-1248 116 22 : : SYM american_scientific_journal-1248 116 23 n n NUM american_scientific_journal-1248 116 24 % % NOUN american_scientific_journal-1248 116 25 boundary boundary ADJ american_scientific_journal-1248 116 26 conditions condition NOUN american_scientific_journal-1248 116 27 at at ADP american_scientific_journal-1248 116 28 j j PROPN american_scientific_journal-1248 116 29 = = PROPN american_scientific_journal-1248 116 30 m+1 m+1 PROPN american_scientific_journal-1248 116 31 and and CCONJ american_scientific_journal-1248 116 32 j j NOUN american_scientific_journal-1248 116 33 = = NOUN american_scientific_journal-1248 117 1 1 1 NUM american_scientific_journal-1248 117 2 6 6 NUM american_scientific_journal-1248 117 3 t(i t(i NUM american_scientific_journal-1248 117 4 , , PUNCT american_scientific_journal-1248 117 5 m+1 m+1 NUM american_scientific_journal-1248 117 6 ) ) PUNCT american_scientific_journal-1248 117 7 = = SYM american_scientific_journal-1248 117 8 t(i t(i PROPN american_scientific_journal-1248 117 9 , , PUNCT american_scientific_journal-1248 117 10 m+1)+ m+1)+ PROPN american_scientific_journal-1248 117 11 r*x(i)*(1 r*x(i)*(1 PROPN american_scientific_journal-1248 117 12 - - PUNCT american_scientific_journal-1248 117 13 x(i x(i PROPN american_scientific_journal-1248 117 14 ) ) PUNCT american_scientific_journal-1248 117 15 ) ) PUNCT american_scientific_journal-1248 117 16 ; ; PUNCT american_scientific_journal-1248 117 17 t(i,1 t(i,1 X american_scientific_journal-1248 117 18 ) ) PUNCT american_scientific_journal-1248 118 1 = = SYM american_scientific_journal-1248 118 2 t(i,1 t(i,1 PROPN american_scientific_journal-1248 118 3 ) ) PUNCT american_scientific_journal-1248 119 1 + + CCONJ american_scientific_journal-1248 119 2 r*x(i)*(x(i)-1 r*x(i)*(x(i)-1 NOUN american_scientific_journal-1248 119 3 ) ) PUNCT american_scientific_journal-1248 119 4 ; ; PUNCT american_scientific_journal-1248 119 5 end end NOUN american_scientific_journal-1248 119 6 ; ; PUNCT american_scientific_journal-1248 119 7 tn tn PROPN american_scientific_journal-1248 119 8 = = SYM american_scientific_journal-1248 119 9 t t PROPN american_scientific_journal-1248 119 10 ; ; PUNCT american_scientific_journal-1248 119 11 % % NOUN american_scientific_journal-1248 119 12 tn tn NOUN american_scientific_journal-1248 120 1 = = PUNCT american_scientific_journal-1248 120 2 new new ADJ american_scientific_journal-1248 120 3 iteration iteration NOUN american_scientific_journal-1248 120 4 for for ADP american_scientific_journal-1248 120 5 solution solution NOUN american_scientific_journal-1248 120 6 err err NOUN american_scientific_journal-1248 120 7 = = SYM american_scientific_journal-1248 120 8 tn tn PROPN american_scientific_journal-1248 120 9 - - PUNCT american_scientific_journal-1248 120 10 t t PROPN american_scientific_journal-1248 120 11 ; ; PUNCT american_scientific_journal-1248 120 12 % % NOUN american_scientific_journal-1248 120 13 parameters parameter NOUN american_scientific_journal-1248 120 14 in in ADP american_scientific_journal-1248 120 15 the the DET american_scientific_journal-1248 120 16 solution solution NOUN american_scientific_journal-1248 120 17 beta beta NOUN american_scientific_journal-1248 120 18 = = SYM american_scientific_journal-1248 120 19 dx dx PROPN american_scientific_journal-1248 120 20 / / SYM american_scientific_journal-1248 120 21 dy dy PROPN american_scientific_journal-1248 120 22 ; ; PUNCT american_scientific_journal-1248 120 23 denom denom PROPN american_scientific_journal-1248 120 24 = = SYM american_scientific_journal-1248 120 25 2*(1+beta^2 2*(1+beta^2 NUM american_scientific_journal-1248 120 26 ) ) PUNCT american_scientific_journal-1248 120 27 ; ; PUNCT american_scientific_journal-1248 121 1 % % NOUN american_scientific_journal-1248 121 2 iterative iterative NOUN american_scientific_journal-1248 121 3 procedure procedure NOUN american_scientific_journal-1248 121 4 epsilon epsilon NOUN american_scientific_journal-1248 121 5 = = SYM american_scientific_journal-1248 121 6 1e-5 