id sid tid token lemma pos american_scientific_journal-11066 1 1 1 1 NUM american_scientific_journal-11066 1 2 american american ADJ american_scientific_journal-11066 1 3 academic academic ADJ american_scientific_journal-11066 1 4 scientific scientific ADJ american_scientific_journal-11066 1 5 research research NOUN american_scientific_journal-11066 1 6 journal journal NOUN american_scientific_journal-11066 1 7 for for ADP american_scientific_journal-11066 1 8 engineering engineering NOUN american_scientific_journal-11066 1 9 , , PUNCT american_scientific_journal-11066 1 10 technology technology NOUN american_scientific_journal-11066 1 11 , , PUNCT american_scientific_journal-11066 1 12 and and CCONJ american_scientific_journal-11066 1 13 sciences sciences PROPN american_scientific_journal-11066 1 14 issn issn PROPN american_scientific_journal-11066 1 15 ( ( PUNCT american_scientific_journal-11066 1 16 print print NOUN american_scientific_journal-11066 1 17 ) ) PUNCT american_scientific_journal-11066 1 18 2313 2313 NUM american_scientific_journal-11066 1 19 - - SYM american_scientific_journal-11066 1 20 4410 4410 NUM american_scientific_journal-11066 1 21 , , PUNCT american_scientific_journal-11066 1 22 issn issn PROPN american_scientific_journal-11066 1 23 ( ( PUNCT american_scientific_journal-11066 1 24 online online ADJ american_scientific_journal-11066 1 25 ) ) PUNCT american_scientific_journal-11066 1 26 2313 2313 NUM american_scientific_journal-11066 1 27 - - SYM american_scientific_journal-11066 1 28 4402 4402 NUM american_scientific_journal-11066 1 29 https://asrjetsjournal.org/index.php/american_scientific_journal/index https://asrjetsjournal.org/index.php/american_scientific_journal/index NOUN american_scientific_journal-11066 1 30 some some DET american_scientific_journal-11066 1 31 solution solution NOUN american_scientific_journal-11066 1 32 methods method NOUN american_scientific_journal-11066 1 33 for for ADP american_scientific_journal-11066 1 34 lane lane NOUN american_scientific_journal-11066 1 35 - - PUNCT american_scientific_journal-11066 1 36 emden emden NOUN american_scientific_journal-11066 1 37 differential differential ADJ american_scientific_journal-11066 1 38 equation equation NOUN american_scientific_journal-11066 1 39 when when SCONJ american_scientific_journal-11066 1 40 the the DET american_scientific_journal-11066 1 41 polytropic polytropic ADJ american_scientific_journal-11066 1 42 index index NOUN american_scientific_journal-11066 1 43 is be AUX american_scientific_journal-11066 1 44 three three NUM american_scientific_journal-11066 1 45 sashinka sashinka NOUN american_scientific_journal-11066 1 46 wimaladharmaa wimaladharmaa VERB american_scientific_journal-11066 1 47 * * PROPN american_scientific_journal-11066 1 48 , , PUNCT american_scientific_journal-11066 1 49 dinuka dinuka PROPN american_scientific_journal-11066 1 50 de de PROPN american_scientific_journal-11066 1 51 silvab silvab PROPN american_scientific_journal-11066 1 52 a a PROPN american_scientific_journal-11066 1 53 , , PUNCT american_scientific_journal-11066 1 54 b b PROPN american_scientific_journal-11066 1 55 department department NOUN american_scientific_journal-11066 1 56 of of ADP american_scientific_journal-11066 1 57 mathematics mathematics PROPN american_scientific_journal-11066 1 58 , , PUNCT american_scientific_journal-11066 1 59 university university NOUN american_scientific_journal-11066 1 60 of of ADP american_scientific_journal-11066 1 61 kelaniya kelaniya PROPN american_scientific_journal-11066 1 62 , , PUNCT american_scientific_journal-11066 1 63 11600 11600 NUM american_scientific_journal-11066 1 64 , , PUNCT american_scientific_journal-11066 1 65 sri sri PROPN american_scientific_journal-11066 1 66 lanka lanka PROPN american_scientific_journal-11066 1 67 aemail aemail NOUN american_scientific_journal-11066 1 68 : : PUNCT american_scientific_journal-11066 1 69 wimaladharma@kln.ac.lk wimaladharma@kln.ac.lk NOUN american_scientific_journal-11066 1 70 bemail bemail NOUN american_scientific_journal-11066 1 71 : : PUNCT american_scientific_journal-11066 1 72 dinuka@kln.ac.lk dinuka@kln.ac.lk PROPN american_scientific_journal-11066 1 73 abstract abstract PROPN american_scientific_journal-11066 1 74 the the DET american_scientific_journal-11066 1 75 lane lane NOUN american_scientific_journal-11066 1 76 - - PUNCT american_scientific_journal-11066 1 77 emden emden NOUN american_scientific_journal-11066 1 78 equation equation NOUN american_scientific_journal-11066 1 79 is be AUX american_scientific_journal-11066 1 80 one one NUM american_scientific_journal-11066 1 81 of of ADP american_scientific_journal-11066 1 82 the the DET american_scientific_journal-11066 1 83 widely widely ADV american_scientific_journal-11066 1 84 used use VERB american_scientific_journal-11066 1 85 and and CCONJ american_scientific_journal-11066 1 86 challenging challenging ADJ american_scientific_journal-11066 1 87 equations equation NOUN american_scientific_journal-11066 1 88 in in ADP american_scientific_journal-11066 1 89 nonlinear nonlinear ADJ american_scientific_journal-11066 1 90 dynamics dynamic NOUN american_scientific_journal-11066 1 91 . . PUNCT american_scientific_journal-11066 2 1 it it PRON american_scientific_journal-11066 2 2 is be AUX american_scientific_journal-11066 2 3 a a DET american_scientific_journal-11066 2 4 central central ADJ american_scientific_journal-11066 2 5 equation equation NOUN american_scientific_journal-11066 2 6 in in ADP american_scientific_journal-11066 2 7 the the DET american_scientific_journal-11066 2 8 theory theory NOUN american_scientific_journal-11066 2 9 of of ADP american_scientific_journal-11066 2 10 steller steller ADJ american_scientific_journal-11066 2 11 structures structure NOUN american_scientific_journal-11066 2 12 . . PUNCT american_scientific_journal-11066 3 1 in in ADP american_scientific_journal-11066 3 2 this this DET american_scientific_journal-11066 3 3 paper paper NOUN american_scientific_journal-11066 3 4 , , PUNCT american_scientific_journal-11066 3 5 lane lane NOUN american_scientific_journal-11066 3 6 - - PUNCT american_scientific_journal-11066 3 7 emden emden NOUN american_scientific_journal-11066 3 8 equation equation NOUN american_scientific_journal-11066 3 9 when when SCONJ american_scientific_journal-11066 3 10 𝑛 𝑛 PROPN american_scientific_journal-11066 3 11 = = SYM american_scientific_journal-11066 3 12 3 3 NUM american_scientific_journal-11066 3 13 has have AUX american_scientific_journal-11066 3 14 been be AUX american_scientific_journal-11066 3 15 solved solve VERB american_scientific_journal-11066 3 16 using use VERB american_scientific_journal-11066 3 17 few few ADJ american_scientific_journal-11066 3 18 different different ADJ american_scientific_journal-11066 3 19 methods method NOUN american_scientific_journal-11066 3 20 . . PUNCT american_scientific_journal-11066 4 1 pade pade PROPN american_scientific_journal-11066 4 2 approximant approximant NOUN american_scientific_journal-11066 4 3 has have AUX american_scientific_journal-11066 4 4 also also ADV american_scientific_journal-11066 4 5 been be AUX american_scientific_journal-11066 4 6 calculated calculate VERB american_scientific_journal-11066 4 7 of of ADP american_scientific_journal-11066 4 8 the the DET american_scientific_journal-11066 4 9 obtained obtain VERB american_scientific_journal-11066 4 10 solution solution NOUN american_scientific_journal-11066 4 11 . . PUNCT american_scientific_journal-11066 5 1 the the DET american_scientific_journal-11066 5 2 results result NOUN american_scientific_journal-11066 5 3 obtained obtain VERB american_scientific_journal-11066 5 4 using use VERB american_scientific_journal-11066 5 5 semi semi ADJ american_scientific_journal-11066 5 6 - - ADJ american_scientific_journal-11066 5 7 analytical analytical ADJ american_scientific_journal-11066 5 8 methods method NOUN american_scientific_journal-11066 5 9 are be AUX american_scientific_journal-11066 5 10 compared compare VERB american_scientific_journal-11066 5 11 with with ADP american_scientific_journal-11066 5 12 the the DET american_scientific_journal-11066 5 13 solutions solution NOUN american_scientific_journal-11066 5 14 obtained obtain VERB american_scientific_journal-11066 5 15 using use VERB american_scientific_journal-11066 5 16 well well ADV american_scientific_journal-11066 5 17 - - PUNCT american_scientific_journal-11066 5 18 known know VERB american_scientific_journal-11066 5 19 numerical numerical ADJ american_scientific_journal-11066 5 20 methods method NOUN american_scientific_journal-11066 5 21 namely namely ADV american_scientific_journal-11066 5 22 runge runge NOUN american_scientific_journal-11066 5 23 - - PUNCT american_scientific_journal-11066 5 24 kutta kutta NOUN american_scientific_journal-11066 5 25 ’s ’s PART american_scientific_journal-11066 5 26 fourth fourth ADJ american_scientific_journal-11066 5 27 order order NOUN american_scientific_journal-11066 5 28 method method NOUN american_scientific_journal-11066 5 29 and and CCONJ american_scientific_journal-11066 5 30 ode ode VERB american_scientific_journal-11066 5 31 45 45 NUM american_scientific_journal-11066 5 32 graphically graphically ADV american_scientific_journal-11066 5 33 . . PUNCT american_scientific_journal-11066 6 1 keywords keyword NOUN american_scientific_journal-11066 6 2 : : PUNCT american_scientific_journal-11066 6 3 adomian adomian NOUN american_scientific_journal-11066 6 4 decomposition decomposition NOUN american_scientific_journal-11066 6 5 method method NOUN american_scientific_journal-11066 6 6 ; ; PUNCT american_scientific_journal-11066 6 7 differential differential NOUN american_scientific_journal-11066 6 8 transform transform NOUN american_scientific_journal-11066 6 9 method method NOUN american_scientific_journal-11066 6 10 ; ; PUNCT american_scientific_journal-11066 6 11 pade pade NOUN american_scientific_journal-11066 6 12 approximation approximation NOUN american_scientific_journal-11066 6 13 ; ; PUNCT american_scientific_journal-11066 6 14 lane lane NOUN american_scientific_journal-11066 6 15 - - PUNCT american_scientific_journal-11066 6 16 emden emden NOUN american_scientific_journal-11066 6 17 differential differential ADJ american_scientific_journal-11066 6 18 equations equation NOUN american_scientific_journal-11066 6 19 . . PUNCT american_scientific_journal-11066 7 1 1 1 X american_scientific_journal-11066 7 2 . . X american_scientific_journal-11066 7 3 introduction introduction NOUN american_scientific_journal-11066 7 4 in in ADP american_scientific_journal-11066 7 5 the the DET american_scientific_journal-11066 7 6 study study NOUN american_scientific_journal-11066 7 7 of of ADP american_scientific_journal-11066 7 8 steller steller ADJ american_scientific_journal-11066 7 9 structure structure NOUN american_scientific_journal-11066 7 10 , , PUNCT american_scientific_journal-11066 7 11 the the DET american_scientific_journal-11066 7 12 lane lane NOUN american_scientific_journal-11066 7 13 - - PUNCT american_scientific_journal-11066 7 14 emden emden NOUN american_scientific_journal-11066 7 15 equation equation NOUN american_scientific_journal-11066 7 16 𝑦′′(π‘₯ 𝑦′′(π‘₯ NUM american_scientific_journal-11066 7 17 ) ) PUNCT american_scientific_journal-11066 7 18 + + CCONJ american_scientific_journal-11066 7 19 2 2 NUM american_scientific_journal-11066 7 20 π‘₯ π‘₯ PRON american_scientific_journal-11066 7 21 𝑦′(π‘₯ 𝑦′(π‘₯ NOUN american_scientific_journal-11066 7 22 ) ) PUNCT american_scientific_journal-11066 8 1 + + CCONJ american_scientific_journal-11066 8 2 𝑦𝑛 𝑦𝑛 NOUN american_scientific_journal-11066 8 3 = = SYM american_scientific_journal-11066 8 4 0 0 NUM american_scientific_journal-11066 8 5 ( ( PUNCT american_scientific_journal-11066 8 6 1 1 NUM american_scientific_journal-11066 8 7 ) ) PUNCT american_scientific_journal-11066 8 8 with with ADP american_scientific_journal-11066 8 9 initial initial ADJ american_scientific_journal-11066 8 10 conditions condition NOUN american_scientific_journal-11066 8 11 𝑦(0 𝑦(0 PROPN american_scientific_journal-11066 8 12 ) ) PUNCT american_scientific_journal-11066 8 13 = = SYM american_scientific_journal-11066 8 14 1 1 NUM american_scientific_journal-11066 8 15 and and CCONJ american_scientific_journal-11066 8 16 𝑦′(0 𝑦′(0 NOUN american_scientific_journal-11066 8 17 ) ) PUNCT american_scientific_journal-11066 8 18 = = SYM american_scientific_journal-11066 8 19 0 0 NUM american_scientific_journal-11066 8 20 , , PUNCT american_scientific_journal-11066 8 21 is be AUX american_scientific_journal-11066 8 22 a a DET american_scientific_journal-11066 8 23 nonlinear nonlinear ADJ american_scientific_journal-11066 8 24 ordinary ordinary ADJ american_scientific_journal-11066 8 25 differential differential ADJ american_scientific_journal-11066 8 26 equation equation NOUN american_scientific_journal-11066 8 27 of of ADP american_scientific_journal-11066 8 28 second second ADJ american_scientific_journal-11066 8 29 order order NOUN american_scientific_journal-11066 8 30 which which PRON american_scientific_journal-11066 8 31 models model VERB american_scientific_journal-11066 8 32 the the DET american_scientific_journal-11066 8 33 thermal thermal ADJ american_scientific_journal-11066 8 34 behavior behavior NOUN american_scientific_journal-11066 8 35 of of ADP american_scientific_journal-11066 8 36 a a DET american_scientific_journal-11066 8 37 spherical spherical ADJ american_scientific_journal-11066 8 38 cloud cloud NOUN american_scientific_journal-11066 8 39 of of ADP american_scientific_journal-11066 8 40 gas gas NOUN american_scientific_journal-11066 8 41 under under ADP american_scientific_journal-11066 8 42 the the DET american_scientific_journal-11066 8 43 mutual mutual ADJ american_scientific_journal-11066 8 44 attraction attraction NOUN american_scientific_journal-11066 8 45 of of ADP american_scientific_journal-11066 8 46 its its PRON american_scientific_journal-11066 8 47 molecules molecule NOUN american_scientific_journal-11066 8 48 and and CCONJ american_scientific_journal-11066 8 49 subject subject NOUN american_scientific_journal-11066 8 50 to to ADP american_scientific_journal-11066 8 51 the the DET american_scientific_journal-11066 8 52 classical classical ADJ american_scientific_journal-11066 8 53 laws law NOUN american_scientific_journal-11066 8 54 of of ADP american_scientific_journal-11066 8 55 thermodynamics thermodynamic NOUN american_scientific_journal-11066 8 56 . . PUNCT american_scientific_journal-11066 9 1 here here ADV american_scientific_journal-11066 9 2 𝑛 𝑛 DET american_scientific_journal-11066 9 3 ia ia PROPN american_scientific_journal-11066 9 4 s s AUX american_scientific_journal-11066 9 5 constant constant ADJ american_scientific_journal-11066 9 6 , , PUNCT american_scientific_journal-11066 9 7 which which PRON american_scientific_journal-11066 9 8 is be AUX american_scientific_journal-11066 9 9 called call VERB american_scientific_journal-11066 9 10 the the DET american_scientific_journal-11066 9 11 polytropic polytropic ADJ american_scientific_journal-11066 9 12 index index NOUN american_scientific_journal-11066 9 13 . . PUNCT american_scientific_journal-11066 10 1 in in ADP american_scientific_journal-11066 10 2 astrophysics astrophysic NOUN american_scientific_journal-11066 10 3 , , PUNCT american_scientific_journal-11066 10 4 the the DET american_scientific_journal-11066 10 5 lane lane NOUN american_scientific_journal-11066 10 6 – – PUNCT american_scientific_journal-11066 10 7 emden emden ADJ american_scientific_journal-11066 10 8 equation equation NOUN american_scientific_journal-11066 10 9 is be AUX american_scientific_journal-11066 10 10 a a DET american_scientific_journal-11066 10 11 dimensionless dimensionless NOUN american_scientific_journal-11066 10 12 form form NOUN american_scientific_journal-11066 10 13 of of ADP american_scientific_journal-11066 10 14 poisson poisson NOUN american_scientific_journal-11066 10 15 's 's PART american_scientific_journal-11066 10 16 equation equation NOUN american_scientific_journal-11066 10 17 for for ADP american_scientific_journal-11066 10 18 the the DET american_scientific_journal-11066 10 19 gravitational gravitational ADJ american_scientific_journal-11066 10 20 potential potential NOUN american_scientific_journal-11066 10 21 of of ADP american_scientific_journal-11066 10 22 a a DET american_scientific_journal-11066 10 23 newtonian newtonian ADJ american_scientific_journal-11066 10 24 self self NOUN american_scientific_journal-11066 10 25 - - PUNCT american_scientific_journal-11066 10 26 gravitating gravitate VERB american_scientific_journal-11066 10 27 , , PUNCT american_scientific_journal-11066 10 28 spherically spherically NOUN american_scientific_journal-11066 10 29 symmetric symmetric NOUN american_scientific_journal-11066 10 30 , , PUNCT american_scientific_journal-11066 10 31 polytrophic polytrophic ADJ american_scientific_journal-11066 10 32 fluid fluid NOUN american_scientific_journal-11066 10 33 . . PUNCT american_scientific_journal-11066 11 1 the the DET american_scientific_journal-11066 11 2 lane lane NOUN american_scientific_journal-11066 11 3 - - PUNCT american_scientific_journal-11066 11 4 emden emden NOUN american_scientific_journal-11066 11 5 equation equation NOUN american_scientific_journal-11066 11 6 is be AUX american_scientific_journal-11066 11 7 proposed propose VERB american_scientific_journal-11066 11 8 by by ADP american_scientific_journal-11066 11 9 astrophysicists astrophysicist NOUN american_scientific_journal-11066 11 10 jonathan jonathan PROPN american_scientific_journal-11066 11 11 homer homer PROPN american_scientific_journal-11066 11 12 lane lane PROPN american_scientific_journal-11066 12 1 [ [ X american_scientific_journal-11066 12 2 1 1 X american_scientific_journal-11066 12 3 ] ] PUNCT american_scientific_journal-11066 12 4 who who PRON american_scientific_journal-11066 12 5 first first ADV american_scientific_journal-11066 12 6 published publish VERB american_scientific_journal-11066 12 7 it it PRON american_scientific_journal-11066 12 8 in in ADP american_scientific_journal-11066 12 9 1870 1870 NUM american_scientific_journal-11066 12 10 and and CCONJ american_scientific_journal-11066 12 11 studied study VERB american_scientific_journal-11066 12 12 in in ADP american_scientific_journal-11066 12 13 detail detail NOUN american_scientific_journal-11066 12 14 by by ADP american_scientific_journal-11066 12 15 robert robert PROPN american_scientific_journal-11066 12 16 emden emden PROPN american_scientific_journal-11066 12 17 . . PUNCT american_scientific_journal-11066 13 1 ----------------------------------------------------------------------- ----------------------------------------------------------------------- PUNCT american_scientific_journal-11066 13 2 received receive VERB american_scientific_journal-11066 13 3 : : PUNCT american_scientific_journal-11066 13 4 8/9/2024 8/9/2024 NUM american_scientific_journal-11066 13 5 accepted accept VERB american_scientific_journal-11066 13 6 : : PUNCT american_scientific_journal-11066 13 7 10/9/2024 10/9/2024 NUM american_scientific_journal-11066 13 8 published publish VERB american_scientific_journal-11066 13 9 : : PUNCT american_scientific_journal-11066 13 10 10/16/2024 10/16/2024 NUM american_scientific_journal-11066 13 11 ----------------------------------------------------------------------- ----------------------------------------------------------------------- PUNCT american_scientific_journal-11066 13 12 * * PUNCT american_scientific_journal-11066 13 13 corresponding correspond VERB american_scientific_journal-11066 13 14 author author NOUN american_scientific_journal-11066 13 15 . . PUNCT american_scientific_journal-11066 14 1 https://asrjetsjournal.org/index.php/american_scientific_journal/index https://asrjetsjournal.org/index.php/american_scientific_journal/index NOUN american_scientific_journal-11066 14 2 american american PROPN american_scientific_journal-11066 14 3 academic academic PROPN american_scientific_journal-11066 14 4 scientific scientific ADJ american_scientific_journal-11066 14 5 research research NOUN american_scientific_journal-11066 14 6 journal journal NOUN american_scientific_journal-11066 14 7 for for ADP american_scientific_journal-11066 14 8 engineering engineering NOUN american_scientific_journal-11066 14 9 , , PUNCT american_scientific_journal-11066 14 10 technology technology NOUN american_scientific_journal-11066 14 11 , , PUNCT american_scientific_journal-11066 14 12 and and CCONJ american_scientific_journal-11066 14 13 sciences science NOUN american_scientific_journal-11066 14 14 ( ( PUNCT american_scientific_journal-11066 14 15 asrjets asrjet NOUN american_scientific_journal-11066 14 16 ) ) PUNCT american_scientific_journal-11066 14 17 volume volume NOUN american_scientific_journal-11066 14 18 99 99 NUM american_scientific_journal-11066 14 19 , , PUNCT american_scientific_journal-11066 14 20 no no DET american_scientific_journal-11066 14 21 1 1 NUM american_scientific_journal-11066 14 22 , , PUNCT american_scientific_journal-11066 14 23 pp pp ADV american_scientific_journal-11066 14 24 1 1 NUM american_scientific_journal-11066 14 25 - - SYM american_scientific_journal-11066 14 26 12 12 NUM american_scientific_journal-11066 14 27 2 2 NUM american_scientific_journal-11066 14 28 the the DET american_scientific_journal-11066 14 29 exact exact ADJ american_scientific_journal-11066 14 30 solutions solution NOUN american_scientific_journal-11066 14 31 have have AUX american_scientific_journal-11066 14 32 been be AUX american_scientific_journal-11066 14 33 found find VERB american_scientific_journal-11066 14 34 for for ADP american_scientific_journal-11066 14 35 𝑛 𝑛 PROPN american_scientific_journal-11066 14 36 = = SYM american_scientific_journal-11066 14 37 0,1 0,1 NUM american_scientific_journal-11066 14 38 and and CCONJ american_scientific_journal-11066 14 39 5 5 NUM american_scientific_journal-11066 14 40 . . X american_scientific_journal-11066 14 41 solutions solution NOUN american_scientific_journal-11066 14 42 for for ADP american_scientific_journal-11066 14 43 all all DET american_scientific_journal-11066 14 44 other other ADJ american_scientific_journal-11066 14 45 values value NOUN american_scientific_journal-11066 14 46 should should AUX american_scientific_journal-11066 14 47 be be AUX american_scientific_journal-11066 14 48 obtained obtain VERB american_scientific_journal-11066 14 49 numerically numerically ADV american_scientific_journal-11066 14 50 . . PUNCT american_scientific_journal-11066 15 1 for for ADP american_scientific_journal-11066 15 2 𝑛 𝑛 NOUN american_scientific_journal-11066 15 3 = = SYM american_scientific_journal-11066 15 4 3 3 NUM american_scientific_journal-11066 15 5 of of ADP american_scientific_journal-11066 15 6 the the DET american_scientific_journal-11066 15 7 equation equation NOUN american_scientific_journal-11066 15 8 ( ( PUNCT american_scientific_journal-11066 15 9 1 1 X american_scientific_journal-11066 15 10 ) ) PUNCT american_scientific_journal-11066 15 11 which which PRON american_scientific_journal-11066 15 12 gives give VERB american_scientific_journal-11066 15 13 𝑦′′(π‘₯ 𝑦′′(π‘₯ PRON american_scientific_journal-11066 15 14 ) ) PUNCT american_scientific_journal-11066 15 15 + + CCONJ american_scientific_journal-11066 15 16 2 2 NUM american_scientific_journal-11066 15 17 π‘₯ π‘₯ PRON american_scientific_journal-11066 15 18 𝑦′(π‘₯ 𝑦′(π‘₯ NOUN american_scientific_journal-11066 15 19 ) ) PUNCT american_scientific_journal-11066 15 20 + + CCONJ american_scientific_journal-11066 15 21 𝑦3 𝑦3 PROPN american_scientific_journal-11066 15 22 = = SYM american_scientific_journal-11066 15 23 0 0 PUNCT american_scientific_journal-11066 15 24 ( ( PUNCT american_scientific_journal-11066 15 25 2 2 NUM american_scientific_journal-11066 15 26 ) ) PUNCT american_scientific_journal-11066 15 27 with with ADP american_scientific_journal-11066 15 28 initial initial ADJ american_scientific_journal-11066 15 29 conditions condition NOUN american_scientific_journal-11066 15 30 𝑦(0 𝑦(0 PROPN american_scientific_journal-11066 15 31 ) ) PUNCT american_scientific_journal-11066 15 32 = = SYM american_scientific_journal-11066 15 33 1 1 NUM american_scientific_journal-11066 15 34 and and CCONJ american_scientific_journal-11066 15 35 𝑦′(0 𝑦′(0 NOUN american_scientific_journal-11066 15 36 ) ) PUNCT american_scientific_journal-11066 15 37 = = SYM american_scientific_journal-11066 15 38 0 0 NUM american_scientific_journal-11066 15 39 is be AUX american_scientific_journal-11066 15 40 called call VERB american_scientific_journal-11066 15 41 the the DET american_scientific_journal-11066 15 42 eddington eddington NOUN american_scientific_journal-11066 15 43 ’s ’s PART american_scientific_journal-11066 15 44 approximation approximation NOUN american_scientific_journal-11066 15 45 and and CCONJ american_scientific_journal-11066 15 46 is be AUX american_scientific_journal-11066 15 47 an an DET american_scientific_journal-11066 15 48 interesting interesting ADJ american_scientific_journal-11066 15 49 case case NOUN american_scientific_journal-11066 15 50 since since SCONJ american_scientific_journal-11066 15 51 it it PRON american_scientific_journal-11066 15 52 corresponds correspond VERB american_scientific_journal-11066 15 53 to to ADP american_scientific_journal-11066 15 54 a a DET american_scientific_journal-11066 15 55 useful useful ADJ american_scientific_journal-11066 15 56 approximation approximation NOUN american_scientific_journal-11066 15 57 of of ADP american_scientific_journal-11066 15 58 the the DET american_scientific_journal-11066 15 59 sun sun NOUN american_scientific_journal-11066 15 60 . . PUNCT american_scientific_journal-11066 16 1 a a DET american_scientific_journal-11066 16 2 number number NOUN american_scientific_journal-11066 16 3 of of ADP american_scientific_journal-11066 16 4 techniques technique NOUN american_scientific_journal-11066 16 5 have have AUX american_scientific_journal-11066 16 6 been be AUX american_scientific_journal-11066 16 7 used use VERB american_scientific_journal-11066 16 8 to to PART american_scientific_journal-11066 16 9 solve solve VERB american_scientific_journal-11066 16 10 lane lane NOUN american_scientific_journal-11066 16 11 - - PUNCT american_scientific_journal-11066 16 12 emden emden NOUN american_scientific_journal-11066 16 13 equations equation NOUN american_scientific_journal-11066 16 14 . . PUNCT american_scientific_journal-11066 17 1 the the DET american_scientific_journal-11066 17 2 differential differential ADJ american_scientific_journal-11066 17 3 transform transform NOUN american_scientific_journal-11066 17 4 method method NOUN american_scientific_journal-11066 17 5 has have AUX american_scientific_journal-11066 17 6 been be AUX american_scientific_journal-11066 17 7 applied apply VERB american_scientific_journal-11066 17 8 to to PART american_scientific_journal-11066 17 9 find find VERB american_scientific_journal-11066 17 10 the the DET american_scientific_journal-11066 17 11 solution solution NOUN american_scientific_journal-11066 17 12 for for ADP american_scientific_journal-11066 17 13 lane lane NOUN american_scientific_journal-11066 17 14 - - PUNCT american_scientific_journal-11066 17 15 emden emden NOUN american_scientific_journal-11066 17 16 equation equation NOUN american_scientific_journal-11066 17 17 with with ADP american_scientific_journal-11066 17 18 polytropic