id sid tid token lemma pos american_scientific_journal-4617 1 1 225 225 NUM american_scientific_journal-4617 1 2 american american ADJ american_scientific_journal-4617 1 3 scientific scientific ADJ american_scientific_journal-4617 1 4 research research NOUN american_scientific_journal-4617 1 5 journal journal NOUN american_scientific_journal-4617 1 6 for for ADP american_scientific_journal-4617 1 7 engineering engineering NOUN american_scientific_journal-4617 1 8 , , PUNCT american_scientific_journal-4617 1 9 technology technology NOUN american_scientific_journal-4617 1 10 , , PUNCT american_scientific_journal-4617 1 11 and and CCONJ american_scientific_journal-4617 1 12 sciences science NOUN american_scientific_journal-4617 1 13 ( ( PUNCT american_scientific_journal-4617 1 14 asrjets asrjet NOUN american_scientific_journal-4617 1 15 ) ) PUNCT american_scientific_journal-4617 1 16 issn issn PROPN american_scientific_journal-4617 1 17 ( ( PUNCT american_scientific_journal-4617 1 18 print print NOUN american_scientific_journal-4617 1 19 ) ) PUNCT american_scientific_journal-4617 1 20 2313 2313 NUM american_scientific_journal-4617 1 21 - - SYM american_scientific_journal-4617 1 22 4410 4410 NUM american_scientific_journal-4617 1 23 , , PUNCT american_scientific_journal-4617 1 24 issn issn PROPN american_scientific_journal-4617 1 25 ( ( PUNCT american_scientific_journal-4617 1 26 online online ADJ american_scientific_journal-4617 1 27 ) ) PUNCT american_scientific_journal-4617 1 28 2313 2313 NUM american_scientific_journal-4617 1 29 - - SYM american_scientific_journal-4617 1 30 4402 4402 NUM american_scientific_journal-4617 1 31 Β© Β© PROPN american_scientific_journal-4617 1 32 global global ADJ american_scientific_journal-4617 1 33 society society NOUN american_scientific_journal-4617 1 34 of of ADP american_scientific_journal-4617 1 35 scientific scientific ADJ american_scientific_journal-4617 1 36 research research NOUN american_scientific_journal-4617 1 37 and and CCONJ american_scientific_journal-4617 1 38 researchers researcher NOUN american_scientific_journal-4617 1 39 http://asrjetsjournal.org/ http://asrjetsjournal.org/ ADJ american_scientific_journal-4617 1 40 solution solution NOUN american_scientific_journal-4617 1 41 of of ADP american_scientific_journal-4617 1 42 second second ADJ american_scientific_journal-4617 1 43 order order NOUN american_scientific_journal-4617 1 44 linear linear ADJ american_scientific_journal-4617 1 45 and and CCONJ american_scientific_journal-4617 1 46 nonlinear nonlinear ADJ american_scientific_journal-4617 1 47 two two NUM american_scientific_journal-4617 1 48 point point NOUN american_scientific_journal-4617 1 49 boundary boundary ADJ american_scientific_journal-4617 1 50 value value NOUN american_scientific_journal-4617 1 51 problems problem NOUN american_scientific_journal-4617 1 52 using use VERB american_scientific_journal-4617 1 53 legendre legendre PROPN american_scientific_journal-4617 1 54 operational operational ADJ american_scientific_journal-4617 1 55 matrix matrix NOUN american_scientific_journal-4617 1 56 of of ADP american_scientific_journal-4617 1 57 differentiation differentiation NOUN american_scientific_journal-4617 1 58 abdurkadir abdurkadir PROPN american_scientific_journal-4617 1 59 edeo edeo PROPN american_scientific_journal-4617 1 60 * * PUNCT american_scientific_journal-4617 1 61 bule bule PROPN american_scientific_journal-4617 1 62 hora hora PROPN american_scientific_journal-4617 1 63 university university PROPN american_scientific_journal-4617 1 64 , , PUNCT american_scientific_journal-4617 1 65 college college NOUN american_scientific_journal-4617 1 66 of of ADP american_scientific_journal-4617 1 67 natural natural ADJ american_scientific_journal-4617 1 68 and and CCONJ american_scientific_journal-4617 1 69 computational computational ADJ american_scientific_journal-4617 1 70 sciences science NOUN american_scientific_journal-4617 1 71 , , PUNCT american_scientific_journal-4617 1 72 department department NOUN american_scientific_journal-4617 1 73 of of ADP american_scientific_journal-4617 1 74 mathematics mathematics PROPN american_scientific_journal-4617 1 75 , , PUNCT american_scientific_journal-4617 1 76 bule bule PROPN american_scientific_journal-4617 1 77 hora hora PROPN american_scientific_journal-4617 1 78 144 144 NUM american_scientific_journal-4617 1 79 , , PUNCT american_scientific_journal-4617 1 80 ethiopia ethiopia PROPN american_scientific_journal-4617 1 81 email email NOUN american_scientific_journal-4617 1 82 : : PUNCT american_scientific_journal-4617 1 83 edeoabdurkadir@gmail.com edeoabdurkadir@gmail.com X american_scientific_journal-4617 1 84 abstract abstract ADJ american_scientific_journal-4617 1 85 in in ADP american_scientific_journal-4617 1 86 this this DET american_scientific_journal-4617 1 87 paper paper NOUN american_scientific_journal-4617 1 88 , , PUNCT american_scientific_journal-4617 1 89 an an DET american_scientific_journal-4617 1 90 approach approach NOUN american_scientific_journal-4617 1 91 using use VERB american_scientific_journal-4617 1 92 tau tau PROPN american_scientific_journal-4617 1 93 method method NOUN american_scientific_journal-4617 1 94 based base VERB american_scientific_journal-4617 1 95 on on ADP american_scientific_journal-4617 1 96 legendre legendre PROPN american_scientific_journal-4617 1 97 operational operational ADJ american_scientific_journal-4617 1 98 matrix matrix NOUN american_scientific_journal-4617 1 99 of of ADP american_scientific_journal-4617 1 100 differentiation differentiation NOUN american_scientific_journal-4617 1 101 [ [ X american_scientific_journal-4617 1 102 2&5 2&5 NOUN american_scientific_journal-4617 1 103 ] ] X american_scientific_journal-4617 1 104 has have AUX american_scientific_journal-4617 1 105 been be AUX american_scientific_journal-4617 1 106 addressed address VERB american_scientific_journal-4617 1 107 to to PART american_scientific_journal-4617 1 108 find find VERB american_scientific_journal-4617 1 109 the the DET american_scientific_journal-4617 1 110 solutions solution NOUN american_scientific_journal-4617 1 111 of of ADP american_scientific_journal-4617 1 112 second second ADJ american_scientific_journal-4617 1 113 order order NOUN american_scientific_journal-4617 1 114 linear linear ADJ american_scientific_journal-4617 1 115 and and CCONJ american_scientific_journal-4617 1 116 nonlinear nonlinear ADJ american_scientific_journal-4617 1 117 two two NUM american_scientific_journal-4617 1 118 point point NOUN american_scientific_journal-4617 1 119 boundary boundary ADJ american_scientific_journal-4617 1 120 value value NOUN american_scientific_journal-4617 1 121 problems problem NOUN american_scientific_journal-4617 1 122 of of ADP american_scientific_journal-4617 1 123 ordinary ordinary ADJ american_scientific_journal-4617 1 124 differential differential ADJ american_scientific_journal-4617 1 125 equations equation NOUN american_scientific_journal-4617 1 126 . . PUNCT american_scientific_journal-4617 2 1 in in ADP american_scientific_journal-4617 2 2 the the DET american_scientific_journal-4617 2 3 implementation implementation NOUN american_scientific_journal-4617 2 4 of of ADP american_scientific_journal-4617 2 5 this this DET american_scientific_journal-4617 2 6 approach approach NOUN american_scientific_journal-4617 2 7 , , PUNCT american_scientific_journal-4617 2 8 the the DET american_scientific_journal-4617 2 9 given give VERB american_scientific_journal-4617 2 10 second second ADJ american_scientific_journal-4617 2 11 order order NOUN american_scientific_journal-4617 2 12 two two NUM american_scientific_journal-4617 2 13 point point NOUN american_scientific_journal-4617 2 14 boundary boundary ADJ american_scientific_journal-4617 2 15 problems problem NOUN american_scientific_journal-4617 2 16 is be AUX american_scientific_journal-4617 2 17 converted convert VERB american_scientific_journal-4617 2 18 into into ADP american_scientific_journal-4617 2 19 a a DET american_scientific_journal-4617 2 20 system system NOUN american_scientific_journal-4617 2 21 of of ADP american_scientific_journal-4617 2 22 algebraic algebraic ADJ american_scientific_journal-4617 2 23 equations equation NOUN american_scientific_journal-4617 2 24 , , PUNCT american_scientific_journal-4617 2 25 whose whose DET american_scientific_journal-4617 2 26 solutions solution NOUN american_scientific_journal-4617 2 27 are be AUX american_scientific_journal-4617 2 28 the the DET american_scientific_journal-4617 2 29 legendre legendre PROPN american_scientific_journal-4617 2 30 coefficients coefficient NOUN american_scientific_journal-4617 2 31 . . PUNCT american_scientific_journal-4617 3 1 the the DET american_scientific_journal-4617 3 2 validity validity NOUN american_scientific_journal-4617 3 3 and and CCONJ american_scientific_journal-4617 3 4 efficiency efficiency NOUN american_scientific_journal-4617 3 5 of of ADP american_scientific_journal-4617 3 6 the the DET american_scientific_journal-4617 3 7 method method NOUN american_scientific_journal-4617 3 8 has have AUX american_scientific_journal-4617 3 9 also also ADV american_scientific_journal-4617 3 10 been be AUX american_scientific_journal-4617 3 11 illustrated illustrate VERB american_scientific_journal-4617 3 12 with with ADP american_scientific_journal-4617 3 13 numerical numerical ADJ american_scientific_journal-4617 3 14 examples example NOUN american_scientific_journal-4617 3 15 supported support VERB american_scientific_journal-4617 3 16 by by ADP american_scientific_journal-4617 3 17 graphs graph NOUN american_scientific_journal-4617 3 18 . . PUNCT american_scientific_journal-4617 4 1 keywords keyword NOUN american_scientific_journal-4617 4 2 : : PUNCT american_scientific_journal-4617 4 3 boundary boundary ADJ american_scientific_journal-4617 4 4 value value NOUN american_scientific_journal-4617 4 5 problems problem NOUN american_scientific_journal-4617 4 6 ( ( PUNCT american_scientific_journal-4617 4 7 bvps bvps PROPN american_scientific_journal-4617 4 8 ) ) PUNCT american_scientific_journal-4617 4 9 ; ; PUNCT american_scientific_journal-4617 4 10 legendre legendre PROPN american_scientific_journal-4617 4 11 operational operational ADJ american_scientific_journal-4617 4 12 matrix matrix NOUN american_scientific_journal-4617 4 13 of of ADP american_scientific_journal-4617 4 14 differentiation differentiation NOUN american_scientific_journal-4617 4 15 ; ; PUNCT american_scientific_journal-4617 4 16 linear linear ADJ american_scientific_journal-4617 4 17 and and CCONJ american_scientific_journal-4617 4 18 nonlinear nonlinear ADJ american_scientific_journal-4617 4 19 ordinary ordinary ADJ american_scientific_journal-4617 4 20 differential differential ADJ american_scientific_journal-4617 4 21 equations equation NOUN american_scientific_journal-4617 4 22 . . PUNCT american_scientific_journal-4617 5 1 1 1 X american_scientific_journal-4617 5 2 . . X american_scientific_journal-4617 5 3 introduction introduction NOUN american_scientific_journal-4617 5 4 according accord VERB american_scientific_journal-4617 5 5 to to ADP american_scientific_journal-4617 5 6 [ [ X american_scientific_journal-4617 5 7 1 1 NUM american_scientific_journal-4617 5 8 ] ] PUNCT american_scientific_journal-4617 5 9 , , PUNCT american_scientific_journal-4617 5 10 second second ADJ american_scientific_journal-4617 5 11 order order NOUN american_scientific_journal-4617 5 12 two two NUM american_scientific_journal-4617 5 13 point point NOUN american_scientific_journal-4617 5 14 boundary boundary ADJ american_scientific_journal-4617 5 15 value value NOUN american_scientific_journal-4617 5 16 problems problem NOUN american_scientific_journal-4617 5 17 of of ADP american_scientific_journal-4617 5 18 ordinary ordinary ADJ american_scientific_journal-4617 5 19 differential differential ADJ american_scientific_journal-4617 5 20 equations equation NOUN american_scientific_journal-4617 5 21 are be AUX american_scientific_journal-4617 5 22 equations equation NOUN american_scientific_journal-4617 5 23 of of ADP american_scientific_journal-4617 5 24 the the DET american_scientific_journal-4617 5 25 form form NOUN american_scientific_journal-4617 5 26 𝑦𝑦′′ 𝑦𝑦′′ NOUN american_scientific_journal-4617 5 27 = = SYM american_scientific_journal-4617 5 28 𝑓𝑓(π‘₯π‘₯,𝑦𝑦,𝑦𝑦′ 𝑓𝑓(π‘₯π‘₯,𝑦𝑦,𝑦𝑦′ X american_scientific_journal-4617 5 29 ) ) PUNCT american_scientific_journal-4617 5 30 , , PUNCT american_scientific_journal-4617 5 31 π‘Žπ‘Ž π‘Žπ‘Ž VERB american_scientific_journal-4617 5 32 ≀ ≀ NOUN american_scientific_journal-4617 5 33 π‘₯π‘₯ π‘₯π‘₯ ADP american_scientific_journal-4617 5 34 ≀ ≀ NUM american_scientific_journal-4617 5 35 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4617 5 36 , , PUNCT american_scientific_journal-4617 5 37 with with ADP american_scientific_journal-4617 5 38 the the DET american_scientific_journal-4617 5 39 boundary boundary ADJ american_scientific_journal-4617 5 40 conditions condition NOUN american_scientific_journal-4617 5 41 on on ADP american_scientific_journal-4617 5 42 the the DET american_scientific_journal-4617 5 43 solution solution NOUN american_scientific_journal-4617 5 44 prescribed prescribe VERB american_scientific_journal-4617 5 45 by by ADP american_scientific_journal-4617 5 46 𝑦𝑦(π‘Žπ‘Ž 𝑦𝑦(π‘Žπ‘Ž PROPN american_scientific_journal-4617 5 47 ) ) PUNCT american_scientific_journal-4617 6 1 = = SYM american_scientific_journal-4617 6 2 𝛼𝛼 𝛼𝛼 PROPN american_scientific_journal-4617 6 3 , , PUNCT american_scientific_journal-4617 6 4 𝑦𝑦(𝑏𝑏 𝑦𝑦(𝑏𝑏 PROPN american_scientific_journal-4617 6 5 ) ) PUNCT american_scientific_journal-4617 6 6 = = PUNCT american_scientific_journal-4617 7 1 𝛽𝛽 𝛽𝛽 NOUN american_scientific_journal-4617 7 2 , , PUNCT american_scientific_journal-4617 7 3 for for ADP american_scientific_journal-4617 7 4 some some DET american_scientific_journal-4617 7 5 constants constant NOUN american_scientific_journal-4617 7 6 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4617 7 7 and and CCONJ american_scientific_journal-4617 7 8 𝛽𝛽. 𝛽𝛽. NOUN american_scientific_journal-4617 7 9 ----------------------------------------------------------------------- ----------------------------------------------------------------------- PUNCT american_scientific_journal-4617 8 1 * * PUNCT american_scientific_journal-4617 8 2 corresponding correspond VERB american_scientific_journal-4617 8 3 author author NOUN american_scientific_journal-4617 8 4 . . PUNCT american_scientific_journal-4617 9 1 http://asrjetsjournal.org/ http://asrjetsjournal.org/ PROPN american_scientific_journal-4617 9 2 american american PROPN american_scientific_journal-4617 9 3 scientific scientific PROPN american_scientific_journal-4617 9 4 research research NOUN american_scientific_journal-4617 9 5 journal journal NOUN american_scientific_journal-4617 9 6 for for ADP american_scientific_journal-4617 9 7 engineering engineering NOUN american_scientific_journal-4617 9 8 , , PUNCT american_scientific_journal-4617 9 9 technology technology NOUN american_scientific_journal-4617 9 10 , , PUNCT american_scientific_journal-4617 9 11 and and CCONJ american_scientific_journal-4617 9 12 sciences science NOUN american_scientific_journal-4617 9 13 ( ( PUNCT american_scientific_journal-4617 9 14 asrjets asrjet NOUN american_scientific_journal-4617 9 15 ) ) PUNCT american_scientific_journal-4617 9 16 ( ( PUNCT american_scientific_journal-4617 9 17 2019 2019 NUM american_scientific_journal-4617 9 18 ) ) PUNCT american_scientific_journal-4617 9 19 volume volume NOUN american_scientific_journal-4617 9 20 51 51 NUM american_scientific_journal-4617 9 21 , , PUNCT american_scientific_journal-4617 9 22 no no DET american_scientific_journal-4617 9 23 1 1 NUM american_scientific_journal-4617 9 24 , , PUNCT american_scientific_journal-4617 9 25 pp pp ADV american_scientific_journal-4617 9 26 225 225 NUM american_scientific_journal-4617 9 27 - - SYM american_scientific_journal-4617 9 28 234 234 NUM american_scientific_journal-4617 9 29 226 226 NUM american_scientific_journal-4617 9 30 second second ADJ american_scientific_journal-4617 9 31 order order NOUN american_scientific_journal-4617 9 32 differential differential ADJ american_scientific_journal-4617 9 33 equations equation NOUN american_scientific_journal-4617 9 34 with with ADP american_scientific_journal-4617 9 35 various various ADJ american_scientific_journal-4617 9 36 types type NOUN american_scientific_journal-4617 9 37 of of ADP american_scientific_journal-4617 9 38 boundary boundary ADJ american_scientific_journal-4617 9 39 conditions condition NOUN american_scientific_journal-4617 9 40 are be AUX american_scientific_journal-4617 9 41 among among ADP american_scientific_journal-4617 9 42 many many ADJ american_scientific_journal-4617 9 43 of of ADP american_scientific_journal-4617 9 44 linear linear ADJ american_scientific_journal-4617 9 45 and and CCONJ american_scientific_journal-4617 9 46 nonlinear nonlinear ADJ american_scientific_journal-4617 9 47 problems problem NOUN american_scientific_journal-4617 9 48 occurring occur VERB american_scientific_journal-4617 9 49 in in ADP american_scientific_journal-4617 9 50 science science NOUN american_scientific_journal-4617 9 51 and and CCONJ american_scientific_journal-4617 9 52 engineering engineering NOUN american_scientific_journal-4617 9 53 which which PRON american_scientific_journal-4617 9 54 can can AUX american_scientific_journal-4617 9 55 be be AUX american_scientific_journal-4617 9 56 solved solve VERB american_scientific_journal-4617 9 57 either either CCONJ american_scientific_journal-4617 9 58 analytically analytically ADV american_scientific_journal-4617 9 59 or or CCONJ american_scientific_journal-4617 9 60 numerically numerically ADV american_scientific_journal-4617 9 61 [ [ X american_scientific_journal-4617 9 62 3 3 NUM american_scientific_journal-4617 9 63 ] ] PUNCT american_scientific_journal-4617 9 64 . . PUNCT american_scientific_journal-4617 