1e-5 NUM american_scientific_journal-1248 121 7 ; ; PUNCT american_scientific_journal-1248 121 8 % % NOUN american_scientific_journal-1248 121 9 tolerance tolerance NOUN american_scientific_journal-1248 121 10 for for ADP american_scientific_journal-1248 121 11 convergence convergence NOUN american_scientific_journal-1248 121 12 imax imax NOUN american_scientific_journal-1248 121 13 = = SYM american_scientific_journal-1248 121 14 1000 1000 NUM american_scientific_journal-1248 121 15 ; ; PUNCT american_scientific_journal-1248 121 16 % % NOUN american_scientific_journal-1248 121 17 maximum maximum ADJ american_scientific_journal-1248 121 18 number number NOUN american_scientific_journal-1248 121 19 of of ADP american_scientific_journal-1248 121 20 iterations iteration NOUN american_scientific_journal-1248 121 21 allowed allow VERB american_scientific_journal-1248 121 22 k k NOUN american_scientific_journal-1248 121 23 = = SYM american_scientific_journal-1248 121 24 1 1 NUM american_scientific_journal-1248 121 25 ; ; PUNCT american_scientific_journal-1248 121 26 % % NOUN american_scientific_journal-1248 121 27 initial initial ADJ american_scientific_journal-1248 121 28 index index NOUN american_scientific_journal-1248 121 29 value value NOUN american_scientific_journal-1248 121 30 for for ADP american_scientific_journal-1248 121 31 iteration iteration NOUN american_scientific_journal-1248 121 32 % % NOUN american_scientific_journal-1248 121 33 calculation calculation NOUN american_scientific_journal-1248 121 34 loop loop NOUN american_scientific_journal-1248 121 35 while while SCONJ american_scientific_journal-1248 121 36 k<= k<= PROPN american_scientific_journal-1248 121 37 imax imax PROPN american_scientific_journal-1248 121 38 189 189 NUM american_scientific_journal-1248 121 39 american american PROPN american_scientific_journal-1248 121 40 scientific scientific ADJ american_scientific_journal-1248 121 41 research research NOUN american_scientific_journal-1248 121 42 journal journal NOUN american_scientific_journal-1248 121 43 for for ADP american_scientific_journal-1248 121 44 engineering engineering NOUN american_scientific_journal-1248 121 45 , , PUNCT american_scientific_journal-1248 121 46 technology technology NOUN american_scientific_journal-1248 121 47 , , PUNCT american_scientific_journal-1248 121 48 and and CCONJ american_scientific_journal-1248 121 49 sciences science NOUN american_scientific_journal-1248 121 50 ( ( PUNCT american_scientific_journal-1248 121 51 asrjets asrjet NOUN american_scientific_journal-1248 121 52 ) ) PUNCT american_scientific_journal-1248 121 53 ( ( PUNCT american_scientific_journal-1248 121 54 2016 2016 NUM american_scientific_journal-1248 121 55 ) ) PUNCT american_scientific_journal-1248 121 56 volume volume NOUN american_scientific_journal-1248 121 57 16 16 NUM american_scientific_journal-1248 121 58 , , PUNCT american_scientific_journal-1248 121 59 no no DET american_scientific_journal-1248 121 60 1 1 NUM american_scientific_journal-1248 121 61 , , PUNCT american_scientific_journal-1248 121 62 pp pp ADV american_scientific_journal-1248 121 63 178 178 NUM american_scientific_journal-1248 121 64 - - SYM american_scientific_journal-1248 121 65 192 192 NUM american_scientific_journal-1248 121 66 for for ADP american_scientific_journal-1248 121 67 i i PRON american_scientific_journal-1248 121 68 = = NOUN american_scientific_journal-1248 121 69 2 2 NUM american_scientific_journal-1248 121 70 : : PUNCT american_scientific_journal-1248 121 71 n n PROPN american_scientific_journal-1248 121 72 for for ADP american_scientific_journal-1248 121 73 j j PROPN american_scientific_journal-1248 121 74 = = SYM american_scientific_journal-1248 121 75 2 2 NUM american_scientific_journal-1248 121 76 : : PUNCT american_scientific_journal-1248 121 77 m m NOUN american_scientific_journal-1248 121 78 tn(i tn(i ADJ american_scientific_journal-1248 121 79 , , PUNCT american_scientific_journal-1248 121 80 j)=(t(i-1,j)+t(i+1,j)+beta^2*(t(i j)=(t(i-1,j)+t(i+1,j)+beta^2*(t(i PROPN