polytropic ADJ american_scientific_journal-11066 17 19 index index NOUN american_scientific_journal-11066 17 20 𝑛 𝑛 VERB american_scientific_journal-11066 17 21 in in ADP american_scientific_journal-11066 17 22 [ [ X american_scientific_journal-11066 17 23 2 2 NUM american_scientific_journal-11066 17 24 ] ] PUNCT american_scientific_journal-11066 17 25 . . PUNCT american_scientific_journal-11066 18 1 a a DET american_scientific_journal-11066 18 2 numerical numerical ADJ american_scientific_journal-11066 18 3 solution solution NOUN american_scientific_journal-11066 18 4 for for ADP american_scientific_journal-11066 18 5 differential differential ADJ american_scientific_journal-11066 18 6 equations equation NOUN american_scientific_journal-11066 18 7 of of ADP american_scientific_journal-11066 18 8 lane lane NOUN american_scientific_journal-11066 18 9 - - PUNCT american_scientific_journal-11066 18 10 emden emden NOUN american_scientific_journal-11066 18 11 type type NOUN american_scientific_journal-11066 18 12 has have AUX american_scientific_journal-11066 18 13 been be AUX american_scientific_journal-11066 18 14 found find VERB american_scientific_journal-11066 18 15 in in ADP american_scientific_journal-11066 18 16 [ [ X american_scientific_journal-11066 18 17 3 3 NUM american_scientific_journal-11066 18 18 ] ] PUNCT american_scientific_journal-11066 18 19 . . PUNCT american_scientific_journal-11066 19 1 perturbative perturbative ADJ american_scientific_journal-11066 19 2 solution solution NOUN american_scientific_journal-11066 19 3 to to ADP american_scientific_journal-11066 19 4 the the DET american_scientific_journal-11066 19 5 lane lane NOUN american_scientific_journal-11066 19 6 - - PUNCT american_scientific_journal-11066 19 7 emden emden NOUN american_scientific_journal-11066 19 8 equation equation NOUN american_scientific_journal-11066 19 9 with with ADP american_scientific_journal-11066 19 10 eigenvalue eigenvalue ADJ american_scientific_journal-11066 19 11 approach approach NOUN american_scientific_journal-11066 19 12 has have AUX american_scientific_journal-11066 19 13 been be AUX american_scientific_journal-11066 19 14 considered consider VERB american_scientific_journal-11066 19 15 in in ADP american_scientific_journal-11066 19 16 [ [ X american_scientific_journal-11066 19 17 4].a 4].a NUM american_scientific_journal-11066 19 18 special special ADJ american_scientific_journal-11066 19 19 second second ADJ american_scientific_journal-11066 19 20 order order NOUN american_scientific_journal-11066 19 21 lane lane NOUN american_scientific_journal-11066 19 22 - - PUNCT american_scientific_journal-11066 19 23 emden emden VERB american_scientific_journal-11066 19 24 differential differential ADJ american_scientific_journal-11066 19 25 equation equation NOUN american_scientific_journal-11066 19 26 has have AUX american_scientific_journal-11066 19 27 been be AUX american_scientific_journal-11066 19 28 solved solve VERB american_scientific_journal-11066 19 29 by by SCONJ american_scientific_journal-11066 19 30 he he PRON american_scientific_journal-11066 19 31 ’s ’ VERB american_scientific_journal-11066 19 32 variational variational ADJ american_scientific_journal-11066 19 33 iteration iteration NOUN american_scientific_journal-11066 19 34 , , PUNCT american_scientific_journal-11066 19 35 adomian adomian NOUN american_scientific_journal-11066 19 36 decomposition decomposition NOUN american_scientific_journal-11066 19 37 method method NOUN american_scientific_journal-11066 19 38 , , PUNCT american_scientific_journal-11066 19 39 homotopy homotopy VERB american_scientific_journal-11066 19 40 analysis analysis NOUN american_scientific_journal-11066 19 41 method method NOUN american_scientific_journal-11066 19 42 , , PUNCT american_scientific_journal-11066 19 43 homotopy homotopy VERB american_scientific_journal-11066 19 44 perturbation perturbation NOUN american_scientific_journal-11066 19 45 method method NOUN american_scientific_journal-11066 19 46 and and CCONJ american_scientific_journal-11066 19 47 finite finite ADJ american_scientific_journal-11066 19 48 difference difference NOUN american_scientific_journal-11066 19 49 method method NOUN american_scientific_journal-11066 19 50 in in ADP american_scientific_journal-11066 19 51 [ [ X american_scientific_journal-11066 19 52 5 5 NUM american_scientific_journal-11066 19 53 ] ] PUNCT american_scientific_journal-11066 19 54 . . PUNCT american_scientific_journal-11066 20 1 an an DET american_scientific_journal-11066 20 2 efficient efficient ADJ american_scientific_journal-11066 20 3 numerical numerical ADJ american_scientific_journal-11066 20 4 method method NOUN american_scientific_journal-11066 20 5 for for ADP american_scientific_journal-11066 20 6 solving solve VERB american_scientific_journal-11066 20 7 lane lane NOUN american_scientific_journal-11066 20 8 – – PUNCT american_scientific_journal-11066 20 9 emden emden ADJ american_scientific_journal-11066 20 10 type type NOUN american_scientific_journal-11066 20 11 equations equation NOUN american_scientific_journal-11066 20 12 was be AUX american_scientific_journal-11066 20 13 applied apply VERB american_scientific_journal-11066 20 14 using use VERB american_scientific_journal-11066 20 15 bernstein bernstein PROPN american_scientific_journal-11066 20 16 polynomials polynomial NOUN american_scientific_journal-11066 20 17 in in ADP american_scientific_journal-11066 20 18 [ [ X american_scientific_journal-11066 20 19 6 6 NUM american_scientific_journal-11066 20 20 ] ] PUNCT american_scientific_journal-11066 20 21 . . PUNCT american_scientific_journal-11066 21 1 an an DET american_scientific_journal-11066 21 2 iterative iterative NOUN american_scientific_journal-11066 21 3 method method NOUN american_scientific_journal-11066 21 4 has have AUX american_scientific_journal-11066 21 5 been be AUX american_scientific_journal-11066 21 6 introduced introduce VERB american_scientific_journal-11066 21 7 for for ADP american_scientific_journal-11066 21 8 solving solve VERB american_scientific_journal-11066 21 9 two two NUM american_scientific_journal-11066 21 10 special special ADJ american_scientific_journal-11066 21 11 cases case NOUN american_scientific_journal-11066 21 12 of of ADP american_scientific_journal-11066 21 13 lane lane NOUN american_scientific_journal-11066 21 14 - - PUNCT american_scientific_journal-11066 21 15 emden emden NOUN american_scientific_journal-11066 21 16 type type NOUN american_scientific_journal-11066 21 17 equation equation NOUN american_scientific_journal-11066 21 18 by by ADP american_scientific_journal-11066 21 19 [ [ X american_scientific_journal-11066 21 20 7 7 NUM american_scientific_journal-11066 21 21 ] ] PUNCT american_scientific_journal-11066 21 22 . . PUNCT american_scientific_journal-11066 22 1 authors author NOUN american_scientific_journal-11066 22 2 in in ADP american_scientific_journal-11066 22 3 [ [ X american_scientific_journal-11066 22 4 2 2 NUM american_scientific_journal-11066 22 5 ] ] PUNCT american_scientific_journal-11066 22 6 and and CCONJ american_scientific_journal-11066 22 7 [ [ X american_scientific_journal-11066 22 8 6 6 NUM american_scientific_journal-11066 22 9 ] ] PUNCT american_scientific_journal-11066 22 10 have have AUX american_scientific_journal-11066 22 11 solved solve VERB american_scientific_journal-11066 22 12 lane lane NOUN american_scientific_journal-11066 22 13 - - PUNCT american_scientific_journal-11066 22 14 emden emden NOUN american_scientific_journal-11066 22 15 type type NOUN american_scientific_journal-11066 22 16 equations equation NOUN american_scientific_journal-11066 22 17 using use VERB american_scientific_journal-11066 22 18 differential differential ADJ american_scientific_journal-11066 22 19 transform transform NOUN american_scientific_journal-11066 22 20 method method NOUN american_scientific_journal-11066 22 21 , , PUNCT american_scientific_journal-11066 22 22 adomian adomian NOUN american_scientific_journal-11066 22 23 method method NOUN american_scientific_journal-11066 22 24 and and CCONJ american_scientific_journal-11066 22 25 modified modify VERB american_scientific_journal-11066 22 26 adomian adomian NOUN american_scientific_journal-11066 22 27 decomposition decomposition NOUN american_scientific_journal-11066 22 28 method method NOUN american_scientific_journal-11066 22 29 respectively respectively ADV american_scientific_journal-11066 22 30 . . PUNCT american_scientific_journal-11066 23 1 but but CCONJ american_scientific_journal-11066 23 2 they they PRON american_scientific_journal-11066 23 3 have have AUX american_scientific_journal-11066 23 4 not not PART american_scientific_journal-11066 23 5 considered consider VERB american_scientific_journal-11066 23 6 about about ADP american_scientific_journal-11066 23 7 this this DET american_scientific_journal-11066 23 8 special special ADJ american_scientific_journal-11066 23 9 case case NOUN american_scientific_journal-11066 23 10 of of ADP american_scientific_journal-11066 23 11 lane lane NOUN american_scientific_journal-11066 23 12 - - PUNCT american_scientific_journal-11066 23 13 emden emden NOUN american_scientific_journal-11066 23 14 equation equation NOUN american_scientific_journal-11066 23 15 which which PRON american_scientific_journal-11066 23 16 is be AUX american_scientific_journal-11066 23 17 a a DET american_scientific_journal-11066 23 18 very very ADV american_scientific_journal-11066 23 19 important important ADJ american_scientific_journal-11066 23 20 equation equation NOUN american_scientific_journal-11066 23 21 in in ADP american_scientific_journal-11066 23 22 steller steller NOUN american_scientific_journal-11066 23 23 structures structure NOUN american_scientific_journal-11066 23 24 in in ADP american_scientific_journal-11066 23 25 physics physics NOUN american_scientific_journal-11066 23 26 . . PUNCT american_scientific_journal-11066 24 1 in in ADP american_scientific_journal-11066 24 2 this this DET american_scientific_journal-11066 24 3 paper paper NOUN american_scientific_journal-11066 24 4 , , PUNCT american_scientific_journal-11066 24 5 lane lane NOUN american_scientific_journal-11066 24 6 - - PUNCT american_scientific_journal-11066 24 7 emden emden NOUN american_scientific_journal-11066 24 8 equation equation NOUN american_scientific_journal-11066 24 9 for for ADP american_scientific_journal-11066 24 10 𝑛 𝑛 NOUN american_scientific_journal-11066 24 11 = = SYM american_scientific_journal-11066 24 12 3 3 NUM american_scientific_journal-11066 24 13 is be AUX american_scientific_journal-11066 24 14 solved solve VERB american_scientific_journal-11066 24 15 using use VERB american_scientific_journal-11066 24 16 few few ADJ american_scientific_journal-11066 24 17 different different ADJ american_scientific_journal-11066 24 18 semi semi ADJ american_scientific_journal-11066 24 19 - - ADJ american_scientific_journal-11066 24 20 analytical analytical ADJ american_scientific_journal-11066 24 21 and and CCONJ american_scientific_journal-11066 24 22 numerical numerical ADJ american_scientific_journal-11066 24 23 methods method NOUN american_scientific_journal-11066 24 24 . . PUNCT american_scientific_journal-11066 25 1 since since SCONJ american_scientific_journal-11066 25 2 the the DET american_scientific_journal-11066 25 3 exact exact ADJ american_scientific_journal-11066 25 4 solution solution NOUN american_scientific_journal-11066 25 5 of of ADP american_scientific_journal-11066 25 6 the the DET american_scientific_journal-11066 25 7 above above ADJ american_scientific_journal-11066 25 8 equation equation NOUN american_scientific_journal-11066 25 9 does do AUX american_scientific_journal-11066 25 10 not not PART american_scientific_journal-11066 25 11 exist exist VERB american_scientific_journal-11066 25 12 , , PUNCT american_scientific_journal-11066 25 13 we we PRON american_scientific_journal-11066 25 14 solve solve VERB american_scientific_journal-11066 25 15 the the DET american_scientific_journal-11066 25 16 equation equation NOUN american_scientific_journal-11066 25 17 using use VERB american_scientific_journal-11066 25 18 well well ADV american_scientific_journal-11066 25 19 - - PUNCT american_scientific_journal-11066 25 20 known know VERB american_scientific_journal-11066 25 21 numerical numerical ADJ american_scientific_journal-11066 25 22 method method PROPN american_scientific_journal-11066 25 23 runge runge NOUN american_scientific_journal-11066 25 24 - - PUNCT american_scientific_journal-11066 25 25 kutta kutta NOUN american_scientific_journal-11066 25 26 ’s ’s PART american_scientific_journal-11066 25 27 fourth fourth ADJ american_scientific_journal-11066 25 28 order order NOUN american_scientific_journal-11066 25 29 method method NOUN american_scientific_journal-11066 25 30 and and CCONJ american_scientific_journal-11066 25 31 ode ode VERB american_scientific_journal-11066 25 32 45 45 NUM american_scientific_journal-11066 25 33 using use VERB american_scientific_journal-11066 25 34 matlab matlab PROPN american_scientific_journal-11066 25 35 software software NOUN american_scientific_journal-11066 25 36 . . PUNCT american_scientific_journal-11066 26 1 the the DET american_scientific_journal-11066 26 2 comparisons comparison NOUN american_scientific_journal-11066 26 3 of of ADP american_scientific_journal-11066 26 4 the the DET american_scientific_journal-11066 26 5 results result NOUN american_scientific_journal-11066 26 6 obtained obtain VERB american_scientific_journal-11066 26 7 using use VERB american_scientific_journal-11066 26 8 different different ADJ american_scientific_journal-11066 26 9 methods method NOUN american_scientific_journal-11066 26 10 will will AUX american_scientific_journal-11066 26 11 be be AUX american_scientific_journal-11066 26 12 done do VERB american_scientific_journal-11066 26 13 graphically graphically ADV american_scientific_journal-11066 26 14 with with ADP american_scientific_journal-11066 26 15 the the DET american_scientific_journal-11066 26 16 solutions solution NOUN american_scientific_journal-11066 26 17 obtained obtain VERB american_scientific_journal-11066 26 18 using use VERB american_scientific_journal-11066 26 19 above above ADP american_scientific_journal-11066 26 20 numerical numerical ADJ american_scientific_journal-11066 26 21 methods method NOUN american_scientific_journal-11066 26 22 . . PUNCT american_scientific_journal-11066 27 1 2 2 X american_scientific_journal-11066 27 2 . . X american_scientific_journal-11066 27 3 material material NOUN american_scientific_journal-11066 27 4 and and CCONJ american_scientific_journal-11066 27 5 the the DET american_scientific_journal-11066 27 6 method method NOUN american_scientific_journal-11066 27 7 in in ADP american_scientific_journal-11066 27 8 this this DET american_scientific_journal-11066 27 9 section section NOUN american_scientific_journal-11066 27 10 , , PUNCT american_scientific_journal-11066 27 11 the the DET american_scientific_journal-11066 27 12 solution solution NOUN american_scientific_journal-11066 27 13 for for ADP american_scientific_journal-11066 27 14 the the DET american_scientific_journal-11066 27 15 lane lane NOUN american_scientific_journal-11066 27 16 - - PUNCT american_scientific_journal-11066 27 17 emden emden NOUN american_scientific_journal-11066 27 18 equation equation NOUN american_scientific_journal-11066 27 19 has have AUX american_scientific_journal-11066 27 20 been be AUX american_scientific_journal-11066 27 21 obtained obtain VERB american_scientific_journal-11066 27 22 using use VERB american_scientific_journal-11066 27 23 the the DET american_scientific_journal-11066 27 24 methods method NOUN american_scientific_journal-11066 27 25 adomian adomian NOUN american_scientific_journal-11066 27 26 decomposition decomposition NOUN american_scientific_journal-11066 27 27 method method NOUN american_scientific_journal-11066 27 28 ( ( PUNCT american_scientific_journal-11066 27 29 adm adm PROPN american_scientific_journal-11066 27 30 ) ) PUNCT american_scientific_journal-11066 27 31 , , PUNCT american_scientific_journal-11066 27 32 a a DET american_scientific_journal-11066 27 33 modified modified ADJ american_scientific_journal-11066 27 34 adomian adomian NOUN american_scientific_journal-11066 27 35 decomposition decomposition NOUN american_scientific_journal-11066 27 36 method method NOUN american_scientific_journal-11066 27 37 and and CCONJ american_scientific_journal-11066 27 38 the the DET american_scientific_journal-11066 27 39 differential differential ADJ american_scientific_journal-11066 27 40 transform transform NOUN american_scientific_journal-11066 27 41 method method NOUN american_scientific_journal-11066 27 42 ( ( PUNCT american_scientific_journal-11066 27 43 dtm dtm PROPN american_scientific_journal-11066 27 44 ) ) PUNCT american_scientific_journal-11066 27 45 . . PUNCT american_scientific_journal-11066 28 1 2.1 2.1 NUM american_scientific_journal-11066 28 2 the the DET american_scientific_journal-11066 28 3 solution solution NOUN american_scientific_journal-11066 28 4 with with ADP american_scientific_journal-11066 28 5 adomian adomian NOUN american_scientific_journal-11066 28 6 decomposition decomposition NOUN american_scientific_journal-11066 28 7 method method NOUN american_scientific_journal-11066 28 8 the the DET american_scientific_journal-11066 28 9 adomian adomian NOUN american_scientific_journal-11066 28 10 decomposition decomposition NOUN american_scientific_journal-11066 28 11 method method NOUN american_scientific_journal-11066 29 1 [ [ X american_scientific_journal-11066 29 2 8 8 NUM american_scientific_journal-11066 29 3 ] ] PUNCT american_scientific_journal-11066 29 4 , , PUNCT american_scientific_journal-11066 29 5 also also ADV american_scientific_journal-11066 29 6 known know VERB american_scientific_journal-11066 29 7 as as ADP american_scientific_journal-11066 29 8 the the DET american_scientific_journal-11066 29 9 inverse inverse NOUN american_scientific_journal-11066 29 10 operator operator NOUN american_scientific_journal-11066 29 11 method method NOUN american_scientific_journal-11066 29 12 , , PUNCT american_scientific_journal-11066 29 13 is be AUX american_scientific_journal-11066 29 14 a a DET american_scientific_journal-11066 29 15 mathematical mathematical ADJ american_scientific_journal-11066 29 16 method method NOUN american_scientific_journal-11066 29 17 for for ADP american_scientific_journal-11066 29 18 solving solve VERB american_scientific_journal-11066 29 19 linear linear NOUN american_scientific_journal-11066 29 20 and and CCONJ american_scientific_journal-11066 29 21 nonlinear nonlinear ADJ american_scientific_journal-11066 29 22 mathematical mathematical ADJ american_scientific_journal-11066 29 23 physics physics NOUN american_scientific_journal-11066 29 24 equations equation NOUN american_scientific_journal-11066 29 25 ; ; PUNCT american_scientific_journal-11066 29 26 it it PRON american_scientific_journal-11066 29 27 was be AUX american_scientific_journal-11066 29 28 proposed propose VERB american_scientific_journal-11066 29 29 by by ADP american_scientific_journal-11066 29 30 george george PROPN american_scientific_journal-11066 29 31 adomian adomian PROPN american_scientific_journal-11066 29 32 . . PUNCT american_scientific_journal-11066 30 1 in in ADP american_scientific_journal-11066 30 2 standard standard ADJ american_scientific_journal-11066 30 3 adomian adomian NOUN american_scientific_journal-11066 30 4 decomposition decomposition NOUN american_scientific_journal-11066 30 5 method method NOUN american_scientific_journal-11066 30 6 , , PUNCT american_scientific_journal-11066 30 7 we we PRON american_scientific_journal-11066 30 8 define define VERB american_scientific_journal-11066 30 9 linear linear ADJ american_scientific_journal-11066 30 10 operator operator NOUN american_scientific_journal-11066 30 11 𝐿 𝐿 PROPN american_scientific_journal-11066 30 12 and and CCONJ american_scientific_journal-11066 30 13 its its PRON american_scientific_journal-11066 30 14 inverse inverse NOUN american_scientific_journal-11066 30 15 operator operator NOUN american_scientific_journal-11066 30 16 by by ADP american_scientific_journal-11066 30 17 𝐿(βˆ™ 𝐿(βˆ™ NOUN american_scientific_journal-11066 30 18 ) ) PUNCT american_scientific_journal-11066 30 19 = = SYM american_scientific_journal-11066 31 1 π‘₯βˆ’2 π‘₯βˆ’2 NOUN american_scientific_journal-11066 31 2 𝑑 𝑑 NOUN american_scientific_journal-11066 31 3 𝑑π‘₯ 𝑑π‘₯ NOUN american_scientific_journal-11066 31 4 ( ( PUNCT american_scientific_journal-11066 31 5 π‘₯2 π‘₯2 NOUN american_scientific_journal-11066 31 6 𝑑 𝑑 NOUN american_scientific_journal-11066 31 7 𝑑π‘₯ 𝑑π‘₯ VERB american_scientific_journal-11066 31 8 ) ) PUNCT american_scientific_journal-11066 31 9 ( ( PUNCT american_scientific_journal-11066 31 10 βˆ™ βˆ™ PROPN american_scientific_journal-11066 31 11 ) ) PUNCT american_scientific_journal-11066 31 12 ( ( PUNCT american_scientific_journal-11066 31 13 3 3 X american_scientific_journal-11066 31 14 ) ) PUNCT american_scientific_journal-11066 31 15 american american ADJ american_scientific_journal-11066 31 16 academic academic ADJ american_scientific_journal-11066 31 17 scientific scientific ADJ american_scientific_journal-11066 31 18 research research NOUN american_scientific_journal-11066 31 19 journal journal NOUN american_scientific_journal-11066 31 20 for for ADP american_scientific_journal-11066 31 21 engineering engineering NOUN american_scientific_journal-11066 31 22 , , PUNCT american_scientific_journal-11066 31 23 technology technology NOUN american_scientific_journal-11066 31 24 , , PUNCT american_scientific_journal-11066 31 25 and and CCONJ american_scientific_journal-11066 31 26 sciences science NOUN american_scientific_journal-11066 31 27 ( ( PUNCT american_scientific_journal-11066 31 28 asrjets asrjet NOUN american_scientific_journal-11066 31 29 ) ) PUNCT american_scientific_journal-11066 31 30 volume volume NOUN american_scientific_journal-11066 31 31 99 99 NUM american_scientific_journal-11066 31 32 , , PUNCT american_scientific_journal-11066 31 33 no no DET american_scientific_journal-11066 31 34 1 1 NUM american_scientific_journal-11066 31 35 , , PUNCT american_scientific_journal-11066 31 36 pp pp ADV american_scientific_journal-11066 31 37 1 1 NUM american_scientific_journal-11066 31 38 - - SYM american_scientific_journal-11066 31 39 12 12 NUM american_scientific_journal-11066 31 40 3 3 NUM american_scientific_journal-11066 31 41 πΏβˆ’1(βˆ™ πΏβˆ’1(βˆ™ NOUN american_scientific_journal-11066 31 42 ) ) PUNCT american_scientific_journal-11066 31 43 = = SYM american_scientific_journal-11066 31 44 ∫ ∫ PROPN american_scientific_journal-11066 32 1 π‘₯βˆ’2 π‘₯βˆ’2 NUM american_scientific_journal-11066 32 2 ∫ ∫ PROPN american_scientific_journal-11066 32 3 π‘₯2(βˆ™ π‘₯2(βˆ™ PROPN american_scientific_journal-11066 32 4 ) ) PUNCT american_scientific_journal-11066 32 5 𝑑π‘₯ 𝑑π‘₯ VERB american_scientific_journal-11066 32 6 𝑑π‘₯ 𝑑π‘₯ ADP american_scientific_journal-11066 32 7 π‘₯ π‘₯ ADP american_scientific_journal-11066 32 8 0 0 NUM american_scientific_journal-11066 32 9 π‘₯ π‘₯ NOUN american_scientific_journal-11066 32 10 0 0 NUM american_scientific_journal-11066 32 11 ( ( PUNCT american_scientific_journal-11066 32 12 4 4 X american_scientific_journal-11066 32 13 ) ) PUNCT american_scientific_journal-11066 32 14 using use VERB american_scientific_journal-11066 32 15 the the DET american_scientific_journal-11066 32 16 operators operator NOUN american_scientific_journal-11066 32 17 given give VERB american_scientific_journal-11066 32 18 in in ADP american_scientific_journal-11066 32 19 ( ( PUNCT american_scientific_journal-11066 32 20 3 3 NUM american_scientific_journal-11066 32 21 ) ) PUNCT american_scientific_journal-11066 32 22 and and CCONJ american_scientific_journal-11066 32 23 ( ( PUNCT american_scientific_journal-11066 32 24 4 4 NUM american_scientific_journal-11066 32 25 ) ) PUNCT american_scientific_journal-11066 32 26 , , PUNCT american_scientific_journal-11066 32 27 the the DET american_scientific_journal-11066 32 28 equation equation NOUN american_scientific_journal-11066 32 29 ( ( PUNCT american_scientific_journal-11066 32 30 2 2 X american_scientific_journal-11066 32 31 ) ) PUNCT american_scientific_journal-11066 32 32 can can AUX american_scientific_journal-11066 32 33 be be AUX american_scientific_journal-11066 32 34 written write VERB american_scientific_journal-11066 32 35 in in ADP american_scientific_journal-11066 32 36 the the DET american_scientific_journal-11066 32 37 form form NOUN american_scientific_journal-11066 33 1 𝐿𝑦 𝐿𝑦 PROPN american_scientific_journal-11066 33 2 = = SYM american_scientific_journal-11066 33 3 βˆ’π‘¦3 βˆ’π‘¦3 PROPN american_scientific_journal-11066 33 4 ( ( PUNCT american_scientific_journal-11066 33 5 5 5 NUM american_scientific_journal-11066 33 6 ) ) PUNCT american_scientific_journal-11066 33 7 then then ADV american_scientific_journal-11066 33 8 solution solution NOUN american_scientific_journal-11066 33 9 of of ADP american_scientific_journal-11066 33 10 the the DET american_scientific_journal-11066 33 11 equation equation NOUN american_scientific_journal-11066 33 12 ( ( PUNCT american_scientific_journal-11066 33 13 2 2 X american_scientific_journal-11066 33 14 ) ) PUNCT american_scientific_journal-11066 33 15 takes take VERB american_scientific_journal-11066 33 16 the the DET american_scientific_journal-11066 33 17 form form NOUN american_scientific_journal-11066 33 18 𝑦(π‘₯ 𝑦(π‘₯ NOUN american_scientific_journal-11066 33 19 ) ) PUNCT american_scientific_journal-11066 33 20 = = SYM american_scientific_journal-11066 33 21 𝑦(0 𝑦(0 PROPN american_scientific_journal-11066 33 22 ) ) PUNCT american_scientific_journal-11066 33 23 βˆ’ βˆ’ PROPN american_scientific_journal-11066 33 24 πΏβˆ’1(𝑦3 πΏβˆ’1(𝑦3 PROPN american_scientific_journal-11066 33 25 ) ) PUNCT american_scientific_journal-11066 33 26 ( ( PUNCT american_scientific_journal-11066 33 27 6 6 X american_scientific_journal-11066 33 28 ) ) PUNCT american_scientific_journal-11066 33 29 taking take VERB american_scientific_journal-11066 33 30 𝑦(π‘₯ 𝑦(π‘₯ NOUN american_scientific_journal-11066 33 31 ) ) PUNCT american_scientific_journal-11066 33 32 = = PUNCT american_scientific_journal-11066 33 33 βˆ‘ βˆ‘ ADP american_scientific_journal-11066 33 34 𝑦𝑛 𝑦𝑛 VERB american_scientific_journal-11066 33 35 ∞ ∞ NUM american_scientific_journal-11066 33 36 𝑛=0 𝑛=0 PROPN american_scientific_journal-11066 33 37 and and CCONJ american_scientific_journal-11066 33 38 writing write VERB american_scientific_journal-11066 33 39 the the DET american_scientific_journal-11066 33 40 nonlinear nonlinear ADJ american_scientific_journal-11066 33 41 term term NOUN american_scientific_journal-11066 33 42 𝑦3 𝑦3 PROPN american_scientific_journal-11066 33 43 using use VERB american_scientific_journal-11066 33 44 adomian adomian NOUN american_scientific_journal-11066 33 45 polynomials polynomial NOUN american_scientific_journal-11066 33 46 as as ADP american_scientific_journal-11066 33 47 𝑦3 𝑦3 PROPN american_scientific_journal-11066 33 48 = = PUNCT american_scientific_journal-11066 33 49 βˆ‘ βˆ‘ PUNCT american_scientific_journal-11066 33 50 𝐴𝑛 𝐴𝑛 PROPN american_scientific_journal-11066 33 51 ∞ ∞ NUM american_scientific_journal-11066 33 52 𝑛=0 𝑛=0 NOUN american_scientific_journal-11066 33 53 , , PUNCT american_scientific_journal-11066 33 54 we we PRON american_scientific_journal-11066 33 55 have have VERB american_scientific_journal-11066 