10 1 second second ADJ american_scientific_journal-4617 10 2 order order NOUN american_scientific_journal-4617 10 3 two two NUM american_scientific_journal-4617 10 4 point point NOUN american_scientific_journal-4617 10 5 boundary boundary ADJ american_scientific_journal-4617 10 6 value value NOUN american_scientific_journal-4617 10 7 problems problem NOUN american_scientific_journal-4617 10 8 are be AUX american_scientific_journal-4617 10 9 encountered encounter VERB american_scientific_journal-4617 10 10 in in ADP american_scientific_journal-4617 10 11 many many ADJ american_scientific_journal-4617 10 12 engineering engineering NOUN american_scientific_journal-4617 10 13 fields field NOUN american_scientific_journal-4617 10 14 including include VERB american_scientific_journal-4617 10 15 optimal optimal ADJ american_scientific_journal-4617 10 16 control control NOUN american_scientific_journal-4617 10 17 , , PUNCT american_scientific_journal-4617 10 18 beam beam NOUN american_scientific_journal-4617 10 19 deflections deflection NOUN american_scientific_journal-4617 10 20 , , PUNCT american_scientific_journal-4617 10 21 heat heat NOUN american_scientific_journal-4617 10 22 flow flow NOUN american_scientific_journal-4617 10 23 , , PUNCT american_scientific_journal-4617 10 24 and and CCONJ american_scientific_journal-4617 10 25 various various ADJ american_scientific_journal-4617 10 26 dynamical dynamical ADJ american_scientific_journal-4617 10 27 systems system NOUN american_scientific_journal-4617 10 28 [ [ X american_scientific_journal-4617 10 29 4 4 NUM american_scientific_journal-4617 10 30 ] ] PUNCT american_scientific_journal-4617 10 31 . . PUNCT american_scientific_journal-4617 11 1 in in ADP american_scientific_journal-4617 11 2 the the DET american_scientific_journal-4617 11 3 literature literature NOUN american_scientific_journal-4617 11 4 of of ADP american_scientific_journal-4617 11 5 numerical numerical ADJ american_scientific_journal-4617 11 6 analysis analysis NOUN american_scientific_journal-4617 11 7 for for ADP american_scientific_journal-4617 11 8 solving solve VERB american_scientific_journal-4617 11 9 a a DET american_scientific_journal-4617 11 10 two two NUM american_scientific_journal-4617 11 11 point point NOUN american_scientific_journal-4617 11 12 second second ADJ american_scientific_journal-4617 11 13 order order NOUN american_scientific_journal-4617 11 14 boundary boundary ADJ american_scientific_journal-4617 11 15 value value NOUN american_scientific_journal-4617 11 16 problem problem NOUN american_scientific_journal-4617 11 17 ( ( PUNCT american_scientific_journal-4617 11 18 bvp bvp PROPN american_scientific_journal-4617 11 19 ) ) PUNCT american_scientific_journal-4617 11 20 of of ADP american_scientific_journal-4617 11 21 differential differential ADJ american_scientific_journal-4617 11 22 equations equation NOUN american_scientific_journal-4617 11 23 , , PUNCT american_scientific_journal-4617 11 24 many many ADJ american_scientific_journal-4617 11 25 authors author NOUN american_scientific_journal-4617 11 26 have have AUX american_scientific_journal-4617 11 27 attempted attempt VERB american_scientific_journal-4617 11 28 to to PART american_scientific_journal-4617 11 29 obtain obtain VERB american_scientific_journal-4617 11 30 higher high ADJ american_scientific_journal-4617 11 31 accuracy accuracy NOUN american_scientific_journal-4617 11 32 rapidly rapidly ADV american_scientific_journal-4617 11 33 by by ADP american_scientific_journal-4617 11 34 using use VERB american_scientific_journal-4617 11 35 a a DET american_scientific_journal-4617 11 36 numerous numerous ADJ american_scientific_journal-4617 11 37 methods method NOUN american_scientific_journal-4617 11 38 . . PUNCT american_scientific_journal-4617 12 1 among among ADP american_scientific_journal-4617 12 2 various various ADJ american_scientific_journal-4617 12 3 numerical numerical ADJ american_scientific_journal-4617 12 4 techniques technique NOUN american_scientific_journal-4617 12 5 , , PUNCT american_scientific_journal-4617 12 6 finite finite ADJ american_scientific_journal-4617 12 7 difference difference NOUN american_scientific_journal-4617 12 8 method method NOUN american_scientific_journal-4617 12 9 has have AUX american_scientific_journal-4617 12 10 been be AUX american_scientific_journal-4617 12 11 widely widely ADV american_scientific_journal-4617 12 12 used use VERB american_scientific_journal-4617 12 13 but but CCONJ american_scientific_journal-4617 12 14 it it PRON american_scientific_journal-4617 12 15 takes take VERB american_scientific_journal-4617 12 16 more more ADJ american_scientific_journal-4617 12 17 computational computational ADJ american_scientific_journal-4617 12 18 costs cost NOUN american_scientific_journal-4617 12 19 to to PART american_scientific_journal-4617 12 20 get get VERB american_scientific_journal-4617 12 21 high high ADJ american_scientific_journal-4617 12 22 accuracy accuracy NOUN american_scientific_journal-4617 13 1 [ [ X american_scientific_journal-4617 13 2 3 3 NUM american_scientific_journal-4617 13 3 ] ] PUNCT american_scientific_journal-4617 13 4 . . PUNCT american_scientific_journal-4617 14 1 in in ADP american_scientific_journal-4617 14 2 [ [ X american_scientific_journal-4617 14 3 4 4 X american_scientific_journal-4617 14 4 ] ] PUNCT american_scientific_journal-4617 14 5 the the DET american_scientific_journal-4617 14 6 author author NOUN american_scientific_journal-4617 14 7 has have AUX american_scientific_journal-4617 14 8 applied apply VERB american_scientific_journal-4617 14 9 a a DET american_scientific_journal-4617 14 10 cubic cubic ADJ american_scientific_journal-4617 14 11 b b NOUN american_scientific_journal-4617 14 12 - - PUNCT american_scientific_journal-4617 14 13 spline spline NOUN american_scientific_journal-4617 14 14 method method NOUN american_scientific_journal-4617 14 15 to to PART american_scientific_journal-4617 14 16 find find VERB american_scientific_journal-4617 14 17 the the DET american_scientific_journal-4617 14 18 solutions solution NOUN american_scientific_journal-4617 14 19 of of ADP american_scientific_journal-4617 14 20 both both CCONJ american_scientific_journal-4617 14 21 linear linear ADJ american_scientific_journal-4617 14 22 and and CCONJ american_scientific_journal-4617 14 23 nonlinear nonlinear ADJ american_scientific_journal-4617 14 24 second second ADJ american_scientific_journal-4617 14 25 order order NOUN american_scientific_journal-4617 14 26 two two NUM american_scientific_journal-4617 14 27 point point NOUN american_scientific_journal-4617 14 28 bvps bvps NOUN american_scientific_journal-4617 14 29 of of ADP american_scientific_journal-4617 14 30 ordinary ordinary ADJ american_scientific_journal-4617 14 31 differential differential ADJ american_scientific_journal-4617 14 32 equations equation NOUN american_scientific_journal-4617 14 33 . . PUNCT american_scientific_journal-4617 15 1 the the DET american_scientific_journal-4617 15 2 authors author NOUN american_scientific_journal-4617 15 3 in in ADP american_scientific_journal-4617 15 4 [ [ X american_scientific_journal-4617 15 5 6 6 NUM american_scientific_journal-4617 15 6 ] ] PUNCT american_scientific_journal-4617 15 7 have have AUX american_scientific_journal-4617 15 8 used use VERB american_scientific_journal-4617 15 9 an an DET american_scientific_journal-4617 15 10 extended extended ADJ american_scientific_journal-4617 15 11 cubic cubic ADJ american_scientific_journal-4617 15 12 b b NOUN american_scientific_journal-4617 15 13 - - PUNCT american_scientific_journal-4617 15 14 spline spline NOUN american_scientific_journal-4617 15 15 method method NOUN american_scientific_journal-4617 15 16 for for ADP american_scientific_journal-4617 15 17 solving solve VERB american_scientific_journal-4617 15 18 linear linear PROPN american_scientific_journal-4617 15 19 two two NUM american_scientific_journal-4617 15 20 point point NOUN american_scientific_journal-4617 15 21 bvps bvps NOUN american_scientific_journal-4617 15 22 . . PUNCT american_scientific_journal-4617 16 1 in in ADP american_scientific_journal-4617 16 2 [ [ X american_scientific_journal-4617 16 3 7 7 X american_scientific_journal-4617 16 4 ] ] PUNCT american_scientific_journal-4617 16 5 the the DET american_scientific_journal-4617 16 6 authors author NOUN american_scientific_journal-4617 16 7 found find VERB american_scientific_journal-4617 16 8 the the DET american_scientific_journal-4617 16 9 solution solution NOUN american_scientific_journal-4617 16 10 of of ADP american_scientific_journal-4617 16 11 two two NUM american_scientific_journal-4617 16 12 point point NOUN american_scientific_journal-4617 16 13 boundary boundary ADJ american_scientific_journal-4617 16 14 value value NOUN american_scientific_journal-4617 16 15 problems problem NOUN american_scientific_journal-4617 16 16 using use VERB american_scientific_journal-4617 16 17 quartic quartic ADJ american_scientific_journal-4617 16 18 b b NOUN american_scientific_journal-4617 16 19 - - PUNCT american_scientific_journal-4617 16 20 spline spline NOUN american_scientific_journal-4617 16 21 method method NOUN american_scientific_journal-4617 16 22 . . PUNCT american_scientific_journal-4617 17 1 but but CCONJ american_scientific_journal-4617 17 2 these these DET american_scientific_journal-4617 17 3 b b X american_scientific_journal-4617 17 4 - - PUNCT american_scientific_journal-4617 17 5 spline spline NOUN american_scientific_journal-4617 17 6 methods method NOUN american_scientific_journal-4617 17 7 require require VERB american_scientific_journal-4617 17 8 dividing divide VERB american_scientific_journal-4617 17 9 the the DET american_scientific_journal-4617 17 10 interval interval NOUN american_scientific_journal-4617 17 11 [ [ X american_scientific_journal-4617 17 12 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4617 17 13 , , PUNCT american_scientific_journal-4617 17 14 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4617 17 15 ] ] PUNCT american_scientific_journal-4617 17 16 into into ADP american_scientific_journal-4617 17 17 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4617 17 18 subintervals subinterval NOUN american_scientific_journal-4617 17 19 and and CCONJ american_scientific_journal-4617 17 20 the the DET american_scientific_journal-4617 17 21 construction construction NOUN american_scientific_journal-4617 17 22 of of ADP american_scientific_journal-4617 17 23 cubic cubic ADJ american_scientific_journal-4617 17 24 or or CCONJ american_scientific_journal-4617 17 25 extended extended ADJ american_scientific_journal-4617 17 26 cubic cubic ADJ american_scientific_journal-4617 17 27 b b NOUN american_scientific_journal-4617 17 28 - - PUNCT american_scientific_journal-4617 17 29 splines spline NOUN american_scientific_journal-4617 17 30 in in ADP american_scientific_journal-4617 17 31 each each DET american_scientific_journal-4617 17 32 subinterval subinterval NOUN american_scientific_journal-4617 17 33 . . PUNCT american_scientific_journal-4617 18 1 in in ADP american_scientific_journal-4617 18 2 addition addition NOUN american_scientific_journal-4617 18 3 with with ADP american_scientific_journal-4617 18 4 these these DET american_scientific_journal-4617 18 5 methods method NOUN american_scientific_journal-4617 18 6 we we PRON american_scientific_journal-4617 18 7 need need VERB american_scientific_journal-4617 18 8 to to PART american_scientific_journal-4617 18 9 solve solve VERB american_scientific_journal-4617 18 10 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4617 18 11 + + CCONJ american_scientific_journal-4617 18 12 1 1 NUM american_scientific_journal-4617 18 13 systems system NOUN american_scientific_journal-4617 18 14 of of ADP american_scientific_journal-4617 18 15 linear linear ADJ american_scientific_journal-4617 18 16 or or CCONJ american_scientific_journal-4617 18 17 nonlinear nonlinear ADJ american_scientific_journal-4617 18 18 equations equation NOUN american_scientific_journal-4617 18 19 to to PART american_scientific_journal-4617 18 20 arrive arrive VERB american_scientific_journal-4617 18 21 at at ADP american_scientific_journal-4617 18 22 better well ADJ american_scientific_journal-4617 18 23 accuracy accuracy NOUN american_scientific_journal-4617 18 24 . . PUNCT american_scientific_journal-4617 19 1 therefore therefore ADV american_scientific_journal-4617 19 2 , , PUNCT american_scientific_journal-4617 19 3 like like ADP american_scientific_journal-4617 19 4 the the DET american_scientific_journal-4617 19 5 finite finite ADJ american_scientific_journal-4617 19 6 difference difference NOUN american_scientific_journal-4617 19 7 method method NOUN american_scientific_journal-4617 19 8 these these DET american_scientific_journal-4617 19 9 methods method NOUN american_scientific_journal-4617 19 10 are be AUX american_scientific_journal-4617 19 11 also also ADV american_scientific_journal-4617 19 12 computational computational ADJ american_scientific_journal-4617 19 13 too too ADV american_scientific_journal-4617 19 14 cost cost NOUN american_scientific_journal-4617 19 15 . . PUNCT american_scientific_journal-4617 20 1 in in ADP american_scientific_journal-4617 20 2 [ [ X american_scientific_journal-4617 20 3 8 8 NUM american_scientific_journal-4617 20 4 ] ] PUNCT american_scientific_journal-4617 20 5 the the DET american_scientific_journal-4617 20 6 authors author NOUN american_scientific_journal-4617 20 7 have have AUX american_scientific_journal-4617 20 8 developed develop VERB american_scientific_journal-4617 20 9 galerkin galerkin ADJ american_scientific_journal-4617 20 10 method method NOUN american_scientific_journal-4617 20 11 to to PART american_scientific_journal-4617 20 12 approximate approximate VERB american_scientific_journal-4617 20 13 the the DET american_scientific_journal-4617 20 14 solution solution NOUN american_scientific_journal-4617 20 15 of of ADP american_scientific_journal-4617 20 16 second second ADJ american_scientific_journal-4617 20 17 order order NOUN american_scientific_journal-4617 20 18 neumann neumann NOUN american_scientific_journal-4617 20 19 and and CCONJ american_scientific_journal-4617 20 20 cauchy cauchy PROPN american_scientific_journal-4617 20 21 linear linear PROPN american_scientific_journal-4617 20 22 boundary boundary ADJ american_scientific_journal-4617 20 23 value value NOUN american_scientific_journal-4617 20 24 problems problem NOUN american_scientific_journal-4617 20 25 . . PUNCT american_scientific_journal-4617 21 1 the the DET american_scientific_journal-4617 21 2 authors author NOUN american_scientific_journal-4617 21 3 in in ADP american_scientific_journal-4617 21 4 [ [ X american_scientific_journal-4617 21 5 9 9 NUM american_scientific_journal-4617 21 6 ] ] PUNCT american_scientific_journal-4617 21 7 derived derive VERB american_scientific_journal-4617 21 8 a a DET american_scientific_journal-4617 21 9 new new ADJ american_scientific_journal-4617 21 10 difference difference NOUN american_scientific_journal-4617 21 11 scheme scheme NOUN american_scientific_journal-4617 21 12 for for ADP american_scientific_journal-4617 21 13 solving solve VERB american_scientific_journal-4617 21 14 the the DET american_scientific_journal-4617 21 15 linear linear ADJ american_scientific_journal-4617 21 16 and and CCONJ american_scientific_journal-4617 21 17 nonlinear nonlinear ADJ american_scientific_journal-4617 21 18 second second ADJ american_scientific_journal-4617 21 19 order order NOUN american_scientific_journal-4617 21 20 two two NUM american_scientific_journal-4617 21 21 - - PUNCT american_scientific_journal-4617 21 22 point point NOUN american_scientific_journal-4617 21 23 boundary boundary ADJ american_scientific_journal-4617 21 24 value value NOUN american_scientific_journal-4617 21 25 problem problem NOUN american_scientific_journal-4617 21 26 by by ADP american_scientific_journal-4617 21 27 using use VERB american_scientific_journal-4617 21 28 the the DET american_scientific_journal-4617 21 29 quartic quartic ADJ american_scientific_journal-4617 21 30 spline spline NOUN american_scientific_journal-4617 21 31 interpolation interpolation NOUN american_scientific_journal-4617 21 32 and and CCONJ american_scientific_journal-4617 21 33 taylor taylor PROPN american_scientific_journal-4617 21 34 expansion expansion NOUN american_scientific_journal-4617 21 35 . . PUNCT american_scientific_journal-4617 22 1 finding find VERB american_scientific_journal-4617 22 2 the the DET american_scientific_journal-4617 22 3 solution solution NOUN american_scientific_journal-4617 22 4 of of ADP american_scientific_journal-4617 22 5 ordinary ordinary ADJ american_scientific_journal-4617 22 6 differential differential ADJ american_scientific_journal-4617 22 7 equations equation NOUN american_scientific_journal-4617 22 8 numerically numerically ADV american_scientific_journal-4617 22 9 is be AUX american_scientific_journal-4617 22 10 not not PART american_scientific_journal-4617 22 11 only only ADV american_scientific_journal-4617 22 12 concerned concern VERB american_scientific_journal-4617 22 13 with with ADP american_scientific_journal-4617 22 14 getting get VERB american_scientific_journal-4617 22 15 better well ADJ american_scientific_journal-4617 22 16 accuracy accuracy NOUN american_scientific_journal-4617 22 17 . . PUNCT american_scientific_journal-4617 23 1 it it PRON american_scientific_journal-4617 23 2 is be AUX american_scientific_journal-4617 23 3 also also ADV american_scientific_journal-4617 23 4 concerned concerned ADJ american_scientific_journal-4617 23 5 with with ADP american_scientific_journal-4617 23 6 saving save VERB american_scientific_journal-4617 23 7 computational computational ADJ american_scientific_journal-4617 23 8 speed speed NOUN american_scientific_journal-4617 23 9 and and CCONJ american_scientific_journal-4617 23 10 effort effort NOUN american_scientific_journal-4617 23 11 . . PUNCT american_scientific_journal-4617 