american_scientific_journal-1248 121 81 , , PUNCT american_scientific_journal-1248 121 82 j-1)+t(i j-1)+t(i PROPN american_scientific_journal-1248 121 83 , , PUNCT american_scientific_journal-1248 121 84 j+1)))/denom j+1)))/denom ADJ american_scientific_journal-1248 121 85 ; ; PUNCT american_scientific_journal-1248 121 86 err(i err(i PROPN american_scientific_journal-1248 121 87 , , PUNCT american_scientific_journal-1248 121 88 j j PROPN american_scientific_journal-1248 121 89 ) ) PUNCT american_scientific_journal-1248 121 90 = = SYM american_scientific_journal-1248 121 91 abs(tn(i abs(tn(i PROPN american_scientific_journal-1248 121 92 , , PUNCT american_scientific_journal-1248 121 93 j)-t(i j)-t(i PROPN american_scientific_journal-1248 121 94 , , PUNCT american_scientific_journal-1248 121 95 j j PROPN american_scientific_journal-1248 121 96 ) ) PUNCT american_scientific_journal-1248 121 97 ) ) PUNCT american_scientific_journal-1248 121 98 ; ; PUNCT american_scientific_journal-1248 122 1 end end NOUN american_scientific_journal-1248 122 2 ; ; PUNCT american_scientific_journal-1248 122 3 end end NOUN american_scientific_journal-1248 122 4 ; ; PUNCT american_scientific_journal-1248 122 5 t t PROPN american_scientific_journal-1248 122 6 = = SYM american_scientific_journal-1248 122 7 tn tn PROPN american_scientific_journal-1248 122 8 ; ; PUNCT american_scientific_journal-1248 122 9 k k PROPN american_scientific_journal-1248 122 10 = = PUNCT american_scientific_journal-1248 122 11 k k PROPN american_scientific_journal-1248 123 1 + + PROPN american_scientific_journal-1248 123 2 1 1 NUM american_scientific_journal-1248 123 3 ; ; PUNCT american_scientific_journal-1248 123 4 errmax errmax NOUN american_scientific_journal-1248 123 5 = = SYM american_scientific_journal-1248 123 6 max(max(err max(max(err NOUN american_scientific_journal-1248 123 7 ) ) PUNCT american_scientific_journal-1248 123 8 ) ) PUNCT american_scientific_journal-1248 123 9 ; ; PUNCT american_scientific_journal-1248 123 10 if if SCONJ american_scientific_journal-1248 123 11 errmax errmax PROPN american_scientific_journal-1248 123 12 < < X american_scientific_journal-1248 123 13 epsilon epsilon PROPN american_scientific_journal-1248 123 14 [ [ X american_scientific_journal-1248 123 15 x x X american_scientific_journal-1248 123 16 , , PUNCT american_scientific_journal-1248 123 17 y y PROPN american_scientific_journal-1248 123 18 ] ] X american_scientific_journal-1248 123 19 = = PUNCT american_scientific_journal-1248 123 20 meshgrid(x meshgrid(x PROPN american_scientific_journal-1248 123 21 , , PUNCT american_scientific_journal-1248 123 22 y y PROPN american_scientific_journal-1248 123 23 ) ) PUNCT american_scientific_journal-1248 123 24 ; ; PUNCT american_scientific_journal-1248 123 25 figure(2);contour(x figure(2);contour(x PROPN american_scientific_journal-1248 123 26 , , PUNCT american_scientific_journal-1248 123 27 y y PROPN american_scientific_journal-1248 123 28 , , PUNCT american_scientific_journal-1248 123 29 t',20);xlabel('x');ylabel('y t',20);xlabel('x');ylabel('y PROPN american_scientific_journal-1248 123 30 ' ' PUNCT american_scientific_journal-1248 123 31 ) ) PUNCT american_scientific_journal-1248 123 32 ; ; PUNCT american_scientific_journal-1248 123 33 title('laplace title('laplace VERB american_scientific_journal-1248 123 34 equation equation NOUN american_scientific_journal-1248 123 35 solution solution NOUN american_scientific_journal-1248 123 36 dirichlet dirichlet PROPN american_scientific_journal-1248 123 37 boundary boundary PROPN american_scientific_journal-1248 123 38 conditions condition NOUN american_scientific_journal-1248 123 39 explicit explicit ADJ american_scientific_journal-1248 123 40 ' ' PUNCT american_scientific_journal-1248 123 41 ) ) PUNCT american_scientific_journal-1248 123 42 ; ; PUNCT american_scientific_journal-1248 123 43 figure(3);surfc(x figure(3);surfc(x PROPN american_scientific_journal-1248 123 44 , , PUNCT american_scientific_journal-1248 123 45 y y PROPN american_scientific_journal-1248 