33 56 βˆ‘ βˆ‘ ADV american_scientific_journal-11066 33 57 𝑦𝑛 𝑦𝑛 VERB american_scientific_journal-11066 33 58 ∞ ∞ PROPN american_scientific_journal-11066 33 59 𝑛=0 𝑛=0 PROPN american_scientific_journal-11066 33 60 = = SYM american_scientific_journal-11066 34 1 1 1 NUM american_scientific_journal-11066 34 2 βˆ’ βˆ’ NOUN american_scientific_journal-11066 34 3 πΏβˆ’1(βˆ‘ πΏβˆ’1(βˆ‘ NOUN american_scientific_journal-11066 35 1 𝐴𝑛 𝐴𝑛 NOUN american_scientific_journal-11066 35 2 ∞ ∞ NUM american_scientific_journal-11066 35 3 𝑛=0 𝑛=0 PROPN american_scientific_journal-11066 35 4 ) ) PUNCT american_scientific_journal-11066 35 5 ( ( PUNCT american_scientific_journal-11066 35 6 7 7 X american_scientific_journal-11066 35 7 ) ) PUNCT american_scientific_journal-11066 35 8 or or CCONJ american_scientific_journal-11066 35 9 𝑦0 𝑦0 NOUN american_scientific_journal-11066 35 10 = = SYM american_scientific_journal-11066 35 11 1 1 NUM american_scientific_journal-11066 35 12 ( ( PUNCT american_scientific_journal-11066 35 13 8) 8) NUM american_scientific_journal-11066 35 14 𝑦𝑛+1 𝑦𝑛+1 NUM american_scientific_journal-11066 35 15 = = SYM american_scientific_journal-11066 35 16 βˆ’πΏβˆ’1(𝐴𝑛 βˆ’πΏβˆ’1(𝐴𝑛 PROPN american_scientific_journal-11066 35 17 ) ) PUNCT american_scientific_journal-11066 35 18 , , PUNCT american_scientific_journal-11066 35 19 𝑛 𝑛 NOUN american_scientific_journal-11066 35 20 = = SYM american_scientific_journal-11066 35 21 0,1,2 0,1,2 NUM american_scientific_journal-11066 35 22 , , PUNCT american_scientific_journal-11066 35 23 … … PUNCT american_scientific_journal-11066 35 24 ( ( PUNCT american_scientific_journal-11066 35 25 9 9 X american_scientific_journal-11066 35 26 ) ) PUNCT american_scientific_journal-11066 35 27 using use VERB american_scientific_journal-11066 35 28 the the DET american_scientific_journal-11066 35 29 definition definition NOUN american_scientific_journal-11066 35 30 of of ADP american_scientific_journal-11066 35 31 the the DET american_scientific_journal-11066 35 32 inverse inverse NOUN american_scientific_journal-11066 35 33 operator operator NOUN american_scientific_journal-11066 35 34 as as ADP american_scientific_journal-11066 35 35 in in ADP american_scientific_journal-11066 35 36 ( ( PUNCT american_scientific_journal-11066 35 37 4 4 NUM american_scientific_journal-11066 35 38 ) ) PUNCT american_scientific_journal-11066 35 39 , , PUNCT american_scientific_journal-11066 35 40 we we PRON american_scientific_journal-11066 35 41 have have VERB american_scientific_journal-11066 35 42 𝑦𝑛+1 𝑦𝑛+1 NUM american_scientific_journal-11066 35 43 = = SYM american_scientific_journal-11066 35 44 βˆ’ βˆ’ NOUN american_scientific_journal-11066 35 45 ∫ ∫ NOUN american_scientific_journal-11066 36 1 π‘₯βˆ’2 π‘₯βˆ’2 NUM american_scientific_journal-11066 36 2 ∫ ∫ NOUN american_scientific_journal-11066 36 3 π‘₯2(𝐴𝑛 π‘₯2(𝐴𝑛 NOUN american_scientific_journal-11066 36 4 ) ) PUNCT american_scientific_journal-11066 36 5 𝑑π‘₯ 𝑑π‘₯ VERB american_scientific_journal-11066 36 6 𝑑π‘₯ 𝑑π‘₯ ADP american_scientific_journal-11066 36 7 π‘₯ π‘₯ ADP american_scientific_journal-11066 36 8 0 0 NUM american_scientific_journal-11066 36 9 π‘₯ π‘₯ NOUN american_scientific_journal-11066 36 10 0 0 NUM american_scientific_journal-11066 36 11 ( ( PUNCT american_scientific_journal-11066 36 12 10 10 NUM american_scientific_journal-11066 36 13 ) ) PUNCT american_scientific_journal-11066 36 14 it it PRON american_scientific_journal-11066 36 15 is be AUX american_scientific_journal-11066 36 16 convenient convenient ADJ american_scientific_journal-11066 36 17 to to PART american_scientific_journal-11066 36 18 list list VERB american_scientific_journal-11066 36 19 the the DET american_scientific_journal-11066 36 20 first first ADJ american_scientific_journal-11066 36 21 few few ADJ american_scientific_journal-11066 36 22 adomian adomian NOUN american_scientific_journal-11066 36 23 polynomials polynomial NOUN american_scientific_journal-11066 36 24 . . PUNCT american_scientific_journal-11066 37 1 following follow VERB american_scientific_journal-11066 37 2 [ [ X american_scientific_journal-11066 37 3 9 9 NUM american_scientific_journal-11066 37 4 ] ] PUNCT american_scientific_journal-11066 37 5 and and CCONJ american_scientific_journal-11066 37 6 after after ADP american_scientific_journal-11066 37 7 some some DET american_scientific_journal-11066 37 8 calculations calculation NOUN american_scientific_journal-11066 37 9 , , PUNCT american_scientific_journal-11066 37 10 the the DET american_scientific_journal-11066 37 11 first first ADJ american_scientific_journal-11066 37 12 few few ADJ american_scientific_journal-11066 37 13 adomian adomian NOUN american_scientific_journal-11066 37 14 polynomials polynomial NOUN american_scientific_journal-11066 37 15 take take VERB american_scientific_journal-11066 37 16 the the DET american_scientific_journal-11066 37 17 form form NOUN american_scientific_journal-11066 37 18 𝐴0 𝐴0 PROPN american_scientific_journal-11066 37 19 = = SYM american_scientific_journal-11066 37 20 1 1 NUM american_scientific_journal-11066 37 21 𝐴1 𝐴1 NOUN american_scientific_journal-11066 37 22 = = SYM american_scientific_journal-11066 37 23 βˆ’ βˆ’ PROPN american_scientific_journal-11066 37 24 π‘₯2 π‘₯2 NOUN american_scientific_journal-11066 37 25 6 6 NUM american_scientific_journal-11066 37 26 𝐴2 𝐴2 NOUN american_scientific_journal-11066 37 27 = = SYM american_scientific_journal-11066 37 28 19 19 NUM american_scientific_journal-11066 37 29 120 120 NUM american_scientific_journal-11066 37 30 π‘₯4 π‘₯4 NOUN american_scientific_journal-11066 37 31 𝐴3 𝐴3 PROPN american_scientific_journal-11066 37 32 = = PROPN american_scientific_journal-11066 37 33 357 357 NUM american_scientific_journal-11066 37 34 5040 5040 NUM american_scientific_journal-11066 37 35 π‘₯6 π‘₯6 NOUN american_scientific_journal-11066 37 36 and and CCONJ american_scientific_journal-11066 37 37 so so ADV american_scientific_journal-11066 37 38 on on ADV american_scientific_journal-11066 37 39 . . PUNCT american_scientific_journal-11066 38 1 now now ADV american_scientific_journal-11066 38 2 inserting insert VERB american_scientific_journal-11066 38 3 these these DET american_scientific_journal-11066 38 4 adomian adomian NOUN american_scientific_journal-11066 38 5 polynomials polynomial NOUN american_scientific_journal-11066 38 6 to to ADP american_scientific_journal-11066 38 7 equation equation NOUN american_scientific_journal-11066 38 8 ( ( PUNCT american_scientific_journal-11066 38 9 8) 8) NUM american_scientific_journal-11066 38 10 and and CCONJ american_scientific_journal-11066 38 11 ( ( PUNCT american_scientific_journal-11066 38 12 9 9 X american_scientific_journal-11066 38 13 ) ) PUNCT american_scientific_journal-11066 38 14 gives give VERB american_scientific_journal-11066 38 15 𝑦0 𝑦0 NOUN american_scientific_journal-11066 38 16 = = SYM american_scientific_journal-11066 38 17 1 1 NUM american_scientific_journal-11066 38 18 , , PUNCT american_scientific_journal-11066 38 19 𝑦1 𝑦1 NOUN american_scientific_journal-11066 38 20 = = SYM american_scientific_journal-11066 38 21 βˆ’πΏβˆ’1(𝐴0 βˆ’πΏβˆ’1(𝐴0 PROPN american_scientific_journal-11066 38 22 ) ) PUNCT american_scientific_journal-11066 39 1 = = PUNCT american_scientific_journal-11066 40 1 βˆ’ βˆ’ PROPN american_scientific_journal-11066 40 2 1 1 NUM american_scientific_journal-11066 40 3 6 6 NUM american_scientific_journal-11066 40 4 π‘₯2 π‘₯2 NOUN american_scientific_journal-11066 40 5 , , PUNCT american_scientific_journal-11066 40 6 𝑦2 𝑦2 PROPN american_scientific_journal-11066 40 7 = = SYM american_scientific_journal-11066 40 8 βˆ’πΏβˆ’1(𝐴1 βˆ’πΏβˆ’1(𝐴1 NOUN american_scientific_journal-11066 40 9 ) ) PUNCT american_scientific_journal-11066 40 10 = = SYM american_scientific_journal-11066 40 11 1 1 NUM american_scientific_journal-11066 40 12 40 40 NUM american_scientific_journal-11066 40 13 π‘₯4 π‘₯4 NOUN american_scientific_journal-11066 40 14 american american ADJ american_scientific_journal-11066 40 15 academic academic ADJ american_scientific_journal-11066 40 16 scientific scientific ADJ american_scientific_journal-11066 40 17 research research NOUN american_scientific_journal-11066 40 18 journal journal NOUN american_scientific_journal-11066 40 19 for for ADP american_scientific_journal-11066 40 20 engineering engineering NOUN american_scientific_journal-11066 40 21 , , PUNCT american_scientific_journal-11066 40 22 technology technology NOUN american_scientific_journal-11066 40 23 , , PUNCT american_scientific_journal-11066 40 24 and and CCONJ american_scientific_journal-11066 40 25 sciences science NOUN american_scientific_journal-11066 40 26 ( ( PUNCT american_scientific_journal-11066 40 27 asrjets asrjet NOUN american_scientific_journal-11066 40 28 ) ) PUNCT american_scientific_journal-11066 40 29 volume volume NOUN american_scientific_journal-11066 40 30 99 99 NUM american_scientific_journal-11066 40 31 , , PUNCT american_scientific_journal-11066 40 32 no no DET american_scientific_journal-11066 40 33 1 1 NUM american_scientific_journal-11066 40 34 , , PUNCT american_scientific_journal-11066 40 35 pp pp ADV american_scientific_journal-11066 40 36 1 1 NUM american_scientific_journal-11066 40 37 - - SYM american_scientific_journal-11066 40 38 12 12 NUM american_scientific_journal-11066 40 39 4 4 NUM american_scientific_journal-11066 40 40 𝑦3 𝑦3 PROPN american_scientific_journal-11066 40 41 = = SYM american_scientific_journal-11066 40 42 βˆ’πΏβˆ’1(𝐴2 βˆ’πΏβˆ’1(𝐴2 PROPN american_scientific_journal-11066 40 43 ) ) PUNCT american_scientific_journal-11066 40 44 = = PUNCT american_scientific_journal-11066 41 1 βˆ’ βˆ’ PROPN american_scientific_journal-11066 41 2 19 19 NUM american_scientific_journal-11066 41 3 5040 5040 NUM american_scientific_journal-11066 41 4 π‘₯6 π‘₯6 NOUN american_scientific_journal-11066 41 5 𝑦4 𝑦4 NOUN american_scientific_journal-11066 41 6 = = SYM american_scientific_journal-11066 41 7 βˆ’πΏβˆ’1(𝐴3 βˆ’πΏβˆ’1(𝐴3 PROPN american_scientific_journal-11066 41 8 ) ) PUNCT american_scientific_journal-11066 42 1 = = PRON american_scientific_journal-11066 42 2 βˆ’357 βˆ’357 NOUN american_scientific_journal-11066 42 3 362880 362880 NUM american_scientific_journal-11066 42 4 π‘₯8 π‘₯8 NOUN american_scientific_journal-11066 42 5 therefore therefore ADV american_scientific_journal-11066 42 6 , , PUNCT american_scientific_journal-11066 42 7 substituting substitute VERB american_scientific_journal-11066 42 8 above above ADP american_scientific_journal-11066 42 9 polynomials polynomial NOUN american_scientific_journal-11066 42 10 to to ADP american_scientific_journal-11066 42 11 the the DET american_scientific_journal-11066 42 12 equation equation NOUN american_scientific_journal-11066 42 13 ( ( PUNCT american_scientific_journal-11066 42 14 7 7 NUM american_scientific_journal-11066 42 15 ) ) PUNCT american_scientific_journal-11066 42 16 , , PUNCT american_scientific_journal-11066 42 17 the the DET american_scientific_journal-11066 42 18 solution solution NOUN american_scientific_journal-11066 42 19 of of ADP american_scientific_journal-11066 42 20 the the DET american_scientific_journal-11066 42 21 lane lane NOUN american_scientific_journal-11066 42 22 - - PUNCT american_scientific_journal-11066 42 23 emden emden NOUN american_scientific_journal-11066 42 24 equation equation NOUN american_scientific_journal-11066 42 25 when when SCONJ american_scientific_journal-11066 42 26 𝑛 𝑛 PROPN american_scientific_journal-11066 42 27 = = SYM american_scientific_journal-11066 42 28 3 3 NUM american_scientific_journal-11066 42 29 can can AUX american_scientific_journal-11066 42 30 be be AUX american_scientific_journal-11066 42 31 written write VERB american_scientific_journal-11066 42 32 in in ADP american_scientific_journal-11066 42 33 the the DET american_scientific_journal-11066 42 34 form form NOUN american_scientific_journal-11066 42 35 𝑦(π‘₯ 𝑦(π‘₯ NOUN american_scientific_journal-11066 42 36 ) ) PUNCT american_scientific_journal-11066 42 37 = = SYM american_scientific_journal-11066 42 38 1 1 NUM american_scientific_journal-11066 42 39 βˆ’ βˆ’ NUM american_scientific_journal-11066 42 40 1 1 NUM american_scientific_journal-11066 42 41 6 6 NUM american_scientific_journal-11066 42 42 π‘₯2 π‘₯2 NOUN american_scientific_journal-11066 42 43 + + CCONJ american_scientific_journal-11066 42 44 1 1 NUM american_scientific_journal-11066 42 45 40 40 NUM american_scientific_journal-11066 42 46 π‘₯4 π‘₯4 NOUN american_scientific_journal-11066 42 47 βˆ’ βˆ’ PROPN american_scientific_journal-11066 42 48 19 19 NUM american_scientific_journal-11066 42 49 5040 5040 NUM american_scientific_journal-11066 42 50 π‘₯6 π‘₯6 NOUN american_scientific_journal-11066 42 51 βˆ’ βˆ’ PROPN american_scientific_journal-11066 42 52 357 357 NUM american_scientific_journal-11066 42 53 362880 362880 NUM american_scientific_journal-11066 42 54 π‘₯8 π‘₯8 NOUN american_scientific_journal-11066 42 55 + + SYM american_scientific_journal-11066 42 56 β‹― β‹― PROPN american_scientific_journal-11066 42 57 ( ( PUNCT american_scientific_journal-11066 42 58 11 11 NUM american_scientific_journal-11066 42 59 ) ) PUNCT american_scientific_journal-11066 42 60 2.2 2.2 NUM american_scientific_journal-11066 42 61 the the DET american_scientific_journal-11066 42 62 solution solution NOUN american_scientific_journal-11066 42 63 using use VERB american_scientific_journal-11066 42 64 a a DET american_scientific_journal-11066 42 65 modified modified ADJ american_scientific_journal-11066 42 66 adomian adomian NOUN american_scientific_journal-11066 42 67 decomposition decomposition NOUN american_scientific_journal-11066 42 68 method method NOUN american_scientific_journal-11066 42 69 in in ADP american_scientific_journal-11066 42 70 this this DET american_scientific_journal-11066 42 71 method method NOUN american_scientific_journal-11066 42 72 , , PUNCT american_scientific_journal-11066 42 73 adomian adomian NOUN american_scientific_journal-11066 42 74 decomposition decomposition NOUN american_scientific_journal-11066 42 75 method method NOUN american_scientific_journal-11066 42 76 was be AUX american_scientific_journal-11066 42 77 applied apply VERB american_scientific_journal-11066 42 78 in in ADP american_scientific_journal-11066 42 79 a a DET american_scientific_journal-11066 42 80 straightforward straightforward ADJ american_scientific_journal-11066 42 81 manner manner NOUN american_scientific_journal-11066 42 82 with with ADP american_scientific_journal-11066 42 83 a a DET american_scientific_journal-11066 42 84 new new ADJ american_scientific_journal-11066 42 85 choice choice NOUN american_scientific_journal-11066 42 86 for for ADP american_scientific_journal-11066 42 87 the the DET american_scientific_journal-11066 42 88 differential differential ADJ american_scientific_journal-11066 42 89 operator operator NOUN american_scientific_journal-11066 42 90 [ [ X american_scientific_journal-11066 42 91 9 9 NUM american_scientific_journal-11066 42 92 ] ] PUNCT american_scientific_journal-11066 42 93 . . PUNCT american_scientific_journal-11066 43 1 the the DET american_scientific_journal-11066 43 2 differential differential ADJ american_scientific_journal-11066 43 3 operator operator NOUN american_scientific_journal-11066 43 4 𝐿1 𝐿1 VERB american_scientific_journal-11066 43 5 together together ADV american_scientific_journal-11066 43 6 with with ADP american_scientific_journal-11066 43 7 its its PRON american_scientific_journal-11066 43 8 corresponding corresponding ADJ american_scientific_journal-11066 43 9 inverse inverse ADJ american_scientific_journal-11066 43 10 integral integral ADJ american_scientific_journal-11066 43 11 operator operator NOUN american_scientific_journal-11066 43 12 𝐿1 𝐿1 ADP american_scientific_journal-11066 43 13 βˆ’1 βˆ’1 NOUN american_scientific_journal-11066 43 14 are be AUX american_scientific_journal-11066 43 15 defined define VERB american_scientific_journal-11066 43 16 as as ADP american_scientific_journal-11066 43 17 𝐿1(βˆ™ 𝐿1(βˆ™ PROPN american_scientific_journal-11066 43 18 ) ) PUNCT american_scientific_journal-11066 43 19 = = SYM american_scientific_journal-11066 44 1 1 1 NUM american_scientific_journal-11066 44 2 π‘₯ π‘₯ PRON american_scientific_journal-11066 44 3 𝑑2 𝑑2 NOUN american_scientific_journal-11066 44 4 𝑑π‘₯2 𝑑π‘₯2 NOUN american_scientific_journal-11066 44 5 π‘₯(βˆ™ π‘₯(βˆ™ ADJ american_scientific_journal-11066 44 6 ) ) PUNCT american_scientific_journal-11066 44 7 ( ( PUNCT american_scientific_journal-11066 44 8 12 12 NUM american_scientific_journal-11066 44 9 ) ) PUNCT american_scientific_journal-11066 44 10 and and CCONJ american_scientific_journal-11066 44 11 𝐿1 𝐿1 VERB american_scientific_journal-11066 44 12 βˆ’1(βˆ™ βˆ’1(βˆ™ ADJ american_scientific_journal-11066 44 13 ) ) PUNCT american_scientific_journal-11066 44 14 = = SYM american_scientific_journal-11066 45 1 1 1 NUM american_scientific_journal-11066 45 2 π‘₯ π‘₯ PRON american_scientific_journal-11066 45 3 ∫ ∫ PROPN american_scientific_journal-11066 45 4 ∫ ∫ PROPN american_scientific_journal-11066 45 5 π‘₯(βˆ™)𝑑π‘₯𝑑π‘₯ π‘₯(βˆ™)𝑑π‘₯𝑑π‘₯ PROPN american_scientific_journal-11066 46 1 π‘₯ π‘₯ INTJ american_scientific_journal-11066 46 2 0 0 NUM american_scientific_journal-11066 46 3 π‘₯ π‘₯ NOUN american_scientific_journal-11066 46 4 0 0 NUM american_scientific_journal-11066 46 5 . . PUNCT american_scientific_journal-11066 47 1 ( ( PUNCT american_scientific_journal-11066 47 2 13 13 NUM american_scientific_journal-11066 47 3 ) ) PUNCT american_scientific_journal-11066 47 4 now now ADV american_scientific_journal-11066 47 5 equation equation NOUN american_scientific_journal-11066 47 6 ( ( PUNCT american_scientific_journal-11066 47 7 2 2 X american_scientific_journal-11066 47 8 ) ) PUNCT american_scientific_journal-11066 47 9 takes take VERB american_scientific_journal-11066 47 10 the the DET american_scientific_journal-11066 47 11 form form NOUN american_scientific_journal-11066 47 12 𝐿1𝑦 𝐿1𝑦 PROPN american_scientific_journal-11066 47 13 = = PUNCT american_scientific_journal-11066 48 1 βˆ’π‘¦3 βˆ’π‘¦3 PROPN american_scientific_journal-11066 48 2 ( ( PUNCT american_scientific_journal-11066 48 3 14 14 NUM american_scientific_journal-11066 48 4 ) ) PUNCT american_scientific_journal-11066 48 5 as as ADP american_scientific_journal-11066 48 6 in in ADP american_scientific_journal-11066 48 7 the the DET american_scientific_journal-11066 48 8 section section NOUN american_scientific_journal-11066 48 9 2.1 2.1 NUM american_scientific_journal-11066 48 10 , , PUNCT american_scientific_journal-11066 48 11 the the DET american_scientific_journal-11066 48 12 solution solution NOUN american_scientific_journal-11066 48 13 of of ADP american_scientific_journal-11066 48 14 the the DET american_scientific_journal-11066 48 15 equation equation NOUN american_scientific_journal-11066 48 16 ( ( PUNCT american_scientific_journal-11066 48 17 2 2 X american_scientific_journal-11066 48 18 ) ) PUNCT american_scientific_journal-11066 48 19 can can AUX american_scientific_journal-11066 48 20 be be AUX american_scientific_journal-11066 48 21 written write VERB american_scientific_journal-11066 48 22 in in ADP american_scientific_journal-11066 48 23 the the DET american_scientific_journal-11066 48 24 form form NOUN american_scientific_journal-11066 48 25 𝑦(π‘₯ 𝑦(π‘₯ NOUN american_scientific_journal-11066 48 26 ) ) PUNCT american_scientific_journal-11066 48 27 = = SYM american_scientific_journal-11066 48 28 𝑦(0 𝑦(0 PROPN american_scientific_journal-11066 48 29 ) ) PUNCT american_scientific_journal-11066 48 30 βˆ’ βˆ’ PROPN american_scientific_journal-11066 48 31 πΏβˆ’1(𝑦3 πΏβˆ’1(𝑦3 PROPN american_scientific_journal-11066 48 32 ) ) PUNCT american_scientific_journal-11066 48 33 ( ( PUNCT american_scientific_journal-11066 48 34 15 15 X american_scientific_journal-11066 48 35 ) ) PUNCT american_scientific_journal-11066 48 36 taking take VERB american_scientific_journal-11066 48 37 𝑦(π‘₯ 𝑦(π‘₯ NOUN american_scientific_journal-11066 48 38 ) ) PUNCT american_scientific_journal-11066 48 39 = = PUNCT american_scientific_journal-11066 48 40 βˆ‘ βˆ‘ ADP american_scientific_journal-11066 48 41 𝑦𝑛 𝑦𝑛 VERB american_scientific_journal-11066 48 42 ∞ ∞ NUM american_scientific_journal-11066 48 43 𝑛=0 𝑛=0 PROPN american_scientific_journal-11066 48 44 and and CCONJ american_scientific_journal-11066 48 45 writing write VERB american_scientific_journal-11066 48 46 the the DET american_scientific_journal-11066 48 47 nonlinear nonlinear ADJ american_scientific_journal-11066 48 48 term term NOUN american_scientific_journal-11066 48 49 𝑦3 𝑦3 PROPN american_scientific_journal-11066 48 50 using use VERB american_scientific_journal-11066 48 51 adomian adomian NOUN american_scientific_journal-11066 48 52 polynomials polynomial NOUN american_scientific_journal-11066 48 53 as as ADP american_scientific_journal-11066 48 54 𝑦3 𝑦3 PROPN american_scientific_journal-11066 48 55 = = PUNCT american_scientific_journal-11066 48 56 βˆ‘ βˆ‘ PUNCT american_scientific_journal-11066 48 57 𝐴𝑛 𝐴𝑛 PROPN american_scientific_journal-11066 48 58 ∞ ∞ NUM american_scientific_journal-11066 48 59 𝑛=0 𝑛=0 NOUN american_scientific_journal-11066 48 60 , , PUNCT american_scientific_journal-11066 48 61 we we PRON american_scientific_journal-11066 48 62 have have AUX american_scientific_journal-11066 48 63 βˆ‘ βˆ‘ ADV american_scientific_journal-11066 48 64 𝑦𝑛 𝑦𝑛 VERB american_scientific_journal-11066 48 65 ∞ ∞ PROPN american_scientific_journal-11066 48 66 𝑛=0 𝑛=0 PROPN american_scientific_journal-11066 48 67 = = PUNCT american_scientific_journal-11066 49 1 1 1 NUM american_scientific_journal-11066 49 2 βˆ’ βˆ’ NOUN american_scientific_journal-11066 49 3 𝐿1 𝐿1 PROPN american_scientific_journal-11066 49 4 βˆ’1 βˆ’1 NOUN american_scientific_journal-11066 49 5 βˆ‘ βˆ‘ PUNCT american_scientific_journal-11066 49 6 𝐴𝑛 𝐴𝑛 PROPN american_scientific_journal-11066 49 7 ∞ ∞ NUM american_scientific_journal-11066 49 8 𝑛=0 𝑛=0 PROPN american_scientific_journal-11066 49 9 ( ( PUNCT american_scientific_journal-11066 49 10 16 16 NUM american_scientific_journal-11066 49 11 ) ) PUNCT american_scientific_journal-11066 49 12 𝑦0 𝑦0 NOUN american_scientific_journal-11066 49 13 = = SYM american_scientific_journal-11066 49 14 1 1 NUM american_scientific_journal-11066 49 15 ( ( PUNCT american_scientific_journal-11066 49 16 17 17 NUM american_scientific_journal-11066 49 17 ) ) PUNCT american_scientific_journal-11066 49 18 𝑦𝑛+1 𝑦𝑛+1 PROPN american_scientific_journal-11066 49 19 = = SYM american_scientific_journal-11066 49 20 βˆ’πΏ1 βˆ’πΏ1 PROPN american_scientific_journal-11066 49 21 βˆ’1(𝐴𝑛 βˆ’1(𝐴𝑛 NUM american_scientific_journal-11066 49 22 ) ) PUNCT american_scientific_journal-11066 49 23 , , PUNCT american_scientific_journal-11066 49 24 𝑛 𝑛 NOUN american_scientific_journal-11066 49 25 = = SYM american_scientific_journal-11066 49 26 0,1,2 0,1,2 NUM american_scientific_journal-11066 49 27 , , PUNCT american_scientific_journal-11066 49 28 … … PUNCT american_scientific_journal-11066 49 29 ( ( PUNCT american_scientific_journal-11066 49 30 18 18 NUM american_scientific_journal-11066 49 31 ) ) PUNCT american_scientific_journal-11066 49 32 using use VERB american_scientific_journal-11066 49 33 the the DET american_scientific_journal-11066 49 34 definition definition NOUN american_scientific_journal-11066 49 35 of of ADP american_scientific_journal-11066 49 36 the the DET american_scientific_journal-11066 49 37 inverse inverse NOUN american_scientific_journal-11066 49 38 operator operator NOUN american_scientific_journal-11066 49 39 as as ADP american_scientific_journal-11066 49 40 in in ADP american_scientific_journal-11066 49 41 ( ( PUNCT american_scientific_journal-11066 49 42 13 13 NUM american_scientific_journal-11066 49 43 ) ) PUNCT american_scientific_journal-11066 49 44 , , PUNCT american_scientific_journal-11066 49 45 we we PRON american_scientific_journal-11066 49 46 have have VERB american_scientific_journal-11066 49 47 𝑦𝑛+1 𝑦𝑛+1 NUM american_scientific_journal-11066 49 48 = = SYM american_scientific_journal-11066 49 49 βˆ’ βˆ’ PROPN american_scientific_journal-11066 49 50 1 1 NUM american_scientific_journal-11066 49 51 π‘₯ π‘₯ DET american_scientific_journal-11066 49 52 ∫ ∫ PROPN american_scientific_journal-11066 49 53 ∫ ∫ PROPN american_scientific_journal-11066 49 54 π‘₯(𝐴𝑛)𝑑π‘₯𝑑π‘₯ π‘₯(𝐴𝑛)𝑑π‘₯𝑑π‘₯ PROPN american_scientific_journal-11066 49 55 π‘₯ π‘₯ NOUN american_scientific_journal-11066 49 56 0 0 NUM american_scientific_journal-11066 50 1 π‘₯ π‘₯ NOUN american_scientific_journal-11066 50 2 0 0 NUM american_scientific_journal-11066 50 3 ( ( PUNCT american_scientific_journal-11066 50 4 19 19 NUM american_scientific_journal-11066 50 5 ) ) PUNCT american_scientific_journal-11066 50 6 as as ADP american_scientific_journal-11066 50 7 in in ADP american_scientific_journal-11066 50 8 the the DET american_scientific_journal-11066 50 9 section section NOUN american_scientific_journal-11066 50 10 2.1 2.1 NUM american_scientific_journal-11066 50 11 , , PUNCT american_scientific_journal-11066 50 12 the the DET american_scientific_journal-11066 50 13 first first ADJ american_scientific_journal-11066 50 14 few few ADJ american_scientific_journal-11066 50 15 adomian adomian NOUN american_scientific_journal-11066 50 16 polynomials polynomial NOUN american_scientific_journal-11066 50 17 take take VERB american_scientific_journal-11066 50 18 the the DET american_scientific_journal-11066 50 19 form form NOUN american_scientific_journal-11066 50 20 american american ADJ american_scientific_journal-11066 50 21 academic academic ADJ american_scientific_journal-11066 50 22 scientific scientific ADJ american_scientific_journal-11066 50 23 research research NOUN american_scientific_journal-11066 50 24 journal journal NOUN american_scientific_journal-11066 50 25 for for ADP american_scientific_journal-11066 50 26 engineering engineering NOUN american_scientific_journal-11066 50 27 , , PUNCT american_scientific_journal-11066 50 28 technology technology NOUN american_scientific_journal-11066 50 29 , , PUNCT american_scientific_journal-11066 50 30 and and CCONJ american_scientific_journal-11066 50 31 sciences science NOUN american_scientific_journal-11066 50 32 ( ( PUNCT american_scientific_journal-11066 50 33 asrjets asrjet NOUN american_scientific_journal-11066 50 34 ) ) PUNCT american_scientific_journal-11066 50 35 volume volume NOUN american_scientific_journal-11066 50 36 99 99 NUM american_scientific_journal-11066 50 37 , , PUNCT american_scientific_journal-11066 50 38 no no DET american_scientific_journal-11066 50 39 1 1 NUM american_scientific_journal-11066 50 40 , , PUNCT american_scientific_journal-11066 50 41 pp pp ADV american_scientific_journal-11066 50 42 1 1 NUM american_scientific_journal-11066 50 43 - - SYM american_scientific_journal-11066 50 44 12 12 NUM american_scientific_journal-11066 50 45 5 5 NUM american_scientific_journal-11066 50 46 𝐴0 𝐴0 NOUN american_scientific_journal-11066 50 47 = = SYM american_scientific_journal-11066 50 48 1 1 NUM american_scientific_journal-11066 50 49 𝐴1 𝐴1 NOUN american_scientific_journal-11066 50 50 = = SYM american_scientific_journal-11066 50 51 βˆ’ βˆ’ PROPN american_scientific_journal-11066 50 52 π‘₯2 π‘₯2 NOUN american_scientific_journal-11066 50 53 6 6 NUM american_scientific_journal-11066 50 54 𝐴2 𝐴2 NOUN american_scientific_journal-11066 50 55 = = SYM american_scientific_journal-11066 50 56 19 19 NUM american_scientific_journal-11066 50 57 120 120 NUM american_scientific_journal-11066 50 58 π‘₯4 π‘₯4 NOUN american_scientific_journal-11066 