24 1 this this DET american_scientific_journal-4617 24 2 paper paper NOUN american_scientific_journal-4617 24 3 is be AUX american_scientific_journal-4617 24 4 therefore therefore ADV american_scientific_journal-4617 24 5 aimed aim VERB american_scientific_journal-4617 24 6 at at ADP american_scientific_journal-4617 24 7 finding find VERB american_scientific_journal-4617 24 8 the the DET american_scientific_journal-4617 24 9 solutions solution NOUN american_scientific_journal-4617 24 10 of of ADP american_scientific_journal-4617 24 11 general general ADJ american_scientific_journal-4617 24 12 second second ADJ american_scientific_journal-4617 24 13 order order NOUN american_scientific_journal-4617 24 14 linear linear ADJ american_scientific_journal-4617 24 15 and and CCONJ american_scientific_journal-4617 24 16 nonlinear nonlinear ADJ american_scientific_journal-4617 24 17 two two NUM american_scientific_journal-4617 24 18 point point NOUN american_scientific_journal-4617 24 19 boundary boundary ADJ american_scientific_journal-4617 24 20 value value NOUN american_scientific_journal-4617 24 21 problems problem NOUN american_scientific_journal-4617 24 22 of of ADP american_scientific_journal-4617 24 23 ordinary ordinary ADJ american_scientific_journal-4617 24 24 differential differential ADJ american_scientific_journal-4617 24 25 equations equation NOUN american_scientific_journal-4617 24 26 of of ADP american_scientific_journal-4617 24 27 the the DET american_scientific_journal-4617 24 28 form form NOUN american_scientific_journal-4617 24 29 : : PUNCT american_scientific_journal-4617 24 30 𝑦𝑦′′(π‘₯π‘₯ 𝑦𝑦′′(π‘₯π‘₯ NUM american_scientific_journal-4617 24 31 ) ) PUNCT american_scientific_journal-4617 24 32 + + CCONJ american_scientific_journal-4617 25 1 𝑝𝑝(π‘₯π‘₯)𝑦𝑦′(π‘₯π‘₯ 𝑝𝑝(π‘₯π‘₯)𝑦𝑦′(π‘₯π‘₯ PROPN american_scientific_journal-4617 25 2 ) ) PUNCT american_scientific_journal-4617 26 1 + + CCONJ american_scientific_journal-4617 26 2 𝑓𝑓(π‘₯π‘₯,𝑦𝑦 𝑓𝑓(π‘₯π‘₯,𝑦𝑦 NOUN american_scientific_journal-4617 26 3 ) ) PUNCT american_scientific_journal-4617 26 4 = = SYM american_scientific_journal-4617 26 5 𝑔𝑔(π‘₯π‘₯ 𝑔𝑔(π‘₯π‘₯ PROPN american_scientific_journal-4617 26 6 ) ) PUNCT american_scientific_journal-4617 26 7 , , PUNCT american_scientific_journal-4617 26 8 0 0 NUM american_scientific_journal-4617 26 9 ≀ ≀ NOUN american_scientific_journal-4617 26 10 π‘₯π‘₯ π‘₯π‘₯ ADP american_scientific_journal-4617 26 11 ≀ ≀ NUM american_scientific_journal-4617 26 12 1 1 NUM american_scientific_journal-4617 26 13 ( ( PUNCT american_scientific_journal-4617 26 14 1.1 1.1 NUM american_scientific_journal-4617 26 15 ) ) PUNCT american_scientific_journal-4617 26 16 subject subject NOUN american_scientific_journal-4617 26 17 to to ADP american_scientific_journal-4617 26 18 the the DET american_scientific_journal-4617 26 19 boundary boundary ADJ american_scientific_journal-4617 26 20 conditions condition NOUN american_scientific_journal-4617 26 21 : : PUNCT american_scientific_journal-4617 26 22 𝑦𝑦(0 𝑦𝑦(0 X american_scientific_journal-4617 26 23 ) ) PUNCT american_scientific_journal-4617 26 24 = = SYM american_scientific_journal-4617 27 1 𝛼𝛼 𝛼𝛼 PROPN american_scientific_journal-4617 27 2 , , PUNCT american_scientific_journal-4617 27 3 𝑦𝑦(1 𝑦𝑦(1 NOUN american_scientific_journal-4617 27 4 ) ) PUNCT american_scientific_journal-4617 27 5 = = PUNCT american_scientific_journal-4617 27 6 𝛽𝛽 𝛽𝛽 X american_scientific_journal-4617 27 7 ( ( PUNCT american_scientific_journal-4617 27 8 1.2 1.2 NUM american_scientific_journal-4617 27 9 ) ) PUNCT american_scientific_journal-4617 27 10 by by ADP american_scientific_journal-4617 27 11 using use VERB american_scientific_journal-4617 27 12 legendre legendre PROPN american_scientific_journal-4617 27 13 operational operational ADJ american_scientific_journal-4617 27 14 matrix matrix NOUN american_scientific_journal-4617 27 15 of of ADP american_scientific_journal-4617 27 16 differentiation differentiation NOUN american_scientific_journal-4617 27 17 . . PUNCT american_scientific_journal-4617 28 1 this this DET american_scientific_journal-4617 28 2 study study NOUN american_scientific_journal-4617 28 3 is be AUX american_scientific_journal-4617 28 4 a a DET american_scientific_journal-4617 28 5 contribution contribution NOUN american_scientific_journal-4617 28 6 towards towards ADP american_scientific_journal-4617 28 7 finding find VERB american_scientific_journal-4617 28 8 the the DET american_scientific_journal-4617 28 9 solutions solution NOUN american_scientific_journal-4617 28 10 of of ADP american_scientific_journal-4617 28 11 second second ADJ american_scientific_journal-4617 28 12 order order NOUN american_scientific_journal-4617 28 13 linear linear ADJ american_scientific_journal-4617 28 14 and and CCONJ american_scientific_journal-4617 28 15 nonlinear nonlinear ADJ american_scientific_journal-4617 28 16 two two NUM american_scientific_journal-4617 28 17 point point NOUN american_scientific_journal-4617 28 18 boundary boundary ADJ american_scientific_journal-4617 28 19 value value NOUN american_scientific_journal-4617 28 20 problems problem NOUN american_scientific_journal-4617 28 21 of of ADP american_scientific_journal-4617 28 22 ordinary ordinary ADJ american_scientific_journal-4617 28 23 differential differential ADJ american_scientific_journal-4617 28 24 equations equation NOUN american_scientific_journal-4617 28 25 , , PUNCT american_scientific_journal-4617 28 26 by by ADP american_scientific_journal-4617 28 27 using use VERB american_scientific_journal-4617 28 28 tau tau PROPN american_scientific_journal-4617 28 29 method method NOUN american_scientific_journal-4617 28 30 based base VERB american_scientific_journal-4617 28 31 on on ADP american_scientific_journal-4617 28 32 legendre legendre PROPN american_scientific_journal-4617 28 33 operational operational ADJ american_scientific_journal-4617 28 34 matrix matrix NOUN american_scientific_journal-4617 28 35 of of ADP american_scientific_journal-4617 28 36 differentiation differentiation NOUN american_scientific_journal-4617 28 37 . . PUNCT american_scientific_journal-4617 29 1 even even ADV american_scientific_journal-4617 29 2 though though SCONJ american_scientific_journal-4617 29 3 many many ADJ american_scientific_journal-4617 29 4 authors author NOUN american_scientific_journal-4617 29 5 have have AUX american_scientific_journal-4617 29 6 attempted attempt VERB american_scientific_journal-4617 29 7 to to PART american_scientific_journal-4617 29 8 obtain obtain VERB american_scientific_journal-4617 29 9 higher high ADJ american_scientific_journal-4617 29 10 accuracy accuracy NOUN american_scientific_journal-4617 29 11 rapidly rapidly ADV american_scientific_journal-4617 29 12 by by ADP american_scientific_journal-4617 29 13 using use VERB american_scientific_journal-4617 29 14 a a DET american_scientific_journal-4617 29 15 numerous numerous ADJ american_scientific_journal-4617 29 16 methods method NOUN american_scientific_journal-4617 29 17 like like ADP american_scientific_journal-4617 29 18 finite finite ADJ american_scientific_journal-4617 29 19 difference difference NOUN american_scientific_journal-4617 29 20 method method NOUN american_scientific_journal-4617 29 21 and and CCONJ american_scientific_journal-4617 29 22 spline spline NOUN american_scientific_journal-4617 29 23 methods method NOUN american_scientific_journal-4617 29 24 such such ADJ american_scientific_journal-4617 29 25 as as ADP american_scientific_journal-4617 29 26 cubic cubic ADJ american_scientific_journal-4617 29 27 b b NOUN american_scientific_journal-4617 29 28 - - PUNCT american_scientific_journal-4617 29 29 spline spline NOUN american_scientific_journal-4617 29 30 method method NOUN american_scientific_journal-4617 29 31 , , PUNCT american_scientific_journal-4617 29 32 an an DET american_scientific_journal-4617 29 33 extended extended ADJ american_scientific_journal-4617 29 34 cubic cubic ADJ american_scientific_journal-4617 29 35 b b NOUN american_scientific_journal-4617 29 36 - - PUNCT american_scientific_journal-4617 29 37 spline spline NOUN american_scientific_journal-4617 29 38 method method NOUN american_scientific_journal-4617 29 39 and and CCONJ american_scientific_journal-4617 29 40 quartic quartic ADJ american_scientific_journal-4617 29 41 b b X american_scientific_journal-4617 29 42 - - PUNCT american_scientific_journal-4617 29 43 spline spline NOUN american_scientific_journal-4617 29 44 method method NOUN american_scientific_journal-4617 29 45 , , PUNCT american_scientific_journal-4617 29 46 each each PRON american_scientific_journal-4617 29 47 of of ADP american_scientific_journal-4617 29 48 these these DET american_scientific_journal-4617 29 49 methods method NOUN american_scientific_journal-4617 29 50 are be AUX american_scientific_journal-4617 29 51 computational computational ADJ american_scientific_journal-4617 29 52 too too ADV american_scientific_journal-4617 29 53 cost cost NOUN american_scientific_journal-4617 29 54 and and CCONJ american_scientific_journal-4617 29 55 slow slow ADJ american_scientific_journal-4617 29 56 when when SCONJ american_scientific_journal-4617 29 57 compared compare VERB american_scientific_journal-4617 29 58 to to ADP american_scientific_journal-4617 29 59 the the DET american_scientific_journal-4617 29 60 present present ADJ american_scientific_journal-4617 29 61 method method NOUN american_scientific_journal-4617 29 62 . . PUNCT american_scientific_journal-4617 30 1 thus thus ADV american_scientific_journal-4617 30 2 , , PUNCT american_scientific_journal-4617 30 3 the the DET american_scientific_journal-4617 30 4 purpose purpose NOUN american_scientific_journal-4617 30 5 of of ADP american_scientific_journal-4617 30 6 this this DET american_scientific_journal-4617 30 7 study study NOUN american_scientific_journal-4617 30 8 was be AUX american_scientific_journal-4617 30 9 to to PART american_scientific_journal-4617 30 10 find find VERB american_scientific_journal-4617 30 11 the the DET american_scientific_journal-4617 30 12 solutions solution NOUN american_scientific_journal-4617 30 13 of of ADP american_scientific_journal-4617 30 14 second second ADJ american_scientific_journal-4617 30 15 order order NOUN american_scientific_journal-4617 30 16 linear linear ADJ american_scientific_journal-4617 30 17 and and CCONJ american_scientific_journal-4617 30 18 nonlinear nonlinear ADJ american_scientific_journal-4617 30 19 two two NUM american_scientific_journal-4617 30 20 point point NOUN american_scientific_journal-4617 30 21 boundary boundary ADJ american_scientific_journal-4617 30 22 value value NOUN american_scientific_journal-4617 30 23 problems problem NOUN american_scientific_journal-4617 30 24 of of ADP american_scientific_journal-4617 30 25 ordinary ordinary ADJ american_scientific_journal-4617 30 26 differential differential ADJ american_scientific_journal-4617 30 27 equations equation NOUN american_scientific_journal-4617 30 28 , , PUNCT american_scientific_journal-4617 30 29 by by ADP american_scientific_journal-4617 30 30 using use VERB american_scientific_journal-4617 30 31 tau tau PROPN american_scientific_journal-4617 30 32 method method NOUN american_scientific_journal-4617 30 33 based base VERB american_scientific_journal-4617 30 34 on on ADP american_scientific_journal-4617 30 35 legendre legendre PROPN american_scientific_journal-4617 30 36 operational operational ADJ american_scientific_journal-4617 30 37 matrix matrix NOUN american_scientific_journal-4617 30 38 of of ADP american_scientific_journal-4617 30 39 differentiation differentiation NOUN american_scientific_journal-4617 30 40 . . PUNCT american_scientific_journal-4617 31 1 the the DET american_scientific_journal-4617 31 2 advantage advantage NOUN american_scientific_journal-4617 31 3 of of ADP american_scientific_journal-4617 31 4 this this DET american_scientific_journal-4617 31 5 method method NOUN american_scientific_journal-4617 31 6 over over ADP american_scientific_journal-4617 31 7 the the DET american_scientific_journal-4617 31 8 other other ADJ american_scientific_journal-4617 31 9 methods method NOUN american_scientific_journal-4617 31 10 is be AUX american_scientific_journal-4617 31 11 therefore therefore ADV american_scientific_journal-4617 31 12 american american ADJ american_scientific_journal-4617 31 13 scientific scientific ADJ american_scientific_journal-4617 31 14 research research PROPN american_scientific_journal-4617 31 15 journal journal NOUN american_scientific_journal-4617 31 16 for for ADP american_scientific_journal-4617 31 17 engineering engineering NOUN american_scientific_journal-4617 31 18 , , PUNCT american_scientific_journal-4617 31 19 technology technology NOUN american_scientific_journal-4617 31 20 , , PUNCT american_scientific_journal-4617 31 21 and and CCONJ american_scientific_journal-4617 31 22 sciences science NOUN american_scientific_journal-4617 31 23 ( ( PUNCT american_scientific_journal-4617 31 24 asrjets asrjet NOUN american_scientific_journal-4617 31 25 ) ) PUNCT american_scientific_journal-4617 31 26 ( ( PUNCT american_scientific_journal-4617 31 27 2019 2019 NUM american_scientific_journal-4617 31 28 ) ) PUNCT american_scientific_journal-4617 31 29 volume volume NOUN american_scientific_journal-4617 31 30 51 51 NUM american_scientific_journal-4617 31 31 , , PUNCT american_scientific_journal-4617 31 32 no no DET american_scientific_journal-4617 31 33 1 1 NUM american_scientific_journal-4617 31 34 , , PUNCT american_scientific_journal-4617 31 35 pp pp ADV american_scientific_journal-4617 31 36 225 225 NUM american_scientific_journal-4617 31 37 - - SYM american_scientific_journal-4617 31 38 234 234 NUM american_scientific_journal-4617 31 39 227 227 NUM american_scientific_journal-4617 31 40 β€’ β€’ NOUN american_scientific_journal-4617 31 41 it it PRON american_scientific_journal-4617 31 42 is be AUX american_scientific_journal-4617 31 43 computational computational ADJ american_scientific_journal-4617 31 44 less less ADJ american_scientific_journal-4617 31 45 cost cost NOUN american_scientific_journal-4617 31 46 , , PUNCT american_scientific_journal-4617 31 47 β€’ β€’ ADP american_scientific_journal-4617 31 48 it it PRON american_scientific_journal-4617 31 49 needs need VERB american_scientific_journal-4617 31 50 less less ADV american_scientific_journal-4617 31 51 computational computational ADJ american_scientific_journal-4617 31 52 time time NOUN american_scientific_journal-4617 31 53 and and CCONJ american_scientific_journal-4617 31 54 effort effort NOUN american_scientific_journal-4617 31 55 and and CCONJ american_scientific_journal-4617 31 56 β€’ β€’ NOUN american_scientific_journal-4617 31 57 it it PRON american_scientific_journal-4617 31 58 has have VERB american_scientific_journal-4617 31 59 better well ADJ american_scientific_journal-4617 31 60 accuracy accuracy NOUN american_scientific_journal-4617 31 61 . . PUNCT american_scientific_journal-4617 32 1 the the DET american_scientific_journal-4617 32 2 remainder remainder NOUN american_scientific_journal-4617 32 3 of of ADP american_scientific_journal-4617 32 4 this this DET american_scientific_journal-4617 32 5 paper paper NOUN american_scientific_journal-4617 32 6 has have AUX american_scientific_journal-4617 32 7 been be AUX american_scientific_journal-4617 32 8 organized organize VERB american_scientific_journal-4617 32 9 in in ADP american_scientific_journal-4617 32 10 the the DET american_scientific_journal-4617 32 11 following follow VERB american_scientific_journal-4617 32 12 order order NOUN american_scientific_journal-4617 32 13 and and CCONJ american_scientific_journal-4617 32 14 procedures procedure NOUN american_scientific_journal-4617 32 15 . . PUNCT american_scientific_journal-4617 33 1 β€’ β€’ NUM american_scientific_journal-4617 33 2 in in ADP american_scientific_journal-4617 33 3 section section NOUN american_scientific_journal-4617 33 4 2 2 NUM american_scientific_journal-4617 33 5 legendre legendre NOUN american_scientific_journal-4617 33 6 polynomial polynomial ADJ american_scientific_journal-4617 33 7 and and CCONJ american_scientific_journal-4617 33 8 its its PRON american_scientific_journal-4617 33 9 operational operational ADJ american_scientific_journal-4617 33 10 matrix matrix NOUN american_scientific_journal-4617 33 11 of of ADP american_scientific_journal-4617 33 12 differentiation differentiation NOUN american_scientific_journal-4617 33 13 has have AUX american_scientific_journal-4617 33 14 been be AUX american_scientific_journal-4617 33 15 discussed discuss VERB american_scientific_journal-4617 33 16 . . PUNCT american_scientific_journal-4617 34 1 β€’ β€’ NUM american_scientific_journal-4617 34 2 in in ADP american_scientific_journal-4617 34 3 section section NOUN american_scientific_journal-4617 34 4 3 3 NUM american_scientific_journal-4617 34 5 the the DET american_scientific_journal-4617 34 6 application application NOUN american_scientific_journal-4617 34 7 of of ADP american_scientific_journal-4617 34 8 this this DET american_scientific_journal-4617 34 9 method method NOUN american_scientific_journal-4617 34 10 has have AUX american_scientific_journal-4617 34 11 been be AUX american_scientific_journal-4617 34 12 discussed discuss VERB american_scientific_journal-4617 34 13 in in ADP american_scientific_journal-4617 34 14 brief brief NOUN american_scientific_journal-4617 34 15 . . PUNCT american_scientific_journal-4617 35 1 β€’ β€’ NUM american_scientific_journal-4617 35 2 in in ADP american_scientific_journal-4617 35 3 section section NOUN american_scientific_journal-4617 35 4 4 4 NUM american_scientific_journal-4617 35 5 this this DET american_scientific_journal-4617 35 6 method method NOUN american_scientific_journal-4617 35 7 has have AUX american_scientific_journal-4617 35 8 been be AUX american_scientific_journal-4617 35 9 used use VERB american_scientific_journal-4617 35 10 to to PART american_scientific_journal-4617 35 11 solve solve VERB american_scientific_journal-4617 