123 46 , , PUNCT american_scientific_journal-1248 123 47 t');xlabel('x');ylabel('y');zlabel('t(x t');xlabel('x');ylabel('y');zlabel('t(x NOUN american_scientific_journal-1248 123 48 , , PUNCT american_scientific_journal-1248 123 49 y y NOUN american_scientific_journal-1248 123 50 ) ) PUNCT american_scientific_journal-1248 123 51 ' ' PUNCT american_scientific_journal-1248 123 52 ) ) PUNCT american_scientific_journal-1248 123 53 ; ; PUNCT american_scientific_journal-1248 123 54 title('laplace title('laplace VERB american_scientific_journal-1248 123 55 equation equation NOUN american_scientific_journal-1248 123 56 solution solution NOUN american_scientific_journal-1248 123 57 dirichlet dirichlet PROPN american_scientific_journal-1248 123 58 boundary boundary PROPN american_scientific_journal-1248 123 59 conditions condition NOUN american_scientific_journal-1248 123 60 explicit explicit ADJ american_scientific_journal-1248 123 61 ' ' PUNCT american_scientific_journal-1248 123 62 ) ) PUNCT american_scientific_journal-1248 123 63 ; ; PUNCT american_scientific_journal-1248 123 64 fprintf('convergence fprintf('convergence AUX american_scientific_journal-1248 123 65 achieved achieve VERB american_scientific_journal-1248 123 66 after after ADP american_scientific_journal-1248 123 67 % % INTJ american_scientific_journal-1248 123 68 i i PRON american_scientific_journal-1248 123 69 iterations.\n',k iterations.\n',k PROPN american_scientific_journal-1248 123 70 ) ) PUNCT american_scientific_journal-1248 123 71 ; ; PUNCT american_scientific_journal-1248 123 72 fprintf('see fprintf('see VERB american_scientific_journal-1248 123 73 the the DET american_scientific_journal-1248 123 74 following follow VERB american_scientific_journal-1248 123 75 figures:\n figures:\n NOUN american_scientific_journal-1248 123 76 ' ' PUNCT american_scientific_journal-1248 123 77 ) ) PUNCT american_scientific_journal-1248 123 78 ; ; PUNCT american_scientific_journal-1248 123 79 fprintf('==========================\n fprintf('==========================\n NOUN american_scientific_journal-1248 123 80 ' ' NUM american_scientific_journal-1248 123 81 ) ) PUNCT american_scientific_journal-1248 123 82 ; ; PUNCT american_scientific_journal-1248 123 83 fprintf('figure fprintf('figure ADJ american_scientific_journal-1248 123 84 1 1 NUM american_scientific_journal-1248 123 85 sketch sketch NOUN american_scientific_journal-1248 123 86 of of ADP american_scientific_journal-1248 123 87 computational computational ADJ american_scientific_journal-1248 123 88 grid grid NOUN american_scientific_journal-1248 123 89 \n \n PROPN american_scientific_journal-1248 123 90 ' ' PUNCT american_scientific_journal-1248 123 91 ) ) PUNCT american_scientific_journal-1248 123 92 ; ; PUNCT american_scientific_journal-1248 123 93 fprintf('figure fprintf('figure NOUN american_scientific_journal-1248 123 94 2 2 NUM american_scientific_journal-1248 123 95 contour contour NOUN american_scientific_journal-1248 123 96 plot plot NOUN american_scientific_journal-1248 123 97 of of ADP american_scientific_journal-1248 123 98 temperature temperature NOUN american_scientific_journal-1248 123 99 \n \n NOUN american_scientific_journal-1248 123 100 ' ' PUNCT american_scientific_journal-1248 123 101 ) ) PUNCT american_scientific_journal-1248 123 102 ; ; PUNCT american_scientific_journal-1248 123 103 fprintf('figure fprintf('figure NOUN american_scientific_journal-1248 123 104 3 3 NUM american_scientific_journal-1248 123 105 surface surface NOUN american_scientific_journal-1248 123 106 plot plot NOUN american_scientific_journal-1248 123 107 of of ADP american_scientific_journal-1248 123 108 temperature temperature NOUN american_scientific_journal-1248 123 109 \n \n NOUN american_scientific_journal-1248 123 110 ' ' PUNCT american_scientific_journal-1248 123 111 ) ) PUNCT american_scientific_journal-1248 123 112 ; ; PUNCT american_scientific_journal-1248 123 113 190 190 NUM american_scientific_journal-1248 123 114 american american ADJ american_scientific_journal-1248 123 115 