50 59 𝐴3 𝐴3 PROPN american_scientific_journal-11066 50 60 = = PROPN american_scientific_journal-11066 50 61 357 357 NUM american_scientific_journal-11066 50 62 5040 5040 NUM american_scientific_journal-11066 50 63 π‘₯6 π‘₯6 NOUN american_scientific_journal-11066 50 64 and and CCONJ american_scientific_journal-11066 50 65 so so ADV american_scientific_journal-11066 50 66 on on ADV american_scientific_journal-11066 50 67 . . PUNCT american_scientific_journal-11066 51 1 now now ADV american_scientific_journal-11066 51 2 inserting insert VERB american_scientific_journal-11066 51 3 these these DET american_scientific_journal-11066 51 4 adomian adomian NOUN american_scientific_journal-11066 51 5 polynomials polynomial NOUN american_scientific_journal-11066 51 6 to to ADP american_scientific_journal-11066 51 7 equation equation NOUN american_scientific_journal-11066 51 8 ( ( PUNCT american_scientific_journal-11066 51 9 8) 8) NUM american_scientific_journal-11066 51 10 and and CCONJ american_scientific_journal-11066 51 11 ( ( PUNCT american_scientific_journal-11066 51 12 9 9 X american_scientific_journal-11066 51 13 ) ) PUNCT american_scientific_journal-11066 51 14 gives give VERB american_scientific_journal-11066 51 15 𝑦0 𝑦0 NOUN american_scientific_journal-11066 51 16 = = SYM american_scientific_journal-11066 51 17 1 1 NUM american_scientific_journal-11066 51 18 , , PUNCT american_scientific_journal-11066 51 19 𝑦1 𝑦1 NOUN american_scientific_journal-11066 51 20 = = SYM american_scientific_journal-11066 51 21 βˆ’πΏβˆ’1(𝐴0 βˆ’πΏβˆ’1(𝐴0 PROPN american_scientific_journal-11066 51 22 ) ) PUNCT american_scientific_journal-11066 52 1 = = PUNCT american_scientific_journal-11066 53 1 βˆ’ βˆ’ PROPN american_scientific_journal-11066 53 2 1 1 NUM american_scientific_journal-11066 53 3 6 6 NUM american_scientific_journal-11066 53 4 π‘₯2 π‘₯2 NOUN american_scientific_journal-11066 53 5 , , PUNCT american_scientific_journal-11066 53 6 𝑦2 𝑦2 PROPN american_scientific_journal-11066 53 7 = = SYM american_scientific_journal-11066 53 8 βˆ’πΏβˆ’1(𝐴1 βˆ’πΏβˆ’1(𝐴1 NOUN american_scientific_journal-11066 53 9 ) ) PUNCT american_scientific_journal-11066 53 10 = = SYM american_scientific_journal-11066 53 11 1 1 NUM american_scientific_journal-11066 53 12 40 40 NUM american_scientific_journal-11066 53 13 π‘₯4 π‘₯4 NOUN american_scientific_journal-11066 53 14 𝑦3 𝑦3 PROPN american_scientific_journal-11066 53 15 = = SYM american_scientific_journal-11066 53 16 βˆ’πΏβˆ’1(𝐴2 βˆ’πΏβˆ’1(𝐴2 PROPN american_scientific_journal-11066 53 17 ) ) PUNCT american_scientific_journal-11066 53 18 = = PUNCT american_scientific_journal-11066 54 1 βˆ’ βˆ’ PROPN american_scientific_journal-11066 54 2 19 19 NUM american_scientific_journal-11066 54 3 5040 5040 NUM american_scientific_journal-11066 54 4 π‘₯6 π‘₯6 NOUN american_scientific_journal-11066 54 5 𝑦4 𝑦4 NOUN american_scientific_journal-11066 54 6 = = SYM american_scientific_journal-11066 54 7 βˆ’πΏβˆ’1(𝐴3 βˆ’πΏβˆ’1(𝐴3 PROPN american_scientific_journal-11066 54 8 ) ) PUNCT american_scientific_journal-11066 55 1 = = PRON american_scientific_journal-11066 55 2 βˆ’357 βˆ’357 NOUN american_scientific_journal-11066 55 3 362880 362880 NUM american_scientific_journal-11066 55 4 π‘₯8 π‘₯8 NOUN american_scientific_journal-11066 55 5 therefore therefore ADV american_scientific_journal-11066 55 6 , , PUNCT american_scientific_journal-11066 55 7 substituting substitute VERB american_scientific_journal-11066 55 8 above above ADP american_scientific_journal-11066 55 9 polynomials polynomial NOUN american_scientific_journal-11066 55 10 to to ADP american_scientific_journal-11066 55 11 the the DET american_scientific_journal-11066 55 12 equation equation NOUN american_scientific_journal-11066 55 13 ( ( PUNCT american_scientific_journal-11066 55 14 15 15 NUM american_scientific_journal-11066 55 15 ) ) PUNCT american_scientific_journal-11066 55 16 , , PUNCT american_scientific_journal-11066 55 17 the the DET american_scientific_journal-11066 55 18 solution solution NOUN american_scientific_journal-11066 55 19 of of ADP american_scientific_journal-11066 55 20 the the DET american_scientific_journal-11066 55 21 lane lane NOUN american_scientific_journal-11066 55 22 - - PUNCT american_scientific_journal-11066 55 23 emden emden NOUN american_scientific_journal-11066 55 24 equation equation NOUN american_scientific_journal-11066 55 25 when when SCONJ american_scientific_journal-11066 55 26 𝑛 𝑛 PROPN american_scientific_journal-11066 55 27 = = SYM american_scientific_journal-11066 55 28 3 3 NUM american_scientific_journal-11066 55 29 can can AUX american_scientific_journal-11066 55 30 be be AUX american_scientific_journal-11066 55 31 written write VERB american_scientific_journal-11066 55 32 in in ADP american_scientific_journal-11066 55 33 the the DET american_scientific_journal-11066 55 34 form form NOUN american_scientific_journal-11066 55 35 𝑦(π‘₯ 𝑦(π‘₯ NOUN american_scientific_journal-11066 55 36 ) ) PUNCT american_scientific_journal-11066 55 37 = = SYM american_scientific_journal-11066 55 38 1 1 NUM american_scientific_journal-11066 55 39 βˆ’ βˆ’ NUM american_scientific_journal-11066 55 40 1 1 NUM american_scientific_journal-11066 55 41 6 6 NUM american_scientific_journal-11066 55 42 π‘₯2 π‘₯2 NOUN american_scientific_journal-11066 55 43 + + CCONJ american_scientific_journal-11066 55 44 1 1 NUM american_scientific_journal-11066 55 45 40 40 NUM american_scientific_journal-11066 55 46 π‘₯4 π‘₯4 NOUN american_scientific_journal-11066 55 47 βˆ’ βˆ’ PROPN american_scientific_journal-11066 55 48 19 19 NUM american_scientific_journal-11066 55 49 5040 5040 NUM american_scientific_journal-11066 55 50 π‘₯6 π‘₯6 NOUN american_scientific_journal-11066 55 51 βˆ’ βˆ’ PROPN american_scientific_journal-11066 55 52 357 357 NUM american_scientific_journal-11066 55 53 362880 362880 NUM american_scientific_journal-11066 55 54 π‘₯8 π‘₯8 NOUN american_scientific_journal-11066 55 55 + + SYM american_scientific_journal-11066 55 56 β‹― β‹― PROPN american_scientific_journal-11066 55 57 ( ( PUNCT american_scientific_journal-11066 55 58 20 20 NUM american_scientific_journal-11066 55 59 ) ) PUNCT american_scientific_journal-11066 55 60 2.3 2.3 NUM american_scientific_journal-11066 55 61 the the DET american_scientific_journal-11066 55 62 solution solution NOUN american_scientific_journal-11066 55 63 using use VERB american_scientific_journal-11066 55 64 differential differential ADJ american_scientific_journal-11066 55 65 transform transform NOUN american_scientific_journal-11066 55 66 method method NOUN american_scientific_journal-11066 55 67 the the DET american_scientific_journal-11066 55 68 differential differential ADJ american_scientific_journal-11066 55 69 transform transform NOUN american_scientific_journal-11066 55 70 technique technique NOUN american_scientific_journal-11066 55 71 is be AUX american_scientific_journal-11066 55 72 one one NUM american_scientific_journal-11066 55 73 of of ADP american_scientific_journal-11066 55 74 the the DET american_scientific_journal-11066 55 75 semianalytical semianalytical ADJ american_scientific_journal-11066 55 76 numerical numerical ADJ american_scientific_journal-11066 55 77 methods method NOUN american_scientific_journal-11066 55 78 for for ADP american_scientific_journal-11066 55 79 ordinary ordinary ADJ american_scientific_journal-11066 55 80 and and CCONJ american_scientific_journal-11066 55 81 partial partial ADJ american_scientific_journal-11066 55 82 differential differential ADJ american_scientific_journal-11066 55 83 equations equation NOUN american_scientific_journal-11066 55 84 that that PRON american_scientific_journal-11066 55 85 use use VERB american_scientific_journal-11066 55 86 the the DET american_scientific_journal-11066 55 87 form form NOUN american_scientific_journal-11066 55 88 of of ADP american_scientific_journal-11066 55 89 polynomials polynomial NOUN american_scientific_journal-11066 55 90 as as ADP american_scientific_journal-11066 55 91 approximations approximation NOUN american_scientific_journal-11066 55 92 of of ADP american_scientific_journal-11066 55 93 the the DET american_scientific_journal-11066 55 94 exact exact ADJ american_scientific_journal-11066 55 95 solutions solution NOUN american_scientific_journal-11066 55 96 that that PRON american_scientific_journal-11066 55 97 are be AUX american_scientific_journal-11066 55 98 sufficiently sufficiently ADV american_scientific_journal-11066 55 99 differentiable differentiable ADJ american_scientific_journal-11066 55 100 which which PRON american_scientific_journal-11066 55 101 is be AUX american_scientific_journal-11066 55 102 first first ADV american_scientific_journal-11066 55 103 introduced introduce VERB american_scientific_journal-11066 55 104 by by ADP american_scientific_journal-11066 55 105 zhou zhou PROPN american_scientific_journal-11066 56 1 [ [ X american_scientific_journal-11066 56 2 10 10 NUM american_scientific_journal-11066 56 3 ] ] PUNCT american_scientific_journal-11066 56 4 in in ADP american_scientific_journal-11066 56 5 1986.the 1986.the DET american_scientific_journal-11066 56 6 method method NOUN american_scientific_journal-11066 56 7 is be AUX american_scientific_journal-11066 56 8 an an DET american_scientific_journal-11066 56 9 analytical analytical ADJ american_scientific_journal-11066 56 10 an an DET american_scientific_journal-11066 56 11 numerical numerical ADJ american_scientific_journal-11066 56 12 method method NOUN american_scientific_journal-11066 56 13 for for ADP american_scientific_journal-11066 56 14 solving solve VERB american_scientific_journal-11066 56 15 a a DET american_scientific_journal-11066 56 16 wide wide ADJ american_scientific_journal-11066 56 17 variety variety NOUN american_scientific_journal-11066 56 18 of of ADP american_scientific_journal-11066 56 19 differential differential ADJ american_scientific_journal-11066 56 20 equations equation NOUN american_scientific_journal-11066 56 21 and and CCONJ american_scientific_journal-11066 56 22 usually usually ADV american_scientific_journal-11066 56 23 gets get VERB american_scientific_journal-11066 56 24 the the DET american_scientific_journal-11066 56 25 solution solution NOUN american_scientific_journal-11066 56 26 in in ADP american_scientific_journal-11066 56 27 series series NOUN american_scientific_journal-11066 56 28 form form NOUN american_scientific_journal-11066 56 29 . . PUNCT american_scientific_journal-11066 57 1 differential differential ADJ american_scientific_journal-11066 57 2 transform transform NOUN american_scientific_journal-11066 57 3 of of ADP american_scientific_journal-11066 57 4 a a DET american_scientific_journal-11066 57 5 function function NOUN american_scientific_journal-11066 57 6 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-11066 57 7 ) ) PUNCT american_scientific_journal-11066 57 8 is be AUX american_scientific_journal-11066 57 9 defined define VERB american_scientific_journal-11066 57 10 by by ADP american_scientific_journal-11066 57 11 𝐹(π‘˜ 𝐹(π‘˜ NUM american_scientific_journal-11066 57 12 ) ) PUNCT american_scientific_journal-11066 57 13 = = NOUN american_scientific_journal-11066 57 14 1 1 NUM american_scientific_journal-11066 57 15 π‘˜ π‘˜ X american_scientific_journal-11066 57 16 [ [ PUNCT american_scientific_journal-11066 57 17 π‘‘π‘˜π‘“ π‘‘π‘˜π‘“ NUM american_scientific_journal-11066 57 18 𝑑π‘₯π‘˜ 𝑑π‘₯π‘˜ PROPN american_scientific_journal-11066 57 19 ] ] X american_scientific_journal-11066 57 20 π‘₯=0 π‘₯=0 PUNCT american_scientific_journal-11066 57 21 ( ( PUNCT american_scientific_journal-11066 57 22 21 21 NUM american_scientific_journal-11066 57 23 ) ) PUNCT american_scientific_journal-11066 57 24 where where SCONJ american_scientific_journal-11066 57 25 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-11066 57 26 ) ) PUNCT american_scientific_journal-11066 57 27 is be AUX american_scientific_journal-11066 57 28 the the DET american_scientific_journal-11066 57 29 original original ADJ american_scientific_journal-11066 57 30 function function NOUN american_scientific_journal-11066 57 31 and and CCONJ american_scientific_journal-11066 57 32 𝐹(π‘˜ 𝐹(π‘˜ NUM american_scientific_journal-11066 57 33 ) ) PUNCT american_scientific_journal-11066 57 34 is be AUX american_scientific_journal-11066 57 35 the the DET american_scientific_journal-11066 57 36 transformed transform VERB american_scientific_journal-11066 57 37 function function NOUN american_scientific_journal-11066 57 38 . . PUNCT american_scientific_journal-11066 58 1 inverse inverse NOUN american_scientific_journal-11066 58 2 differential differential NOUN american_scientific_journal-11066 58 3 transform transform NOUN american_scientific_journal-11066 58 4 is be AUX american_scientific_journal-11066 58 5 defined define VERB american_scientific_journal-11066 58 6 by by ADP american_scientific_journal-11066 58 7 american american ADJ american_scientific_journal-11066 58 8 academic academic ADJ american_scientific_journal-11066 58 9 scientific scientific ADJ american_scientific_journal-11066 58 10 research research NOUN american_scientific_journal-11066 58 11 journal journal NOUN american_scientific_journal-11066 58 12 for for ADP american_scientific_journal-11066 58 13 engineering engineering NOUN american_scientific_journal-11066 58 14 , , PUNCT american_scientific_journal-11066 58 15 technology technology NOUN american_scientific_journal-11066 58 16 , , PUNCT american_scientific_journal-11066 58 17 and and CCONJ american_scientific_journal-11066 58 18 sciences science NOUN american_scientific_journal-11066 58 19 ( ( PUNCT american_scientific_journal-11066 58 20 asrjets asrjet NOUN american_scientific_journal-11066 58 21 ) ) PUNCT american_scientific_journal-11066 58 22 volume volume NOUN american_scientific_journal-11066 58 23 99 99 NUM american_scientific_journal-11066 58 24 , , PUNCT american_scientific_journal-11066 58 25 no no DET american_scientific_journal-11066 58 26 1 1 NUM american_scientific_journal-11066 58 27 , , PUNCT american_scientific_journal-11066 58 28 pp pp ADV american_scientific_journal-11066 58 29 1 1 NUM american_scientific_journal-11066 58 30 - - SYM american_scientific_journal-11066 58 31 12 12 NUM american_scientific_journal-11066 58 32 6 6 NUM american_scientific_journal-11066 58 33 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-11066 58 34 ) ) PUNCT american_scientific_journal-11066 58 35 = = SYM american_scientific_journal-11066 59 1 βˆ‘ βˆ‘ PUNCT american_scientific_journal-11066 59 2 [ [ PUNCT american_scientific_journal-11066 59 3 π‘‘π‘˜π‘“ π‘‘π‘˜π‘“ NUM american_scientific_journal-11066 59 4 𝑑π‘₯π‘˜ 𝑑π‘₯π‘˜ PROPN american_scientific_journal-11066 59 5 ] ] X american_scientific_journal-11066 59 6 π‘₯=0 π‘₯=0 NUM american_scientific_journal-11066 59 7 π‘₯π‘˜ π‘₯π‘˜ X american_scientific_journal-11066 59 8 π‘˜ π‘˜ X american_scientific_journal-11066 59 9 ! ! PUNCT american_scientific_journal-11066 60 1 ∞ ∞ NUM american_scientific_journal-11066 60 2 π‘˜=0 π‘˜=0 PROPN american_scientific_journal-11066 60 3 ( ( PUNCT american_scientific_journal-11066 60 4 22 22 NUM american_scientific_journal-11066 60 5 ) ) PUNCT american_scientific_journal-11066 60 6 a a DET american_scientific_journal-11066 60 7 few few ADJ american_scientific_journal-11066 60 8 basic basic ADJ american_scientific_journal-11066 60 9 results result NOUN american_scientific_journal-11066 60 10 which which PRON american_scientific_journal-11066 60 11 have have AUX american_scientific_journal-11066 60 12 been be AUX american_scientific_journal-11066 60 13 proved prove VERB american_scientific_journal-11066 60 14 and and CCONJ american_scientific_journal-11066 60 15 can can AUX american_scientific_journal-11066 60 16 be be AUX american_scientific_journal-11066 60 17 found find VERB american_scientific_journal-11066 60 18 in in ADP american_scientific_journal-11066 60 19 the the DET american_scientific_journal-11066 60 20 literature literature NOUN american_scientific_journal-11066 60 21 are be AUX american_scientific_journal-11066 60 22 given give VERB american_scientific_journal-11066 60 23 by by ADP american_scientific_journal-11066 60 24 following follow VERB american_scientific_journal-11066 60 25 table table NOUN american_scientific_journal-11066 60 26 . . PUNCT american_scientific_journal-11066 61 1 table table NOUN american_scientific_journal-11066 61 2 1 1 NUM american_scientific_journal-11066 61 3 : : PUNCT american_scientific_journal-11066 61 4 basic basic ADJ american_scientific_journal-11066 61 5 results result NOUN american_scientific_journal-11066 61 6 of of ADP american_scientific_journal-11066 61 7 differential differential ADJ american_scientific_journal-11066 61 8 transform transform NOUN american_scientific_journal-11066 61 9 method method NOUN american_scientific_journal-11066 61 10 the the DET american_scientific_journal-11066 61 11 function function NOUN american_scientific_journal-11066 61 12 𝒇(𝒙 𝒇(𝒙 NOUN american_scientific_journal-11066 61 13 ) ) PUNCT american_scientific_journal-11066 61 14 the the DET american_scientific_journal-11066 61 15 transformed transform VERB american_scientific_journal-11066 61 16 function function NOUN american_scientific_journal-11066 61 17 𝑭(π’Œ 𝑭(π’Œ NOUN american_scientific_journal-11066 61 18 ) ) PUNCT american_scientific_journal-11066 61 19 𝑔(π‘₯ 𝑔(π‘₯ PROPN american_scientific_journal-11066 61 20 ) ) PUNCT american_scientific_journal-11066 61 21 Β± Β± NUM american_scientific_journal-11066 61 22 β„Ž(π‘₯ β„Ž(π‘₯ NOUN american_scientific_journal-11066 61 23 ) ) PUNCT american_scientific_journal-11066 61 24 𝐺(π‘˜ 𝐺(π‘˜ NOUN american_scientific_journal-11066 61 25 ) ) PUNCT american_scientific_journal-11066 61 26 Β± Β± NOUN american_scientific_journal-11066 61 27 𝐻(π‘˜ 𝐻(π‘˜ NOUN american_scientific_journal-11066 61 28 ) ) PUNCT american_scientific_journal-11066 61 29 𝑐𝑔(π‘₯ 𝑐𝑔(π‘₯ PUNCT american_scientific_journal-11066 61 30 ) ) PUNCT american_scientific_journal-11066 62 1 𝑐𝐺(π‘˜ 𝑐𝐺(π‘˜ X american_scientific_journal-11066 62 2 ) ) PUNCT american_scientific_journal-11066 62 3 𝑑𝑛𝑔(π‘₯ 𝑑𝑛𝑔(π‘₯ PROPN american_scientific_journal-11066 62 4 ) ) PUNCT american_scientific_journal-11066 62 5 𝑑π‘₯𝑛 𝑑π‘₯𝑛 VERB american_scientific_journal-11066 62 6 ( ( PUNCT american_scientific_journal-11066 62 7 π‘˜+𝑛 π‘˜+𝑛 NOUN american_scientific_journal-11066 62 8 ) ) PUNCT american_scientific_journal-11066 62 9 ! ! PUNCT american_scientific_journal-11066 63 1 π‘˜ π‘˜ X american_scientific_journal-11066 63 2 ! ! PUNCT american_scientific_journal-11066 64 1 𝐺(π‘˜ 𝐺(π‘˜ X american_scientific_journal-11066 64 2 + + NUM american_scientific_journal-11066 64 3 𝑛 𝑛 NOUN american_scientific_journal-11066 64 4 ) ) PUNCT american_scientific_journal-11066 64 5 𝑔(π‘₯)β„Ž(π‘₯ 𝑔(π‘₯)β„Ž(π‘₯ NOUN american_scientific_journal-11066 64 6 ) ) PUNCT american_scientific_journal-11066 64 7 βˆ‘ βˆ‘ PUNCT american_scientific_journal-11066 64 8 𝐺(π‘˜)𝐻(π‘˜ 𝐺(π‘˜)𝐻(π‘˜ PRON american_scientific_journal-11066 64 9 βˆ’ βˆ’ NOUN american_scientific_journal-11066 64 10 𝑙)π‘˜ 𝑙)π‘˜ NUM american_scientific_journal-11066 64 11 𝑙=0 𝑙=0 PUNCT american_scientific_journal-11066 65 1 π‘₯𝑛 π‘₯𝑛 PROPN american_scientific_journal-11066 65 2 𝛿(π‘˜ 𝛿(π‘˜ PROPN american_scientific_journal-11066 65 3 βˆ’ βˆ’ ADP american_scientific_journal-11066 65 4 𝑛 𝑛 PROPN american_scientific_journal-11066 65 5 ) ) PUNCT american_scientific_journal-11066 65 6 ∫ ∫ PROPN american_scientific_journal-11066 65 7 𝑔(𝑑)𝑑𝑑 𝑔(𝑑)𝑑𝑑 PART american_scientific_journal-11066 65 8 π‘₯ π‘₯ NOUN american_scientific_journal-11066 65 9 0 0 NUM american_scientific_journal-11066 65 10 𝐺(π‘˜βˆ’1 𝐺(π‘˜βˆ’1 NOUN american_scientific_journal-11066 65 11 ) ) PUNCT american_scientific_journal-11066 65 12 π‘˜ π‘˜ ADP american_scientific_journal-11066 65 13 π‘˜ π‘˜ DET american_scientific_journal-11066 65 14 β‰₯ β‰₯ NUM american_scientific_journal-11066 65 15 1 1 NUM american_scientific_journal-11066 65 16 𝑒(π‘₯)𝑣(π‘₯)𝑀(π‘₯ 𝑒(π‘₯)𝑣(π‘₯)𝑀(π‘₯ NOUN american_scientific_journal-11066 65 17 ) ) PUNCT american_scientific_journal-11066 65 18 βˆ‘ βˆ‘ ADP american_scientific_journal-11066 65 19 βˆ‘ βˆ‘ PROPN american_scientific_journal-11066 65 20 βˆ‘ βˆ‘ PUNCT american_scientific_journal-11066 65 21 π‘ˆ(π‘Ÿ)𝑉(𝑠)π‘Š(π‘š)π‘˜βˆ’π‘Ÿβˆ’π‘  π‘ˆ(π‘Ÿ)𝑉(𝑠)π‘Š(π‘š)π‘˜βˆ’π‘Ÿβˆ’π‘  NOUN american_scientific_journal-11066 65 22 π‘š=0 π‘š=0 PUNCT american_scientific_journal-11066 65 23 π‘˜βˆ’π‘Ÿ π‘˜βˆ’π‘Ÿ NOUN american_scientific_journal-11066 65 24 𝑠=0 𝑠=0 PUNCT american_scientific_journal-11066 65 25 π‘˜ π‘˜ NOUN american_scientific_journal-11066 65 26 π‘Ÿ=0 π‘Ÿ=0 PUNCT american_scientific_journal-11066 65 27 where where SCONJ american_scientific_journal-11066 65 28 𝐺(π‘˜ 𝐺(π‘˜ NOUN american_scientific_journal-11066 65 29 ) ) PUNCT american_scientific_journal-11066 65 30 , , PUNCT american_scientific_journal-11066 65 31 𝐻(π‘˜ 𝐻(π‘˜ NOUN american_scientific_journal-11066 65 32 ) ) PUNCT american_scientific_journal-11066 65 33 , , PUNCT american_scientific_journal-11066 65 34 π‘ˆ(π‘Ÿ π‘ˆ(π‘Ÿ ADP american_scientific_journal-11066 65 35 ) ) PUNCT american_scientific_journal-11066 65 36 , , PUNCT american_scientific_journal-11066 65 37 𝑉(𝑠 𝑉(𝑠 NOUN american_scientific_journal-11066 65 38 ) ) PUNCT american_scientific_journal-11066 65 39 and and CCONJ american_scientific_journal-11066 65 40 π‘Š(π‘š π‘Š(π‘š ADV american_scientific_journal-11066 65 41 ) ) PUNCT american_scientific_journal-11066 65 42 are be AUX american_scientific_journal-11066 65 43 the the DET american_scientific_journal-11066 65 44 transformed transform VERB american_scientific_journal-11066 65 45 functions function NOUN american_scientific_journal-11066 65 46 of of ADP american_scientific_journal-11066 65 47 𝑔(π‘₯ 𝑔(π‘₯ PROPN american_scientific_journal-11066 65 48 ) ) PUNCT american_scientific_journal-11066 65 49 , , PUNCT american_scientific_journal-11066 65 50 β„Ž(π‘₯ β„Ž(π‘₯ NUM american_scientific_journal-11066 65 51 ) ) PUNCT american_scientific_journal-11066 65 52 , , PUNCT american_scientific_journal-11066 65 53 𝑒(π‘₯ 𝑒(π‘₯ PROPN american_scientific_journal-11066 65 54 ) ) PUNCT american_scientific_journal-11066 65 55 𝑣(π‘₯ 𝑣(π‘₯ PROPN american_scientific_journal-11066 65 56 ) ) PUNCT american_scientific_journal-11066 65 57 and and CCONJ american_scientific_journal-11066 65 58 𝑀(π‘₯ 𝑀(π‘₯ NOUN american_scientific_journal-11066 65 59 ) ) PUNCT american_scientific_journal-11066 65 60 , , PUNCT american_scientific_journal-11066 65 61 respectively respectively ADV american_scientific_journal-11066 65 62 and and CCONJ american_scientific_journal-11066 65 63 𝛿(π‘Ÿ 𝛿(π‘Ÿ PROPN american_scientific_journal-11066 65 64 ) ) PUNCT american_scientific_journal-11066 66 1 is be AUX american_scientific_journal-11066 66 2 the the DET american_scientific_journal-11066 66 3 kronecker kronecker NOUN american_scientific_journal-11066 66 4 delta delta NOUN american_scientific_journal-11066 66 5 function function NOUN american_scientific_journal-11066 66 6 . . PUNCT american_scientific_journal-11066 67 1 after after ADP american_scientific_journal-11066 67 2 rewriting rewrite VERB american_scientific_journal-11066 67 3 the the DET american_scientific_journal-11066 67 4 equation equation NOUN american_scientific_journal-11066 67 5 ( ( PUNCT american_scientific_journal-11066 67 6 2 2 NUM american_scientific_journal-11066 67 7 ) ) PUNCT american_scientific_journal-11066 67 8 , , PUNCT american_scientific_journal-11066 67 9 we we PRON american_scientific_journal-11066 67 10 get get VERB american_scientific_journal-11066 67 11 π‘₯𝑦′′(π‘₯ π‘₯𝑦′′(π‘₯ ADJ american_scientific_journal-11066 67 12 ) ) PUNCT american_scientific_journal-11066 68 1 + + NUM american_scientific_journal-11066 69 1 2𝑦′(π‘₯ 2𝑦′(π‘₯ NUM american_scientific_journal-11066 69 2 ) ) PUNCT american_scientific_journal-11066 69 3 + + NUM american_scientific_journal-11066 69 4 π‘₯𝑦3 π‘₯𝑦3 NOUN american_scientific_journal-11066 69 5 = = SYM american_scientific_journal-11066 69 6 0 0 NUM american_scientific_journal-11066 69 7 ( ( PUNCT american_scientific_journal-11066 69 8 23 23 NUM american_scientific_journal-11066 69 9 ) ) PUNCT american_scientific_journal-11066 69 10 applying apply VERB american_scientific_journal-11066 69 11 the the DET american_scientific_journal-11066 69 12 results result NOUN american_scientific_journal-11066 69 13 given give VERB american_scientific_journal-11066 69 14 in in ADP american_scientific_journal-11066 69 15 the the DET american_scientific_journal-11066 69 16 table table NOUN american_scientific_journal-11066 69 17 01 01 NUM american_scientific_journal-11066 69 18 to to ADP american_scientific_journal-11066 69 19 the the DET american_scientific_journal-11066 69 20 equation equation NOUN american_scientific_journal-11066 69 21 ( ( PUNCT american_scientific_journal-11066 69 22 3 3 NUM american_scientific_journal-11066 69 23 ) ) PUNCT american_scientific_journal-11066 69 24 , , PUNCT american_scientific_journal-11066 69 25 we we PRON american_scientific_journal-11066 69 26 have have VERB american_scientific_journal-11066 69 27 βˆ‘ βˆ‘ ADV american_scientific_journal-11066 69 28 𝛿(𝑙 𝛿(𝑙 VERB american_scientific_journal-11066 69 29 βˆ’ βˆ’ ADP american_scientific_journal-11066 69 30 1 1 NUM american_scientific_journal-11066 69 31 ) ) PUNCT american_scientific_journal-11066 69 32 π‘˜ π‘˜ X american_scientific_journal-11066 69 33 𝑙=0 𝑙=0 PUNCT american_scientific_journal-11066 70 1 ( ( PUNCT american_scientific_journal-11066 70 2 π‘˜ π‘˜ PROPN american_scientific_journal-11066 70 3 + + ADJ american_scientific_journal-11066 70 4 2 2 NUM american_scientific_journal-11066 70 5 βˆ’ βˆ’ NOUN american_scientific_journal-11066 70 6 𝑙)π‘Œ(π‘˜ 𝑙)π‘Œ(π‘˜ NOUN american_scientific_journal-11066 71 1 + + CCONJ american_scientific_journal-11066 71 2 2 2 NUM american_scientific_journal-11066 71 3 βˆ’ βˆ’ NOUN american_scientific_journal-11066 71 4 𝑙 𝑙 NUM american_scientific_journal-11066 71 5 ) ) PUNCT american_scientific_journal-11066 71 6 + + CCONJ american_scientific_journal-11066 71 7 2(π‘˜ 2(π‘˜ NUM american_scientific_journal-11066 71 8 + + CCONJ american_scientific_journal-11066 71 9 1)π‘Œ(π‘˜ 1)π‘Œ(π‘˜ NUM american_scientific_journal-11066 71 10 + + CCONJ american_scientific_journal-11066 71 11 1 1 NUM american_scientific_journal-11066 71 12 ) ) PUNCT american_scientific_journal-11066 71 13 + + CCONJ american_scientific_journal-11066 71 14 βˆ‘ βˆ‘ PROPN american_scientific_journal-11066 71 15 βˆ‘ βˆ‘ PROPN american_scientific_journal-11066 71 16 βˆ‘ βˆ‘ PUNCT american_scientific_journal-11066 71 17 𝛿(π‘Ÿ 𝛿(π‘Ÿ PROPN american_scientific_journal-11066 71 18 βˆ’ βˆ’ PROPN american_scientific_journal-11066 71 19 1)π‘Œ(𝑠)π‘Œ(π‘š)π‘Œ(π‘˜ 1)π‘Œ(𝑠)π‘Œ(π‘š)π‘Œ(π‘˜ NOUN american_scientific_journal-11066 71 20 βˆ’ βˆ’ NOUN american_scientific_journal-11066 72 1 π‘Ÿ π‘Ÿ PRON american_scientific_journal-11066 73 1 βˆ’ βˆ’ X american_scientific_journal-11066 73 2 𝑠 𝑠 INTJ american_scientific_journal-11066 73 3 βˆ’ βˆ’ PROPN american_scientific_journal-11066 73 4 π‘š)π‘˜βˆ’π‘Ÿβˆ’π‘  π‘š)π‘˜βˆ’π‘Ÿβˆ’π‘  PROPN american_scientific_journal-11066 73 5 π‘š=0 π‘š=0 PUNCT american_scientific_journal-11066 73 6 π‘˜βˆ’π‘Ÿ π‘˜βˆ’π‘Ÿ NOUN american_scientific_journal-11066 73 7 𝑠=0 𝑠=0 PUNCT american_scientific_journal-11066 73 8 π‘˜ π‘˜ X american_scientific_journal-11066 73 9 π‘Ÿ=0 π‘Ÿ=0 SYM american_scientific_journal-11066 73 10 = = SYM american_scientific_journal-11066 73 11 0 0 NUM american_scientific_journal-11066 73 12 ( ( PUNCT american_scientific_journal-11066 73 13 24 24 NUM american_scientific_journal-11066 73 14 ) ) PUNCT american_scientific_journal-11066 73 15 where where SCONJ american_scientific_journal-11066 73 16 π‘Œ(π‘˜ π‘Œ(π‘˜ X american_scientific_journal-11066 73 17 ) ) PUNCT american_scientific_journal-11066 73 18 is be AUX american_scientific_journal-11066 73 19 the the DET american_scientific_journal-11066 73 20 transformed transform VERB american_scientific_journal-11066 73 21 function function NOUN american_scientific_journal-11066 73 22 of of ADP american_scientific_journal-11066 73 23 𝑦(π‘₯ 𝑦(π‘₯ NOUN american_scientific_journal-11066 73 24 ) ) PUNCT american_scientific_journal-11066 73 25 . . PUNCT american_scientific_journal-11066 74 1 transformed transform VERB american_scientific_journal-11066 74 2 initial initial ADJ american_scientific_journal-11066 74 3 conditions condition NOUN american_scientific_journal-11066 74 4 gives give VERB american_scientific_journal-11066 74 5 π‘Œ(0 π‘Œ(0 PRON american_scientific_journal-11066 74 6 ) ) PUNCT american_scientific_journal-11066 74 7 = = SYM american_scientific_journal-11066 74 8 1 1 NUM american_scientific_journal-11066 74 9 and and CCONJ american_scientific_journal-11066 74 10 π‘Œ(1 π‘Œ(1 NUM american_scientific_journal-11066 74 11 ) ) PUNCT american_scientific_journal-11066 75 1 = = SYM american_scientific_journal-11066 75 2 0 0 X american_scientific_journal-11066 75 3 . . PUNCT american_scientific_journal-11066 76 1 ( ( PUNCT american_scientific_journal-11066 76 2 25 25 NUM american_scientific_journal-11066 76 3 ) ) PUNCT american_scientific_journal-11066 76 4 using use VERB american_scientific_journal-11066 76 5 the the DET american_scientific_journal-11066 76 6 results result NOUN american_scientific_journal-11066 76 7 in in ADP american_scientific_journal-11066 76 8 ( ( PUNCT american_scientific_journal-11066 76 9 25 25 NUM american_scientific_journal-11066 76 10 ) ) PUNCT american_scientific_journal-11066 76 11 with with ADP american_scientific_journal-11066 76 12 the the DET american_scientific_journal-11066 76 13 equation equation NOUN american_scientific_journal-11066 76 14 ( ( PUNCT american_scientific_journal-11066 76 15 24 24 NUM american_scientific_journal-11066 76 16 ) ) PUNCT american_scientific_journal-11066 76 17 and and CCONJ american_scientific_journal-11066 76 18 solving solve VERB american_scientific_journal-11066 76 19 the the DET american_scientific_journal-11066 76 20 iterative iterative NOUN american_scientific_journal-11066 76 21 equations equation NOUN american_scientific_journal-11066 76 22 , , PUNCT american_scientific_journal-11066 76 23 gives give VERB american_scientific_journal-11066 76 24 π‘Œ(2 π‘Œ(2 NOUN american_scientific_journal-11066 76 25 ) ) PUNCT american_scientific_journal-11066 76 26 = = SYM american_scientific_journal-11066 77 1 βˆ’ βˆ’ PROPN american_scientific_journal-11066 77 2 1 1 NUM american_scientific_journal-11066 77 3 6 6 NUM american_scientific_journal-11066 77 4 , , PUNCT american_scientific_journal-11066 77 5 american american ADJ american_scientific_journal-11066 77 6 academic academic ADJ american_scientific_journal-11066 77 7 scientific scientific ADJ american_scientific_journal-11066 77 8 research research NOUN american_scientific_journal-11066 77 9 journal journal NOUN american_scientific_journal-11066 77 10 for for ADP american_scientific_journal-11066 77 11 engineering engineering NOUN american_scientific_journal-11066 77 12 , , PUNCT american_scientific_journal-11066 77 13 technology technology NOUN american_scientific_journal-11066 77 14 , , PUNCT american_scientific_journal-11066 77 15 and and CCONJ american_scientific_journal-11066 77 16 sciences science NOUN american_scientific_journal-11066 77 17 ( ( PUNCT american_scientific_journal-11066 77 18 asrjets asrjet NOUN american_scientific_journal-11066 77 19 ) ) PUNCT american_scientific_journal-11066 77 20 volume volume NOUN american_scientific_journal-11066 77 21 99 99 NUM american_scientific_journal-11066 77 22 , , PUNCT american_scientific_journal-11066 77 23 no no DET american_scientific_journal-11066 77 24 1 1 NUM american_scientific_journal-11066 77 25 , , PUNCT american_scientific_journal-11066 77 26 pp pp ADV american_scientific_journal-11066 77 27 1 1 NUM american_scientific_journal-11066 77 28 - - SYM american_scientific_journal-11066 77 29 12 12 NUM american_scientific_journal-11066 77 30 7 7 NUM american_scientific_journal-11066 77 31 π‘Œ(4 π‘Œ(4 NUM american_scientific_journal-11066 77 32 ) ) PUNCT american_scientific_journal-11066 77 33 = = SYM american_scientific_journal-11066 77 34 1 1 NUM american_scientific_journal-11066 77 35 40 40 NUM american_scientific_journal-11066 77 36 , , PUNCT american_scientific_journal-11066 77 37 π‘Œ(6 π‘Œ(6 ADP american_scientific_journal-11066 77 38 ) ) PUNCT american_scientific_journal-11066 77 39 = = SYM american_scientific_journal-11066 77 40 βˆ’19 βˆ’19 NOUN american_scientific_journal-11066 77 41 5040 5040 NUM american_scientific_journal-11066 77 42 , , PUNCT american_scientific_journal-11066 77 43 π‘Œ(8 π‘Œ(8 NOUN american_scientific_journal-11066 77 44 ) ) PUNCT american_scientific_journal-11066 77 45 = = NOUN american_scientific_journal-11066 77 46 619 619 NUM american_scientific_journal-11066 77 47 1088640 1088640 NUM american_scientific_journal-11066 77 48 . . PUNCT american_scientific_journal-11066 78 1 and and CCONJ american_scientific_journal-11066 78 2 π‘Œ(3 π‘Œ(3 NUM american_scientific_journal-11066 78 3 ) ) PUNCT american_scientific_journal-11066 78 4 = = SYM american_scientific_journal-11066 78 5 π‘Œ(5 π‘Œ(5 X american_scientific_journal-11066 78 6 ) ) PUNCT american_scientific_journal-11066 78 7 = = SYM american_scientific_journal-11066 79 1 π‘Œ(7 π‘Œ(7 X american_scientific_journal-11066 79 2 ) ) PUNCT american_scientific_journal-11066 80 1 = = SYM american_scientific_journal-11066 80 2 0 0 X american_scientific_journal-11066 80 3 . . PUNCT american_scientific_journal-11066 81 1 now now ADV american_scientific_journal-11066 81 2 using use VERB american_scientific_journal-11066 81 3 the the DET american_scientific_journal-11066 81 4 definition definition NOUN american_scientific_journal-11066 81 5 of of ADP american_scientific_journal-11066 81 6 inverse inverse NOUN american_scientific_journal-11066 81 7 differential differential NOUN american_scientific_journal-11066 81 8 transform transform NOUN american_scientific_journal-11066 81 9 ( ( PUNCT american_scientific_journal-11066 81 10 22 22 NUM american_scientific_journal-11066 81 11 ) ) PUNCT american_scientific_journal-11066 81 12 , , PUNCT american_scientific_journal-11066 81 13 the the DET american_scientific_journal-11066 81 14 solution solution NOUN american_scientific_journal-11066 81 15 takes take VERB american_scientific_journal-11066 81 16 the the DET american_scientific_journal-11066 81 17 form form NOUN american_scientific_journal-11066 81 18 , , PUNCT american_scientific_journal-11066 81 19 𝑦(π‘₯ 𝑦(π‘₯ PROPN american_scientific_journal-11066 81 20 ) ) PUNCT american_scientific_journal-11066 81 21 = = SYM american_scientific_journal-11066 81 22 1 1 NUM american_scientific_journal-11066 81 23 βˆ’ βˆ’ NUM american_scientific_journal-11066 81 24 1 1 NUM american_scientific_journal-11066 81 25 6 6 NUM american_scientific_journal-11066 81 26 π‘₯2 π‘₯2 NOUN american_scientific_journal-11066 81 27 + + CCONJ american_scientific_journal-11066 81 28 1 1 NUM american_scientific_journal-11066 81 29 40 40 NUM american_scientific_journal-11066 81 30 π‘₯4 π‘₯4 NOUN american_scientific_journal-11066 81 31 βˆ’ βˆ’ PROPN american_scientific_journal-11066 81 32 19 19 NUM american_scientific_journal-11066 81 33 5040 5040 NUM american_scientific_journal-11066 81 34 π‘₯6 π‘₯6 NOUN american_scientific_journal-11066 81 35 + + CCONJ american_scientific_journal-11066 81 36 619 619 NUM american_scientific_journal-11066 81 37 1088640 1088640 NUM american_scientific_journal-11066 81 38 π‘₯8 π‘₯8 NOUN american_scientific_journal-11066 81 39 βˆ’ βˆ’ PROPN american_scientific_journal-11066 81 40 β‹― β‹― PROPN american_scientific_journal-11066 81 41 . . PUNCT american_scientific_journal-11066 82 1 ( ( PUNCT american_scientific_journal-11066 82 2 26 26 NUM american_scientific_journal-11066 82 3 ) ) PUNCT american_scientific_journal-11066 82 4 here here ADV american_scientific_journal-11066 82 5 we we PRON american_scientific_journal-11066 82 6 can can AUX american_scientific_journal-11066 82 7 see see VERB american_scientific_journal-11066 82 8 from from ADP american_scientific_journal-11066 82 9 equations equation NOUN american_scientific_journal-11066 82 10 ( ( PUNCT american_scientific_journal-11066 82 11 11 11 NUM american_scientific_journal-11066 82 12 ) ) PUNCT american_scientific_journal-11066 82 13 , , PUNCT american_scientific_journal-11066 82 14 ( ( PUNCT american_scientific_journal-11066 82 15 20 20 NUM american_scientific_journal-11066 82 16 ) ) PUNCT american_scientific_journal-11066 82 17 and and CCONJ american_scientific_journal-11066 82 18 ( ( PUNCT american_scientific_journal-11066 82 19 26 26 NUM american_scientific_journal-11066 82 20 ) ) PUNCT american_scientific_journal-11066 82 21 that that SCONJ american_scientific_journal-11066 82 22 the the DET american_scientific_journal-11066 82 23 solution solution NOUN american_scientific_journal-11066 82 24 of of ADP american_scientific_journal-11066 82 25 the the DET american_scientific_journal-11066 82 26 lane lane NOUN american_scientific_journal-11066 82 27 - - PUNCT american_scientific_journal-11066 82 28 emden emden NOUN american_scientific_journal-11066 82 29 equation equation NOUN american_scientific_journal-11066 82 30 when when SCONJ american_scientific_journal-11066 82 31 𝑛 𝑛 PROPN american_scientific_journal-11066 82 32 = = SYM american_scientific_journal-11066 82 33 3 3 NUM american_scientific_journal-11066 82 34 are be AUX american_scientific_journal-11066 82 35 the the DET american_scientific_journal-11066 82 36 same same ADJ american_scientific_journal-11066 82 37 for for ADP american_scientific_journal-11066 82 38 all all DET american_scientific_journal-11066 82 39 the the DET american_scientific_journal-11066 82 40 methods method NOUN american_scientific_journal-11066 82 41 we we PRON american_scientific_journal-11066 82 42 used use VERB american_scientific_journal-11066 82 43 . . PUNCT american_scientific_journal-11066 83 1 since since SCONJ american_scientific_journal-11066 83 2 the the DET american_scientific_journal-11066 83 3 padΓ© padΓ© NOUN american_scientific_journal-11066 83 4 approximant approximant ADJ american_scientific_journal-11066 83 5 [ [ X american_scientific_journal-11066 83 6 11 11 NUM american_scientific_journal-11066 83 7 ] ] PUNCT american_scientific_journal-11066 83 8 often often ADV american_scientific_journal-11066 83 9 gives give VERB american_scientific_journal-11066 83 10 better well ADJ american_scientific_journal-11066 83 11 approximation approximation NOUN american_scientific_journal-11066 83 12 of of ADP american_scientific_journal-11066 83 13 the the DET american_scientific_journal-11066 83 14 function function NOUN american_scientific_journal-11066 83 15 than than ADP american_scientific_journal-11066 83 16 truncating truncate VERB american_scientific_journal-11066 83 17 its its PRON american_scientific_journal-11066 83 18 taylor taylor PROPN american_scientific_journal-11066 83 19 series series NOUN american_scientific_journal-11066 83 20 and and CCONJ american_scientific_journal-11066 83 21 it it PRON american_scientific_journal-11066 83 22 may may AUX american_scientific_journal-11066 83 23 still still ADV american_scientific_journal-11066 83 24 work work VERB american_scientific_journal-11066 83 25 where where SCONJ american_scientific_journal-11066 83 26 the the DET american_scientific_journal-11066 83 27 taylor taylor PROPN american_scientific_journal-11066 83 28 series series NOUN american_scientific_journal-11066 83 29 does do AUX american_scientific_journal-11066 83 30 not not PART american_scientific_journal-11066 83 31 converge converge VERB american_scientific_journal-11066 83 32 , , PUNCT american_scientific_journal-11066 83 33 to to PART american_scientific_journal-11066 83 34 increase increase VERB american_scientific_journal-11066 83 35 the the DET american_scientific_journal-11066 83 36 accuracy accuracy NOUN american_scientific_journal-11066 83 37 of of ADP american_scientific_journal-11066 83 38 the the DET american_scientific_journal-11066 83 39 solution solution NOUN american_scientific_journal-11066 83 40 , , PUNCT american_scientific_journal-11066 83 41 pade pade NOUN american_scientific_journal-11066 83 42 approximant approximant ADJ american_scientific_journal-11066 84 1 [ [ X american_scientific_journal-11066 84 2 4/4 4/4 NUM american_scientific_journal-11066 84 3 ] ] PUNCT american_scientific_journal-11066 84 4 have have AUX american_scientific_journal-11066 84 5 been be AUX american_scientific_journal-11066 84 6 used use VERB american_scientific_journal-11066 84 7 . . PUNCT american_scientific_journal-11066 85 1 after after ADP american_scientific_journal-11066 85 2 calculating calculate VERB american_scientific_journal-11066 85 3 pade pade NOUN american_scientific_journal-11066 85 4 approximant approximant ADJ american_scientific_journal-11066 85 5 [ [ X american_scientific_journal-11066 85 6 4/4 4/4 NUM american_scientific_journal-11066 85 7 ] ] PUNCT american_scientific_journal-11066 85 8 , , PUNCT american_scientific_journal-11066 85 9 the the DET american_scientific_journal-11066 85 10 solution solution NOUN american_scientific_journal-11066 85 11 takes take VERB american_scientific_journal-11066 85 12 the the DET american_scientific_journal-11066 85 13 form form NOUN american_scientific_journal-11066 85 14 𝑦(π‘₯ 𝑦(π‘₯ NOUN american_scientific_journal-11066 85 15 ) ) PUNCT american_scientific_journal-11066 85 16 = = NOUN american_scientific_journal-11066 86 1 1βˆ’0.009259259233694π‘₯2βˆ’0.0002425044095963438π‘₯4 1βˆ’0.009259259233694π‘₯2βˆ’0.0002425044095963438π‘₯4 NUM american_scientific_journal-11066 86 2 1 1 NUM american_scientific_journal-11066 86 3 + + NOUN american_scientific_journal-11066 86 4 0.157407407432973π‘₯2 0.157407407432973π‘₯2 NOUN american_scientific_journal-11066 86 5 + + NOUN american_scientific_journal-11066 86 6 0.000992063495899π‘₯4 0.000992063495899π‘₯4 NUM american_scientific_journal-11066 86 7 ( ( PUNCT american_scientific_journal-11066 86 8 27 27 NUM american_scientific_journal-11066 86 9 ) ) PUNCT american_scientific_journal-11066 86 10 since since SCONJ american_scientific_journal-11066 86 11 there there PRON american_scientific_journal-11066 86 12 are be VERB american_scientific_journal-11066 86 13 no no DET american_scientific_journal-11066 86 14 exact exact ADJ american_scientific_journal-11066 86 15 solution solution NOUN american_scientific_journal-11066 86 16 for for ADP american_scientific_journal-11066 86 17 this this DET american_scientific_journal-11066 86 18 equation equation NOUN american_scientific_journal-11066 86 19 , , PUNCT american_scientific_journal-11066 86 20 in in ADP american_scientific_journal-11066 86 21 the the DET american_scientific_journal-11066 86 22 next next ADJ american_scientific_journal-11066 86 23 section section NOUN american_scientific_journal-11066 86 24 , , PUNCT american_scientific_journal-11066 86 25 runge runge NOUN american_scientific_journal-11066 86 26 - - PUNCT american_scientific_journal-11066 86 27 kutta kutta NOUN american_scientific_journal-11066 86 28 ’s ’s PART american_scientific_journal-11066 86 29 fourth fourth ADJ american_scientific_journal-11066 86 30 order order NOUN american_scientific_journal-11066 86 31 method method NOUN american_scientific_journal-11066 86 32 and and CCONJ american_scientific_journal-11066 86 33 ode ode ADJ american_scientific_journal-11066 86 34 45 45 NUM american_scientific_journal-11066 86 35 will will AUX american_scientific_journal-11066 86 36 be be AUX american_scientific_journal-11066 86 37 used use VERB american_scientific_journal-11066 86 38 to to PART american_scientific_journal-11066 86 39 find find VERB american_scientific_journal-11066 86 40 the the DET american_scientific_journal-11066 86 41 solution solution NOUN american_scientific_journal-11066 86 42 of of ADP american_scientific_journal-11066 86 43 the the DET american_scientific_journal-11066 86 44 equation equation NOUN american_scientific_journal-11066 86 45 ( ( PUNCT american_scientific_journal-11066 86 46 2 2 NUM american_scientific_journal-11066 86 47 ) ) PUNCT american_scientific_journal-11066 86 48 . . PUNCT american_scientific_journal-11066 87 1 2.4 2.4 NUM american_scientific_journal-11066 87 2 the the DET american_scientific_journal-11066 87 3 solution solution NOUN american_scientific_journal-11066 87 4 using use VERB american_scientific_journal-11066 87 5 runge runge NOUN american_scientific_journal-11066 87 6 - - PUNCT american_scientific_journal-11066 87 7 kutta kutta NOUN american_scientific_journal-11066 87 8 ’s ’s PART american_scientific_journal-11066 87 9 fourth fourth ADJ american_scientific_journal-11066 87 10 order order NOUN american_scientific_journal-11066 87 11 method method NOUN american_scientific_journal-11066 87 12 runge runge NOUN american_scientific_journal-11066 87 13 - - PUNCT american_scientific_journal-11066 87 14 kutta kutta NOUN american_scientific_journal-11066 87 15 method method NOUN american_scientific_journal-11066 87 16 was be AUX american_scientific_journal-11066 87 17 formulated formulate VERB american_scientific_journal-11066 87 18 by by ADP american_scientific_journal-11066 87 19 the the DET american_scientific_journal-11066 87 20 german german ADJ american_scientific_journal-11066 87 21 mathematicians mathematician NOUN american_scientific_journal-11066 87 22 , , PUNCT american_scientific_journal-11066 87 23 runge runge NOUN american_scientific_journal-11066 87 24 and and CCONJ american_scientific_journal-11066 87 25 kutta kutta NOUN american_scientific_journal-11066 87 26 . . PUNCT american_scientific_journal-11066 88 1 the the DET american_scientific_journal-11066 88 2 4th 4th ADJ american_scientific_journal-11066 88 3 order order NOUN american_scientific_journal-11066 88 4 rungekutta rungekutta NOUN american_scientific_journal-11066 88 5 method method NOUN american_scientific_journal-11066 88 6 approximates approximate VERB american_scientific_journal-11066 88 7 the the DET american_scientific_journal-11066 88 8 curve curve NOUN american_scientific_journal-11066 88 9 of of ADP american_scientific_journal-11066 88 10 the the DET american_scientific_journal-11066 88 11 function function NOUN american_scientific_journal-11066 88 12 f f PROPN american_scientific_journal-11066 88 13 between between ADP american_scientific_journal-11066 88 14 two two NUM american_scientific_journal-11066 88 15 points point NOUN american_scientific_journal-11066 88 16 by by ADP american_scientific_journal-11066 88 17 a a DET american_scientific_journal-11066 88 18 4th 4th ADJ american_scientific_journal-11066 88 19 degree degree NOUN american_scientific_journal-11066 88 20 polynomial polynomial ADJ american_scientific_journal-11066 88 21 . . PUNCT american_scientific_journal-11066 89 1 it it PRON american_scientific_journal-11066 89 2 requires require VERB american_scientific_journal-11066 89 3 4 4 NUM american_scientific_journal-11066 89 4 slope slope NOUN american_scientific_journal-11066 89 5 estimates estimate NOUN american_scientific_journal-11066 89 6 at at ADP american_scientific_journal-11066 89 7 each each DET american_scientific_journal-11066 89 8 step step NOUN american_scientific_journal-11066 89 9 . . PUNCT american_scientific_journal-11066 90 1 geometrically geometrically ADV american_scientific_journal-11066 90 2 , , PUNCT american_scientific_journal-11066 90 3 we we PRON american_scientific_journal-11066 90 4 can can AUX american_scientific_journal-11066 90 5 visualize visualize VERB american_scientific_journal-11066 90 6 the the DET american_scientific_journal-11066 90 7 curve curve NOUN american_scientific_journal-11066 90 8 between between ADP american_scientific_journal-11066 90 9 two two NUM american_scientific_journal-11066 90 10 points point NOUN american_scientific_journal-11066 90 11 as as ADP american_scientific_journal-11066 90 12 part part NOUN american_scientific_journal-11066 90 13 of of ADP american_scientific_journal-11066 90 14 a a DET american_scientific_journal-11066 90 15 4th 4th ADJ american_scientific_journal-11066 90 16 degree degree NOUN american_scientific_journal-11066 90 17 polynomial polynomial ADJ american_scientific_journal-11066 90 18 . . PUNCT american_scientific_journal-11066 91 1 equations equation NOUN american_scientific_journal-11066 91 2 for for ADP american_scientific_journal-11066 91 3 the the DET american_scientific_journal-11066 91 4 fourth fourth ADJ american_scientific_journal-11066 91 5 order order NOUN american_scientific_journal-11066 91 6 runge runge NOUN american_scientific_journal-11066 91 7 - - PUNCT american_scientific_journal-11066 91 8 kutta kutta NOUN american_scientific_journal-11066 91 9 ’s ’s PART american_scientific_journal-11066 91 10 method method NOUN american_scientific_journal-11066 91 11 for for ADP american_scientific_journal-11066 91 12 a a DET american_scientific_journal-11066 91 13 single single ADJ american_scientific_journal-11066 91 14 ordinary ordinary ADJ american_scientific_journal-11066 91 15 differential differential ADJ american_scientific_journal-11066 91 16 equation equation NOUN american_scientific_journal-11066 91 17 are be AUX american_scientific_journal-11066 91 18 given give VERB american_scientific_journal-11066 91 19 by by ADP american_scientific_journal-11066 91 20 𝑦𝑛+1(π‘₯ 𝑦𝑛+1(π‘₯ PROPN american_scientific_journal-11066 91 21 ) ) PUNCT american_scientific_journal-11066 91 22 = = PUNCT american_scientific_journal-11066 91 23 𝑦𝑛(π‘₯ 𝑦𝑛(π‘₯ NOUN american_scientific_journal-11066 91 24 ) ) PUNCT american_scientific_journal-11066 92 1 + + CCONJ american_scientific_journal-11066 92 2 1 1 NUM american_scientific_journal-11066 92 3 6 6 NUM american_scientific_journal-11066 92 4 ( ( PUNCT american_scientific_journal-11066 92 5 π‘˜1 π‘˜1 PROPN american_scientific_journal-11066 92 6 + + CCONJ american_scientific_journal-11066 92 7 2π‘˜2 2π‘˜2 NUM american_scientific_journal-11066 92 8 + + CCONJ american_scientific_journal-11066 92 9 2π‘˜3 2π‘˜3 NUM american_scientific_journal-11066 92 10 + + NUM american_scientific_journal-11066 92 11 π‘˜4 π‘˜4 PROPN american_scientific_journal-11066 92 12 ) ) PUNCT american_scientific_journal-11066 92 13 , , PUNCT american_scientific_journal-11066 92 14 𝑛 𝑛 NOUN american_scientific_journal-11066 92 15 = = SYM american_scientific_journal-11066 92 16 0,1,2,3 0,1,2,3 NUM american_scientific_journal-11066 92 17 … … PUNCT american_scientific_journal-11066 92 18 ( ( PUNCT american_scientific_journal-11066 92 19 28 28 NUM american_scientific_journal-11066 92 20 ) ) PUNCT american_scientific_journal-11066 92 21 where where SCONJ american_scientific_journal-11066 92 22 π‘˜1 π‘˜1 NOUN american_scientific_journal-11066 92 23 = = PUNCT american_scientific_journal-11066 92 24 β„Ž β„Ž PROPN american_scientific_journal-11066 92 25 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-11066 92 26 , , PUNCT american_scientific_journal-11066 92 27 𝑦 𝑦 NOUN american_scientific_journal-11066 92 28 ) ) PUNCT american_scientific_journal-11066 92 29 π‘˜2 π‘˜2 NOUN american_scientific_journal-11066 92 30 = = PUNCT american_scientific_journal-11066 92 31 β„Žπ‘“ β„Žπ‘“ X american_scientific_journal-11066 92 32 ( ( PUNCT american_scientific_journal-11066 92 33 π‘₯ π‘₯ NOUN american_scientific_journal-11066 92 34 + + CCONJ american_scientific_journal-11066 92 35 β„Ž β„Ž X american_scientific_journal-11066 92 36 2 2 NUM american_scientific_journal-11066 92 37 , , PUNCT american_scientific_journal-11066 92 38 𝑦 𝑦 NOUN american_scientific_journal-11066 92 39 + + CCONJ american_scientific_journal-11066 92 40 π‘˜1 π‘˜1 ADJ american_scientific_journal-11066 92 41 2 2 NUM american_scientific_journal-11066 92 42 ) ) PUNCT american_scientific_journal-11066 92 43 american american ADJ american_scientific_journal-11066 92 44 academic academic ADJ american_scientific_journal-11066 92 45 scientific scientific ADJ american_scientific_journal-11066 92 46 research research NOUN american_scientific_journal-11066 92 47 journal journal NOUN american_scientific_journal-11066 92 48 for for ADP american_scientific_journal-11066 92 49 engineering engineering NOUN american_scientific_journal-11066 92 50 , , PUNCT american_scientific_journal-11066 92 51 technology technology NOUN american_scientific_journal-11066 92 52 , , PUNCT american_scientific_journal-11066 92 53 and and CCONJ american_scientific_journal-11066 92 54 sciences science NOUN american_scientific_journal-11066 92 55 ( ( PUNCT american_scientific_journal-11066 92 56 asrjets asrjet NOUN american_scientific_journal-11066 92 57 ) ) PUNCT american_scientific_journal-11066 92 58 volume volume NOUN american_scientific_journal-11066 92 59 99 99 NUM american_scientific_journal-11066 92 60 , , PUNCT american_scientific_journal-11066 92 61 no no DET american_scientific_journal-11066 92 62 1 1 NUM american_scientific_journal-11066 92 63 , , PUNCT american_scientific_journal-11066 92 64 pp pp ADV american_scientific_journal-11066 92 65 1 1 NUM american_scientific_journal-11066 92 66 - - SYM american_scientific_journal-11066 92 67 12 12 NUM american_scientific_journal-11066 92 68 8 8 NUM american_scientific_journal-11066 92 69 π‘˜3 π‘˜3 PROPN american_scientific_journal-11066 92 70 = = SYM american_scientific_journal-11066 92 71 β„Žπ‘“ β„Žπ‘“ X american_scientific_journal-11066 92 72 ( ( PUNCT american_scientific_journal-11066 92 73 π‘₯ π‘₯ PROPN american_scientific_journal-11066 92 74 + + CCONJ american_scientific_journal-11066 92 75 β„Ž β„Ž X american_scientific_journal-11066 92 76 2 2 NUM american_scientific_journal-11066 92 77 , , PUNCT american_scientific_journal-11066 92 78 𝑦 𝑦 NOUN american_scientific_journal-11066 92 79 + + CCONJ american_scientific_journal-11066 92 80 π‘˜2 π‘˜2 PROPN american_scientific_journal-11066 92 81 2 2 X american_scientific_journal-11066 92 82 ) ) PUNCT american_scientific_journal-11066 92 83 π‘˜4 π‘˜4 PROPN american_scientific_journal-11066 92 84 = = PUNCT american_scientific_journal-11066 92 85 β„Žπ‘“(π‘₯ β„Žπ‘“(π‘₯ PROPN american_scientific_journal-11066 92 86 + + NUM american_scientific_journal-11066 92 87 β„Ž β„Ž NOUN american_scientific_journal-11066 92 88 , , PUNCT american_scientific_journal-11066 92 89 𝑦 𝑦 NOUN american_scientific_journal-11066 92 90 + + X american_scientific_journal-11066 92 91 π‘˜3 π‘˜3 PROPN american_scientific_journal-11066 92 92 ) ) PUNCT american_scientific_journal-11066 92 93 . . PUNCT american_scientific_journal-11066 93 1 the the DET american_scientific_journal-11066 93 2 table table NOUN american_scientific_journal-11066 93 3 of of ADP american_scientific_journal-11066 93 4 obtained obtain VERB american_scientific_journal-11066 93 5 values value NOUN american_scientific_journal-11066 93 6 the the DET american_scientific_journal-11066 93 7 𝑦(π‘₯ 𝑦(π‘₯ NOUN american_scientific_journal-11066 93 8 ) ) PUNCT american_scientific_journal-11066 93 9 of of ADP american_scientific_journal-11066 93 10 equation equation NOUN american_scientific_journal-11066 93 11 ( ( PUNCT american_scientific_journal-11066 93 12 2 2 NUM american_scientific_journal-11066 93 13 ) ) PUNCT american_scientific_journal-11066 93 14 for for ADP american_scientific_journal-11066 93 15 different different ADJ american_scientific_journal-11066 