35 12 some some DET american_scientific_journal-4617 35 13 numerical numerical ADJ american_scientific_journal-4617 35 14 examples example NOUN american_scientific_journal-4617 35 15 supported support VERB american_scientific_journal-4617 35 16 by by ADP american_scientific_journal-4617 35 17 graphs graph NOUN american_scientific_journal-4617 35 18 . . PUNCT american_scientific_journal-4617 36 1 β€’ β€’ ADP american_scientific_journal-4617 36 2 the the DET american_scientific_journal-4617 36 3 paper paper NOUN american_scientific_journal-4617 36 4 has have AUX american_scientific_journal-4617 36 5 been be AUX american_scientific_journal-4617 36 6 concluded conclude VERB american_scientific_journal-4617 36 7 and and CCONJ american_scientific_journal-4617 36 8 recommended recommend VERB american_scientific_journal-4617 36 9 in in ADP american_scientific_journal-4617 36 10 section section NOUN american_scientific_journal-4617 36 11 5 5 NUM american_scientific_journal-4617 36 12 . . NOUN american_scientific_journal-4617 37 1 2 2 NUM american_scientific_journal-4617 37 2 . . PUNCT american_scientific_journal-4617 37 3 legendre legendre PROPN american_scientific_journal-4617 37 4 polynomials polynomial NOUN american_scientific_journal-4617 37 5 and and CCONJ american_scientific_journal-4617 37 6 its its PRON american_scientific_journal-4617 37 7 operational operational ADJ american_scientific_journal-4617 37 8 matrix matrix NOUN american_scientific_journal-4617 37 9 of of ADP american_scientific_journal-4617 37 10 differentiation differentiation NOUN american_scientific_journal-4617 37 11 legendre legendre NOUN american_scientific_journal-4617 37 12 polynomials polynomial NOUN american_scientific_journal-4617 37 13 are be AUX american_scientific_journal-4617 37 14 defined define VERB american_scientific_journal-4617 37 15 on on ADP american_scientific_journal-4617 37 16 the the DET american_scientific_journal-4617 37 17 interval interval NOUN american_scientific_journal-4617 37 18 [ [ X american_scientific_journal-4617 37 19 βˆ’1,1 βˆ’1,1 X american_scientific_journal-4617 37 20 ] ] PUNCT american_scientific_journal-4617 37 21 and and CCONJ american_scientific_journal-4617 37 22 can can AUX american_scientific_journal-4617 37 23 be be AUX american_scientific_journal-4617 37 24 determined determine VERB american_scientific_journal-4617 37 25 with with ADP american_scientific_journal-4617 37 26 the the DET american_scientific_journal-4617 37 27 aids aid NOUN american_scientific_journal-4617 37 28 of of ADP american_scientific_journal-4617 37 29 the the DET american_scientific_journal-4617 37 30 following follow VERB american_scientific_journal-4617 37 31 recurrence recurrence NOUN american_scientific_journal-4617 37 32 formulae formulae VERB american_scientific_journal-4617 37 33 [ [ X american_scientific_journal-4617 37 34 5 5 NUM american_scientific_journal-4617 37 35 ] ] PUNCT american_scientific_journal-4617 37 36 . . PUNCT american_scientific_journal-4617 37 37 𝐿𝐿0(𝑧𝑧 𝐿𝐿0(𝑧𝑧 PROPN american_scientific_journal-4617 37 38 ) ) PUNCT american_scientific_journal-4617 38 1 = = SYM american_scientific_journal-4617 38 2 1 1 NUM american_scientific_journal-4617 38 3 , , PUNCT american_scientific_journal-4617 38 4 𝐿𝐿1(𝑧𝑧 𝐿𝐿1(𝑧𝑧 PROPN american_scientific_journal-4617 38 5 ) ) PUNCT american_scientific_journal-4617 38 6 = = PUNCT american_scientific_journal-4617 38 7 𝑧𝑧 𝑧𝑧 ADP american_scientific_journal-4617 38 8 πΏπΏπ‘Ÿπ‘Ÿ+1(𝑧𝑧 πΏπΏπ‘Ÿπ‘Ÿ+1(𝑧𝑧 NOUN american_scientific_journal-4617 38 9 ) ) PUNCT american_scientific_journal-4617 39 1 = = PUNCT american_scientific_journal-4617 40 1 2π‘Ÿπ‘Ÿ+1 2π‘Ÿπ‘Ÿ+1 NUM american_scientific_journal-4617 40 2 π‘Ÿπ‘Ÿ+1 π‘Ÿπ‘Ÿ+1 NUM american_scientific_journal-4617 40 3 ( ( PUNCT american_scientific_journal-4617 40 4 𝑧𝑧)πΏπΏπ‘Ÿπ‘Ÿ(𝑧𝑧 𝑧𝑧)πΏπΏπ‘Ÿπ‘Ÿ(𝑧𝑧 NOUN american_scientific_journal-4617 40 5 ) ) PUNCT american_scientific_journal-4617 40 6 βˆ’ βˆ’ PROPN american_scientific_journal-4617 41 1 π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿ ADP american_scientific_journal-4617 41 2 π‘Ÿπ‘Ÿ+1 π‘Ÿπ‘Ÿ+1 NUM american_scientific_journal-4617 41 3 πΏπΏπ‘Ÿπ‘Ÿβˆ’1(𝑧𝑧 πΏπΏπ‘Ÿπ‘Ÿβˆ’1(𝑧𝑧 NOUN american_scientific_journal-4617 41 4 ) ) PUNCT american_scientific_journal-4617 41 5 ; ; PUNCT american_scientific_journal-4617 41 6 π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿ X american_scientific_journal-4617 41 7 = = SYM american_scientific_journal-4617 41 8 1,2,3 1,2,3 NUM american_scientific_journal-4617 41 9 , , PUNCT american_scientific_journal-4617 41 10 . . PUNCT american_scientific_journal-4617 41 11 . . PUNCT american_scientific_journal-4617 41 12 . . PUNCT american_scientific_journal-4617 42 1 ( ( PUNCT american_scientific_journal-4617 42 2 2.1 2.1 NUM american_scientific_journal-4617 42 3 ) ) PUNCT american_scientific_journal-4617 42 4 in in ADP american_scientific_journal-4617 42 5 order order NOUN american_scientific_journal-4617 42 6 to to PART american_scientific_journal-4617 42 7 use use VERB american_scientific_journal-4617 42 8 this this DET american_scientific_journal-4617 42 9 polynomials polynomial NOUN american_scientific_journal-4617 42 10 on on ADP american_scientific_journal-4617 42 11 the the DET american_scientific_journal-4617 42 12 interval interval NOUN american_scientific_journal-4617 42 13 [ [ X american_scientific_journal-4617 42 14 0 0 NUM american_scientific_journal-4617 42 15 , , PUNCT american_scientific_journal-4617 42 16 1 1 NUM american_scientific_journal-4617 42 17 ] ] PUNCT american_scientific_journal-4617 42 18 , , PUNCT american_scientific_journal-4617 42 19 the the DET american_scientific_journal-4617 42 20 so so ADV american_scientific_journal-4617 42 21 - - PUNCT american_scientific_journal-4617 42 22 called call VERB american_scientific_journal-4617 42 23 shifted shift VERB american_scientific_journal-4617 42 24 legendre legendre PROPN american_scientific_journal-4617 42 25 polynomials polynomial NOUN american_scientific_journal-4617 42 26 is be AUX american_scientific_journal-4617 42 27 defined define VERB american_scientific_journal-4617 42 28 by by ADP american_scientific_journal-4617 42 29 introducing introduce VERB american_scientific_journal-4617 42 30 𝑧𝑧 𝑧𝑧 PRON american_scientific_journal-4617 42 31 = = PROPN american_scientific_journal-4617 42 32 2π‘₯π‘₯ 2π‘₯π‘₯ NOUN american_scientific_journal-4617 42 33 βˆ’ βˆ’ NOUN american_scientific_journal-4617 43 1 1 1 X american_scientific_journal-4617 43 2 . . PUNCT american_scientific_journal-4617 44 1 let let VERB american_scientific_journal-4617 44 2 the the DET american_scientific_journal-4617 44 3 shifted shift VERB american_scientific_journal-4617 44 4 legendre legendre PROPN american_scientific_journal-4617 44 5 polynomial polynomial PROPN american_scientific_journal-4617 44 6 πΏπΏπ‘Ÿπ‘Ÿ(2π‘₯π‘₯ πΏπΏπ‘Ÿπ‘Ÿ(2π‘₯π‘₯ NOUN american_scientific_journal-4617 45 1 βˆ’ βˆ’ NOUN american_scientific_journal-4617 45 2 1 1 X american_scientific_journal-4617 45 3 ) ) PUNCT american_scientific_journal-4617 45 4 be be AUX american_scientific_journal-4617 45 5 denoted denote VERB american_scientific_journal-4617 45 6 by by ADP american_scientific_journal-4617 45 7 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ NOUN american_scientific_journal-4617 45 8 ) ) PUNCT american_scientific_journal-4617 45 9 . . PUNCT american_scientific_journal-4617 46 1 then then ADV american_scientific_journal-4617 46 2 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ X american_scientific_journal-4617 46 3 ) ) PUNCT american_scientific_journal-4617 46 4 can can AUX american_scientific_journal-4617 46 5 be be AUX american_scientific_journal-4617 46 6 obtained obtain VERB american_scientific_journal-4617 46 7 as as ADP american_scientific_journal-4617 46 8 follows follow VERB american_scientific_journal-4617 46 9 : : PUNCT american_scientific_journal-4617 46 10 𝑃𝑃0(π‘₯π‘₯ 𝑃𝑃0(π‘₯π‘₯ X american_scientific_journal-4617 46 11 ) ) PUNCT american_scientific_journal-4617 46 12 = = SYM american_scientific_journal-4617 46 13 1 1 NUM american_scientific_journal-4617 46 14 , , PUNCT american_scientific_journal-4617 46 15 𝑃𝑃1(π‘₯π‘₯ 𝑃𝑃1(π‘₯π‘₯ NUM american_scientific_journal-4617 46 16 ) ) PUNCT american_scientific_journal-4617 46 17 = = SYM american_scientific_journal-4617 47 1 2π‘₯π‘₯ 2π‘₯π‘₯ NOUN american_scientific_journal-4617 47 2 βˆ’ βˆ’ NOUN american_scientific_journal-4617 47 3 1 1 NUM american_scientific_journal-4617 47 4 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ+1(π‘₯π‘₯ π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ+1(π‘₯π‘₯ NOUN american_scientific_journal-4617 47 5 ) ) PUNCT american_scientific_journal-4617 47 6 = = PUNCT american_scientific_journal-4617 48 1 2π‘Ÿπ‘Ÿ+1 2π‘Ÿπ‘Ÿ+1 NUM american_scientific_journal-4617 48 2 π‘Ÿπ‘Ÿ+1 π‘Ÿπ‘Ÿ+1 NUM american_scientific_journal-4617 48 3 ( ( PUNCT american_scientific_journal-4617 48 4 2π‘₯π‘₯ 2π‘₯π‘₯ NOUN american_scientific_journal-4617 48 5 βˆ’ βˆ’ PROPN american_scientific_journal-4617 48 6 1)π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ 1)π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ NUM american_scientific_journal-4617 48 7 ) ) PUNCT american_scientific_journal-4617 48 8 βˆ’ βˆ’ PROPN american_scientific_journal-4617 48 9 π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿ ADP american_scientific_journal-4617 48 10 π‘Ÿπ‘Ÿ+1 π‘Ÿπ‘Ÿ+1 NUM american_scientific_journal-4617 48 11 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿβˆ’1(π‘₯π‘₯ π‘ƒπ‘ƒπ‘Ÿπ‘Ÿβˆ’1(π‘₯π‘₯ NOUN american_scientific_journal-4617 48 12 ) ) PUNCT american_scientific_journal-4617 48 13 ; ; PUNCT american_scientific_journal-4617 48 14 π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿ X american_scientific_journal-4617 48 15 = = SYM american_scientific_journal-4617 48 16 1,2,3 1,2,3 NUM american_scientific_journal-4617 48 17 , , PUNCT american_scientific_journal-4617 48 18 . . PUNCT american_scientific_journal-4617 48 19 . . PUNCT american_scientific_journal-4617 48 20 . . PUNCT american_scientific_journal-4617 49 1 ( ( PUNCT american_scientific_journal-4617 49 2 2.2 2.2 NUM american_scientific_journal-4617 49 3 ) ) PUNCT american_scientific_journal-4617 49 4 the the DET american_scientific_journal-4617 49 5 analytical analytical ADJ american_scientific_journal-4617 49 6 form form NOUN american_scientific_journal-4617 49 7 of of ADP american_scientific_journal-4617 49 8 the the DET american_scientific_journal-4617 49 9 shifted shift VERB american_scientific_journal-4617 49 10 legendre legendre PROPN american_scientific_journal-4617 49 11 polynomial polynomial PROPN american_scientific_journal-4617 49 12 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ PROPN american_scientific_journal-4617 49 13 ) ) PUNCT american_scientific_journal-4617 49 14 of of ADP american_scientific_journal-4617 49 15 degree degree NOUN american_scientific_journal-4617 49 16 r r NOUN american_scientific_journal-4617 49 17 is be AUX american_scientific_journal-4617 49 18 given give VERB american_scientific_journal-4617 49 19 by by ADP american_scientific_journal-4617 49 20 ; ; PUNCT american_scientific_journal-4617 49 21 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ X american_scientific_journal-4617 49 22 ) ) PUNCT american_scientific_journal-4617 49 23 = = PUNCT american_scientific_journal-4617 49 24 βˆ‘ βˆ‘ PUNCT american_scientific_journal-4617 49 25 ( ( PUNCT american_scientific_journal-4617 49 26 βˆ’1)π‘Ÿπ‘Ÿ+π‘˜π‘˜π‘Ÿπ‘Ÿ βˆ’1)π‘Ÿπ‘Ÿ+π‘˜π‘˜π‘Ÿπ‘Ÿ NOUN american_scientific_journal-4617 49 27 π‘˜π‘˜=0 π‘˜π‘˜=0 NOUN american_scientific_journal-4617 49 28 ( ( PUNCT american_scientific_journal-4617 49 29 π‘Ÿπ‘Ÿ+π‘˜π‘˜ π‘Ÿπ‘Ÿ+π‘˜π‘˜ NUM american_scientific_journal-4617 49 30 ) ) PUNCT american_scientific_journal-4617 49 31 ! ! PUNCT american_scientific_journal-4617 50 1 ( ( PUNCT american_scientific_journal-4617 50 2 π‘Ÿπ‘Ÿβˆ’π‘˜π‘˜ π‘Ÿπ‘Ÿβˆ’π‘˜π‘˜ ADV american_scientific_journal-4617 50 3 ) ) PUNCT american_scientific_journal-4617 50 4 ! ! PUNCT american_scientific_journal-4617 51 1 π‘₯π‘₯π‘˜π‘˜ π‘₯π‘₯π‘˜π‘˜ INTJ american_scientific_journal-4617 51 2 ( ( PUNCT american_scientific_journal-4617 51 3 π‘˜π‘˜!)2 π‘˜π‘˜!)2 X american_scientific_journal-4617 51 4 ( ( PUNCT american_scientific_journal-4617 51 5 2.3 2.3 NUM american_scientific_journal-4617 51 6 ) ) PUNCT american_scientific_journal-4617 51 7 nb nb NOUN american_scientific_journal-4617 51 8 : : PUNCT american_scientific_journal-4617 51 9 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(0 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(0 X american_scientific_journal-4617 51 10 ) ) PUNCT american_scientific_journal-4617 51 11 = = SYM american_scientific_journal-4617 51 12 ( ( PUNCT american_scientific_journal-4617 51 13 βˆ’1)π‘Ÿπ‘Ÿ βˆ’1)π‘Ÿπ‘Ÿ INTJ american_scientific_journal-4617 51 14 and and CCONJ american_scientific_journal-4617 51 15 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(1 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(1 NOUN american_scientific_journal-4617 51 16 ) ) PUNCT american_scientific_journal-4617 51 17 = = PUNCT american_scientific_journal-4617 52 1 1 1 NUM american_scientific_journal-4617 52 2 the the DET american_scientific_journal-4617 52 3 orthogonality orthogonality NOUN american_scientific_journal-4617 52 4 condition condition NOUN american_scientific_journal-4617 52 5 for for ADP american_scientific_journal-4617 52 6 these these DET american_scientific_journal-4617 52 7 shifted shift VERB american_scientific_journal-4617 52 8 legendre legendre PROPN american_scientific_journal-4617 52 9 polynomials polynomial NOUN american_scientific_journal-4617 52 10 is be AUX american_scientific_journal-4617 52 11 : : PUNCT american_scientific_journal-4617 52 12 ∫ ∫ PROPN american_scientific_journal-4617 52 13 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)𝑃𝑃𝑠𝑠(π‘₯π‘₯)1 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)𝑃𝑃𝑠𝑠(π‘₯π‘₯)1 PROPN american_scientific_journal-4617 52 14 0 0 NUM american_scientific_journal-4617 52 15 𝑑𝑑π‘₯π‘₯ 𝑑𝑑π‘₯π‘₯ NOUN american_scientific_journal-4617 52 16 = = SYM american_scientific_journal-4617 52 17 οΏ½ οΏ½ PROPN american_scientific_journal-4617 52 18 1 1 NUM american_scientific_journal-4617 52 19 2π‘Ÿπ‘Ÿ+1 2π‘Ÿπ‘Ÿ+1 NUM american_scientific_journal-4617 52 20 π‘“π‘“π‘“π‘“π‘Ÿπ‘Ÿ π‘“π‘“π‘“π‘“π‘Ÿπ‘Ÿ NOUN american_scientific_journal-4617 52 21 π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿ NOUN american_scientific_journal-4617 52 22 = = PUNCT american_scientific_journal-4617 52 23 𝑠𝑠 𝑠𝑠 ADP american_scientific_journal-4617 52 24 0 0 NUM american_scientific_journal-4617 52 25 π‘“π‘“π‘“π‘“π‘Ÿπ‘Ÿ π‘“π‘“π‘“π‘“π‘Ÿπ‘Ÿ NOUN american_scientific_journal-4617 52 26 π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿ ADP american_scientific_journal-4617 52 27 β‰  β‰  PROPN american_scientific_journal-4617 52 28 𝑠𝑠 𝑠𝑠 ADP american_scientific_journal-4617 52 29 ( ( PUNCT american_scientific_journal-4617 52 30 2.4 2.4 NUM american_scientific_journal-4617 52 31 ) ) PUNCT american_scientific_journal-4617 52 32 any any DET american_scientific_journal-4617 52 33 function function NOUN american_scientific_journal-4617 52 34 𝑦𝑦(π‘₯π‘₯ 𝑦𝑦(π‘₯π‘₯ PROPN american_scientific_journal-4617 52 35 ) ) PUNCT american_scientific_journal-4617 52 36 ∈ ∈ PROPN american_scientific_journal-4617 52 37 𝐿𝐿2[0,1 𝐿𝐿2[0,1 NOUN american_scientific_journal-4617 52 38 ] ] PUNCT american_scientific_journal-4617 52 39 can can AUX american_scientific_journal-4617 52 40 be be AUX american_scientific_journal-4617 52 41 approximated approximate VERB american_scientific_journal-4617 52 42 in in ADP american_scientific_journal-4617 52 43 terms term NOUN american_scientific_journal-4617 52 44 of of ADP american_scientific_journal-4617 52 45 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ NOUN american_scientific_journal-4617 52 46 ) ) PUNCT american_scientific_journal-4617 52 47 by by ADP american_scientific_journal-4617 52 48 : : PUNCT american_scientific_journal-4617 52 49 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-4617 52 50 οΏ½ οΏ½ PROPN american_scientific_journal-4617 52 51 (π‘₯π‘₯ (π‘₯π‘₯ PUNCT american_scientific_journal-4617 52 52 ) ) PUNCT american_scientific_journal-4617 52 53 = = PUNCT american_scientific_journal-4617 53 1 βˆ‘ βˆ‘ PUNCT american_scientific_journal-4617 53 2 π‘π‘π‘Ÿπ‘Ÿπ‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)∞ π‘π‘π‘Ÿπ‘Ÿπ‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)∞ ADP american_scientific_journal-4617 53 3 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PROPN american_scientific_journal-4617 53 4 ( ( PUNCT american_scientific_journal-4617 53 5 2.5 2.5 NUM american_scientific_journal-4617 53 6 ) ) PUNCT american_scientific_journal-4617 53 7 american american ADJ american_scientific_journal-4617 53 8 scientific scientific ADJ american_scientific_journal-4617 53 9 research research NOUN american_scientific_journal-4617 53 10 journal journal NOUN american_scientific_journal-4617 53 11 for for ADP american_scientific_journal-4617 53 12 engineering engineering NOUN american_scientific_journal-4617 53 13 , , PUNCT american_scientific_journal-4617 53 14 technology technology NOUN american_scientific_journal-4617 53 15 , , PUNCT american_scientific_journal-4617 53 16 and and CCONJ american_scientific_journal-4617 53 17 sciences science NOUN american_scientific_journal-4617 53 18 ( ( PUNCT american_scientific_journal-4617 53 19 asrjets asrjet NOUN american_scientific_journal-4617 53 20 ) ) PUNCT american_scientific_journal-4617 53 21 ( ( PUNCT american_scientific_journal-4617 53 22 2019 2019 NUM american_scientific_journal-4617 53 23 ) ) PUNCT american_scientific_journal-4617 53 24 volume volume NOUN american_scientific_journal-4617 53 25 51 51 NUM american_scientific_journal-4617 53 26 , , PUNCT american_scientific_journal-4617 53 27 no no DET american_scientific_journal-4617 53 28 1 1 NUM american_scientific_journal-4617 53 29 , , PUNCT american_scientific_journal-4617 53 30 pp pp ADV american_scientific_journal-4617 53 31 225 225 NUM american_scientific_journal-4617 53 