scientific scientific ADJ american_scientific_journal-1248 123 116 research research NOUN american_scientific_journal-1248 123 117 journal journal NOUN american_scientific_journal-1248 123 118 for for ADP american_scientific_journal-1248 123 119 engineering engineering NOUN american_scientific_journal-1248 123 120 , , PUNCT american_scientific_journal-1248 123 121 technology technology NOUN american_scientific_journal-1248 123 122 , , PUNCT american_scientific_journal-1248 123 123 and and CCONJ american_scientific_journal-1248 123 124 sciences science NOUN american_scientific_journal-1248 123 125 ( ( PUNCT american_scientific_journal-1248 123 126 asrjets asrjet NOUN american_scientific_journal-1248 123 127 ) ) PUNCT american_scientific_journal-1248 123 128 ( ( PUNCT american_scientific_journal-1248 123 129 2016 2016 NUM american_scientific_journal-1248 123 130 ) ) PUNCT american_scientific_journal-1248 123 131 volume volume NOUN american_scientific_journal-1248 123 132 16 16 NUM american_scientific_journal-1248 123 133 , , PUNCT american_scientific_journal-1248 123 134 no no DET american_scientific_journal-1248 123 135 1 1 NUM american_scientific_journal-1248 123 136 , , PUNCT american_scientific_journal-1248 123 137 pp pp ADV american_scientific_journal-1248 123 138 178 178 NUM american_scientific_journal-1248 123 139 - - SYM american_scientific_journal-1248 123 140 192 192 NUM american_scientific_journal-1248 123 141 return return NOUN american_scientific_journal-1248 123 142 end end NOUN american_scientific_journal-1248 123 143 ; ; PUNCT american_scientific_journal-1248 123 144 end end NOUN american_scientific_journal-1248 123 145 ; ; PUNCT american_scientific_journal-1248 123 146 fprintf('\n fprintf('\n X american_scientific_journal-1248 123 147 no no DET american_scientific_journal-1248 123 148 convergence convergence NOUN american_scientific_journal-1248 123 149 after after ADP american_scientific_journal-1248 123 150 % % INTJ american_scientific_journal-1248 123 151 i i PRON american_scientific_journal-1248 123 152 iterations iteration NOUN american_scientific_journal-1248 123 153 . . PUNCT american_scientific_journal-1248 123 154 ',k ',k NUM american_scientific_journal-1248 123 155 ) ) PUNCT american_scientific_journal-1248 123 156 ; ; PUNCT american_scientific_journal-1248 123 157 to to PART american_scientific_journal-1248 123 158 activate activate VERB american_scientific_journal-1248 123 159 the the DET american_scientific_journal-1248 123 160 function function NOUN american_scientific_journal-1248 123 161 for for ADP american_scientific_journal-1248 123 162 the the DET american_scientific_journal-1248 123 163 case case NOUN american_scientific_journal-1248 123 164 illustrated illustrate VERB american_scientific_journal-1248 123 165 in in ADP american_scientific_journal-1248 123 166 the the DET american_scientific_journal-1248 123 167 figure figure NOUN american_scientific_journal-1248 123 168 above above ADP american_scientific_journal-1248 123 169 we we PRON american_scientific_journal-1248 123 170 use use VERB american_scientific_journal-1248 123 171 : : PUNCT american_scientific_journal-1248 123 172 ≫ ≫ PROPN american_scientific_journal-1248 123 173 [ [ X american_scientific_journal-1248 123 174 𝑋,π‘Œ,𝑇 𝑋,π‘Œ,𝑇 X american_scientific_journal-1248 123 175 ] ] X american_scientific_journal-1248 123 176 = = SYM american_scientific_journal-1248 123 177 πΏπ‘Žπ‘π‘™π‘Žπ‘π‘’πΈπ‘₯𝑝𝑙𝑖𝑐𝑖𝑑(5,7,1,0.5 πΏπ‘Žπ‘π‘™π‘Žπ‘π‘’πΈπ‘₯𝑝𝑙𝑖𝑐𝑖𝑑(5,7,1,0.5 PROPN american_scientific_journal-1248 123 178 ) ) PUNCT american_scientific_journal-1248 123 179 the the DET american_scientific_journal-1248 123 180 solution solution NOUN american_scientific_journal-1248 123 181 is be AUX american_scientific_journal-1248 123 182 returned return VERB american_scientific_journal-1248 123 183 in in ADP american_scientific_journal-1248 123 184 the the DET american_scientific_journal-1248 123 185 vectors vector NOUN american_scientific_journal-1248 123 186 π‘₯ π‘₯ PROPN american_scientific_journal-1248 123 187 and and CCONJ american_scientific_journal-1248 123 188 𝑦 𝑦 NOUN american_scientific_journal-1248 123 189 , , PUNCT american_scientific_journal-1248 123 190 and and CCONJ american_scientific_journal-1248 123 191 in in ADP american_scientific_journal-1248 123 192 matrix matrix NOUN american_scientific_journal-1248 123 193 𝑇 𝑇 PROPN american_scientific_journal-1248 123 194 .the .the PRON american_scientific_journal-1248 123 195 function function NOUN american_scientific_journal-1248 123 196 produces produce VERB american_scientific_journal-1248 123 197 three three NUM american_scientific_journal-1248 123 198 plots plot NOUN american_scientific_journal-1248 123 199 : : PUNCT american_scientific_journal-1248 123 200 a a DET american_scientific_journal-1248 123 201 sketch sketch NOUN american_scientific_journal-1248 123 202 of of ADP american_scientific_journal-1248 123 203 the the DET american_scientific_journal-1248 123 204 grid grid NOUN american_scientific_journal-1248 123 205 ( ( PUNCT american_scientific_journal-1248 123 206 similar similar ADJ american_scientific_journal-1248 123 207 to to ADP american_scientific_journal-1248 123 208 the the DET american_scientific_journal-1248 123 209 figure figure NOUN american_scientific_journal-1248 123 210 above),the above),the NOUN american_scientific_journal-1248 123 211 solution solution NOUN american_scientific_journal-1248 123 212 as as ADP american_scientific_journal-1248 123 213 a a DET american_scientific_journal-1248 123 214 contours contours NOUN american_scientific_journal-1248 123 215 , , PUNCT american_scientific_journal-1248 123 216 and and CCONJ american_scientific_journal-1248 123 217 the the DET american_scientific_journal-1248 123 218 solution solution NOUN american_scientific_journal-1248 123 219 as as ADP american_scientific_journal-1248 123 220 a a DET american_scientific_journal-1248 123 221 surface surface NOUN american_scientific_journal-1248 123 222 .the .the INTJ american_scientific_journal-1248 124 1 last last ADJ american_scientific_journal-1248 124 2 two two NUM american_scientific_journal-1248 124 3 figures figure NOUN american_scientific_journal-1248 124 4 are be AUX american_scientific_journal-1248 124 5 shown show VERB american_scientific_journal-1248 124 6 next next ADV american_scientific_journal-1248 124 7 : : PUNCT american_scientific_journal-1248 124 8 laplace laplace NOUN american_scientific_journal-1248 124 9 equation equation NOUN american_scientific_journal-1248 124 10 solution solution NOUN american_scientific_journal-1248 124 11 – – PUNCT american_scientific_journal-1248 124 12 dirichletboundary dirichletboundary ADJ american_scientific_journal-1248 124 13 conditions condition NOUN american_scientific_journal-1248 124 14 - - PUNCT american_scientific_journal-1248 124 15 explicit explicit ADJ american_scientific_journal-1248 124 16 laplace laplace NOUN american_scientific_journal-1248 124 17 equation equation NOUN american_scientific_journal-1248 124 18 solution solution NOUN american_scientific_journal-1248 124 19 dirichletboundary dirichletboundary VERB american_scientific_journal-1248 124 20 conditions condition NOUN american_scientific_journal-1248 124 21 explicit explicit ADJ american_scientific_journal-1248 124 22 191 191 NUM american_scientific_journal-1248 124 23 american american ADJ american_scientific_journal-1248 124 24 scientific scientific ADJ american_scientific_journal-1248 124 25 research research NOUN american_scientific_journal-1248 124 26 journal journal NOUN american_scientific_journal-1248 124 27 for for ADP american_scientific_journal-1248 124 28 engineering engineering NOUN american_scientific_journal-1248 124 29 , , PUNCT american_scientific_journal-1248 124 30 technology technology NOUN american_scientific_journal-1248 124 31 , , PUNCT american_scientific_journal-1248 124 32 and and CCONJ american_scientific_journal-1248 124 33 sciences science NOUN american_scientific_journal-1248 124 34 ( ( PUNCT american_scientific_journal-1248 124 35 asrjets asrjet NOUN american_scientific_journal-1248 124 