93 16 values value NOUN american_scientific_journal-11066 93 17 of of ADP american_scientific_journal-11066 93 18 π‘₯ π‘₯ NOUN american_scientific_journal-11066 93 19 by by ADP american_scientific_journal-11066 93 20 using use VERB american_scientific_journal-11066 93 21 runge runge NOUN american_scientific_journal-11066 93 22 - - PUNCT american_scientific_journal-11066 93 23 kutta kutta NOUN american_scientific_journal-11066 93 24 ’s ’s PART american_scientific_journal-11066 93 25 fourth fourth ADJ american_scientific_journal-11066 93 26 order order NOUN american_scientific_journal-11066 93 27 method method NOUN american_scientific_journal-11066 93 28 will will AUX american_scientific_journal-11066 93 29 be be AUX american_scientific_journal-11066 93 30 given give VERB american_scientific_journal-11066 93 31 in in ADP american_scientific_journal-11066 93 32 section section NOUN american_scientific_journal-11066 93 33 3 3 NUM american_scientific_journal-11066 93 34 . . NOUN american_scientific_journal-11066 93 35 2.5 2.5 NUM american_scientific_journal-11066 94 1 the the DET american_scientific_journal-11066 94 2 solution solution NOUN american_scientific_journal-11066 94 3 using use VERB american_scientific_journal-11066 94 4 ode ode ADJ american_scientific_journal-11066 94 5 45 45 NUM american_scientific_journal-11066 94 6 method method NOUN american_scientific_journal-11066 94 7 ode ode PROPN american_scientific_journal-11066 94 8 45 45 NUM american_scientific_journal-11066 94 9 is be AUX american_scientific_journal-11066 94 10 a a DET american_scientific_journal-11066 94 11 solver solver NOUN american_scientific_journal-11066 94 12 of of ADP american_scientific_journal-11066 94 13 ordinary ordinary ADJ american_scientific_journal-11066 94 14 differential differential ADJ american_scientific_journal-11066 94 15 equations equation NOUN american_scientific_journal-11066 94 16 which which PRON american_scientific_journal-11066 94 17 is be AUX american_scientific_journal-11066 94 18 using use VERB american_scientific_journal-11066 94 19 in in ADP american_scientific_journal-11066 94 20 matlab matlab PROPN american_scientific_journal-11066 94 21 software software NOUN american_scientific_journal-11066 94 22 . . PUNCT american_scientific_journal-11066 95 1 it it PRON american_scientific_journal-11066 95 2 is be AUX american_scientific_journal-11066 95 3 constructed construct VERB american_scientific_journal-11066 95 4 with with ADP american_scientific_journal-11066 95 5 runge runge NOUN american_scientific_journal-11066 95 6 - - PUNCT american_scientific_journal-11066 95 7 kutta kutta NOUN american_scientific_journal-11066 95 8 ’s ’s PART american_scientific_journal-11066 95 9 method method NOUN american_scientific_journal-11066 95 10 . . PUNCT american_scientific_journal-11066 96 1 ode ode PROPN american_scientific_journal-11066 96 2 45 45 NUM american_scientific_journal-11066 96 3 can can AUX american_scientific_journal-11066 96 4 be be AUX american_scientific_journal-11066 96 5 used use VERB american_scientific_journal-11066 96 6 to to PART american_scientific_journal-11066 96 7 obtain obtain VERB american_scientific_journal-11066 96 8 solutions solution NOUN american_scientific_journal-11066 96 9 with with ADP american_scientific_journal-11066 96 10 high high ADJ american_scientific_journal-11066 96 11 accuracy accuracy NOUN american_scientific_journal-11066 96 12 . . PUNCT american_scientific_journal-11066 97 1 the the DET american_scientific_journal-11066 97 2 table table NOUN american_scientific_journal-11066 97 3 of of ADP american_scientific_journal-11066 97 4 obtained obtain VERB american_scientific_journal-11066 97 5 values value NOUN american_scientific_journal-11066 97 6 for for ADP american_scientific_journal-11066 97 7 the the DET american_scientific_journal-11066 97 8 𝑦(π‘₯ 𝑦(π‘₯ NOUN american_scientific_journal-11066 97 9 ) ) PUNCT american_scientific_journal-11066 97 10 of of ADP american_scientific_journal-11066 97 11 equation equation NOUN american_scientific_journal-11066 97 12 ( ( PUNCT american_scientific_journal-11066 97 13 2 2 NUM american_scientific_journal-11066 97 14 ) ) PUNCT american_scientific_journal-11066 97 15 for for ADP american_scientific_journal-11066 97 16 different different ADJ american_scientific_journal-11066 97 17 values value NOUN american_scientific_journal-11066 97 18 of of ADP american_scientific_journal-11066 97 19 π‘₯ π‘₯ NOUN american_scientific_journal-11066 97 20 by by ADP american_scientific_journal-11066 97 21 ode ode PROPN american_scientific_journal-11066 97 22 45 45 NUM american_scientific_journal-11066 97 23 method method NOUN american_scientific_journal-11066 97 24 will will AUX american_scientific_journal-11066 97 25 be be AUX american_scientific_journal-11066 97 26 given give VERB american_scientific_journal-11066 97 27 in in ADP american_scientific_journal-11066 97 28 results result NOUN american_scientific_journal-11066 97 29 section section NOUN american_scientific_journal-11066 97 30 . . PUNCT american_scientific_journal-11066 98 1 3 3 X american_scientific_journal-11066 98 2 . . NOUN american_scientific_journal-11066 98 3 results result VERB american_scientific_journal-11066 98 4 the the DET american_scientific_journal-11066 98 5 values value NOUN american_scientific_journal-11066 98 6 for for ADP american_scientific_journal-11066 98 7 the the DET american_scientific_journal-11066 98 8 solution solution NOUN american_scientific_journal-11066 98 9 of of ADP american_scientific_journal-11066 98 10 equation equation NOUN american_scientific_journal-11066 98 11 ( ( PUNCT american_scientific_journal-11066 98 12 2 2 X american_scientific_journal-11066 98 13 ) ) PUNCT american_scientific_journal-11066 98 14 using use VERB american_scientific_journal-11066 98 15 different different ADJ american_scientific_journal-11066 98 16 methods method NOUN american_scientific_journal-11066 98 17 have have AUX american_scientific_journal-11066 98 18 been be AUX american_scientific_journal-11066 98 19 tabulated tabulate VERB american_scientific_journal-11066 98 20 as as SCONJ american_scientific_journal-11066 98 21 follows follow VERB american_scientific_journal-11066 98 22 : : PUNCT american_scientific_journal-11066 98 23 american american ADJ american_scientific_journal-11066 98 24 academic academic ADJ american_scientific_journal-11066 98 25 scientific scientific ADJ american_scientific_journal-11066 98 26 research research NOUN american_scientific_journal-11066 98 27 journal journal NOUN american_scientific_journal-11066 98 28 for for ADP american_scientific_journal-11066 98 29 engineering engineering NOUN american_scientific_journal-11066 98 30 , , PUNCT american_scientific_journal-11066 98 31 technology technology NOUN american_scientific_journal-11066 98 32 , , PUNCT american_scientific_journal-11066 98 33 and and CCONJ american_scientific_journal-11066 98 34 sciences science NOUN american_scientific_journal-11066 98 35 ( ( PUNCT american_scientific_journal-11066 98 36 asrjets asrjet NOUN american_scientific_journal-11066 98 37 ) ) PUNCT american_scientific_journal-11066 98 38 volume volume NOUN american_scientific_journal-11066 98 39 99 99 NUM american_scientific_journal-11066 98 40 , , PUNCT american_scientific_journal-11066 98 41 no no DET american_scientific_journal-11066 98 42 1 1 NUM american_scientific_journal-11066 98 43 , , PUNCT american_scientific_journal-11066 98 44 pp pp ADV american_scientific_journal-11066 98 45 1 1 NUM american_scientific_journal-11066 98 46 - - SYM american_scientific_journal-11066 98 47 12 12 NUM american_scientific_journal-11066 98 48 9 9 NUM american_scientific_journal-11066 98 49 table table NOUN american_scientific_journal-11066 98 50 2 2 NUM american_scientific_journal-11066 98 51 : : PUNCT american_scientific_journal-11066 99 1 solutions solution NOUN american_scientific_journal-11066 99 2 obtained obtain VERB american_scientific_journal-11066 99 3 using use VERB american_scientific_journal-11066 99 4 adm adm PROPN american_scientific_journal-11066 99 5 or or CCONJ american_scientific_journal-11066 99 6 dtm dtm PROPN american_scientific_journal-11066 99 7 with with ADP american_scientific_journal-11066 99 8 pade pade NOUN american_scientific_journal-11066 99 9 approximation approximation NOUN american_scientific_journal-11066 99 10 , , PUNCT american_scientific_journal-11066 99 11 runge runge NOUN american_scientific_journal-11066 99 12 - - PUNCT american_scientific_journal-11066 99 13 kutta kutta NOUN american_scientific_journal-11066 99 14 ’s ’s PART american_scientific_journal-11066 99 15 fourth fourth ADJ american_scientific_journal-11066 99 16 order order NOUN american_scientific_journal-11066 99 17 method method NOUN american_scientific_journal-11066 99 18 and and CCONJ american_scientific_journal-11066 99 19 ode ode VERB american_scientific_journal-11066 99 20 45 45 NUM american_scientific_journal-11066 99 21 method method NOUN american_scientific_journal-11066 99 22 π‘₯ π‘₯ PROPN american_scientific_journal-11066 99 23 adm adm PROPN american_scientific_journal-11066 99 24 and and CCONJ american_scientific_journal-11066 99 25 dtm dtm PROPN american_scientific_journal-11066 99 26 with with ADP american_scientific_journal-11066 99 27 pade pade NOUN american_scientific_journal-11066 99 28 runge runge NOUN american_scientific_journal-11066 99 29 - - PUNCT american_scientific_journal-11066 99 30 kutta kutta NOUN american_scientific_journal-11066 99 31 ’s ’s PART american_scientific_journal-11066 99 32 ode ode PROPN american_scientific_journal-11066 99 33 45 45 NUM american_scientific_journal-11066 99 34 approximation approximation NOUN american_scientific_journal-11066 99 35 fourth fourth ADJ american_scientific_journal-11066 99 36 order order NOUN american_scientific_journal-11066 99 37 0.0 0.0 NUM american_scientific_journal-11066 99 38 1.000000000000000 1.000000000000000 NUM american_scientific_journal-11066 99 39 1.000000000000000 1.000000000000000 NUM american_scientific_journal-11066 99 40 1.000000000000000 1.000000000000000 NUM american_scientific_journal-11066 99 41 0.5 0.5 NUM american_scientific_journal-11066 99 42 0.95983906946246 0.95983906946246 NUM american_scientific_journal-11066 99 43 0.882334391276042 0.882334391276042 NUM american_scientific_journal-11066 99 44 0.959829596023597 0.959829596023597 NUM american_scientific_journal-11066 99 45 1.0 1.0 NUM american_scientific_journal-11066 99 46 0.855057569702160 0.855057569702160 NUM american_scientific_journal-11066 99 47 0.818324571511985 0.818324571511985 NUM american_scientific_journal-11066 99 48 0.855038804145293 0.855038804145293 NUM american_scientific_journal-11066 99 49 1.5 1.5 NUM american_scientific_journal-11066 99 50 0.719501847543459 0.719501847543459 NUM american_scientific_journal-11066 99 51 0.708922682687295 0.708922682687295 NUM american_scientific_journal-11066 99 52 0.719520205360974 0.719520205360974 NUM american_scientific_journal-11066 99 53 2.0 2.0 NUM american_scientific_journal-11066 99 54 0.582851018220845 0.582851018220845 NUM american_scientific_journal-11066 99 55 0.588530418166383 0.588530418166383 NUM american_scientific_journal-11066 99 56 0.582887896307385 0.582887896307385 NUM american_scientific_journal-11066 100 1 2.5 2.5 NUM american_scientific_journal-11066 100 2 0.461129448151790 0.461129448151790 NUM american_scientific_journal-11066 100 3 0.475572766200929 0.475572766200929 NUM american_scientific_journal-11066 100 4 0.461230453835357 0.461230453835357 NUM american_scientific_journal-11066 100 5 3.0 3.0 NUM american_scientific_journal-11066 100 6 0.359237187128084 0.359237187128084 NUM american_scientific_journal-11066 100 7 0.377621800786575 0.377621800786575 NUM american_scientific_journal-11066 100 8 0.359326962683592 0.359326962683592 NUM american_scientific_journal-11066 100 9 3.5 3.5 NUM american_scientific_journal-11066 100 10 0.276292569751981 0.276292569751981 NUM american_scientific_journal-11066 100 11 0.295897586463492 0.295897586463492 NUM 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-0.042093399286705 -0.042093399286705 NOUN american_scientific_journal-11066 101 4 -0.055024532465237 -0.055024532465237 PROPN american_scientific_journal-11066 102 1 9.0 9.0 NUM american_scientific_journal-11066 102 2 -0.066196562308791 -0.066196562308791 NOUN american_scientific_journal-11066 102 3 -0.055656119406371 -0.055656119406371 NOUN american_scientific_journal-11066 102 4 -0.068156949384751 -0.068156949384751 NOUN american_scientific_journal-11066 102 5 9.5 9.5 NUM american_scientific_journal-11066 102 6 -0.077764651766161 -0.077764651766161 NOUN american_scientific_journal-11066 102 7 -0.067749292650195 -0.067749292650195 X american_scientific_journal-11066 103 1 -0.079830913010358 -0.079830913010358 PROPN american_scientific_journal-11066 103 2 10.0 10.0 NUM american_scientific_journal-11066 103 3 -0.088178871397516 -0.088178871397516 NOUN american_scientific_journal-11066 103 4 -0.078558271426851 -0.078558271426851 NOUN american_scientific_journal-11066 103 5 -.090215526101679 -.090215526101679 PUNCT american_scientific_journal-11066 103 6 for for ADP american_scientific_journal-11066 103 7 the the DET american_scientific_journal-11066 103 8 sake sake NOUN american_scientific_journal-11066 103 9 of of ADP american_scientific_journal-11066 103 10 comparison comparison NOUN american_scientific_journal-11066 103 11 , , PUNCT american_scientific_journal-11066 103 12 the the DET american_scientific_journal-11066 103 13 all all DET american_scientific_journal-11066 103 14 three three NUM american_scientific_journal-11066 103 15 graphs graph NOUN american_scientific_journal-11066 103 16 have have AUX american_scientific_journal-11066 103 17 been be AUX american_scientific_journal-11066 103 18 plotted plot VERB american_scientific_journal-11066 103 19 in in ADP american_scientific_journal-11066 103 20 the the DET american_scientific_journal-11066 103 21 same same ADJ american_scientific_journal-11066 103 22 plot plot NOUN american_scientific_journal-11066 103 23 using use VERB american_scientific_journal-11066 103 24 matlab matlab PROPN american_scientific_journal-11066 103 25 software software NOUN american_scientific_journal-11066 103 26 . . PUNCT american_scientific_journal-11066 104 1 american american ADJ american_scientific_journal-11066 104 2 academic academic ADJ american_scientific_journal-11066 104 3 scientific scientific ADJ american_scientific_journal-11066 104 4 research research NOUN american_scientific_journal-11066 104 5 journal journal NOUN american_scientific_journal-11066 104 6 for for ADP american_scientific_journal-11066 104 7 engineering engineering NOUN american_scientific_journal-11066 104 8 , , PUNCT american_scientific_journal-11066 104 9 technology technology NOUN american_scientific_journal-11066 104 10 , , PUNCT american_scientific_journal-11066 104 11 and and CCONJ american_scientific_journal-11066 104 12 sciences science NOUN american_scientific_journal-11066 104 13 ( ( PUNCT american_scientific_journal-11066 104 14 asrjets asrjet NOUN american_scientific_journal-11066 104 15 ) ) PUNCT american_scientific_journal-11066 104 16 volume volume NOUN american_scientific_journal-11066 104 17 99 99 NUM american_scientific_journal-11066 104 18 , , PUNCT american_scientific_journal-11066 104 19 no no DET american_scientific_journal-11066 104 20 1 1 NUM american_scientific_journal-11066 104 21 , , PUNCT american_scientific_journal-11066 104 22 pp pp ADV american_scientific_journal-11066 104 23 1 1 NUM american_scientific_journal-11066 104 24 - - SYM american_scientific_journal-11066 104 25 12 12 NUM american_scientific_journal-11066 104 26 10 10 NUM american_scientific_journal-11066 104 27 figure figure NOUN american_scientific_journal-11066 104 28 1 1 NUM american_scientific_journal-11066 104 29 : : PUNCT american_scientific_journal-11066 104 30 the the DET american_scientific_journal-11066 104 31 solutions solution NOUN american_scientific_journal-11066 104 32 of of ADP american_scientific_journal-11066 104 33 lane lane NOUN american_scientific_journal-11066 104 34 - - PUNCT american_scientific_journal-11066 104 35 emden emden NOUN american_scientific_journal-11066 104 36 equation equation NOUN american_scientific_journal-11066 104 37 when when SCONJ american_scientific_journal-11066 104 38 𝑛 𝑛 PROPN american_scientific_journal-11066 104 39 = = SYM american_scientific_journal-11066 104 40 3 3 NUM american_scientific_journal-11066 104 41 using use VERB american_scientific_journal-11066 104 42 different different ADJ american_scientific_journal-11066 104 43 methods method NOUN american_scientific_journal-11066 104 44 4 4 NUM american_scientific_journal-11066 104 45 . . PUNCT american_scientific_journal-11066 104 46 discussion discussion NOUN american_scientific_journal-11066 104 47 in in ADP american_scientific_journal-11066 104 48 section section NOUN american_scientific_journal-11066 104 49 3.1 3.1 NUM american_scientific_journal-11066 104 50 , , PUNCT american_scientific_journal-11066 104 51 using use VERB american_scientific_journal-11066 104 52 standard standard ADJ american_scientific_journal-11066 104 53 adomian adomian NOUN american_scientific_journal-11066 104 54 decomposition decomposition NOUN american_scientific_journal-11066 104 55 method method NOUN american_scientific_journal-11066 104 56 , , PUNCT american_scientific_journal-11066 104 57 a a DET american_scientific_journal-11066 104 58 solution solution NOUN american_scientific_journal-11066 104 59 has have AUX american_scientific_journal-11066 104 60 been be AUX american_scientific_journal-11066 104 61 obtained obtain VERB american_scientific_journal-11066 104 62 . . PUNCT american_scientific_journal-11066 105 1 in in ADP american_scientific_journal-11066 105 2 section section NOUN american_scientific_journal-11066 105 3 3.2 3.2 NUM american_scientific_journal-11066 105 4 , , PUNCT american_scientific_journal-11066 105 5 changing change VERB american_scientific_journal-11066 105 6 the the DET american_scientific_journal-11066 105 7 differential differential ADJ american_scientific_journal-11066 105 8 operator operator NOUN american_scientific_journal-11066 105 9 , , PUNCT american_scientific_journal-11066 105 10 a a DET american_scientific_journal-11066 105 11 solution solution NOUN american_scientific_journal-11066 105 12 has have AUX american_scientific_journal-11066 105 13 been be AUX american_scientific_journal-11066 105 14 obtained obtain VERB american_scientific_journal-11066 105 15 . . PUNCT american_scientific_journal-11066 106 1 the the DET american_scientific_journal-11066 106 2 solutions solution NOUN american_scientific_journal-11066 106 3 obtained obtain VERB american_scientific_journal-11066 106 4 using use VERB american_scientific_journal-11066 106 5 standard standard ADJ american_scientific_journal-11066 106 6 adomian adomian NOUN american_scientific_journal-11066 106 7 method method NOUN american_scientific_journal-11066 106 8 as as ADV american_scientific_journal-11066 106 9 well well ADV american_scientific_journal-11066 106 10 as as ADP american_scientific_journal-11066 106 11 a a DET american_scientific_journal-11066 106 12 new new ADJ american_scientific_journal-11066 106 13 method method NOUN american_scientific_journal-11066 106 14 which which PRON american_scientific_journal-11066 106 15 has have VERB american_scientific_journal-11066 106 16 a a DET american_scientific_journal-11066 106 17 new new ADJ american_scientific_journal-11066 106 18 form form NOUN american_scientific_journal-11066 106 19 of of ADP american_scientific_journal-11066 106 20 differential differential ADJ american_scientific_journal-11066 106 21 operator operator NOUN american_scientific_journal-11066 106 22 takes take VERB american_scientific_journal-11066 106 23 the the DET american_scientific_journal-11066 106 24 same same ADJ american_scientific_journal-11066 106 25 form form NOUN american_scientific_journal-11066 106 26 for for ADP american_scientific_journal-11066 106 27 the the DET american_scientific_journal-11066 106 28 range range NOUN american_scientific_journal-11066 106 29 of of ADP american_scientific_journal-11066 106 30 values value NOUN american_scientific_journal-11066 106 31 we we PRON american_scientific_journal-11066 106 32 considered consider VERB american_scientific_journal-11066 106 33 . . PUNCT american_scientific_journal-11066 107 1 the the DET american_scientific_journal-11066 107 2 differential differential ADJ american_scientific_journal-11066 107 3 transform transform NOUN american_scientific_journal-11066 107 4 method method NOUN american_scientific_journal-11066 107 5 has have AUX american_scientific_journal-11066 107 6 been be AUX american_scientific_journal-11066 107 7 used use VERB american_scientific_journal-11066 107 8 to to PART american_scientific_journal-11066 107 9 solve solve VERB american_scientific_journal-11066 107 10 lane lane NOUN american_scientific_journal-11066 107 11 - - PUNCT american_scientific_journal-11066 107 12 emden emden NOUN american_scientific_journal-11066 107 13 equation equation NOUN american_scientific_journal-11066 107 14 when when SCONJ american_scientific_journal-11066 107 15 𝑛 𝑛 PROPN american_scientific_journal-11066 107 16 = = SYM american_scientific_journal-11066 107 17 3 3 NUM american_scientific_journal-11066 107 18 in in ADP american_scientific_journal-11066 107 19 section section NOUN american_scientific_journal-11066 107 20 3.3 3.3 NUM american_scientific_journal-11066 107 21 . . PUNCT american_scientific_journal-11066 108 1 the the DET american_scientific_journal-11066 108 2 terms term NOUN american_scientific_journal-11066 108 3 of of ADP american_scientific_journal-11066 108 4 the the DET american_scientific_journal-11066 108 5 infinite infinite ADJ american_scientific_journal-11066 108 6 series series NOUN american_scientific_journal-11066 108 7 has have AUX american_scientific_journal-11066 108 8 been be AUX american_scientific_journal-11066 108 9 obtained obtain VERB american_scientific_journal-11066 108 10 by by ADP american_scientific_journal-11066 108 11 an an DET american_scientific_journal-11066 108 12 iterative iterative NOUN american_scientific_journal-11066 108 13 relation relation NOUN american_scientific_journal-11066 108 14 using use VERB american_scientific_journal-11066 108 15 matlab matlab PROPN american_scientific_journal-11066 108 16 software software NOUN american_scientific_journal-11066 108 17 . . PUNCT american_scientific_journal-11066 109 1 to to PART american_scientific_journal-11066 109 2 increase increase VERB american_scientific_journal-11066 109 3 the the DET american_scientific_journal-11066 109 4 accuracy accuracy NOUN american_scientific_journal-11066 109 5 of of ADP american_scientific_journal-11066 109 6 the the DET american_scientific_journal-11066 109 7 solution solution NOUN american_scientific_journal-11066 109 8 , , PUNCT american_scientific_journal-11066 109 9 pade pade VERB american_scientific_journal-11066 109 10 approximation[4/4 approximation[4/4 PROPN american_scientific_journal-11066 109 11 ] ] PUNCT american_scientific_journal-11066 109 12 of of ADP american_scientific_journal-11066 109 13 the the DET american_scientific_journal-11066 109 14 solution solution NOUN american_scientific_journal-11066 109 15 has have AUX american_scientific_journal-11066 109 16 been be AUX american_scientific_journal-11066 109 17 considered consider VERB american_scientific_journal-11066 109 18 . . PUNCT american_scientific_journal-11066 110 1 the the DET american_scientific_journal-11066 110 2 solution solution NOUN american_scientific_journal-11066 110 3 obtained obtain VERB american_scientific_journal-11066 110 4 here here ADV american_scientific_journal-11066 110 5 , , PUNCT american_scientific_journal-11066 110 6 also also ADV american_scientific_journal-11066 110 7 takes take VERB american_scientific_journal-11066 110 8 the the DET american_scientific_journal-11066 110 9 same same ADJ american_scientific_journal-11066 110 10 form form NOUN american_scientific_journal-11066 110 11 of of ADP american_scientific_journal-11066 110 12 the the DET american_scientific_journal-11066 110 13 solution solution NOUN american_scientific_journal-11066 110 14 we we PRON american_scientific_journal-11066 110 15 have have VERB american_scientific_journal-11066 110 16 in in ADP american_scientific_journal-11066 110 17 the the DET american_scientific_journal-11066 110 18 two two NUM american_scientific_journal-11066 110 19 pervious pervious ADJ american_scientific_journal-11066 110 20 sections section NOUN american_scientific_journal-11066 110 21 . . PUNCT american_scientific_journal-11066 111 1 so so ADV american_scientific_journal-11066 111 2 for for ADP american_scientific_journal-11066 111 3 the the DET american_scientific_journal-11066 111 4 considered consider VERB american_scientific_journal-11066 111 5 range range NOUN american_scientific_journal-11066 111 6 of of ADP american_scientific_journal-11066 111 7 values value NOUN american_scientific_journal-11066 111 8 , , PUNCT american_scientific_journal-11066 111 9 the the DET american_scientific_journal-11066 111 10 solutions solution NOUN american_scientific_journal-11066 111 11 obtained obtain VERB american_scientific_journal-11066 111 12 for for ADP american_scientific_journal-11066 111 13 the the DET american_scientific_journal-11066 111 14 lane lane NOUN american_scientific_journal-11066 111 15 - - PUNCT american_scientific_journal-11066 111 16 emden emden NOUN american_scientific_journal-11066 111 17 equation equation NOUN american_scientific_journal-11066 111 18 when when SCONJ american_scientific_journal-11066 111 19 𝑛 𝑛 PROPN american_scientific_journal-11066 111 20 = = SYM american_scientific_journal-11066 111 21 3 3 NUM american_scientific_journal-11066 111 22 using use VERB american_scientific_journal-11066 111 23 the the DET american_scientific_journal-11066 111 24 above above ADJ american_scientific_journal-11066 111 25 three three NUM american_scientific_journal-11066 111 26 different different ADJ american_scientific_journal-11066 111 27 methods method NOUN american_scientific_journal-11066 111 28 take take VERB american_scientific_journal-11066 111 29 the the DET american_scientific_journal-11066 111 30 same same ADJ american_scientific_journal-11066 111 31 form form NOUN american_scientific_journal-11066 111 32 . . PUNCT american_scientific_journal-11066 112 1 since since SCONJ american_scientific_journal-11066 112 2 there there PRON american_scientific_journal-11066 112 3 is be VERB american_scientific_journal-11066 112 4 no no DET american_scientific_journal-11066 112 5 exact exact ADJ american_scientific_journal-11066 112 6 solution solution NOUN american_scientific_journal-11066 112 7 for for ADP american_scientific_journal-11066 112 8 lane lane NOUN american_scientific_journal-11066 112 9 - - PUNCT american_scientific_journal-11066 112 10 emden emden NOUN american_scientific_journal-11066 112 11 equation equation NOUN american_scientific_journal-11066 112 12 when when SCONJ american_scientific_journal-11066 112 13 𝑛 𝑛 PROPN american_scientific_journal-11066 112 14 = = SYM american_scientific_journal-11066 112 15 3 3 NUM american_scientific_journal-11066 112 16 , , PUNCT american_scientific_journal-11066 112 17 to to PART american_scientific_journal-11066 112 18 check check VERB american_scientific_journal-11066 112 19 the the DET american_scientific_journal-11066 112 20 accuracy accuracy NOUN american_scientific_journal-11066 