32 - - SYM american_scientific_journal-4617 53 33 234 234 NUM american_scientific_journal-4617 53 34 228 228 NUM american_scientific_journal-4617 53 35 where where SCONJ american_scientific_journal-4617 53 36 the the DET american_scientific_journal-4617 53 37 coefficients coefficient NOUN american_scientific_journal-4617 53 38 π‘π‘π‘Ÿπ‘Ÿ π‘π‘π‘Ÿπ‘Ÿ VERB american_scientific_journal-4617 53 39 are be AUX american_scientific_journal-4617 53 40 given give VERB american_scientific_journal-4617 53 41 by by ADP american_scientific_journal-4617 53 42 π‘π‘π‘Ÿπ‘Ÿ π‘π‘π‘Ÿπ‘Ÿ NOUN american_scientific_journal-4617 53 43 = = SYM american_scientific_journal-4617 53 44 ( ( PUNCT american_scientific_journal-4617 53 45 2π‘Ÿπ‘Ÿ 2π‘Ÿπ‘Ÿ ADJ american_scientific_journal-4617 53 46 + + CCONJ american_scientific_journal-4617 53 47 1)∫ 1)∫ NUM american_scientific_journal-4617 53 48 𝑦𝑦(π‘₯π‘₯)1 𝑦𝑦(π‘₯π‘₯)1 NOUN american_scientific_journal-4617 53 49 0 0 NUM american_scientific_journal-4617 53 50 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)𝑑𝑑π‘₯π‘₯ π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)𝑑𝑑π‘₯π‘₯ ADJ american_scientific_journal-4617 53 51 ; ; PUNCT american_scientific_journal-4617 53 52 π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿ X american_scientific_journal-4617 53 53 = = NOUN american_scientific_journal-4617 53 54 1,2,3 1,2,3 NUM american_scientific_journal-4617 53 55 , , PUNCT american_scientific_journal-4617 53 56 . . PUNCT american_scientific_journal-4617 53 57 . . PUNCT american_scientific_journal-4617 53 58 . . PUNCT american_scientific_journal-4617 54 1 ( ( PUNCT american_scientific_journal-4617 54 2 2.6 2.6 NUM american_scientific_journal-4617 54 3 ) ) PUNCT american_scientific_journal-4617 54 4 by by ADP american_scientific_journal-4617 54 5 considering consider VERB american_scientific_journal-4617 54 6 only only ADV american_scientific_journal-4617 54 7 the the DET american_scientific_journal-4617 54 8 first first ADJ american_scientific_journal-4617 54 9 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 54 10 + + ADJ american_scientific_journal-4617 54 11 1 1 NUM american_scientific_journal-4617 54 12 terms term NOUN american_scientific_journal-4617 54 13 of of ADP american_scientific_journal-4617 54 14 the the DET american_scientific_journal-4617 54 15 series series NOUN american_scientific_journal-4617 54 16 ( ( PUNCT american_scientific_journal-4617 54 17 2.5 2.5 NUM american_scientific_journal-4617 54 18 ) ) PUNCT american_scientific_journal-4617 54 19 we we PRON american_scientific_journal-4617 54 20 get get VERB american_scientific_journal-4617 54 21 ; ; PUNCT american_scientific_journal-4617 54 22 π‘¦π‘¦π‘šπ‘š(π‘₯π‘₯ π‘¦π‘¦π‘šπ‘š(π‘₯π‘₯ PROPN american_scientific_journal-4617 54 23 ) ) PUNCT american_scientific_journal-4617 54 24 = = PUNCT american_scientific_journal-4617 55 1 βˆ‘ βˆ‘ PUNCT american_scientific_journal-4617 55 2 π‘π‘π‘Ÿπ‘Ÿπ‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)π‘šπ‘š π‘π‘π‘Ÿπ‘Ÿπ‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)π‘šπ‘š PUNCT american_scientific_journal-4617 55 3 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PROPN american_scientific_journal-4617 55 4 = = PUNCT american_scientific_journal-4617 55 5 πΆπΆπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ πΆπΆπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ X american_scientific_journal-4617 55 6 ) ) PUNCT american_scientific_journal-4617 55 7 ( ( PUNCT american_scientific_journal-4617 55 8 2.7 2.7 NUM american_scientific_journal-4617 55 9 ) ) PUNCT american_scientific_journal-4617 55 10 where where SCONJ american_scientific_journal-4617 55 11 𝐢𝐢𝑇𝑇 𝐢𝐢𝑇𝑇 PROPN american_scientific_journal-4617 56 1 = = PUNCT american_scientific_journal-4617 57 1 [ [ X american_scientific_journal-4617 57 2 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 57 3 , , PUNCT american_scientific_journal-4617 57 4 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 57 5 , , PUNCT american_scientific_journal-4617 57 6 . . PUNCT american_scientific_journal-4617 57 7 . . PUNCT american_scientific_journal-4617 57 8 . . PUNCT american_scientific_journal-4617 58 1 , , PUNCT american_scientific_journal-4617 58 2 π‘π‘π‘šπ‘š π‘π‘π‘šπ‘š PROPN american_scientific_journal-4617 58 3 ] ] PUNCT american_scientific_journal-4617 58 4 is be AUX american_scientific_journal-4617 58 5 the the DET american_scientific_journal-4617 58 6 shifted shift VERB american_scientific_journal-4617 58 7 legendre legendre PROPN american_scientific_journal-4617 58 8 coefficient coefficient PROPN american_scientific_journal-4617 58 9 and and CCONJ american_scientific_journal-4617 58 10 πœ‘πœ‘(π‘₯π‘₯ πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 58 11 ) ) PUNCT american_scientific_journal-4617 59 1 = = PUNCT american_scientific_journal-4617 60 1 [ [ X american_scientific_journal-4617 60 2 𝑝𝑝0(π‘₯π‘₯ 𝑝𝑝0(π‘₯π‘₯ X american_scientific_journal-4617 60 3 ) ) PUNCT american_scientific_journal-4617 60 4 , , PUNCT american_scientific_journal-4617 60 5 𝑝𝑝1(π‘₯π‘₯ 𝑝𝑝1(π‘₯π‘₯ PROPN american_scientific_journal-4617 60 6 ) ) PUNCT american_scientific_journal-4617 60 7 , , PUNCT american_scientific_journal-4617 60 8 … … PUNCT american_scientific_journal-4617 60 9 , , PUNCT american_scientific_journal-4617 60 10 π‘π‘π‘šπ‘š(π‘₯π‘₯)]𝑇𝑇 π‘π‘π‘šπ‘š(π‘₯π‘₯)]𝑇𝑇 NOUN american_scientific_journal-4617 60 11 is be AUX american_scientific_journal-4617 60 12 the the DET american_scientific_journal-4617 60 13 shifted shift VERB american_scientific_journal-4617 60 14 legendre legendre PROPN american_scientific_journal-4617 60 15 vector vector PROPN american_scientific_journal-4617 60 16 . . PUNCT american_scientific_journal-4617 61 1 the the DET american_scientific_journal-4617 61 2 derivative derivative NOUN american_scientific_journal-4617 61 3 of of ADP american_scientific_journal-4617 61 4 the the DET american_scientific_journal-4617 61 5 vector vector NOUN american_scientific_journal-4617 61 6 πœ‘πœ‘(π‘₯π‘₯ πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 61 7 ) ) PUNCT american_scientific_journal-4617 61 8 can can AUX american_scientific_journal-4617 61 9 be be AUX american_scientific_journal-4617 61 10 expressed express VERB american_scientific_journal-4617 61 11 as as ADP american_scientific_journal-4617 61 12 : : PUNCT american_scientific_journal-4617 61 13 𝑑𝑑𝑑𝑑(π‘₯π‘₯ 𝑑𝑑𝑑𝑑(π‘₯π‘₯ NOUN american_scientific_journal-4617 61 14 ) ) PUNCT american_scientific_journal-4617 61 15 𝑑𝑑π‘₯π‘₯ 𝑑𝑑π‘₯π‘₯ NOUN american_scientific_journal-4617 61 16 = = SYM american_scientific_journal-4617 61 17 𝐷𝐷(1)πœ‘πœ‘(π‘₯π‘₯ 𝐷𝐷(1)πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 61 18 ) ) PUNCT american_scientific_journal-4617 61 19 ( ( PUNCT american_scientific_journal-4617 61 20 2.8 2.8 NUM american_scientific_journal-4617 61 21 ) ) PUNCT american_scientific_journal-4617 61 22 where where SCONJ american_scientific_journal-4617 61 23 𝐷𝐷(1 𝐷𝐷(1 PROPN american_scientific_journal-4617 61 24 ) ) PUNCT american_scientific_journal-4617 61 25 is be AUX american_scientific_journal-4617 61 26 ( ( PUNCT american_scientific_journal-4617 61 27 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 61 28 + + NOUN american_scientific_journal-4617 61 29 1 1 NUM american_scientific_journal-4617 61 30 ) ) PUNCT american_scientific_journal-4617 61 31 Γ— Γ— NOUN american_scientific_journal-4617 61 32 ( ( PUNCT american_scientific_journal-4617 61 33 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 61 34 + + ADJ american_scientific_journal-4617 61 35 1 1 X american_scientific_journal-4617 61 36 ) ) PUNCT american_scientific_journal-4617 61 37 operational operational ADJ american_scientific_journal-4617 61 38 matrix matrix NOUN american_scientific_journal-4617 61 39 of of ADP american_scientific_journal-4617 61 40 derivative derivative NOUN american_scientific_journal-4617 61 41 which which PRON american_scientific_journal-4617 61 42 is be AUX american_scientific_journal-4617 61 43 given give VERB american_scientific_journal-4617 61 44 by by ADP american_scientific_journal-4617 61 45 𝐷𝐷(1 𝐷𝐷(1 PROPN american_scientific_journal-4617 61 46 ) ) PUNCT american_scientific_journal-4617 61 47 = = PUNCT american_scientific_journal-4617 61 48 οΏ½ οΏ½ PROPN american_scientific_journal-4617 61 49 𝑑𝑑𝑖𝑖𝑖𝑖 𝑑𝑑𝑖𝑖𝑖𝑖 NOUN american_scientific_journal-4617 61 50 οΏ½ οΏ½ PROPN american_scientific_journal-4617 61 51 = = SYM american_scientific_journal-4617 61 52 οΏ½ οΏ½ PROPN american_scientific_journal-4617 61 53 2(2𝑗𝑗 2(2𝑗𝑗 NUM american_scientific_journal-4617 61 54 + + CCONJ american_scientific_journal-4617 61 55 1 1 NUM american_scientific_journal-4617 61 56 ) ) PUNCT american_scientific_journal-4617 61 57 π‘“π‘“π‘“π‘“π‘Ÿπ‘Ÿ π‘“π‘“π‘“π‘“π‘Ÿπ‘Ÿ NOUN american_scientific_journal-4617 61 58 𝑗𝑗 𝑗𝑗 NOUN american_scientific_journal-4617 62 1 = = PUNCT american_scientific_journal-4617 62 2 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-4617 62 3 βˆ’ βˆ’ PROPN american_scientific_journal-4617 62 4 π‘˜π‘˜ π‘˜π‘˜ NOUN american_scientific_journal-4617 62 5 0 0 NUM american_scientific_journal-4617 62 6 π‘“π‘“π‘œπ‘œβ„Žπ‘’π‘’π‘Ÿπ‘Ÿπ‘’π‘’π‘–π‘–π‘ π‘ π‘’π‘’ π‘“π‘“π‘œπ‘œβ„Žπ‘’π‘’π‘Ÿπ‘Ÿπ‘’π‘’π‘–π‘–π‘ π‘ π‘’π‘’ NOUN american_scientific_journal-4617 62 7 ; ; PUNCT american_scientific_journal-4617 62 8 οΏ½ οΏ½ PROPN american_scientific_journal-4617 62 9 π‘˜π‘˜ π‘˜π‘˜ NOUN american_scientific_journal-4617 62 10 = = SYM american_scientific_journal-4617 62 11 1,3 1,3 NUM american_scientific_journal-4617 62 12 , , PUNCT american_scientific_journal-4617 62 13 … … PUNCT american_scientific_journal-4617 62 14 , , PUNCT american_scientific_journal-4617 62 15 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 62 16 𝑖𝑖𝑓𝑓 𝑖𝑖𝑓𝑓 NOUN american_scientific_journal-4617 62 17 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 62 18 𝑖𝑖𝑠𝑠 𝑖𝑖𝑠𝑠 NOUN american_scientific_journal-4617 62 19 𝑓𝑓𝑑𝑑𝑑𝑑 𝑓𝑓𝑑𝑑𝑑𝑑 ADJ american_scientific_journal-4617 62 20 π‘˜π‘˜ π‘˜π‘˜ PROPN american_scientific_journal-4617 62 21 = = SYM american_scientific_journal-4617 62 22 1,3 1,3 NUM american_scientific_journal-4617 62 23 , , PUNCT american_scientific_journal-4617 62 24 … … PUNCT american_scientific_journal-4617 62 25 , , PUNCT american_scientific_journal-4617 62 26 π‘šπ‘šβˆ’ π‘šπ‘šβˆ’ PROPN american_scientific_journal-4617 62 27 1 1 NUM american_scientific_journal-4617 62 28 𝑖𝑖𝑓𝑓 𝑖𝑖𝑓𝑓 NOUN american_scientific_journal-4617 62 29 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 62 30 𝑖𝑖𝑠𝑠 𝑖𝑖𝑠𝑠 NOUN american_scientific_journal-4617 62 31 𝑒𝑒𝑒𝑒𝑒𝑒𝑛𝑛 𝑒𝑒𝑒𝑒𝑒𝑒𝑛𝑛 NOUN american_scientific_journal-4617 62 32 for for ADP american_scientific_journal-4617 62 33 example example NOUN american_scientific_journal-4617 62 34 for for ADP american_scientific_journal-4617 62 35 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 62 36 even even ADV american_scientific_journal-4617 62 37 we we PRON american_scientific_journal-4617 62 38 have have VERB american_scientific_journal-4617 62 39 ; ; PUNCT american_scientific_journal-4617 62 40 𝐷𝐷(1 𝐷𝐷(1 PROPN american_scientific_journal-4617 62 41 ) ) PUNCT american_scientific_journal-4617 62 42 = = SYM american_scientific_journal-4617 62 43 2 2 NUM american_scientific_journal-4617 62 44 ⎝ ⎝ NUM american_scientific_journal-4617 62 45 ⎜ ⎜ PROPN american_scientific_journal-4617 62 46 ⎜ ⎜ PROPN american_scientific_journal-4617 62 47 ⎜ ⎜ PROPN american_scientific_journal-4617 62 48 ⎜ ⎜ PROPN american_scientific_journal-4617 62 49 ⎜ ⎜ PROPN american_scientific_journal-4617 62 50 ⎜ ⎜ PROPN american_scientific_journal-4617 62 51 ⎜ ⎜ PROPN american_scientific_journal-4617 62 52 βŽ› βŽ› X american_scientific_journal-4617 62 53 0 0 NUM american_scientific_journal-4617 62 54 0 0 NUM american_scientific_journal-4617 62 55 0 0 NUM american_scientific_journal-4617 62 56 0 0 NUM american_scientific_journal-4617 62 57 ... ... PUNCT american_scientific_journal-4617 63 1 0 0 NUM american_scientific_journal-4617 63 2 0 0 NUM american_scientific_journal-4617 63 3 0 0 NUM american_scientific_journal-4617 63 4 1 1 NUM american_scientific_journal-4617 63 5 0 0 NUM american_scientific_journal-4617 63 6 0 0 NUM american_scientific_journal-4617 63 7 0 0 NUM american_scientific_journal-4617 63 8 ... ... PUNCT american_scientific_journal-4617 64 1 0 0 NUM american_scientific_journal-4617 64 2 0 0 NUM american_scientific_journal-4617 64 3 0 0 NUM american_scientific_journal-4617 64 4 0 0 NUM american_scientific_journal-4617 64 5 3 3 NUM american_scientific_journal-4617 64 6 0 0 NUM american_scientific_journal-4617 64 7 0 0 NUM american_scientific_journal-4617 64 8 ... ... PUNCT american_scientific_journal-4617 65 1 0 0 NUM american_scientific_journal-4617 65 2 0 0 NUM american_scientific_journal-4617 65 3 0 0 NUM american_scientific_journal-4617 65 4 1 1 NUM american_scientific_journal-4617 65 5 0 0 NUM american_scientific_journal-4617 65 6 5 5 NUM american_scientific_journal-4617 65 7 0 0 NUM american_scientific_journal-4617 65 8 ... ... PUNCT american_scientific_journal-4617 66 1 0 0 NUM american_scientific_journal-4617 66 2 0 0 NUM american_scientific_journal-4617 66 3 0 0 NUM american_scientific_journal-4617 66 4 . . PUNCT american_scientific_journal-4617 66 5 . . PUNCT american_scientific_journal-4617 66 6 . . PUNCT american_scientific_journal-4617 66 7 . . PUNCT american_scientific_journal-4617 66 8 ... ... PUNCT american_scientific_journal-4617 66 9 . . PUNCT american_scientific_journal-4617 66 10 . . PUNCT american_scientific_journal-4617 66 11 . . PUNCT american_scientific_journal-4617 67 1 1 1 NUM american_scientific_journal-4617 67 2 0 0 NUM american_scientific_journal-4617 67 3 5 5 NUM american_scientific_journal-4617 67 4 0 0 NUM american_scientific_journal-4617 67 5 ... ... PUNCT american_scientific_journal-4617 67 6 0 0 NUM american_scientific_journal-4617 67 7 0 0 NUM american_scientific_journal-4617 67 8 0 0 NUM american_scientific_journal-4617 67 9 3 3 NUM american_scientific_journal-4617 67 10 0 0 NUM american_scientific_journal-4617 67 11 7 7 NUM american_scientific_journal-4617 67 12 ... ... SYM american_scientific_journal-4617 67 13 0 0 NUM american_scientific_journal-4617 67 14 0 0 NUM american_scientific_journal-4617 67 15 Ο„ Ο„ PROPN american_scientific_journal-4617 67 16 Ο‰ Ο‰ NUM american_scientific_journal-4617 67 17 ⎠ ⎠ X american_scientific_journal-4617 67 18 ⎟ ⎟ X american_scientific_journal-4617 67 19 ⎟ ⎟ X american_scientific_journal-4617 67 20 ⎟ ⎟ X american_scientific_journal-4617 67 21 ⎟ ⎟ X american_scientific_journal-4617 67 22 ⎟ ⎟ PROPN american_scientific_journal-4617 67 23 ⎟ ⎟ PROPN american_scientific_journal-4617 67 24 ⎟ ⎟ PROPN american_scientific_journal-4617 67 25 ⎞ ⎞ VERB american_scientific_journal-4617 67 26 where where SCONJ american_scientific_journal-4617 67 27 𝜏𝜏 𝜏𝜏 ADJ american_scientific_journal-4617 67 28 = = PUNCT american_scientific_journal-4617 68 1 2π‘šπ‘š 2π‘šπ‘š ADJ american_scientific_journal-4617 68 2 βˆ’ βˆ’ NOUN american_scientific_journal-4617 68 3 3,πœ”πœ” 3,πœ”πœ” NUM american_scientific_journal-4617 68 4 = = SYM american_scientific_journal-4617 68 5 2π‘šπ‘š 2π‘šπ‘š ADJ american_scientific_journal-4617 68 6 βˆ’ βˆ’ NOUN american_scientific_journal-4617 68 7 1 1 NUM american_scientific_journal-4617 68 8 ( ( PUNCT american_scientific_journal-4617 68 9 2.9 2.9 NUM american_scientific_journal-4617 68 10 ) ) PUNCT american_scientific_journal-4617 68 11 from from ADP american_scientific_journal-4617 68 12 equation equation NOUN american_scientific_journal-4617 68 13 ( ( PUNCT american_scientific_journal-4617 68 14 2.8 2.8 NUM american_scientific_journal-4617 68 15 ) ) PUNCT american_scientific_journal-4617 68 16 it it PRON american_scientific_journal-4617 68 17 can can AUX american_scientific_journal-4617 68 18 be be AUX american_scientific_journal-4617 68 19 generalized generalize VERB american_scientific_journal-4617 68 20 for for ADP american_scientific_journal-4617 68 21 any any DET american_scientific_journal-4617 68 22 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4617 68 23 ∈ ∈ PROPN american_scientific_journal-4617 68 24 𝑁𝑁 𝑁𝑁 PROPN american_scientific_journal-4617 68 25 as as ADP american_scientific_journal-4617 68 26 : : PUNCT american_scientific_journal-4617 68 27 𝑑𝑑𝑛𝑛𝑑𝑑(π‘₯π‘₯ 𝑑𝑑𝑛𝑛𝑑𝑑(π‘₯π‘₯ NOUN american_scientific_journal-4617 68 28 ) ) PUNCT american_scientific_journal-4617 68 29 𝑑𝑑π‘₯π‘₯𝑛𝑛 𝑑𝑑π‘₯π‘₯𝑛𝑛 NOUN american_scientific_journal-4617 68 30 = = SYM american_scientific_journal-4617 68 31 ( ( PUNCT american_scientific_journal-4617 68 32 𝐷𝐷(1))π‘›π‘›πœ‘πœ‘(π‘₯π‘₯ 𝐷𝐷(1))π‘›π‘›πœ‘πœ‘(π‘₯π‘₯ ADJ american_scientific_journal-4617 68 33 ) ) PUNCT american_scientific_journal-4617 68 34 = = SYM american_scientific_journal-4617 68 35 𝐷𝐷(𝑛𝑛)πœ‘πœ‘(π‘₯π‘₯ 𝐷𝐷(𝑛𝑛)πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 68 36 ) ) PUNCT american_scientific_journal-4617 68 37 , , PUNCT american_scientific_journal-4617 68 38 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4617 68 39 = = SYM american_scientific_journal-4617 68 40 1 1 NUM american_scientific_journal-4617 68 41 , , PUNCT american_scientific_journal-4617 68 42 2 2 NUM american_scientific_journal-4617 68 43 , , PUNCT american_scientific_journal-4617 68 44 3 3 NUM american_scientific_journal-4617 68 45 , , PUNCT american_scientific_journal-4617 68 46 … … PUNCT american_scientific_journal-4617 68 47 ( ( PUNCT american_scientific_journal-4617 68 48 2.10 2.10 NUM american_scientific_journal-4617 68 49 ) ) PUNCT american_scientific_journal-4617 69 1 where where SCONJ american_scientific_journal-4617 69 2 , , PUNCT american_scientific_journal-4617 69 3 ( ( PUNCT american_scientific_journal-4617 69 4 𝐷𝐷(1))𝑛𝑛 𝐷𝐷(1))𝑛𝑛 PROPN american_scientific_journal-4617 69 5 denotes denote NOUN american_scientific_journal-4617 69 6 matrix matrix NOUN american_scientific_journal-4617 69 7 powers power NOUN american_scientific_journal-4617 69 8 . . PUNCT american_scientific_journal-4617 70 1 3 3 X american_scientific_journal-4617 70 2 . . X american_scientific_journal-4617 70 3 methods method NOUN american_scientific_journal-4617 70 4 and and CCONJ american_scientific_journal-4617 70 5 materials