36 ) ) PUNCT american_scientific_journal-1248 124 37 ( ( PUNCT american_scientific_journal-1248 124 38 2016 2016 NUM american_scientific_journal-1248 124 39 ) ) PUNCT american_scientific_journal-1248 124 40 volume volume NOUN american_scientific_journal-1248 124 41 16 16 NUM american_scientific_journal-1248 124 42 , , PUNCT american_scientific_journal-1248 124 43 no no DET american_scientific_journal-1248 124 44 1 1 NUM american_scientific_journal-1248 124 45 , , PUNCT american_scientific_journal-1248 124 46 pp pp ADV american_scientific_journal-1248 124 47 178 178 NUM american_scientific_journal-1248 124 48 - - SYM american_scientific_journal-1248 124 49 192 192 NUM american_scientific_journal-1248 124 50 7 7 NUM american_scientific_journal-1248 124 51 . . PUNCT american_scientific_journal-1248 124 52 conclusion conclusion NOUN american_scientific_journal-1248 124 53 and and CCONJ american_scientific_journal-1248 124 54 recommendation recommendation NOUN american_scientific_journal-1248 124 55 this this DET american_scientific_journal-1248 124 56 paper paper NOUN american_scientific_journal-1248 124 57 showed show VERB american_scientific_journal-1248 124 58 that that SCONJ american_scientific_journal-1248 124 59 numerical numerical ADJ american_scientific_journal-1248 124 60 methods method NOUN american_scientific_journal-1248 124 61 are be AUX american_scientific_journal-1248 124 62 usually usually ADV american_scientific_journal-1248 124 63 easier easy ADJ american_scientific_journal-1248 124 64 to to PART american_scientific_journal-1248 124 65 use use VERB american_scientific_journal-1248 124 66 in in ADP american_scientific_journal-1248 124 67 the the DET american_scientific_journal-1248 124 68 solution solution NOUN american_scientific_journal-1248 124 69 of of ADP american_scientific_journal-1248 124 70 laplace laplace NOUN american_scientific_journal-1248 124 71 's 's PART american_scientific_journal-1248 124 72 equation equation NOUN american_scientific_journal-1248 124 73 . . PUNCT american_scientific_journal-1248 125 1 we we PRON american_scientific_journal-1248 125 2 notice notice VERB american_scientific_journal-1248 125 3 that that SCONJ american_scientific_journal-1248 125 4 our our PRON american_scientific_journal-1248 125 5 results result NOUN american_scientific_journal-1248 125 6 agree agree VERB american_scientific_journal-1248 125 7 with with ADP american_scientific_journal-1248 125 8 other other ADJ american_scientific_journal-1248 125 9 works work NOUN american_scientific_journal-1248 125 10 cited cite VERB american_scientific_journal-1248 125 11 in in ADP american_scientific_journal-1248 125 12 article article NOUN american_scientific_journal-1248 125 13 . . PUNCT american_scientific_journal-1248 126 1 it it PRON american_scientific_journal-1248 126 2 is be AUX american_scientific_journal-1248 126 3 aim aim NOUN american_scientific_journal-1248 126 4 to to PART american_scientific_journal-1248 126 5 continue continue VERB american_scientific_journal-1248 126 6 higher high ADJ american_scientific_journal-1248 126 7 order order NOUN american_scientific_journal-1248 126 8 for for ADP american_scientific_journal-1248 126 9 two two NUM american_scientific_journal-1248 126 10 dimensional dimensional ADJ american_scientific_journal-1248 126 11 . . PUNCT american_scientific_journal-1248 127 1 references reference NOUN american_scientific_journal-1248 127 2 [ [ X american_scientific_journal-1248 127 3 1 1 NUM american_scientific_journal-1248 127 4 ] ] X american_scientific_journal-1248 127 5 partial partial ADJ american_scientific_journal-1248 127 6 differential differential NOUN american_scientific_journal-1248 127 7 equations equation NOUN american_scientific_journal-1248 127 8 of of ADP american_scientific_journal-1248 127 9 mathematical mathematical ADJ american_scientific_journal-1248 127 10 physics.1969 physics.1969 PROPN american_scientific_journal-1248 127 11 . . PUNCT american_scientific_journal-1248 128 1 pp pp CCONJ american_scientific_journal-1248 128 2 120 