112 21 of of ADP american_scientific_journal-11066 112 22 the the DET american_scientific_journal-11066 112 23 obtained obtain VERB american_scientific_journal-11066 112 24 solution solution NOUN american_scientific_journal-11066 112 25 , , PUNCT american_scientific_journal-11066 112 26 we we PRON american_scientific_journal-11066 112 27 solve solve VERB american_scientific_journal-11066 112 28 the the DET american_scientific_journal-11066 112 29 equation equation NOUN american_scientific_journal-11066 112 30 using use VERB american_scientific_journal-11066 112 31 well well ADV american_scientific_journal-11066 112 32 known know VERB american_scientific_journal-11066 112 33 numerical numerical ADJ american_scientific_journal-11066 112 34 methods method NOUN american_scientific_journal-11066 112 35 runge runge NOUN american_scientific_journal-11066 112 36 - - PUNCT american_scientific_journal-11066 112 37 kutta kutta NOUN american_scientific_journal-11066 112 38 ’s ’s PART american_scientific_journal-11066 112 39 fourth fourth ADJ american_scientific_journal-11066 112 40 order order NOUN american_scientific_journal-11066 112 41 method method NOUN american_scientific_journal-11066 112 42 and and CCONJ american_scientific_journal-11066 112 43 ode ode VERB american_scientific_journal-11066 112 44 45 45 NUM american_scientific_journal-11066 112 45 method method NOUN american_scientific_journal-11066 112 46 with with ADP american_scientific_journal-11066 112 47 the the DET american_scientific_journal-11066 112 48 help help NOUN american_scientific_journal-11066 112 49 of of ADP american_scientific_journal-11066 112 50 matlab matlab PROPN american_scientific_journal-11066 112 51 software software NOUN american_scientific_journal-11066 112 52 . . PUNCT american_scientific_journal-11066 113 1 to to ADP american_scientific_journal-11066 113 2 the the DET american_scientific_journal-11066 113 3 clarification clarification NOUN american_scientific_journal-11066 113 4 of of ADP american_scientific_journal-11066 113 5 the the DET american_scientific_journal-11066 113 6 results result NOUN american_scientific_journal-11066 113 7 , , PUNCT american_scientific_journal-11066 113 8 the the DET american_scientific_journal-11066 113 9 solutions solution NOUN american_scientific_journal-11066 113 10 obtained obtain VERB american_scientific_journal-11066 113 11 using use VERB american_scientific_journal-11066 113 12 all all DET american_scientific_journal-11066 113 13 methods method NOUN american_scientific_journal-11066 113 14 have have AUX american_scientific_journal-11066 113 15 been be AUX american_scientific_journal-11066 113 16 graphed graph VERB american_scientific_journal-11066 113 17 in in ADP american_scientific_journal-11066 113 18 the the DET american_scientific_journal-11066 113 19 same same ADJ american_scientific_journal-11066 113 20 plot plot NOUN american_scientific_journal-11066 113 21 in in ADP american_scientific_journal-11066 113 22 figure figure NOUN american_scientific_journal-11066 113 23 1 1 NUM american_scientific_journal-11066 113 24 . . PUNCT american_scientific_journal-11066 114 1 according accord VERB american_scientific_journal-11066 114 2 to to ADP american_scientific_journal-11066 114 3 the the DET american_scientific_journal-11066 114 4 plot plot NOUN american_scientific_journal-11066 114 5 , , PUNCT american_scientific_journal-11066 114 6 all all DET american_scientific_journal-11066 114 7 the the DET american_scientific_journal-11066 114 8 graphs graph NOUN american_scientific_journal-11066 114 9 are be AUX american_scientific_journal-11066 114 10 going go VERB american_scientific_journal-11066 114 11 together together ADV american_scientific_journal-11066 114 12 except except SCONJ american_scientific_journal-11066 114 13 some some DET american_scientific_journal-11066 114 14 points point NOUN american_scientific_journal-11066 114 15 which which PRON american_scientific_journal-11066 114 16 means mean VERB american_scientific_journal-11066 114 17 that that SCONJ american_scientific_journal-11066 114 18 the the DET american_scientific_journal-11066 114 19 solutions solution NOUN american_scientific_journal-11066 114 20 obtained obtain VERB american_scientific_journal-11066 114 21 using use VERB american_scientific_journal-11066 114 22 few few ADJ american_scientific_journal-11066 114 23 different different ADJ american_scientific_journal-11066 114 24 semi semi ADJ american_scientific_journal-11066 114 25 - - ADJ american_scientific_journal-11066 114 26 analytical analytical ADJ american_scientific_journal-11066 114 27 and and CCONJ american_scientific_journal-11066 114 28 numerical numerical ADJ american_scientific_journal-11066 114 29 are be AUX american_scientific_journal-11066 114 30 agreed agree VERB american_scientific_journal-11066 114 31 for for ADP american_scientific_journal-11066 114 32 the the DET american_scientific_journal-11066 114 33 range range NOUN american_scientific_journal-11066 114 34 of of ADP american_scientific_journal-11066 114 35 values value NOUN american_scientific_journal-11066 114 36 we we PRON american_scientific_journal-11066 114 37 considered consider VERB american_scientific_journal-11066 114 38 . . PUNCT american_scientific_journal-11066 115 1 in in ADP american_scientific_journal-11066 115 2 this this DET american_scientific_journal-11066 115 3 study study NOUN american_scientific_journal-11066 115 4 , , PUNCT american_scientific_journal-11066 115 5 only only ADV american_scientific_journal-11066 115 6 a a DET american_scientific_journal-11066 115 7 special special ADJ american_scientific_journal-11066 115 8 type type NOUN american_scientific_journal-11066 115 9 of of ADP american_scientific_journal-11066 115 10 lane lane NOUN american_scientific_journal-11066 115 11 - - PUNCT american_scientific_journal-11066 115 12 emden emden NOUN american_scientific_journal-11066 115 13 equation equation NOUN american_scientific_journal-11066 115 14 has have AUX american_scientific_journal-11066 115 15 been be AUX american_scientific_journal-11066 115 16 taken take VERB american_scientific_journal-11066 115 17 into into ADP american_scientific_journal-11066 115 18 account account NOUN american_scientific_journal-11066 115 19 and and CCONJ american_scientific_journal-11066 115 20 solved solve VERB american_scientific_journal-11066 115 21 by by ADP american_scientific_journal-11066 115 22 different different ADJ american_scientific_journal-11066 115 23 methods method NOUN american_scientific_journal-11066 115 24 . . PUNCT american_scientific_journal-11066 116 1 the the DET american_scientific_journal-11066 116 2 constraints constraint NOUN american_scientific_journal-11066 116 3 we we PRON american_scientific_journal-11066 116 4 met meet VERB american_scientific_journal-11066 116 5 in in ADP american_scientific_journal-11066 116 6 this this DET american_scientific_journal-11066 116 7 study study NOUN american_scientific_journal-11066 116 8 are be AUX american_scientific_journal-11066 116 9 as as SCONJ american_scientific_journal-11066 116 10 follows follow NOUN american_scientific_journal-11066 116 11 . . PUNCT american_scientific_journal-11066 117 1 american american ADJ american_scientific_journal-11066 117 2 academic academic ADJ american_scientific_journal-11066 117 3 scientific scientific ADJ american_scientific_journal-11066 117 4 research research NOUN american_scientific_journal-11066 117 5 journal journal NOUN american_scientific_journal-11066 117 6 for for ADP american_scientific_journal-11066 117 7 engineering engineering NOUN american_scientific_journal-11066 117 8 , , PUNCT american_scientific_journal-11066 117 9 technology technology NOUN american_scientific_journal-11066 117 10 , , PUNCT american_scientific_journal-11066 117 11 and and CCONJ american_scientific_journal-11066 117 12 sciences science NOUN american_scientific_journal-11066 117 13 ( ( PUNCT american_scientific_journal-11066 117 14 asrjets asrjet NOUN american_scientific_journal-11066 117 15 ) ) PUNCT american_scientific_journal-11066 117 16 volume volume NOUN american_scientific_journal-11066 117 17 99 99 NUM american_scientific_journal-11066 117 18 , , PUNCT american_scientific_journal-11066 117 19 no no DET american_scientific_journal-11066 117 20 1 1 NUM american_scientific_journal-11066 117 21 , , PUNCT american_scientific_journal-11066 117 22 pp pp ADV american_scientific_journal-11066 117 23 1 1 NUM american_scientific_journal-11066 117 24 - - SYM american_scientific_journal-11066 117 25 12 12 NUM american_scientific_journal-11066 117 26 11 11 NUM american_scientific_journal-11066 117 27 the the DET american_scientific_journal-11066 117 28 solutions solution NOUN american_scientific_journal-11066 117 29 for for ADP american_scientific_journal-11066 117 30 the the DET american_scientific_journal-11066 117 31 lane lane ADJ american_scientific_journal-11066 117 32 -emden -emden ADJ american_scientific_journal-11066 117 33 equation equation NOUN american_scientific_journal-11066 117 34 for for ADP american_scientific_journal-11066 117 35 𝑛 𝑛 NOUN american_scientific_journal-11066 117 36 = = SYM american_scientific_journal-11066 117 37 3 3 NUM american_scientific_journal-11066 117 38 has have AUX american_scientific_journal-11066 117 39 been be AUX american_scientific_journal-11066 117 40 calculated calculate VERB american_scientific_journal-11066 117 41 for for ADP american_scientific_journal-11066 117 42 a a DET american_scientific_journal-11066 117 43 small small ADJ american_scientific_journal-11066 117 44 range range NOUN american_scientific_journal-11066 117 45 of of ADP american_scientific_journal-11066 117 46 values value NOUN american_scientific_journal-11066 117 47 for for ADP american_scientific_journal-11066 117 48 . . PUNCT american_scientific_journal-11066 118 1 when when SCONJ american_scientific_journal-11066 118 2 the the DET american_scientific_journal-11066 118 3 range range NOUN american_scientific_journal-11066 118 4 of of ADP american_scientific_journal-11066 118 5 values value NOUN american_scientific_journal-11066 118 6 is be AUX american_scientific_journal-11066 118 7 larger large ADJ american_scientific_journal-11066 118 8 , , PUNCT american_scientific_journal-11066 118 9 the the DET american_scientific_journal-11066 118 10 solutions solution NOUN american_scientific_journal-11066 118 11 obtained obtain VERB american_scientific_journal-11066 118 12 by by ADP american_scientific_journal-11066 118 13 methods method NOUN american_scientific_journal-11066 118 14 we we PRON american_scientific_journal-11066 118 15 used use VERB american_scientific_journal-11066 118 16 will will AUX american_scientific_journal-11066 118 17 not not PART american_scientific_journal-11066 118 18 be be AUX american_scientific_journal-11066 118 19 gone go VERB american_scientific_journal-11066 118 20 with with ADP american_scientific_journal-11066 118 21 the the DET american_scientific_journal-11066 118 22 solutions solution NOUN american_scientific_journal-11066 118 23 obtained obtain VERB american_scientific_journal-11066 118 24 by by ADP american_scientific_journal-11066 118 25 the the DET american_scientific_journal-11066 118 26 well well ADV american_scientific_journal-11066 118 27 - - PUNCT american_scientific_journal-11066 118 28 known know VERB american_scientific_journal-11066 118 29 method method NOUN american_scientific_journal-11066 118 30 such such ADJ american_scientific_journal-11066 118 31 as as ADP american_scientific_journal-11066 118 32 runge runge NOUN american_scientific_journal-11066 118 33 - - PUNCT american_scientific_journal-11066 118 34 kutta kutta NOUN american_scientific_journal-11066 118 35 ’s ’s PART american_scientific_journal-11066 118 36 fourth fourth ADJ american_scientific_journal-11066 118 37 order order NOUN american_scientific_journal-11066 118 38 method method NOUN american_scientific_journal-11066 118 39 . . PUNCT american_scientific_journal-11066 119 1 also also ADV american_scientific_journal-11066 119 2 when when SCONJ american_scientific_journal-11066 119 3 the the DET american_scientific_journal-11066 119 4 number number NOUN american_scientific_journal-11066 119 5 of of ADP american_scientific_journal-11066 119 6 terms term NOUN american_scientific_journal-11066 119 7 of of ADP american_scientific_journal-11066 119 8 the the DET american_scientific_journal-11066 119 9 infinite infinite ADJ american_scientific_journal-11066 119 10 series series NOUN american_scientific_journal-11066 119 11 is be AUX american_scientific_journal-11066 119 12 increasing increase VERB american_scientific_journal-11066 119 13 , , PUNCT american_scientific_journal-11066 119 14 calculations calculation NOUN american_scientific_journal-11066 119 15 will will AUX american_scientific_journal-11066 119 16 be be AUX american_scientific_journal-11066 119 17 more more ADV american_scientific_journal-11066 119 18 harder hard ADJ american_scientific_journal-11066 119 19 than than ADP american_scientific_journal-11066 119 20 considering consider VERB american_scientific_journal-11066 119 21 only only ADV american_scientific_journal-11066 119 22 first first ADJ american_scientific_journal-11066 119 23 few few ADJ american_scientific_journal-11066 119 24 terms term NOUN american_scientific_journal-11066 119 25 . . PUNCT american_scientific_journal-11066 120 1 but but CCONJ american_scientific_journal-11066 120 2 for for ADP american_scientific_journal-11066 120 3 the the DET american_scientific_journal-11066 120 4 sake sake NOUN american_scientific_journal-11066 120 5 of of ADP american_scientific_journal-11066 120 6 the the DET american_scientific_journal-11066 120 7 accuracy accuracy NOUN american_scientific_journal-11066 120 8 , , PUNCT american_scientific_journal-11066 120 9 we we PRON american_scientific_journal-11066 120 10 have have VERB american_scientific_journal-11066 120 11 to to PART american_scientific_journal-11066 120 12 increase increase VERB american_scientific_journal-11066 120 13 the the DET american_scientific_journal-11066 120 14 number number NOUN american_scientific_journal-11066 120 15 of of ADP american_scientific_journal-11066 120 16 terms term NOUN american_scientific_journal-11066 120 17 of of ADP american_scientific_journal-11066 120 18 the the DET american_scientific_journal-11066 120 19 infinite infinite ADJ american_scientific_journal-11066 120 20 series series NOUN american_scientific_journal-11066 120 21 . . PUNCT american_scientific_journal-11066 121 1 5 5 X american_scientific_journal-11066 121 2 . . X american_scientific_journal-11066 121 3 conclusion conclusion NOUN american_scientific_journal-11066 121 4 since since SCONJ american_scientific_journal-11066 121 5 runge runge NOUN american_scientific_journal-11066 121 6 - - PUNCT american_scientific_journal-11066 121 7 kutta kutta NOUN american_scientific_journal-11066 121 8 ’s ’s PART american_scientific_journal-11066 121 9 method method NOUN american_scientific_journal-11066 121 10 is be AUX american_scientific_journal-11066 121 11 a a DET american_scientific_journal-11066 121 12 numerical numerical ADJ american_scientific_journal-11066 121 13 method method NOUN american_scientific_journal-11066 121 14 for for ADP american_scientific_journal-11066 121 15 solving solve VERB american_scientific_journal-11066 121 16 ordinary ordinary ADJ american_scientific_journal-11066 121 17 differential differential ADJ american_scientific_journal-11066 121 18 equations equation NOUN american_scientific_journal-11066 121 19 with with ADP american_scientific_journal-11066 121 20 high high ADJ american_scientific_journal-11066 121 21 accuracy accuracy NOUN american_scientific_journal-11066 121 22 , , PUNCT american_scientific_journal-11066 121 23 the the DET american_scientific_journal-11066 121 24 solution solution NOUN american_scientific_journal-11066 121 25 obtained obtain VERB american_scientific_journal-11066 121 26 by by ADP american_scientific_journal-11066 121 27 using use VERB american_scientific_journal-11066 121 28 adomian adomian NOUN american_scientific_journal-11066 121 29 decomposition decomposition NOUN american_scientific_journal-11066 121 30 method method NOUN american_scientific_journal-11066 121 31 , , PUNCT american_scientific_journal-11066 121 32 a a DET american_scientific_journal-11066 121 33 modified modify VERB american_scientific_journal-11066 121 34 - - PUNCT american_scientific_journal-11066 121 35 adomian adomian NOUN american_scientific_journal-11066 121 36 decomposition decomposition NOUN american_scientific_journal-11066 121 37 method method NOUN american_scientific_journal-11066 121 38 and and CCONJ american_scientific_journal-11066 121 39 differential differential ADJ american_scientific_journal-11066 121 40 transform transform NOUN american_scientific_journal-11066 121 41 method method NOUN american_scientific_journal-11066 121 42 is be AUX american_scientific_journal-11066 121 43 a a DET american_scientific_journal-11066 121 44 good good ADJ american_scientific_journal-11066 121 45 approximation approximation NOUN american_scientific_journal-11066 121 46 for for ADP american_scientific_journal-11066 121 47 the the DET american_scientific_journal-11066 121 48 lane lane NOUN american_scientific_journal-11066 121 49 - - PUNCT american_scientific_journal-11066 121 50 emden emden NOUN american_scientific_journal-11066 121 51 equation equation NOUN american_scientific_journal-11066 121 52 with with ADP american_scientific_journal-11066 121 53 𝑛 𝑛 PROPN american_scientific_journal-11066 121 54 = = SYM american_scientific_journal-11066 121 55 3 3 X american_scientific_journal-11066 121 56 . . PUNCT american_scientific_journal-11066 122 1 therefore therefore ADV american_scientific_journal-11066 122 2 , , PUNCT american_scientific_journal-11066 122 3 with with ADP american_scientific_journal-11066 122 4 the the DET american_scientific_journal-11066 122 5 proper proper ADJ american_scientific_journal-11066 122 6 investment investment NOUN american_scientific_journal-11066 122 7 of of ADP american_scientific_journal-11066 122 8 adomian adomian ADJ american_scientific_journal-11066 122 9 decomposition decomposition NOUN american_scientific_journal-11066 122 10 method method NOUN american_scientific_journal-11066 122 11 , , PUNCT american_scientific_journal-11066 122 12 it it PRON american_scientific_journal-11066 122 13 is be AUX american_scientific_journal-11066 122 14 possible possible ADJ american_scientific_journal-11066 122 15 to to PART american_scientific_journal-11066 122 16 attain attain VERB american_scientific_journal-11066 122 17 an an DET american_scientific_journal-11066 122 18 good good ADJ american_scientific_journal-11066 122 19 approximation approximation NOUN american_scientific_journal-11066 122 20 to to ADP american_scientific_journal-11066 122 21 lane lane NOUN american_scientific_journal-11066 122 22 - - PUNCT american_scientific_journal-11066 122 23 emden emden NOUN american_scientific_journal-11066 122 24 equation equation NOUN american_scientific_journal-11066 122 25 with with ADP american_scientific_journal-11066 122 26 𝑛 𝑛 PROPN american_scientific_journal-11066 122 27 = = SYM american_scientific_journal-11066 122 28 3 3 X american_scientific_journal-11066 122 29 . . PUNCT american_scientific_journal-11066 123 1 the the DET american_scientific_journal-11066 123 2 solution solution NOUN american_scientific_journal-11066 123 3 has have AUX american_scientific_journal-11066 123 4 been be AUX american_scientific_journal-11066 123 5 obtained obtain VERB american_scientific_journal-11066 123 6 in in ADP american_scientific_journal-11066 123 7 the the DET american_scientific_journal-11066 123 8 form form NOUN american_scientific_journal-11066 123 9 of of ADP american_scientific_journal-11066 123 10 a a DET american_scientific_journal-11066 123 11 series series NOUN american_scientific_journal-11066 123 12 with with ADP american_scientific_journal-11066 123 13 comparably comparably ADV american_scientific_journal-11066 123 14 easy easy ADJ american_scientific_journal-11066 123 15 calculations calculation NOUN american_scientific_journal-11066 123 16 and and CCONJ american_scientific_journal-11066 123 17 rapid rapid ADJ american_scientific_journal-11066 123 18 convergence convergence NOUN american_scientific_journal-11066 123 19 in in ADP american_scientific_journal-11066 123 20 the the DET american_scientific_journal-11066 123 21 interval interval NOUN american_scientific_journal-11066 123 22 considered consider VERB american_scientific_journal-11066 123 23 . . PUNCT american_scientific_journal-11066 124 1 a a DET american_scientific_journal-11066 124 2 different different ADJ american_scientific_journal-11066 124 3 form form NOUN american_scientific_journal-11066 124 4 of of ADP american_scientific_journal-11066 124 5 inverse inverse ADJ american_scientific_journal-11066 124 6 integral integral ADJ american_scientific_journal-11066 124 7 operator operator NOUN american_scientific_journal-11066 124 8 based base VERB american_scientific_journal-11066 124 9 on on ADP american_scientific_journal-11066 124 10 the the DET american_scientific_journal-11066 124 11 standard standard ADJ american_scientific_journal-11066 124 12 adomian adomian NOUN american_scientific_journal-11066 124 13 decomposition decomposition NOUN american_scientific_journal-11066 124 14 method method NOUN american_scientific_journal-11066 124 15 considered consider VERB american_scientific_journal-11066 124 16 and and CCONJ american_scientific_journal-11066 124 17 effectively effectively ADV american_scientific_journal-11066 124 18 applied apply VERB american_scientific_journal-11066 124 19 them they PRON american_scientific_journal-11066 124 20 to to ADP american_scientific_journal-11066 124 21 lane lane NOUN american_scientific_journal-11066 124 22 - - PUNCT american_scientific_journal-11066 124 23 emden emden NOUN american_scientific_journal-11066 124 24 equation equation NOUN american_scientific_journal-11066 124 25 considered consider VERB american_scientific_journal-11066 124 26 . . PUNCT american_scientific_journal-11066 125 1 the the DET american_scientific_journal-11066 125 2 solution solution NOUN american_scientific_journal-11066 125 3 was be AUX american_scientific_journal-11066 125 4 the the DET american_scientific_journal-11066 125 5 same same ADJ american_scientific_journal-11066 125 6 with with ADP american_scientific_journal-11066 125 7 the the DET american_scientific_journal-11066 125 8 standard standard ADJ american_scientific_journal-11066 125 9 adm adm PROPN american_scientific_journal-11066 125 10 and and CCONJ american_scientific_journal-11066 125 11 it it PRON american_scientific_journal-11066 125 12 can can AUX american_scientific_journal-11066 125 13 be be AUX american_scientific_journal-11066 125 14 seen see VERB american_scientific_journal-11066 125 15 that that SCONJ american_scientific_journal-11066 125 16 using use VERB american_scientific_journal-11066 125 17 this this DET american_scientific_journal-11066 125 18 method method NOUN american_scientific_journal-11066 125 19 also also ADV american_scientific_journal-11066 125 20 , , PUNCT american_scientific_journal-11066 125 21 a a DET american_scientific_journal-11066 125 22 good good ADJ american_scientific_journal-11066 125 23 approximation approximation NOUN american_scientific_journal-11066 125 24 can can AUX american_scientific_journal-11066 125 25 be be AUX american_scientific_journal-11066 125 26 obtained obtain VERB american_scientific_journal-11066 125 27 . . PUNCT american_scientific_journal-11066 126 1 according accord VERB american_scientific_journal-11066 126 2 to to ADP american_scientific_journal-11066 126 3 the the DET american_scientific_journal-11066 126 4 figure figure NOUN american_scientific_journal-11066 126 5 1 1 NUM american_scientific_journal-11066 126 6 in in ADP american_scientific_journal-11066 126 7 the the DET american_scientific_journal-11066 126 8 section section NOUN american_scientific_journal-11066 126 9 3 3 NUM american_scientific_journal-11066 126 10 , , PUNCT american_scientific_journal-11066 126 11 differential differential ADJ american_scientific_journal-11066 126 12 transform transform NOUN american_scientific_journal-11066 126 13 method method NOUN american_scientific_journal-11066 126 14 is be AUX american_scientific_journal-11066 126 15 a a DET american_scientific_journal-11066 126 16 very very ADJ american_scientific_journal-11066 126 17 fact fact NOUN american_scientific_journal-11066 126 18 convergent convergent NOUN american_scientific_journal-11066 126 19 , , PUNCT american_scientific_journal-11066 126 20 precise precise ADJ american_scientific_journal-11066 126 21 and and CCONJ american_scientific_journal-11066 126 22 time time NOUN american_scientific_journal-11066 126 23 saving save VERB american_scientific_journal-11066 126 24 method method NOUN american_scientific_journal-11066 126 25 for for ADP american_scientific_journal-11066 126 26 solving solve VERB american_scientific_journal-11066 126 27 lane lane NOUN american_scientific_journal-11066 126 28 - - PUNCT american_scientific_journal-11066 126 29 emden emden NOUN american_scientific_journal-11066 126 30 equation equation NOUN american_scientific_journal-11066 126 31 considered consider VERB american_scientific_journal-11066 126 32 . . PUNCT american_scientific_journal-11066 127 1 thus thus ADV american_scientific_journal-11066 127 2 , , PUNCT american_scientific_journal-11066 127 3 differential differential ADJ american_scientific_journal-11066 127 4 transform transform NOUN american_scientific_journal-11066 127 5 is be AUX american_scientific_journal-11066 127 6 an an DET american_scientific_journal-11066 127 7 important important ADJ american_scientific_journal-11066 127 8 tool tool NOUN american_scientific_journal-11066 127 9 in in ADP american_scientific_journal-11066 127 10 handling handle VERB american_scientific_journal-11066 127 11 highly highly ADV american_scientific_journal-11066 127 12 nonlinear nonlinear ADJ american_scientific_journal-11066 127 13 differential differential ADJ american_scientific_journal-11066 127 14 equations equation NOUN american_scientific_journal-11066 127 15 with with ADP american_scientific_journal-11066 127 16 a a DET american_scientific_journal-11066 127 17 minimum minimum ADJ american_scientific_journal-11066 127 18 size size NOUN american_scientific_journal-11066 127 19 of of ADP american_scientific_journal-11066 127 20 computations computation NOUN american_scientific_journal-11066 127 21 . . PUNCT american_scientific_journal-11066 128 1 comparing compare VERB american_scientific_journal-11066 128 2 with with ADP american_scientific_journal-11066 128 3 the the DET american_scientific_journal-11066 128 4 differential differential ADJ american_scientific_journal-11066 128 5 transform transform NOUN american_scientific_journal-11066 128 6 method method NOUN american_scientific_journal-11066 128 7 , , PUNCT american_scientific_journal-11066 128 8 adomian adomian NOUN american_scientific_journal-11066 128 9 decomposition decomposition NOUN american_scientific_journal-11066 128 10 method method NOUN american_scientific_journal-11066 128 11 and and CCONJ american_scientific_journal-11066 128 12 modified modify VERB american_scientific_journal-11066 128 13 adomian adomian NOUN american_scientific_journal-11066 128 14 decomposition decomposition NOUN american_scientific_journal-11066 128 15 method method NOUN american_scientific_journal-11066 