material NOUN american_scientific_journal-4617 70 6 consider consider VERB american_scientific_journal-4617 70 7 the the DET american_scientific_journal-4617 70 8 general general ADJ american_scientific_journal-4617 70 9 second second ADJ american_scientific_journal-4617 70 10 order order NOUN american_scientific_journal-4617 70 11 two two NUM american_scientific_journal-4617 70 12 point point NOUN american_scientific_journal-4617 70 13 boundary boundary ADJ american_scientific_journal-4617 70 14 value value NOUN american_scientific_journal-4617 70 15 problem problem NOUN american_scientific_journal-4617 70 16 of of ADP american_scientific_journal-4617 70 17 ordinary ordinary ADJ american_scientific_journal-4617 70 18 differential differential ADJ american_scientific_journal-4617 70 19 equation equation NOUN american_scientific_journal-4617 70 20 american american PROPN american_scientific_journal-4617 70 21 scientific scientific PROPN american_scientific_journal-4617 70 22 research research PROPN american_scientific_journal-4617 70 23 journal journal NOUN american_scientific_journal-4617 70 24 for for ADP american_scientific_journal-4617 70 25 engineering engineering NOUN american_scientific_journal-4617 70 26 , , PUNCT american_scientific_journal-4617 70 27 technology technology NOUN american_scientific_journal-4617 70 28 , , PUNCT american_scientific_journal-4617 70 29 and and CCONJ american_scientific_journal-4617 70 30 sciences science NOUN american_scientific_journal-4617 70 31 ( ( PUNCT american_scientific_journal-4617 70 32 asrjets asrjet NOUN american_scientific_journal-4617 70 33 ) ) PUNCT american_scientific_journal-4617 70 34 ( ( PUNCT american_scientific_journal-4617 70 35 2019 2019 NUM american_scientific_journal-4617 70 36 ) ) PUNCT american_scientific_journal-4617 70 37 volume volume NOUN american_scientific_journal-4617 70 38 51 51 NUM american_scientific_journal-4617 70 39 , , PUNCT american_scientific_journal-4617 70 40 no no DET american_scientific_journal-4617 70 41 1 1 NUM american_scientific_journal-4617 70 42 , , PUNCT american_scientific_journal-4617 70 43 pp pp ADV american_scientific_journal-4617 70 44 225 225 NUM american_scientific_journal-4617 70 45 - - SYM american_scientific_journal-4617 70 46 234 234 NUM american_scientific_journal-4617 70 47 229 229 NUM american_scientific_journal-4617 70 48 𝑦𝑦′′(π‘₯π‘₯ 𝑦𝑦′′(π‘₯π‘₯ NOUN american_scientific_journal-4617 70 49 ) ) PUNCT american_scientific_journal-4617 70 50 + + SYM american_scientific_journal-4617 70 51 𝑝𝑝(π‘₯π‘₯)𝑦𝑦′(π‘₯π‘₯ 𝑝𝑝(π‘₯π‘₯)𝑦𝑦′(π‘₯π‘₯ PROPN american_scientific_journal-4617 70 52 ) ) PUNCT american_scientific_journal-4617 71 1 + + CCONJ american_scientific_journal-4617 71 2 𝑓𝑓(π‘₯π‘₯,𝑦𝑦 𝑓𝑓(π‘₯π‘₯,𝑦𝑦 NOUN american_scientific_journal-4617 71 3 ) ) PUNCT american_scientific_journal-4617 71 4 = = SYM american_scientific_journal-4617 71 5 𝑔𝑔(π‘₯π‘₯ 𝑔𝑔(π‘₯π‘₯ PROPN american_scientific_journal-4617 71 6 ) ) PUNCT american_scientific_journal-4617 71 7 , , PUNCT american_scientific_journal-4617 71 8 0 0 NUM american_scientific_journal-4617 71 9 ≀ ≀ NOUN american_scientific_journal-4617 71 10 π‘₯π‘₯ π‘₯π‘₯ ADP american_scientific_journal-4617 71 11 ≀ ≀ NUM american_scientific_journal-4617 71 12 1 1 NUM american_scientific_journal-4617 71 13 subject subject NOUN american_scientific_journal-4617 71 14 to to ADP american_scientific_journal-4617 71 15 the the DET american_scientific_journal-4617 71 16 boundary boundary ADJ american_scientific_journal-4617 71 17 conditions condition NOUN american_scientific_journal-4617 71 18 : : PUNCT american_scientific_journal-4617 71 19 𝑦𝑦(0 𝑦𝑦(0 X american_scientific_journal-4617 71 20 ) ) PUNCT american_scientific_journal-4617 71 21 = = SYM american_scientific_journal-4617 72 1 𝛼𝛼 𝛼𝛼 PROPN american_scientific_journal-4617 72 2 , , PUNCT american_scientific_journal-4617 72 3 𝑦𝑦(1 𝑦𝑦(1 NOUN american_scientific_journal-4617 72 4 ) ) PUNCT american_scientific_journal-4617 72 5 = = PUNCT american_scientific_journal-4617 72 6 𝛽𝛽 𝛽𝛽 VERB american_scientific_journal-4617 72 7 as as ADP american_scientific_journal-4617 72 8 in in ADP american_scientific_journal-4617 72 9 [ [ X american_scientific_journal-4617 72 10 2 2 NUM american_scientific_journal-4617 72 11 ] ] PUNCT american_scientific_journal-4617 72 12 , , PUNCT american_scientific_journal-4617 72 13 let let VERB american_scientific_journal-4617 72 14 us we PRON american_scientific_journal-4617 72 15 approximate approximate VERB american_scientific_journal-4617 72 16 𝑦𝑦(π‘₯π‘₯ 𝑦𝑦(π‘₯π‘₯ PROPN american_scientific_journal-4617 72 17 ) ) PUNCT american_scientific_journal-4617 72 18 , , PUNCT american_scientific_journal-4617 72 19 𝑝𝑝(π‘₯π‘₯ 𝑝𝑝(π‘₯π‘₯ PROPN american_scientific_journal-4617 72 20 ) ) PUNCT american_scientific_journal-4617 72 21 , , PUNCT american_scientific_journal-4617 72 22 𝑓𝑓(π‘₯π‘₯,𝑦𝑦 𝑓𝑓(π‘₯π‘₯,𝑦𝑦 PROPN american_scientific_journal-4617 72 23 ) ) PUNCT american_scientific_journal-4617 72 24 and and CCONJ american_scientific_journal-4617 72 25 𝑔𝑔(π‘₯π‘₯ 𝑔𝑔(π‘₯π‘₯ PROPN american_scientific_journal-4617 72 26 ) ) PUNCT american_scientific_journal-4617 72 27 by by ADP american_scientific_journal-4617 72 28 the the DET american_scientific_journal-4617 72 29 shifted shift VERB american_scientific_journal-4617 72 30 legendre legendre PROPN american_scientific_journal-4617 72 31 polynomials polynomial NOUN american_scientific_journal-4617 72 32 as as ADP american_scientific_journal-4617 72 33 𝑦𝑦(π‘₯π‘₯ 𝑦𝑦(π‘₯π‘₯ PROPN american_scientific_journal-4617 72 34 ) ) PUNCT american_scientific_journal-4617 72 35 = = PUNCT american_scientific_journal-4617 73 1 βˆ‘ βˆ‘ PUNCT american_scientific_journal-4617 73 2 π‘π‘π‘Ÿπ‘Ÿπ‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)π‘šπ‘š π‘π‘π‘Ÿπ‘Ÿπ‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)π‘šπ‘š PUNCT american_scientific_journal-4617 73 3 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PROPN american_scientific_journal-4617 73 4 = = PUNCT american_scientific_journal-4617 73 5 πΆπΆπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ πΆπΆπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ X american_scientific_journal-4617 73 6 ) ) PUNCT american_scientific_journal-4617 73 7 , , PUNCT american_scientific_journal-4617 73 8 ( ( PUNCT american_scientific_journal-4617 73 9 3.1 3.1 NUM american_scientific_journal-4617 73 10 ) ) PUNCT american_scientific_journal-4617 73 11 𝑝𝑝(π‘₯π‘₯ 𝑝𝑝(π‘₯π‘₯ NOUN american_scientific_journal-4617 73 12 ) ) PUNCT american_scientific_journal-4617 73 13 = = PUNCT american_scientific_journal-4617 74 1 βˆ‘ βˆ‘ PUNCT american_scientific_journal-4617 74 2 π‘π‘π‘Ÿπ‘Ÿπ‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)π‘šπ‘š π‘π‘π‘Ÿπ‘Ÿπ‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)π‘šπ‘š NUM american_scientific_journal-4617 74 3 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PROPN american_scientific_journal-4617 74 4 = = PUNCT american_scientific_journal-4617 74 5 π‘ƒπ‘ƒπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ π‘ƒπ‘ƒπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 74 6 ) ) PUNCT american_scientific_journal-4617 74 7 , , PUNCT american_scientific_journal-4617 74 8 ( ( PUNCT american_scientific_journal-4617 74 9 3.2 3.2 NUM american_scientific_journal-4617 74 10 ) ) PUNCT american_scientific_journal-4617 74 11 𝑓𝑓(π‘₯π‘₯,𝑦𝑦 𝑓𝑓(π‘₯π‘₯,𝑦𝑦 NOUN american_scientific_journal-4617 74 12 ) ) PUNCT american_scientific_journal-4617 74 13 = = SYM american_scientific_journal-4617 74 14 𝑓𝑓(π‘₯π‘₯,πΆπΆπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯,πΆπΆπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 74 15 ) ) PUNCT american_scientific_journal-4617 74 16 ) ) PUNCT american_scientific_journal-4617 74 17 , , PUNCT american_scientific_journal-4617 74 18 ( ( PUNCT american_scientific_journal-4617 74 19 3.3 3.3 NUM american_scientific_journal-4617 74 20 ) ) PUNCT american_scientific_journal-4617 74 21 𝑔𝑔(π‘₯π‘₯ 𝑔𝑔(π‘₯π‘₯ PROPN american_scientific_journal-4617 74 22 ) ) PUNCT american_scientific_journal-4617 74 23 = = SYM american_scientific_journal-4617 75 1 βˆ‘ βˆ‘ PROPN american_scientific_journal-4617 75 2 π‘”π‘”π‘Ÿπ‘Ÿπ‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)π‘šπ‘š π‘”π‘”π‘Ÿπ‘Ÿπ‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)π‘šπ‘š X american_scientific_journal-4617 75 3 π‘Ÿπ‘Ÿ=0 π‘Ÿπ‘Ÿ=0 PROPN american_scientific_journal-4617 75 4 = = SYM american_scientific_journal-4617 75 5 πΊπΊπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ πΊπΊπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 75 6 ) ) PUNCT american_scientific_journal-4617 75 7 ( ( PUNCT american_scientific_journal-4617 75 8 3.4 3.4 NUM american_scientific_journal-4617 75 9 ) ) PUNCT american_scientific_journal-4617 75 10 where where SCONJ american_scientific_journal-4617 75 11 the the DET american_scientific_journal-4617 75 12 unknowns unknown NOUN american_scientific_journal-4617 75 13 are be AUX american_scientific_journal-4617 75 14 𝐢𝐢 𝐢𝐢 NOUN american_scientific_journal-4617 75 15 = = PUNCT american_scientific_journal-4617 76 1 [ [ X american_scientific_journal-4617 76 2 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 76 3 , , PUNCT american_scientific_journal-4617 76 4 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 76 5 , , PUNCT american_scientific_journal-4617 76 6 . . PUNCT american_scientific_journal-4617 76 7 . . PUNCT american_scientific_journal-4617 77 1 . . PUNCT american_scientific_journal-4617 78 1 , , PUNCT american_scientific_journal-4617 78 2 π‘π‘π‘šπ‘š]𝑇𝑇 π‘π‘π‘šπ‘š]𝑇𝑇 ADP american_scientific_journal-4617 78 3 using use VERB american_scientific_journal-4617 78 4 legendre legendre PROPN american_scientific_journal-4617 78 5 operational operational ADJ american_scientific_journal-4617 78 6 matrix matrix NOUN american_scientific_journal-4617 78 7 of of ADP american_scientific_journal-4617 78 8 differentiation differentiation NOUN american_scientific_journal-4617 78 9 , , PUNCT american_scientific_journal-4617 78 10 equation equation NOUN american_scientific_journal-4617 78 11 ( ( PUNCT american_scientific_journal-4617 78 12 1.1 1.1 NUM american_scientific_journal-4617 78 13 ) ) PUNCT american_scientific_journal-4617 78 14 can can AUX american_scientific_journal-4617 78 15 be be AUX american_scientific_journal-4617 78 16 written write VERB american_scientific_journal-4617 78 17 as as ADP american_scientific_journal-4617 78 18 𝐢𝐢𝑇𝑇𝐷𝐷2πœ‘πœ‘(π‘₯π‘₯ 𝐢𝐢𝑇𝑇𝐷𝐷2πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 78 19 ) ) PUNCT american_scientific_journal-4617 78 20 + + NUM american_scientific_journal-4617 78 21 𝑃𝑃𝑇𝑇𝐷𝐷1πœ‘πœ‘(π‘₯π‘₯ 𝑃𝑃𝑇𝑇𝐷𝐷1πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 78 22 ) ) PUNCT american_scientific_journal-4617 78 23 + + NOUN american_scientific_journal-4617 78 24 𝑓𝑓(π‘₯π‘₯,πΆπΆπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯,πΆπΆπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ NOUN american_scientific_journal-4617 78 25 ) ) PUNCT american_scientific_journal-4617 78 26 ) ) PUNCT american_scientific_journal-4617 79 1 β‰ˆ β‰ˆ PROPN american_scientific_journal-4617 79 2 πΊπΊπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ πΊπΊπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 79 3 ) ) PUNCT american_scientific_journal-4617 79 4 ( ( PUNCT american_scientific_journal-4617 79 5 3.5 3.5 NUM american_scientific_journal-4617 79 6 ) ) PUNCT american_scientific_journal-4617 79 7 the the DET american_scientific_journal-4617 79 8 residual residual ADJ american_scientific_journal-4617 79 9 π‘…π‘…π‘šπ‘š(π‘₯π‘₯ π‘…π‘…π‘šπ‘š(π‘₯π‘₯ NOUN american_scientific_journal-4617 79 10 ) ) PUNCT american_scientific_journal-4617 79 11 for for ADP american_scientific_journal-4617 79 12 equation equation NOUN american_scientific_journal-4617 79 13 ( ( PUNCT american_scientific_journal-4617 79 14 3.5 3.5 NUM american_scientific_journal-4617 79 15 ) ) PUNCT american_scientific_journal-4617 79 16 can can AUX american_scientific_journal-4617 79 17 be be AUX american_scientific_journal-4617 79 18 written write VERB american_scientific_journal-4617 79 19 as as ADP american_scientific_journal-4617 79 20 π‘…π‘…π‘šπ‘š(π‘₯π‘₯ π‘…π‘…π‘šπ‘š(π‘₯π‘₯ NOUN american_scientific_journal-4617 79 21 ) ) PUNCT american_scientific_journal-4617 79 22 = = SYM american_scientific_journal-4617 79 23 𝐢𝐢𝑇𝑇𝐷𝐷2πœ‘πœ‘(π‘₯π‘₯ 𝐢𝐢𝑇𝑇𝐷𝐷2πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 79 24 ) ) PUNCT american_scientific_journal-4617 79 25 + + NUM american_scientific_journal-4617 79 26 𝑃𝑃𝑇𝑇𝐷𝐷1πœ‘πœ‘(π‘₯π‘₯ 𝑃𝑃𝑇𝑇𝐷𝐷1πœ‘πœ‘(π‘₯π‘₯ X american_scientific_journal-4617 79 27 ) ) PUNCT american_scientific_journal-4617 79 28 + + CCONJ american_scientific_journal-4617 79 29 𝑓𝑓 𝑓𝑓 PROPN american_scientific_journal-4617 79 30 οΏ½ οΏ½ PROPN american_scientific_journal-4617 79 31 π‘₯π‘₯,πΆπΆπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ π‘₯π‘₯,πΆπΆπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 79 32 ) ) PUNCT american_scientific_journal-4617 79 33 οΏ½ οΏ½ PROPN american_scientific_journal-4617 79 34 βˆ’ βˆ’ PROPN american_scientific_journal-4617 79 35 πΊπΊπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ πΊπΊπ‘‡π‘‡πœ‘πœ‘(π‘₯π‘₯ PROPN american_scientific_journal-4617 79 36 ) ) PUNCT american_scientific_journal-4617 79 37 ( ( PUNCT american_scientific_journal-4617 79 38 3.6 3.6 NUM american_scientific_journal-4617 79 39 ) ) PUNCT american_scientific_journal-4617 79 40 applying apply VERB american_scientific_journal-4617 79 41 typical typical ADJ american_scientific_journal-4617 79 42 tau tau PROPN american_scientific_journal-4617 79 43 method method NOUN american_scientific_journal-4617 79 44 , , PUNCT american_scientific_journal-4617 79 45 which which PRON american_scientific_journal-4617 79 46 is be AUX american_scientific_journal-4617 79 47 used use VERB american_scientific_journal-4617 79 48 in in ADP american_scientific_journal-4617 79 49 the the DET american_scientific_journal-4617 79 50 sense sense NOUN american_scientific_journal-4617 79 51 of of ADP american_scientific_journal-4617 79 52 particular particular ADJ american_scientific_journal-4617 79 53 form form NOUN american_scientific_journal-4617 79 54 of of ADP american_scientific_journal-4617 79 55 the the DET american_scientific_journal-4617 79 56 petrov petrov PROPN american_scientific_journal-4617 79 57 - - PUNCT american_scientific_journal-4617 79 58 galerkin galerkin PROPN american_scientific_journal-4617 79 59 method method NOUN american_scientific_journal-4617 79 60 as as SCONJ american_scientific_journal-4617 79 61 cited cite VERB american_scientific_journal-4617 79 62 in in ADP american_scientific_journal-4617 79 63 [ [ X american_scientific_journal-4617 79 64 2 2 NUM american_scientific_journal-4617 79 65 ] ] PUNCT american_scientific_journal-4617 79 66 , , PUNCT american_scientific_journal-4617 79 67 [ [ X american_scientific_journal-4617 79 68 5 5 NUM american_scientific_journal-4617 79 69 ] ] PUNCT american_scientific_journal-4617 79 70 , , PUNCT american_scientific_journal-4617 79 71 equation equation NOUN american_scientific_journal-4617 79 72 ( ( PUNCT american_scientific_journal-4617 79 73 3.5 3.5 NUM american_scientific_journal-4617 79 74 ) ) PUNCT american_scientific_journal-4617 79 75 can can AUX american_scientific_journal-4617 79 76 be be AUX american_scientific_journal-4617 79 77 transformed transform VERB american_scientific_journal-4617 79 78 into into ADP american_scientific_journal-4617 79 79 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 79 80 βˆ’ βˆ’ PROPN american_scientific_journal-4617 79 81 1 1 NUM american_scientific_journal-4617 79 82 linear linear ADJ american_scientific_journal-4617 79 83 or or CCONJ american_scientific_journal-4617 79 84 nonlinear nonlinear ADJ american_scientific_journal-4617 79 85 equations equation NOUN american_scientific_journal-4617 79 86 by by ADP american_scientific_journal-4617 79 87 applying apply VERB american_scientific_journal-4617 79 88 〈 〈 PROPN american_scientific_journal-4617 79 89 π‘…π‘…π‘šπ‘š(π‘₯π‘₯),π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ π‘…π‘…π‘šπ‘š(π‘₯π‘₯),π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯ PROPN american_scientific_journal-4617 79 90 ) ) PUNCT american_scientific_journal-4617 79 91 βŒͺ βŒͺ NOUN american_scientific_journal-4617 79 92 = = SYM american_scientific_journal-4617 79 93 ∫ ∫ NOUN american_scientific_journal-4617 79 94 π‘…π‘…π‘šπ‘š(π‘₯π‘₯)π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)1 π‘…π‘…π‘šπ‘š(π‘₯π‘₯)π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ(π‘₯π‘₯)1 NOUN american_scientific_journal-4617 79 95 0 0 NUM american_scientific_journal-4617 80 1 𝑑𝑑π‘₯π‘₯ 𝑑𝑑π‘₯π‘₯ NOUN american_scientific_journal-4617 80 2 = = SYM american_scientific_journal-4617 80 3 0 0 NUM american_scientific_journal-4617 80 4 ; ; PUNCT american_scientific_journal-4617 80 5 π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿ ADP american_scientific_journal-4617 80 6 = = NOUN american_scientific_journal-4617 80 7 0,1,2 0,1,2 NUM american_scientific_journal-4617 80 8 , , PUNCT american_scientific_journal-4617 80 9 . . PUNCT american_scientific_journal-4617 80 10 . . PUNCT american_scientific_journal-4617 81 1 . . PUNCT american_scientific_journal-4617 82 1 , , PUNCT american_scientific_journal-4617 82 2 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 82 3 βˆ’ βˆ’ PROPN american_scientific_journal-4617 82 4 2 2 NUM american_scientific_journal-4617 82 5 ( ( PUNCT american_scientific_journal-4617 82 6 3.7 3.7 NUM american_scientific_journal-4617 82 7 ) ) PUNCT american_scientific_journal-4617 82 8 the the DET american_scientific_journal-4617 82 9 boundary boundary ADJ american_scientific_journal-4617 82 10 conditions condition NOUN american_scientific_journal-4617 82 11 are be AUX american_scientific_journal-4617 82 12 given give VERB american_scientific_journal-4617 82 13 by by ADP american_scientific_journal-4617 82 14 𝑦𝑦(0 𝑦𝑦(0 NOUN american_scientific_journal-4617 82 15 ) ) PUNCT american_scientific_journal-4617 82 16 = = SYM american_scientific_journal-4617 82 17 πΆπΆπ‘‡π‘‡πœ‘πœ‘(0 πΆπΆπ‘‡π‘‡πœ‘πœ‘(0 