120 NUM american_scientific_journal-1248 128 3 - - SYM american_scientific_journal-1248 128 4 126 126 NUM american_scientific_journal-1248 128 5 . . PUNCT american_scientific_journal-1248 129 1 [ [ X american_scientific_journal-1248 129 2 2 2 X american_scientific_journal-1248 129 3 ] ] X american_scientific_journal-1248 129 4 paul paul PROPN american_scientific_journal-1248 129 5 w. w. PROPN american_scientific_journal-1248 129 6 berg berg PROPN american_scientific_journal-1248 129 7 & & CCONJ american_scientific_journal-1248 129 8 james james PROPN american_scientific_journal-1248 129 9 l. l. PROPN american_scientific_journal-1248 129 10 mc mc PROPN american_scientific_journal-1248 129 11 gregor gregor PROPN american_scientific_journal-1248 129 12 elementary elementary PROPN american_scientific_journal-1248 129 13 differential differential PROPN american_scientific_journal-1248 129 14 equation equation NOUN american_scientific_journal-1248 129 15 1960 1960 NUM american_scientific_journal-1248 129 16 , , PUNCT american_scientific_journal-1248 129 17 pp55 pp55 PROPN american_scientific_journal-1248 129 18 . . PUNCT american_scientific_journal-1248 130 1 [ [ X american_scientific_journal-1248 130 2 3 3 NUM american_scientific_journal-1248 130 3 ] ] X american_scientific_journal-1248 130 4 steven steven PROPN american_scientific_journal-1248 130 5 c. c. PROPN american_scientific_journal-1248 130 6 chapra chapra PROPN american_scientific_journal-1248 130 7 & & CCONJ american_scientific_journal-1248 130 8 roymond roymond PROPN american_scientific_journal-1248 130 9 pcanale pcanale PROPN american_scientific_journal-1248 130 10 numerical numerical ADJ american_scientific_journal-1248 130 11 methods method NOUN american_scientific_journal-1248 130 12 for for ADP american_scientific_journal-1248 130 13 engineers engineer NOUN american_scientific_journal-1248 130 14 2002.pp 2002.pp NOUN american_scientific_journal-1248 130 15 40 40 NUM american_scientific_journal-1248 130 16 . . PUNCT american_scientific_journal-1248 131 1 [ [ X american_scientific_journal-1248 131 2 4 4 NUM american_scientific_journal-1248 131 3 ] ] X american_scientific_journal-1248 131 4 grewal grewal PROPN american_scientific_journal-1248 131 5 m.a m.a PROPN american_scientific_journal-1248 131 6 . . PROPN american_scientific_journal-1248 131 7 ph ph PROPN american_scientific_journal-1248 131 8 . . PROPN american_scientific_journal-1248 131 9 d. d. PROPN american_scientific_journal-1248 131 10 higher high ADJ american_scientific_journal-1248 131 11 engineering engineering PROPN american_scientific_journal-1248 131 12 methematics methematic NOUN american_scientific_journal-1248 131 13 1990.pp 1990.pp NUM american_scientific_journal-1248 131 14 86 86 NUM american_scientific_journal-1248 131 15 - - SYM american_scientific_journal-1248 131 16 99 99 NUM american_scientific_journal-1248 131 17 . . PUNCT american_scientific_journal-1248 132 1 [ [ X american_scientific_journal-1248 132 2 5 5 NUM american_scientific_journal-1248 132 3 ] ] X american_scientific_journal-1248 132 4 numerical numerical ADJ american_scientific_journal-1248 132 5 solution solution NOUN american_scientific_journal-1248 132 6 of of ADP american_scientific_journal-1248 132 7 laplace laplace NOUN american_scientific_journal-1248 132 8 equation equation NOUN american_scientific_journal-1248 132 9 by by ADP american_scientific_journal-1248 132 10 gilberto gilberto PROPN american_scientific_journal-1248 132 11 . . PUNCT american_scientific_journal-1248 133 1 urroz urroz PROPN american_scientific_journal-1248 133 2 , , PUNCT american_scientific_journal-1248 133 3 october october PROPN american_scientific_journal-1248 133 4 2004 2004 NUM american_scientific_journal-1248 133 5 . . PUNCT american_scientific_journal-1248 134 1 pp pp ADP american_scientific_journal-1248 134 2 4 4 NUM american_scientific_journal-1248 134 3 - - SYM american_scientific_journal-1248 134 4 7 7 NUM american_scientific_journal-1248 134 5 . . NUM american_scientific_journal-1248 134 6 192 192 NUM