128 16 need need VERB american_scientific_journal-11066 128 17 large large ADJ american_scientific_journal-11066 128 18 quantity quantity NOUN american_scientific_journal-11066 128 19 of of ADP american_scientific_journal-11066 128 20 computations computation NOUN american_scientific_journal-11066 128 21 . . PUNCT american_scientific_journal-11066 129 1 although although SCONJ american_scientific_journal-11066 129 2 the the DET american_scientific_journal-11066 129 3 solutions solution NOUN american_scientific_journal-11066 129 4 agree agree VERB american_scientific_journal-11066 129 5 with with ADP american_scientific_journal-11066 129 6 the the DET american_scientific_journal-11066 129 7 numerical numerical ADJ american_scientific_journal-11066 129 8 solutions solution NOUN american_scientific_journal-11066 129 9 in in ADP american_scientific_journal-11066 129 10 the the DET american_scientific_journal-11066 129 11 small small ADJ american_scientific_journal-11066 129 12 interval interval NOUN american_scientific_journal-11066 129 13 considered consider VERB american_scientific_journal-11066 129 14 , , PUNCT american_scientific_journal-11066 129 15 generally generally ADV american_scientific_journal-11066 129 16 for for ADP american_scientific_journal-11066 129 17 large large ADJ american_scientific_journal-11066 129 18 π‘₯ π‘₯ PROPN american_scientific_journal-11066 129 19 values value NOUN american_scientific_journal-11066 129 20 , , PUNCT american_scientific_journal-11066 129 21 the the DET american_scientific_journal-11066 129 22 solution solution NOUN american_scientific_journal-11066 129 23 will will AUX american_scientific_journal-11066 129 24 be be AUX american_scientific_journal-11066 129 25 diverging diverge VERB american_scientific_journal-11066 129 26 . . PUNCT american_scientific_journal-11066 130 1 therefore therefore ADV american_scientific_journal-11066 130 2 , , PUNCT american_scientific_journal-11066 130 3 as as ADP american_scientific_journal-11066 130 4 a a DET american_scientific_journal-11066 130 5 solution solution NOUN american_scientific_journal-11066 130 6 for for ADP american_scientific_journal-11066 130 7 this this DET american_scientific_journal-11066 130 8 problem problem NOUN american_scientific_journal-11066 130 9 the the DET american_scientific_journal-11066 130 10 differential differential ADJ american_scientific_journal-11066 130 11 transform transform NOUN american_scientific_journal-11066 130 12 method method NOUN american_scientific_journal-11066 130 13 with with ADP american_scientific_journal-11066 130 14 laplace laplace NOUN american_scientific_journal-11066 130 15 transform transform NOUN american_scientific_journal-11066 130 16 and and CCONJ american_scientific_journal-11066 130 17 pade pade NOUN american_scientific_journal-11066 130 18 approximation approximation NOUN american_scientific_journal-11066 130 19 will will AUX american_scientific_journal-11066 130 20 be be AUX american_scientific_journal-11066 130 21 considered.as considered.as PRON american_scientific_journal-11066 130 22 a a DET american_scientific_journal-11066 130 23 future future ADJ american_scientific_journal-11066 130 24 work work NOUN american_scientific_journal-11066 130 25 . . PUNCT american_scientific_journal-11066 131 1 references reference NOUN american_scientific_journal-11066 131 2 [ [ X american_scientific_journal-11066 131 3 1 1 X american_scientific_journal-11066 131 4 ] ] X american_scientific_journal-11066 131 5 h.j h.j PROPN american_scientific_journal-11066 131 6 . . PROPN american_scientific_journal-11066 131 7 lane lane PROPN american_scientific_journal-11066 131 8 . . PUNCT american_scientific_journal-11066 132 1 β€œ " PUNCT american_scientific_journal-11066 132 2 on on ADP american_scientific_journal-11066 132 3 the the DET american_scientific_journal-11066 132 4 theoretical theoretical ADJ american_scientific_journal-11066 132 5 temperature temperature NOUN american_scientific_journal-11066 132 6 of of ADP american_scientific_journal-11066 132 7 the the DET american_scientific_journal-11066 132 8 sun sun NOUN american_scientific_journal-11066 132 9 , , PUNCT american_scientific_journal-11066 132 10 under under ADP american_scientific_journal-11066 132 11 the the DET american_scientific_journal-11066 132 12 hypothesis hypothesis NOUN american_scientific_journal-11066 132 13 of of ADP american_scientific_journal-11066 132 14 a a DET american_scientific_journal-11066 132 15 gaseous gaseous ADJ american_scientific_journal-11066 132 16 mass mass NOUN american_scientific_journal-11066 132 17 maintaining maintain VERB american_scientific_journal-11066 132 18 its its PRON american_scientific_journal-11066 132 19 volume volume NOUN american_scientific_journal-11066 132 20 by by ADP american_scientific_journal-11066 132 21 its its PRON american_scientific_journal-11066 132 22 internal internal ADJ american_scientific_journal-11066 132 23 heat heat NOUN american_scientific_journal-11066 132 24 , , PUNCT american_scientific_journal-11066 132 25 and and CCONJ american_scientific_journal-11066 132 26 depending depend VERB american_scientific_journal-11066 132 27 on on ADP american_scientific_journal-11066 132 28 the the DET american_scientific_journal-11066 132 29 laws law NOUN american_scientific_journal-11066 132 30 of of ADP american_scientific_journal-11066 132 31 gases gas NOUN american_scientific_journal-11066 132 32 as as SCONJ american_scientific_journal-11066 132 33 known know VERB american_scientific_journal-11066 132 34 to to ADP american_scientific_journal-11066 132 35 terrestrial terrestrial ADJ american_scientific_journal-11066 132 36 experiment experiment NOUN american_scientific_journal-11066 132 37 . . PUNCT american_scientific_journal-11066 132 38 ” " PUNCT american_scientific_journal-11066 132 39 american american PROPN american_scientific_journal-11066 132 40 journal journal PROPN american_scientific_journal-11066 132 41 of of ADP american_scientific_journal-11066 132 42 science science NOUN american_scientific_journal-11066 132 43 , , PUNCT american_scientific_journal-11066 132 44 vol.s2 vol.s2 PROPN american_scientific_journal-11066 132 45 - - PUNCT american_scientific_journal-11066 132 46 50,issue 50,issue NUM american_scientific_journal-11066 132 47 148 148 NUM american_scientific_journal-11066 132 48 , , PUNCT american_scientific_journal-11066 132 49 1870 1870 NUM american_scientific_journal-11066 132 50 . . PUNCT american_scientific_journal-11066 133 1 [ [ X american_scientific_journal-11066 133 2 2 2 NUM american_scientific_journal-11066 133 3 ] ] PUNCT american_scientific_journal-11066 133 4 s.mukkerjee s.mukkerjee NOUN american_scientific_journal-11066 133 5 , , PUNCT american_scientific_journal-11066 133 6 b.roy b.roy ADJ american_scientific_journal-11066 133 7 , , PUNCT american_scientific_journal-11066 133 8 p.k.chatterjee p.k.chatterjee NOUN american_scientific_journal-11066 133 9 . . PUNCT american_scientific_journal-11066 134 1 β€œ " PUNCT american_scientific_journal-11066 134 2 solution solution NOUN american_scientific_journal-11066 134 3 to to ADP american_scientific_journal-11066 134 4 lane lane NOUN american_scientific_journal-11066 134 5 - - PUNCT american_scientific_journal-11066 134 6 emden emden NOUN american_scientific_journal-11066 134 7 by by ADP american_scientific_journal-11066 134 8 dtm dtm PROPN american_scientific_journal-11066 134 9 ” " PUNCT american_scientific_journal-11066 134 10 . . PUNCT american_scientific_journal-11066 135 1 international international ADJ american_scientific_journal-11066 135 2 journal journal PROPN american_scientific_journal-11066 135 3 of of ADP american_scientific_journal-11066 135 4 nonlinear nonlinear ADJ american_scientific_journal-11066 135 5 science science NOUN american_scientific_journal-11066 135 6 , , PUNCT american_scientific_journal-11066 135 7 vol.12 vol.12 NOUN american_scientific_journal-11066 135 8 , , PUNCT american_scientific_journal-11066 135 9 no no DET american_scientific_journal-11066 135 10 4,pp-478 4,pp-478 NOUN american_scientific_journal-11066 135 11 - - ADJ american_scientific_journal-11066 135 12 484 484 NUM american_scientific_journal-11066 135 13 , , PUNCT american_scientific_journal-11066 135 14 2011 2011 NUM american_scientific_journal-11066 135 15 . . PUNCT american_scientific_journal-11066 136 1 american american ADJ american_scientific_journal-11066 136 2 academic academic ADJ american_scientific_journal-11066 136 3 scientific scientific ADJ american_scientific_journal-11066 136 4 research research NOUN american_scientific_journal-11066 136 5 journal journal NOUN american_scientific_journal-11066 136 6 for for ADP american_scientific_journal-11066 136 7 engineering engineering NOUN american_scientific_journal-11066 136 8 , , PUNCT american_scientific_journal-11066 136 9 technology technology NOUN american_scientific_journal-11066 136 10 , , PUNCT american_scientific_journal-11066 136 11 and and CCONJ american_scientific_journal-11066 136 12 sciences science NOUN american_scientific_journal-11066 136 13 ( ( PUNCT american_scientific_journal-11066 136 14 asrjets asrjet NOUN american_scientific_journal-11066 136 15 ) ) PUNCT american_scientific_journal-11066 136 16 volume volume NOUN american_scientific_journal-11066 136 17 99 99 NUM american_scientific_journal-11066 136 18 , , PUNCT american_scientific_journal-11066 136 19 no no DET american_scientific_journal-11066 136 20 1 1 NUM american_scientific_journal-11066 136 21 , , PUNCT american_scientific_journal-11066 136 22 pp pp ADV american_scientific_journal-11066 136 23 1 1 NUM american_scientific_journal-11066 136 24 - - SYM american_scientific_journal-11066 136 25 12 12 NUM american_scientific_journal-11066 136 26 12 12 NUM american_scientific_journal-11066 136 27 [ [ X american_scientific_journal-11066 136 28 3 3 NUM american_scientific_journal-11066 136 29 ] ] X american_scientific_journal-11066 136 30 s.k.vanani s.k.vanani NOUN american_scientific_journal-11066 136 31 , , PUNCT american_scientific_journal-11066 136 32 a.aminataei a.aminataei NOUN american_scientific_journal-11066 136 33 . . PUNCT american_scientific_journal-11066 136 34 ”on ”on PROPN american_scientific_journal-11066 137 1 the the DET american_scientific_journal-11066 137 2 numerical numerical ADJ american_scientific_journal-11066 137 3 solution solution NOUN american_scientific_journal-11066 137 4 of of ADP american_scientific_journal-11066 137 5 differential differential ADJ american_scientific_journal-11066 137 6 equations equation NOUN american_scientific_journal-11066 137 7 of of ADP american_scientific_journal-11066 137 8 lane lane NOUN american_scientific_journal-11066 137 9 - - PUNCT american_scientific_journal-11066 137 10 emden emden NOUN american_scientific_journal-11066 137 11 type type NOUN american_scientific_journal-11066 137 12 ” " PUNCT american_scientific_journal-11066 137 13 computers computer NOUN american_scientific_journal-11066 137 14 and and CCONJ american_scientific_journal-11066 137 15 mathematics mathematic NOUN american_scientific_journal-11066 137 16 with with ADP american_scientific_journal-11066 137 17 applications application NOUN american_scientific_journal-11066 137 18 , , PUNCT american_scientific_journal-11066 137 19 volume volume NOUN american_scientific_journal-11066 137 20 59,issue 59,issue NOUN american_scientific_journal-11066 137 21 8 8 NUM american_scientific_journal-11066 137 22 , , PUNCT american_scientific_journal-11066 137 23 pp pp ADP american_scientific_journal-11066 137 24 2815 2815 NUM american_scientific_journal-11066 137 25 - - SYM american_scientific_journal-11066 137 26 2820 2820 NUM american_scientific_journal-11066 137 27 , , PUNCT american_scientific_journal-11066 137 28 april april PROPN american_scientific_journal-11066 137 29 2010 2010 NUM american_scientific_journal-11066 137 30 . . PUNCT american_scientific_journal-11066 138 1 [ [ X american_scientific_journal-11066 138 2 4 4 X american_scientific_journal-11066 138 3 ] ] PUNCT american_scientific_journal-11066 138 4 k.l.s.yip k.l.s.yip NOUN american_scientific_journal-11066 138 5 , , PUNCT american_scientific_journal-11066 138 6 t.k t.k PROPN american_scientific_journal-11066 138 7 chan chan PROPN american_scientific_journal-11066 138 8 , , PUNCT american_scientific_journal-11066 138 9 p.t.leung p.t.leung PROPN american_scientific_journal-11066 138 10 . . PUNCT american_scientific_journal-11066 139 1 β€œ " PUNCT american_scientific_journal-11066 139 2 perturbative perturbative ADJ american_scientific_journal-11066 139 3 solution solution NOUN american_scientific_journal-11066 139 4 to to ADP american_scientific_journal-11066 139 5 the the DET american_scientific_journal-11066 139 6 lane lane NOUN american_scientific_journal-11066 139 7 - - PUNCT american_scientific_journal-11066 139 8 emden emden NOUN american_scientific_journal-11066 139 9 equation equation NOUN american_scientific_journal-11066 139 10 : : PUNCT american_scientific_journal-11066 139 11 an an DET american_scientific_journal-11066 139 12 eigenvalue eigenvalue PROPN american_scientific_journal-11066 139 13 approach approach NOUN american_scientific_journal-11066 139 14 ” " PUNCT american_scientific_journal-11066 139 15 monthly monthly ADJ american_scientific_journal-11066 139 16 notices notice NOUN american_scientific_journal-11066 139 17 of of ADP american_scientific_journal-11066 139 18 the the DET american_scientific_journal-11066 139 19 royal royal ADJ american_scientific_journal-11066 139 20 academic academic ADJ american_scientific_journal-11066 139 21 society society NOUN american_scientific_journal-11066 139 22 , , PUNCT american_scientific_journal-11066 139 23 vol vol NOUN american_scientific_journal-11066 139 24 . . PUNCT american_scientific_journal-11066 140 1 465,issue 465,issue NUM american_scientific_journal-11066 140 2 4 4 NUM american_scientific_journal-11066 140 3 , , PUNCT american_scientific_journal-11066 140 4 pp pp ADP american_scientific_journal-11066 140 5 4265 4265 NUM american_scientific_journal-11066 140 6 - - SYM american_scientific_journal-11066 140 7 4280 4280 NUM american_scientific_journal-11066 140 8 , , PUNCT american_scientific_journal-11066 140 9 march march PROPN american_scientific_journal-11066 140 10 2017 2017 NUM american_scientific_journal-11066 140 11 . . PUNCT american_scientific_journal-11066 141 1 [ [ X american_scientific_journal-11066 141 2 5 5 NUM american_scientific_journal-11066 141 3 ] ] X american_scientific_journal-11066 141 4 k.k.ali k.k.ali PROPN american_scientific_journal-11066 141 5 , , PUNCT american_scientific_journal-11066 141 6 m.s.mehanna m.s.mehanna NOUN american_scientific_journal-11066 141 7 , , PUNCT american_scientific_journal-11066 141 8 a.m.wazwaz a.m.wazwaz NOUN american_scientific_journal-11066 141 9 and and CCONJ american_scientific_journal-11066 141 10 m.a.shaalan m.a.shaalan PROPN american_scientific_journal-11066 141 11 , , PUNCT american_scientific_journal-11066 141 12 β€œ " PUNCT american_scientific_journal-11066 141 13 solve solve VERB american_scientific_journal-11066 141 14 third third ADJ american_scientific_journal-11066 141 15 order order NOUN american_scientific_journal-11066 141 16 lane lane NOUN american_scientific_journal-11066 141 17 - - PUNCT american_scientific_journal-11066 141 18 emden emden VERB american_scientific_journal-11066 141 19 - - PUNCT american_scientific_journal-11066 141 20 fawler fawler NOUN american_scientific_journal-11066 141 21 equation equation NOUN american_scientific_journal-11066 141 22 by by ADP american_scientific_journal-11066 141 23 adomian adomian NOUN american_scientific_journal-11066 141 24 decomposition decomposition NOUN american_scientific_journal-11066 141 25 method method NOUN american_scientific_journal-11066 141 26 and and CCONJ american_scientific_journal-11066 141 27 quadratic quadratic ADJ american_scientific_journal-11066 141 28 - - PUNCT american_scientific_journal-11066 141 29 trigonometric trigonometric NOUN american_scientific_journal-11066 141 30 b b PROPN american_scientific_journal-11066 141 31 spline spline NOUN american_scientific_journal-11066 141 32 method method NOUN american_scientific_journal-11066 141 33 ” " PUNCT american_scientific_journal-11066 141 34 partial partial ADJ american_scientific_journal-11066 141 35 differential differential NOUN american_scientific_journal-11066 141 36 equations equation NOUN american_scientific_journal-11066 141 37 in in ADP american_scientific_journal-11066 141 38 applied applied ADJ american_scientific_journal-11066 141 39 mathematics mathematic NOUN american_scientific_journal-11066 141 40 , , PUNCT american_scientific_journal-11066 141 41 vol.10 vol.10 NOUN american_scientific_journal-11066 141 42 , , PUNCT american_scientific_journal-11066 141 43 june june PROPN american_scientific_journal-11066 141 44 2024,100676 2024,100676 NUM american_scientific_journal-11066 141 45 . . PUNCT american_scientific_journal-11066 142 1 [ [ X american_scientific_journal-11066 142 2 6 6 NUM american_scientific_journal-11066 142 3 ] ] PUNCT american_scientific_journal-11066 142 4 r.k.pandey r.k.pandey NOUN american_scientific_journal-11066 142 5 , , PUNCT american_scientific_journal-11066 142 6 n.kumar n.kumar NOUN american_scientific_journal-11066 142 7 , , PUNCT american_scientific_journal-11066 142 8 β€œ " PUNCT american_scientific_journal-11066 142 9 solution solution NOUN american_scientific_journal-11066 142 10 of of ADP american_scientific_journal-11066 142 11 lane lane NOUN american_scientific_journal-11066 142 12 - - PUNCT american_scientific_journal-11066 142 13 emden emden NOUN american_scientific_journal-11066 142 14 type type NOUN american_scientific_journal-11066 142 15 equations equation NOUN american_scientific_journal-11066 142 16 using use VERB american_scientific_journal-11066 142 17 bernstein bernstein PROPN american_scientific_journal-11066 142 18 operational operational ADJ american_scientific_journal-11066 142 19 matrix matrix NOUN american_scientific_journal-11066 142 20 for for ADP american_scientific_journal-11066 142 21 differentiation differentiation NOUN american_scientific_journal-11066 142 22 , , PUNCT american_scientific_journal-11066 142 23 new new ADJ american_scientific_journal-11066 142 24 astronomy astronomy NOUN american_scientific_journal-11066 142 25 , , PUNCT american_scientific_journal-11066 142 26 vol.17,issue vol.17,issue ADV american_scientific_journal-11066 142 27 03 03 NUM american_scientific_journal-11066 142 28 , , PUNCT american_scientific_journal-11066 142 29 pp303 pp303 NOUN american_scientific_journal-11066 142 30 - - SYM american_scientific_journal-11066 142 31 308 308 NUM american_scientific_journal-11066 142 32 , , PUNCT american_scientific_journal-11066 142 33 april april PROPN american_scientific_journal-11066 142 34 2012 2012 NUM american_scientific_journal-11066 142 35 . . PUNCT american_scientific_journal-11066 143 1 [ [ X american_scientific_journal-11066 143 2 7 7 NUM american_scientific_journal-11066 143 3 ] ] X american_scientific_journal-11066 143 4 p.p.a.alzate p.p.a.alzate ADJ american_scientific_journal-11066 143 5 , , PUNCT american_scientific_journal-11066 143 6 β€œ " PUNCT american_scientific_journal-11066 143 7 an an DET american_scientific_journal-11066 143 8 iterative iterative NOUN american_scientific_journal-11066 143 9 method method NOUN american_scientific_journal-11066 143 10 for for ADP american_scientific_journal-11066 143 11 solving solve VERB american_scientific_journal-11066 143 12 two two NUM american_scientific_journal-11066 143 13 special special ADJ american_scientific_journal-11066 143 14 cases case NOUN american_scientific_journal-11066 143 15 of of ADP american_scientific_journal-11066 143 16 lane lane NOUN american_scientific_journal-11066 143 17 - - PUNCT american_scientific_journal-11066 143 18 emden emden NOUN american_scientific_journal-11066 143 19 type type NOUN american_scientific_journal-11066 143 20 equations”,american equations”,american ADJ american_scientific_journal-11066 143 21 journal journal NOUN american_scientific_journal-11066 143 22 of of ADP american_scientific_journal-11066 143 23 computatioanl computatioanl PROPN american_scientific_journal-11066 143 24 mathematics mathematics PROPN american_scientific_journal-11066 143 25 , , PUNCT american_scientific_journal-11066 143 26 vol.4,no vol.4,no PROPN american_scientific_journal-11066 143 27 3 3 NUM american_scientific_journal-11066 143 28 , , PUNCT american_scientific_journal-11066 143 29 june june PROPN american_scientific_journal-11066 143 30 2014 2014 NUM american_scientific_journal-11066 143 31 . . PUNCT american_scientific_journal-11066 144 1 [ [ X american_scientific_journal-11066 144 2 8 8 NUM american_scientific_journal-11066 144 3 ] ] X american_scientific_journal-11066 144 4 l.zheng l.zheng X american_scientific_journal-11066 144 5 , , PUNCT american_scientific_journal-11066 144 6 x.zhang.in x.zhang.in PROPN american_scientific_journal-11066 144 7 modelling modelling NOUN american_scientific_journal-11066 144 8 and and CCONJ american_scientific_journal-11066 144 9 analysis analysis NOUN american_scientific_journal-11066 144 10 of of ADP american_scientific_journal-11066 144 11 modern modern ADJ american_scientific_journal-11066 144 12 fluid fluid ADJ american_scientific_journal-11066 144 13 problems problem NOUN american_scientific_journal-11066 144 14 . . PUNCT american_scientific_journal-11066 145 1 elsevier elsevier PROPN american_scientific_journal-11066 145 2 , , PUNCT american_scientific_journal-11066 145 3 united united ADJ american_scientific_journal-11066 145 4 kingdom kingdom PROPN american_scientific_journal-11066 145 5 , , PUNCT american_scientific_journal-11066 145 6 pp pp ADV american_scientific_journal-11066 145 7 79 79 NUM american_scientific_journal-11066 145 8 - - SYM american_scientific_journal-11066 145 9 114 114 NUM american_scientific_journal-11066 145 10 , , PUNCT american_scientific_journal-11066 145 11 2017 2017 NUM american_scientific_journal-11066 145 12 . . PUNCT american_scientific_journal-11066 146 1 [ [ X american_scientific_journal-11066 146 2 9 9 NUM american_scientific_journal-11066 146 3 ] ] X american_scientific_journal-11066 146 4 m.al m.al PROPN american_scientific_journal-11066 146 5 - - PROPN american_scientific_journal-11066 146 6 mazmumy mazmumy PROPN american_scientific_journal-11066 146 7 , , PUNCT american_scientific_journal-11066 146 8 a.a a.a PROPN american_scientific_journal-11066 146 9 . . PROPN american_scientific_journal-11066 146 10 alsulami alsulami PROPN american_scientific_journal-11066 146 11 , , PUNCT american_scientific_journal-11066 146 12 h.o.bakodah h.o.bakodah PROPN american_scientific_journal-11066 146 13 and and CCONJ american_scientific_journal-11066 146 14 n.alzaid n.alzaid NOUN american_scientific_journal-11066 146 15 , , PUNCT american_scientific_journal-11066 146 16 β€œ " PUNCT american_scientific_journal-11066 146 17 modified modified ADJ american_scientific_journal-11066 146 18 adomian adomian NOUN american_scientific_journal-11066 146 19 method method NOUN american_scientific_journal-11066 146 20 for for ADP american_scientific_journal-11066 146 21 the the DET american_scientific_journal-11066 146 22 generalized generalize VERB american_scientific_journal-11066 146 23 inhomogeneous inhomogeneous ADJ american_scientific_journal-11066 146 24 lane lane NOUN american_scientific_journal-11066 146 25 - - PUNCT american_scientific_journal-11066 146 26 emden emden NOUN american_scientific_journal-11066 146 27 type type NOUN american_scientific_journal-11066 146 28 equations equation NOUN american_scientific_journal-11066 146 29 ” " PUNCT american_scientific_journal-11066 146 30 nonlinear nonlinear ADJ american_scientific_journal-11066 146 31 analysis analysis NOUN american_scientific_journal-11066 146 32 and and CCONJ american_scientific_journal-11066 146 33 differential differential ADJ american_scientific_journal-11066 146 34 equations equation NOUN american_scientific_journal-11066 146 35 , , PUNCT american_scientific_journal-11066 146 36 vol vol NOUN american_scientific_journal-11066 146 37 . . PROPN american_scientific_journal-11066 146 38 10 10 NUM american_scientific_journal-11066 146 39 , , PUNCT american_scientific_journal-11066 146 40 no.1,pp no.1,pp ADJ american_scientific_journal-11066 146 41 15 15 NUM american_scientific_journal-11066 146 42 - - SYM american_scientific_journal-11066 146 43 35 35 NUM american_scientific_journal-11066 146 44 , , PUNCT american_scientific_journal-11066 146 45 2022 2022 NUM american_scientific_journal-11066 146 46 . . PUNCT american_scientific_journal-11066 147 1 [ [ X american_scientific_journal-11066 147 2 10 10 NUM american_scientific_journal-11066 147 3 ] ] PUNCT american_scientific_journal-11066 147 4 j.h.zhou(1986 j.h.zhou(1986 NUM american_scientific_journal-11066 147 5 ) ) PUNCT american_scientific_journal-11066 147 6 , , PUNCT american_scientific_journal-11066 147 7 differential differential ADJ american_scientific_journal-11066 147 8 transformation transformation NOUN american_scientific_journal-11066 147 9 and and CCONJ american_scientific_journal-11066 147 10 its its PRON american_scientific_journal-11066 147 11 applications application NOUN american_scientific_journal-11066 147 12 for for ADP american_scientific_journal-11066 147 13 electrical electrical ADJ american_scientific_journal-11066 147 14 circuits circuit NOUN american_scientific_journal-11066 147 15 ( ( PUNCT american_scientific_journal-11066 147 16 in in ADP american_scientific_journal-11066 147 17 chinese)huazhung chinese)huazhung PROPN american_scientific_journal-11066 147 18 university university NOUN american_scientific_journal-11066 147 19 press press NOUN american_scientific_journal-11066 147 20 , , PUNCT american_scientific_journal-11066 147 21 wuhan wuhan PROPN american_scientific_journal-11066 147 22 , , PUNCT american_scientific_journal-11066 147 23 china china PROPN american_scientific_journal-11066 147 24 . . PUNCT american_scientific_journal-11066 148 1 [ [ X american_scientific_journal-11066 148 2 11 11 NUM american_scientific_journal-11066 148 3 ] ] PUNCT american_scientific_journal-11066 148 4 i.v.andrianov i.v.andrianov NOUN american_scientific_journal-11066 148 5 , , PUNCT american_scientific_journal-11066 148 6 a.v.shatrov a.v.shatrov NOUN american_scientific_journal-11066 148 7 , , PUNCT american_scientific_journal-11066 148 8 β€œ " PUNCT american_scientific_journal-11066 148 9 pade pade NOUN american_scientific_journal-11066 148 10 approximants approximant NOUN american_scientific_journal-11066 148 11 , , PUNCT american_scientific_journal-11066 148 12 their their PRON american_scientific_journal-11066 148 13 properties property NOUN american_scientific_journal-11066 148 14 and and CCONJ american_scientific_journal-11066 148 15 applications application NOUN american_scientific_journal-11066 148 16 to to ADP american_scientific_journal-11066 148 17 hydrodynamic hydrodynamic ADJ american_scientific_journal-11066 148 18 problems problem NOUN american_scientific_journal-11066 148 19 ” " PUNCT american_scientific_journal-11066 148 20 symmetry symmetry NOUN american_scientific_journal-11066 148 21 , , PUNCT american_scientific_journal-11066 148 22 2021 2021 NUM american_scientific_journal-11066 148 23 , , PUNCT american_scientific_journal-11066 148 24 13 13 NUM american_scientific_journal-11066 148 25 , , PUNCT american_scientific_journal-11066 148 26 1869 1869 NUM american_scientific_journal-11066 148 27 . . PUNCT