PROPN american_scientific_journal-4617 82 18 ) ) PUNCT american_scientific_journal-4617 82 19 = = PUNCT american_scientific_journal-4617 82 20 𝑑𝑑0 𝑑𝑑0 NOUN american_scientific_journal-4617 82 21 , , PUNCT american_scientific_journal-4617 82 22 𝑦𝑦(1 𝑦𝑦(1 NOUN american_scientific_journal-4617 82 23 ) ) PUNCT american_scientific_journal-4617 82 24 = = SYM american_scientific_journal-4617 82 25 πΆπΆπ‘‡π‘‡πœ‘πœ‘(1 πΆπΆπ‘‡π‘‡πœ‘πœ‘(1 PROPN american_scientific_journal-4617 82 26 ) ) PUNCT american_scientific_journal-4617 82 27 = = SYM american_scientific_journal-4617 82 28 𝑑𝑑1 𝑑𝑑1 NOUN american_scientific_journal-4617 82 29 ( ( PUNCT american_scientific_journal-4617 82 30 3.8 3.8 NUM american_scientific_journal-4617 82 31 ) ) PUNCT american_scientific_journal-4617 82 32 equations equation NOUN american_scientific_journal-4617 82 33 ( ( PUNCT american_scientific_journal-4617 82 34 3.7 3.7 NUM american_scientific_journal-4617 82 35 ) ) PUNCT american_scientific_journal-4617 82 36 and and CCONJ american_scientific_journal-4617 82 37 ( ( PUNCT american_scientific_journal-4617 82 38 3.8 3.8 NUM american_scientific_journal-4617 82 39 ) ) PUNCT american_scientific_journal-4617 82 40 generate generate VERB american_scientific_journal-4617 82 41 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 82 42 + + CCONJ american_scientific_journal-4617 82 43 1 1 NUM american_scientific_journal-4617 82 44 linear linear ADJ american_scientific_journal-4617 82 45 or or CCONJ american_scientific_journal-4617 82 46 nonlinear nonlinear ADJ american_scientific_journal-4617 82 47 systems system NOUN american_scientific_journal-4617 82 48 of of ADP american_scientific_journal-4617 82 49 algebraic algebraic ADJ american_scientific_journal-4617 82 50 equations equation NOUN american_scientific_journal-4617 82 51 . . PUNCT american_scientific_journal-4617 83 1 after after ADP american_scientific_journal-4617 83 2 solving solve VERB american_scientific_journal-4617 83 3 these these DET american_scientific_journal-4617 83 4 equations equation NOUN american_scientific_journal-4617 83 5 we we PRON american_scientific_journal-4617 83 6 obtain obtain VERB american_scientific_journal-4617 83 7 the the DET american_scientific_journal-4617 83 8 unknowns unknown NOUN american_scientific_journal-4617 83 9 in in ADP american_scientific_journal-4617 83 10 vector vector NOUN american_scientific_journal-4617 83 11 𝐢𝐢 𝐢𝐢 NOUN american_scientific_journal-4617 83 12 and and CCONJ american_scientific_journal-4617 83 13 use use VERB american_scientific_journal-4617 83 14 them they PRON american_scientific_journal-4617 83 15 to to PART american_scientific_journal-4617 83 16 find find VERB american_scientific_journal-4617 83 17 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-4617 83 18 οΏ½ οΏ½ PROPN american_scientific_journal-4617 83 19 (π‘₯π‘₯ (π‘₯π‘₯ PUNCT american_scientific_journal-4617 83 20 ) ) PUNCT american_scientific_journal-4617 83 21 . . PUNCT american_scientific_journal-4617 84 1 4 4 X american_scientific_journal-4617 84 2 . . X american_scientific_journal-4617 84 3 numerical numerical ADJ american_scientific_journal-4617 84 4 examples example NOUN american_scientific_journal-4617 84 5 example example NOUN american_scientific_journal-4617 84 6 1 1 NUM american_scientific_journal-4617 84 7 : : PUNCT american_scientific_journal-4617 84 8 consider consider VERB american_scientific_journal-4617 84 9 the the DET american_scientific_journal-4617 84 10 second second ADJ american_scientific_journal-4617 84 11 order order NOUN american_scientific_journal-4617 84 12 boundary boundary ADJ american_scientific_journal-4617 84 13 value value NOUN american_scientific_journal-4617 84 14 problem problem NOUN american_scientific_journal-4617 84 15 𝑦𝑦′′ 𝑦𝑦′′ VERB american_scientific_journal-4617 85 1 + + CCONJ american_scientific_journal-4617 85 2 𝑦𝑦′ 𝑦𝑦′ NUM american_scientific_journal-4617 85 3 = = SYM american_scientific_journal-4617 85 4 π‘₯π‘₯ π‘₯π‘₯ ADP american_scientific_journal-4617 85 5 ( ( PUNCT american_scientific_journal-4617 85 6 4.1 4.1 NUM american_scientific_journal-4617 85 7 ) ) PUNCT american_scientific_journal-4617 85 8 american american ADJ american_scientific_journal-4617 85 9 scientific scientific ADJ american_scientific_journal-4617 85 10 research research NOUN american_scientific_journal-4617 85 11 journal journal NOUN american_scientific_journal-4617 85 12 for for ADP american_scientific_journal-4617 85 13 engineering engineering NOUN american_scientific_journal-4617 85 14 , , PUNCT american_scientific_journal-4617 85 15 technology technology NOUN american_scientific_journal-4617 85 16 , , PUNCT american_scientific_journal-4617 85 17 and and CCONJ american_scientific_journal-4617 85 18 sciences science NOUN american_scientific_journal-4617 85 19 ( ( PUNCT american_scientific_journal-4617 85 20 asrjets asrjet NOUN american_scientific_journal-4617 85 21 ) ) PUNCT american_scientific_journal-4617 85 22 ( ( PUNCT american_scientific_journal-4617 85 23 2019 2019 NUM american_scientific_journal-4617 85 24 ) ) PUNCT american_scientific_journal-4617 85 25 volume volume NOUN american_scientific_journal-4617 85 26 51 51 NUM american_scientific_journal-4617 85 27 , , PUNCT american_scientific_journal-4617 85 28 no no DET american_scientific_journal-4617 85 29 1 1 NUM american_scientific_journal-4617 85 30 , , PUNCT american_scientific_journal-4617 85 31 pp pp ADV american_scientific_journal-4617 85 32 225 225 NUM american_scientific_journal-4617 85 33 - - SYM american_scientific_journal-4617 85 34 234 234 NUM american_scientific_journal-4617 85 35 230 230 NUM american_scientific_journal-4617 85 36 with with ADP american_scientific_journal-4617 85 37 boundary boundary ADJ american_scientific_journal-4617 85 38 conditions condition NOUN american_scientific_journal-4617 85 39 𝑦𝑦(0 𝑦𝑦(0 NOUN american_scientific_journal-4617 85 40 ) ) PUNCT american_scientific_journal-4617 86 1 = = SYM american_scientific_journal-4617 86 2 0 0 NUM american_scientific_journal-4617 86 3 , , PUNCT american_scientific_journal-4617 86 4 𝑦𝑦(1 𝑦𝑦(1 NOUN american_scientific_journal-4617 86 5 ) ) PUNCT american_scientific_journal-4617 86 6 = = SYM american_scientific_journal-4617 86 7 1 1 NUM american_scientific_journal-4617 86 8 ( ( PUNCT american_scientific_journal-4617 86 9 4.2 4.2 NUM american_scientific_journal-4617 86 10 ) ) PUNCT american_scientific_journal-4617 86 11 the the DET american_scientific_journal-4617 86 12 exact exact ADJ american_scientific_journal-4617 86 13 solution solution NOUN american_scientific_journal-4617 86 14 is be AUX american_scientific_journal-4617 86 15 𝑦𝑦(π‘₯π‘₯ 𝑦𝑦(π‘₯π‘₯ PROPN american_scientific_journal-4617 86 16 ) ) PUNCT american_scientific_journal-4617 87 1 = = NOUN american_scientific_journal-4617 87 2 π‘₯π‘₯ π‘₯π‘₯ ADP american_scientific_journal-4617 87 3 substituting substitute VERB american_scientific_journal-4617 87 4 equations equation NOUN american_scientific_journal-4617 87 5 ( ( PUNCT american_scientific_journal-4617 87 6 4.2 4.2 NUM american_scientific_journal-4617 87 7 ) ) PUNCT american_scientific_journal-4617 87 8 in in ADP american_scientific_journal-4617 87 9 equation equation NOUN american_scientific_journal-4617 87 10 ( ( PUNCT american_scientific_journal-4617 87 11 2.7 2.7 NUM american_scientific_journal-4617 87 12 ) ) PUNCT american_scientific_journal-4617 87 13 we we PRON american_scientific_journal-4617 87 14 get get VERB american_scientific_journal-4617 87 15 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 87 16 βˆ’ βˆ’ NOUN american_scientific_journal-4617 87 17 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 87 18 + + CCONJ american_scientific_journal-4617 87 19 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-4617 88 1 = = SYM american_scientific_journal-4617 88 2 0 0 NUM american_scientific_journal-4617 88 3 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 88 4 + + X american_scientific_journal-4617 88 5 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 89 1 + + CCONJ american_scientific_journal-4617 89 2 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-4617 89 3 = = SYM american_scientific_journal-4617 89 4 1 1 NUM american_scientific_journal-4617 89 5 ( ( PUNCT american_scientific_journal-4617 89 6 4.3 4.3 NUM american_scientific_journal-4617 89 7 ) ) PUNCT american_scientific_journal-4617 89 8 for for ADP american_scientific_journal-4617 89 9 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 89 10 = = SYM american_scientific_journal-4617 89 11 2 2 NUM american_scientific_journal-4617 89 12 from from ADP american_scientific_journal-4617 89 13 equation equation NOUN american_scientific_journal-4617 89 14 ( ( PUNCT american_scientific_journal-4617 89 15 3.7 3.7 NUM american_scientific_journal-4617 89 16 ) ) PUNCT american_scientific_journal-4617 89 17 we we PRON american_scientific_journal-4617 89 18 get get VERB american_scientific_journal-4617 89 19 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 90 1 + + CCONJ american_scientific_journal-4617 90 2 13𝑐𝑐2 13𝑐𝑐2 NUM american_scientific_journal-4617 90 3 = = SYM american_scientific_journal-4617 90 4 1 1 NUM american_scientific_journal-4617 90 5 2 2 NUM american_scientific_journal-4617 90 6 ( ( PUNCT american_scientific_journal-4617 90 7 4.4 4.4 NUM american_scientific_journal-4617 90 8 ) ) PUNCT american_scientific_journal-4617 90 9 solving solve VERB american_scientific_journal-4617 90 10 a a DET american_scientific_journal-4617 90 11 3 3 NUM american_scientific_journal-4617 90 12 Γ— Γ— NOUN american_scientific_journal-4617 90 13 3 3 NUM american_scientific_journal-4617 90 14 system system NOUN american_scientific_journal-4617 90 15 of of ADP american_scientific_journal-4617 90 16 algebraic algebraic ADJ american_scientific_journal-4617 90 17 equations equation NOUN american_scientific_journal-4617 90 18 ( ( PUNCT american_scientific_journal-4617 90 19 4.3 4.3 NUM american_scientific_journal-4617 90 20 ) ) PUNCT american_scientific_journal-4617 90 21 and and CCONJ american_scientific_journal-4617 90 22 ( ( PUNCT american_scientific_journal-4617 90 23 4.4 4.4 NUM american_scientific_journal-4617 90 24 ) ) PUNCT american_scientific_journal-4617 90 25 to to PART american_scientific_journal-4617 90 26 gives give VERB american_scientific_journal-4617 90 27 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 90 28 = = NOUN american_scientific_journal-4617 90 29 1 1 NUM american_scientific_journal-4617 90 30 2 2 NUM american_scientific_journal-4617 90 31 , , PUNCT american_scientific_journal-4617 90 32 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 90 33 = = SYM american_scientific_journal-4617 90 34 1 1 NUM american_scientific_journal-4617 90 35 2 2 NUM american_scientific_journal-4617 90 36 and and CCONJ american_scientific_journal-4617 90 37 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-4617 91 1 = = SYM american_scientific_journal-4617 91 2 0 0 PUNCT american_scientific_journal-4617 92 1 thus thus ADV american_scientific_journal-4617 92 2 the the DET american_scientific_journal-4617 92 3 approximate approximate ADJ american_scientific_journal-4617 92 4 solution solution NOUN american_scientific_journal-4617 92 5 is be AUX american_scientific_journal-4617 92 6 now now ADV american_scientific_journal-4617 92 7 becomes become VERB american_scientific_journal-4617 92 8 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-4617 92 9 οΏ½ οΏ½ X american_scientific_journal-4617 92 10 (π‘₯π‘₯ (π‘₯π‘₯ PUNCT american_scientific_journal-4617 92 11 ) ) PUNCT american_scientific_journal-4617 92 12 = = PROPN american_scientific_journal-4617 92 13 π‘₯π‘₯ π‘₯π‘₯ ADP american_scientific_journal-4617 92 14 which which PRON american_scientific_journal-4617 92 15 is be AUX american_scientific_journal-4617 92 16 the the DET american_scientific_journal-4617 92 17 exact exact ADJ american_scientific_journal-4617 92 18 solution solution NOUN american_scientific_journal-4617 92 19 . . PUNCT american_scientific_journal-4617 93 1 figure figure NOUN american_scientific_journal-4617 93 2 1 1 NUM american_scientific_journal-4617 93 3 : : PUNCT american_scientific_journal-4617 93 4 graphical graphical ADJ american_scientific_journal-4617 93 5 illustration illustration NOUN american_scientific_journal-4617 93 6 of of ADP american_scientific_journal-4617 93 7 example example NOUN american_scientific_journal-4617 93 8 1 1 NUM american_scientific_journal-4617 93 9 0 0 NUM american_scientific_journal-4617 93 10 0.1 0.1 NUM american_scientific_journal-4617 93 11 0.2 0.2 NUM american_scientific_journal-4617 93 12 0.3 0.3 NUM american_scientific_journal-4617 93 13 0.4 0.4 NUM american_scientific_journal-4617 93 14 0.5 0.5 NUM american_scientific_journal-4617 93 15 0.6 0.6 NUM american_scientific_journal-4617 93 16 0.7 0.7 NUM american_scientific_journal-4617 93 17 0.8 0.8 NUM american_scientific_journal-4617 93 18 0.9 0.9 NUM american_scientific_journal-4617 93 19 1 1 NUM american_scientific_journal-4617 93 20 0 0 NUM american_scientific_journal-4617 93 21 0.1 0.1 NUM american_scientific_journal-4617 93 22 0.2 0.2 NUM american_scientific_journal-4617 93 23 0.3 0.3 NUM american_scientific_journal-4617 93 24 0.4 0.4 NUM american_scientific_journal-4617 93 25 0.5 0.5 NUM american_scientific_journal-4617 93 26 0.6 0.6 NUM american_scientific_journal-4617 93 27 0.7 0.7 NUM american_scientific_journal-4617 93 28 0.8 0.8 NUM american_scientific_journal-4617 93 29 0.9 0.9 NUM american_scientific_journal-4617 93 30 1 1 NUM american_scientific_journal-4617 93 31 x x SYM american_scientific_journal-4617 93 32 y y PROPN american_scientific_journal-4617 93 33 comparison comparison NOUN american_scientific_journal-4617 93 34 with with ADP american_scientific_journal-4617 93 35 different different ADJ american_scientific_journal-4617 93 36 models model NOUN american_scientific_journal-4617 93 37 exact exact ADJ american_scientific_journal-4617 93 38 solution solution NOUN american_scientific_journal-4617 93 39 approximate approximate ADJ american_scientific_journal-4617 93 40 solution solution NOUN american_scientific_journal-4617 93 41 with with ADP american_scientific_journal-4617 93 42 current current ADJ american_scientific_journal-4617 93 43 method method NOUN american_scientific_journal-4617 93 44 approximate approximate ADJ american_scientific_journal-4617 93 45 solution solution NOUN american_scientific_journal-4617 93 46 with with ADP american_scientific_journal-4617 93 47 fdm fdm NOUN american_scientific_journal-4617 93 48 error error NOUN american_scientific_journal-4617 93 49 with with ADP american_scientific_journal-4617 93 50 current current ADJ american_scientific_journal-4617 93 51 method method NOUN american_scientific_journal-4617 93 52 error error NOUN american_scientific_journal-4617 93 53 with with ADP american_scientific_journal-4617 93 54 fdm fdm ADJ american_scientific_journal-4617 93 55 american american ADJ american_scientific_journal-4617 93 56 scientific scientific ADJ american_scientific_journal-4617 93 57 research research NOUN american_scientific_journal-4617 93 58 journal journal NOUN american_scientific_journal-4617 93 59 for for ADP american_scientific_journal-4617 93 60 engineering engineering NOUN american_scientific_journal-4617 93 61 , , PUNCT american_scientific_journal-4617 93 62 technology technology NOUN american_scientific_journal-4617 93 63 , , PUNCT american_scientific_journal-4617 93 64 and and CCONJ american_scientific_journal-4617 93 65 sciences science NOUN american_scientific_journal-4617 93 66 ( ( PUNCT american_scientific_journal-4617 93 67 asrjets asrjet NOUN american_scientific_journal-4617 93 68 ) ) PUNCT american_scientific_journal-4617 93 69 ( ( PUNCT american_scientific_journal-4617 93 70 2019 2019 NUM american_scientific_journal-4617 93 71 ) ) PUNCT american_scientific_journal-4617 93 72 volume volume NOUN american_scientific_journal-4617 93 73 51 51 NUM american_scientific_journal-4617 93 74 , , PUNCT american_scientific_journal-4617 93 75 no no DET american_scientific_journal-4617 93 76 1 1 NUM american_scientific_journal-4617 93 77 , , PUNCT american_scientific_journal-4617 93 78 pp pp ADV american_scientific_journal-4617 93 79 225 225 NUM american_scientific_journal-4617 93 80 - - SYM american_scientific_journal-4617 93 81 234 234 NUM american_scientific_journal-4617 93 82 231 231 NUM american_scientific_journal-4617 93 83 example example NOUN american_scientific_journal-4617 93 84 2 2 NUM american_scientific_journal-4617 93 85 : : PUNCT american_scientific_journal-4617 93 86 consider consider VERB american_scientific_journal-4617 93 87 the the DET american_scientific_journal-4617 93 88 second second ADJ american_scientific_journal-4617 93 89 order order NOUN american_scientific_journal-4617 93 90 boundary boundary ADJ american_scientific_journal-4617 93 91 value value NOUN american_scientific_journal-4617 93 92 problem problem NOUN american_scientific_journal-4617 93 93 𝑦𝑦′′ 𝑦𝑦′′ VERB american_scientific_journal-4617 94 1 + + CCONJ american_scientific_journal-4617 94 2 𝑦𝑦′ 𝑦𝑦′ NUM american_scientific_journal-4617 94 3 + + CCONJ american_scientific_journal-4617 94 4 πœ‹πœ‹2𝑦𝑦 πœ‹πœ‹2𝑦𝑦 NOUN american_scientific_journal-4617 94 5 = = SYM american_scientific_journal-4617 94 6 βˆ’πœ‹πœ‹sin βˆ’πœ‹πœ‹sin NOUN american_scientific_journal-4617 94 7 ( ( PUNCT american_scientific_journal-4617 94 8 πœ‹πœ‹π‘₯π‘₯ πœ‹πœ‹π‘₯π‘₯ PROPN american_scientific_journal-4617 94 9 ) ) PUNCT american_scientific_journal-4617 94 10 , , PUNCT american_scientific_journal-4617 94 11 ( ( PUNCT american_scientific_journal-4617 94 12 4.5 4.5 NUM american_scientific_journal-4617 94 13 ) ) PUNCT american_scientific_journal-4617 94 14 with with ADP american_scientific_journal-4617 94 15 boundary boundary ADJ american_scientific_journal-4617 94 16 conditions condition NOUN american_scientific_journal-4617 94 17 𝑦𝑦(0 𝑦𝑦(0 NOUN american_scientific_journal-4617 94 18 ) ) PUNCT american_scientific_journal-4617 94 19 = = SYM american_scientific_journal-4617 94 20 1 1 NUM american_scientific_journal-4617 94 21 , , PUNCT american_scientific_journal-4617 94 22 𝑦𝑦(1 𝑦𝑦(1 NOUN american_scientific_journal-4617 94 23 ) ) PUNCT american_scientific_journal-4617 94 24 = = SYM american_scientific_journal-4617 94 25 βˆ’1 βˆ’1 NOUN american_scientific_journal-4617 94 26 ( ( PUNCT american_scientific_journal-4617 94 27 4.6 4.6 NUM american_scientific_journal-4617 94 28 ) ) PUNCT american_scientific_journal-4617 94 29 the the DET american_scientific_journal-4617 94 30 exact exact ADJ american_scientific_journal-4617 94 31 solution solution NOUN american_scientific_journal-4617 94 32 is be AUX american_scientific_journal-4617 94 33 𝑦𝑦(π‘₯π‘₯ 𝑦𝑦(π‘₯π‘₯ PROPN american_scientific_journal-4617 94 34 ) ) PUNCT american_scientific_journal-4617 95 1 = = SYM american_scientific_journal-4617 95 2 cos cos PROPN american_scientific_journal-4617 95 3 ( ( PUNCT american_scientific_journal-4617 95 4 πœ‹πœ‹π‘₯π‘₯ πœ‹πœ‹π‘₯π‘₯ PROPN american_scientific_journal-4617 95 5 ) ) PUNCT american_scientific_journal-4617 95 6 substituting substitute VERB american_scientific_journal-4617 95 7 equations equation NOUN american_scientific_journal-4617 95 8 ( ( PUNCT american_scientific_journal-4617 95 9 4.6 4.6 NUM american_scientific_journal-4617 95 10 ) ) PUNCT american_scientific_journal-4617 95 11 in in ADP american_scientific_journal-4617 95 12 equation equation NOUN american_scientific_journal-4617 95 13 ( ( PUNCT american_scientific_journal-4617 95 14 2.7 2.7 NUM american_scientific_journal-4617 95 15 ) ) PUNCT american_scientific_journal-4617 95 16 we we PRON american_scientific_journal-4617 95 17 get get VERB american_scientific_journal-4617 95 18 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 95 19 βˆ’ βˆ’ NOUN american_scientific_journal-4617 95 20 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 95 21 + + CCONJ american_scientific_journal-4617 95 22 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-4617 96 1 = = SYM american_scientific_journal-4617 96 2 1 1 NUM american_scientific_journal-4617 96 3 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 96 4 + + CCONJ american_scientific_journal-4617 96 5 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 96 6 + + CCONJ american_scientific_journal-4617 96 7 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-4617 96 8 = = SYM american_scientific_journal-4617 96 9 βˆ’1 βˆ’1 NOUN american_scientific_journal-4617 96 10 ( ( PUNCT american_scientific_journal-4617 96 11 4.7 4.7 NUM american_scientific_journal-4617 96 12 ) ) PUNCT american_scientific_journal-4617 96 13 for for ADP american_scientific_journal-4617 96 14 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 96 15 = = SYM american_scientific_journal-4617 96 16 2 2 NUM american_scientific_journal-4617 96 17 from from ADP american_scientific_journal-4617 96 18 equation equation NOUN american_scientific_journal-4617 96 19 ( ( PUNCT american_scientific_journal-4617 96 20 3.7 3.7 NUM american_scientific_journal-4617 96 21 ) ) PUNCT american_scientific_journal-4617 96 22 we we PRON american_scientific_journal-4617 96 23 get get VERB american_scientific_journal-4617 96 24 πœ‹πœ‹2𝑐𝑐0 πœ‹πœ‹2𝑐𝑐0 VERB american_scientific_journal-4617 96 25 + + CCONJ american_scientific_journal-4617 96 26 2𝑐𝑐1 2𝑐𝑐1 NUM american_scientific_journal-4617 97 1 + + CCONJ american_scientific_journal-4617 97 2 12𝑐𝑐2 12𝑐𝑐2 NUM american_scientific_journal-4617 97 3 = = SYM american_scientific_journal-4617 97 4 βˆ’2 βˆ’2 NOUN american_scientific_journal-4617 97 5 ( ( PUNCT american_scientific_journal-4617 97 6 4.8 4.8 NUM american_scientific_journal-4617 97 7 ) ) PUNCT american_scientific_journal-4617 97 8 solving solve VERB american_scientific_journal-4617 97 9 a a DET american_scientific_journal-4617 97 10 3 3 NUM american_scientific_journal-4617 97 11 Γ— Γ— NOUN american_scientific_journal-4617 97 12 3 3 NUM american_scientific_journal-4617 97 13 system system NOUN american_scientific_journal-4617 97 14 of of ADP american_scientific_journal-4617 97 15 algebraic algebraic ADJ american_scientific_journal-4617 97 16 equations equation NOUN american_scientific_journal-4617 97 17 ( ( PUNCT american_scientific_journal-4617 97 18 4.7 4.7 NUM american_scientific_journal-4617 97 19 ) ) PUNCT american_scientific_journal-4617 97 20 and and CCONJ american_scientific_journal-4617 97 21 ( ( PUNCT american_scientific_journal-4617 97 22 4.8 4.8 NUM american_scientific_journal-4617 97 23 ) ) PUNCT american_scientific_journal-4617 97 24 to to PART american_scientific_journal-4617 97 25 gives give VERB american_scientific_journal-4617 97 26 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 97 27 = = NOUN american_scientific_journal-4617 97 28 0 0 NUM american_scientific_journal-4617 97 29 , , PUNCT american_scientific_journal-4617 97 30 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 97 31 = = SYM american_scientific_journal-4617 97 32 βˆ’1 βˆ’1 NOUN american_scientific_journal-4617 97 33 and and CCONJ american_scientific_journal-4617 97 34 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-4617 98 1 = = SYM american_scientific_journal-4617 98 2 0 0 PUNCT american_scientific_journal-4617 99 1 thus thus ADV american_scientific_journal-4617 99 2 the the DET american_scientific_journal-4617 99 3 approximate approximate ADJ american_scientific_journal-4617 99 4 solution solution NOUN american_scientific_journal-4617 99 5 is be AUX american_scientific_journal-4617 99 6 now now ADV american_scientific_journal-4617 99 7 becomes become VERB american_scientific_journal-4617 99 8 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-4617 99 9 οΏ½ οΏ½ X american_scientific_journal-4617 99 10 (π‘₯π‘₯ (π‘₯π‘₯ PUNCT american_scientific_journal-4617 99 11 ) ) PUNCT american_scientific_journal-4617 100 1 = = PUNCT american_scientific_journal-4617 101 1 βˆ’2π‘₯π‘₯ βˆ’2π‘₯π‘₯ SYM american_scientific_journal-4617 101 2 + + CCONJ american_scientific_journal-4617 101 3 1 1 NUM american_scientific_journal-4617 101 4 figure figure NOUN american_scientific_journal-4617 101 5 2 2 NUM american_scientific_journal-4617 101 6 : : PUNCT american_scientific_journal-4617 101 7 graphical graphical ADJ american_scientific_journal-4617 101 8 illustration illustration NOUN american_scientific_journal-4617 101 9 of of ADP american_scientific_journal-4617 101 10 example example NOUN american_scientific_journal-4617 101 11 2 2 NUM american_scientific_journal-4617 101 12 0 0 NUM american_scientific_journal-4617 101 13 0.1 0.1 NUM american_scientific_journal-4617 101 14 0.2 0.2 NUM american_scientific_journal-4617 101 15 0.3 0.3 NUM american_scientific_journal-4617 101 16 0.4 0.4 NUM american_scientific_journal-4617 101 17 0.5 0.5 NUM american_scientific_journal-4617 101 18 0.6 0.6 NUM american_scientific_journal-4617 101 19 0.7 0.7 NUM american_scientific_journal-4617 101 20 0.8 0.8 NUM american_scientific_journal-4617 101 21 0.9 0.9 NUM american_scientific_journal-4617 101 22 1 1 NUM american_scientific_journal-4617 101 23 0 0 NUM american_scientific_journal-4617 101 24 0.1 0.1 NUM american_scientific_journal-4617 101 25 0.2 0.2 NUM american_scientific_journal-4617 101 26 0.3 0.3 NUM american_scientific_journal-4617 101 27 0.4 0.4 NUM american_scientific_journal-4617 101 28 0.5 0.5 NUM american_scientific_journal-4617 101 29 0.6 0.6 NUM american_scientific_journal-4617 101 30 0.7 0.7 NUM american_scientific_journal-4617 101 31 0.8 0.8 NUM american_scientific_journal-4617 101 32 0.9 0.9 NUM american_scientific_journal-4617 101 33 1 1 NUM american_scientific_journal-4617 101 34 x x SYM american_scientific_journal-4617 101 35 y y PROPN american_scientific_journal-4617 101 36 comparison comparison NOUN american_scientific_journal-4617 101 37 with with ADP american_scientific_journal-4617 101 38 different different ADJ american_scientific_journal-4617 101 39 models model NOUN american_scientific_journal-4617 101 40 exact exact ADJ american_scientific_journal-4617 101 41 solution solution NOUN american_scientific_journal-4617 101 42 approximate approximate ADJ american_scientific_journal-4617 101 43 solution solution NOUN american_scientific_journal-4617 101 44 with with ADP american_scientific_journal-4617 101 45 current current ADJ american_scientific_journal-4617 101 46 method method NOUN american_scientific_journal-4617 101 47 approximate approximate ADJ american_scientific_journal-4617 101 48 solution solution NOUN american_scientific_journal-4617 101 49 with with ADP american_scientific_journal-4617 101 50 fdm fdm NOUN american_scientific_journal-4617 101 51 error error NOUN american_scientific_journal-4617 101 52 with with ADP american_scientific_journal-4617 101 53 current current ADJ american_scientific_journal-4617 101 54 method method NOUN american_scientific_journal-4617 101 55 error error NOUN american_scientific_journal-4617 101 56 with with ADP american_scientific_journal-4617 101 57 fdm fdm ADJ american_scientific_journal-4617 101 58 american american ADJ american_scientific_journal-4617 101 59 scientific scientific ADJ american_scientific_journal-4617 101 60 research research NOUN american_scientific_journal-4617 101 61 journal journal NOUN american_scientific_journal-4617 101 62 for for ADP american_scientific_journal-4617 101 63 engineering engineering NOUN american_scientific_journal-4617 101 64 , , PUNCT american_scientific_journal-4617 101 65 technology technology NOUN american_scientific_journal-4617 101 66 , , PUNCT american_scientific_journal-4617 101 67 and and CCONJ american_scientific_journal-4617 101 68 sciences science NOUN american_scientific_journal-4617 101 69 ( ( PUNCT american_scientific_journal-4617 101 70 asrjets asrjet NOUN american_scientific_journal-4617 101 71 ) ) PUNCT american_scientific_journal-4617 101 72 ( ( PUNCT american_scientific_journal-4617 101 73 2019 2019 NUM american_scientific_journal-4617 101 74 ) ) PUNCT american_scientific_journal-4617 101 75 volume volume NOUN american_scientific_journal-4617 101 76 51 51 NUM american_scientific_journal-4617 101 77 , , PUNCT american_scientific_journal-4617 101 78 no no DET american_scientific_journal-4617 101 79 1 1 NUM american_scientific_journal-4617 101 80 , , PUNCT american_scientific_journal-4617 101 81 pp pp ADV american_scientific_journal-4617 101 82 225 225 NUM american_scientific_journal-4617 101 83 - - SYM american_scientific_journal-4617 101 84 234 234 NUM american_scientific_journal-4617 101 85 232 232 NUM american_scientific_journal-4617 101 86 example example NOUN american_scientific_journal-4617 101 87 3 3 NUM american_scientific_journal-4617 101 88 : : PUNCT american_scientific_journal-4617 101 89 consider consider VERB american_scientific_journal-4617 101 90 the the DET american_scientific_journal-4617 101 91 boundary boundary ADJ american_scientific_journal-4617 101 92 value value NOUN american_scientific_journal-4617 101 93 problem problem NOUN american_scientific_journal-4617 101 94 [ [ X american_scientific_journal-4617 101 95 4 4 X american_scientific_journal-4617 101 96 ] ] X american_scientific_journal-4617 101 97 𝑦𝑦′′ 𝑦𝑦′′ NOUN american_scientific_journal-4617 101 98 + + CCONJ american_scientific_journal-4617 101 99 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-4617 101 100 = = SYM american_scientific_journal-4617 101 101 π‘₯π‘₯4 π‘₯π‘₯4 NOUN american_scientific_journal-4617 101 102 + + NOUN american_scientific_journal-4617 101 103 2 2 NUM american_scientific_journal-4617 101 104 , , PUNCT american_scientific_journal-4617 101 105 ( ( PUNCT american_scientific_journal-4617 101 106 4.9 4.9 NUM american_scientific_journal-4617 101 107 ) ) PUNCT american_scientific_journal-4617 101 108 with with ADP american_scientific_journal-4617 101 109 boundary boundary ADJ american_scientific_journal-4617 101 110 conditions condition NOUN american_scientific_journal-4617 101 111 𝑦𝑦(0 𝑦𝑦(0 NOUN american_scientific_journal-4617 101 112 ) ) PUNCT american_scientific_journal-4617 101 113 = = SYM american_scientific_journal-4617 101 114 0 0 NUM american_scientific_journal-4617 101 115 , , PUNCT american_scientific_journal-4617 101 116 𝑦𝑦(1 𝑦𝑦(1 NOUN american_scientific_journal-4617 101 117 ) ) PUNCT american_scientific_journal-4617 101 118 = = SYM american_scientific_journal-4617 101 119 1 1 NUM american_scientific_journal-4617 101 120 ( ( PUNCT american_scientific_journal-4617 101 121 4.10 4.10 NUM american_scientific_journal-4617 101 122 ) ) PUNCT american_scientific_journal-4617 101 123 the the DET american_scientific_journal-4617 101 124 exact exact ADJ american_scientific_journal-4617 101 125 solution solution NOUN american_scientific_journal-4617 101 126 is be AUX american_scientific_journal-4617 101 127 𝑦𝑦(π‘₯π‘₯ 𝑦𝑦(π‘₯π‘₯ PROPN american_scientific_journal-4617 101 128 ) ) PUNCT american_scientific_journal-4617 102 1 = = SYM american_scientific_journal-4617 103 1 x2 x2 ADJ american_scientific_journal-4617 103 2 substituting substitute VERB american_scientific_journal-4617 103 3 equations equation NOUN american_scientific_journal-4617 103 4 ( ( PUNCT american_scientific_journal-4617 103 5 4.10 4.10 NUM american_scientific_journal-4617 103 6 ) ) PUNCT american_scientific_journal-4617 103 7 in in ADP american_scientific_journal-4617 103 8 equation equation NOUN american_scientific_journal-4617 103 9 ( ( PUNCT american_scientific_journal-4617 103 10 2.7 2.7 NUM american_scientific_journal-4617 103 11 ) ) PUNCT american_scientific_journal-4617 103 12 we we PRON american_scientific_journal-4617 103 13 get get VERB american_scientific_journal-4617 103 14 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 103 15 βˆ’ βˆ’ NOUN american_scientific_journal-4617 103 16 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 103 17 + + CCONJ american_scientific_journal-4617 103 18 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-4617 104 1 = = SYM american_scientific_journal-4617 104 2 0 0 NUM american_scientific_journal-4617 104 3 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 104 4 + + X american_scientific_journal-4617 104 5 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 105 1 + + CCONJ american_scientific_journal-4617 105 2 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-4617 106 1 = = SYM american_scientific_journal-4617 106 2 1 1 NUM american_scientific_journal-4617 106 3 ( ( PUNCT american_scientific_journal-4617 106 4 4.11 4.11 NUM american_scientific_journal-4617 106 5 ) ) PUNCT american_scientific_journal-4617 106 6 for for ADP american_scientific_journal-4617 106 7 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 106 8 = = SYM american_scientific_journal-4617 106 9 2 2 NUM american_scientific_journal-4617 106 10 from from ADP american_scientific_journal-4617 106 11 equation equation NOUN american_scientific_journal-4617 106 12 ( ( PUNCT american_scientific_journal-4617 106 13 3.7 3.7 NUM american_scientific_journal-4617 106 14 ) ) PUNCT american_scientific_journal-4617 106 15 we we PRON american_scientific_journal-4617 106 16 get get VERB american_scientific_journal-4617 106 17 𝑐𝑐02 𝑐𝑐02 PROPN american_scientific_journal-4617 106 18 + + CCONJ american_scientific_journal-4617 106 19 1 1 NUM american_scientific_journal-4617 106 20 3 3 NUM american_scientific_journal-4617 106 21 𝑐𝑐12 𝑐𝑐12 PROPN american_scientific_journal-4617 106 22 + + CCONJ american_scientific_journal-4617 106 23 1 1 NUM american_scientific_journal-4617 106 24 5 5 X american_scientific_journal-4617 106 25 𝑐𝑐22 𝑐𝑐22 PROPN american_scientific_journal-4617 106 26 + + NUM american_scientific_journal-4617 106 27 12𝑐𝑐2 12𝑐𝑐2 NUM american_scientific_journal-4617 106 28 = = SYM american_scientific_journal-4617 106 29 11 11 NUM american_scientific_journal-4617 106 30 5 5 NUM american_scientific_journal-4617 106 31 ( ( PUNCT american_scientific_journal-4617 106 32 4.12 4.12 NUM american_scientific_journal-4617 106 33 ) ) PUNCT american_scientific_journal-4617 106 34 solving solve VERB american_scientific_journal-4617 106 35 a a DET american_scientific_journal-4617 106 36 3 3 NUM american_scientific_journal-4617 106 37 Γ— Γ— NOUN american_scientific_journal-4617 106 38 3 3 NUM american_scientific_journal-4617 106 39 system system NOUN american_scientific_journal-4617 106 40 of of ADP american_scientific_journal-4617 106 41 algebraic algebraic ADJ american_scientific_journal-4617 106 42 equations equation NOUN american_scientific_journal-4617 106 43 ( ( PUNCT american_scientific_journal-4617 106 44 4.11 4.11 NUM american_scientific_journal-4617 106 45 ) ) PUNCT american_scientific_journal-4617 106 46 and and CCONJ american_scientific_journal-4617 106 47 ( ( PUNCT american_scientific_journal-4617 106 48 4.12 4.12 NUM american_scientific_journal-4617 106 49 ) ) PUNCT american_scientific_journal-4617 106 50 gives give VERB american_scientific_journal-4617 106 51 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 106 52 = = NOUN american_scientific_journal-4617 106 53 1 1 NUM american_scientific_journal-4617 106 54 3 3 NUM american_scientific_journal-4617 106 55 , , PUNCT american_scientific_journal-4617 106 56 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 107 1 = = SYM american_scientific_journal-4617 108 1 1 1 NUM american_scientific_journal-4617 108 2 2 2 NUM american_scientific_journal-4617 108 3 and and CCONJ american_scientific_journal-4617 108 4 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-4617 109 1 = = NOUN american_scientific_journal-4617 109 2 1 1 NUM american_scientific_journal-4617 109 3 6 6 NUM american_scientific_journal-4617 109 4 hence hence ADV american_scientific_journal-4617 109 5 the the DET american_scientific_journal-4617 109 6 approximate approximate ADJ american_scientific_journal-4617 109 7 solution solution NOUN american_scientific_journal-4617 109 8 is be AUX american_scientific_journal-4617 109 9 now now ADV american_scientific_journal-4617 109 10 becomes become VERB american_scientific_journal-4617 109 11 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-4617 109 12 οΏ½ οΏ½ X american_scientific_journal-4617 109 13 (π‘₯π‘₯ (π‘₯π‘₯ PUNCT american_scientific_journal-4617 109 14 ) ) PUNCT american_scientific_journal-4617 110 1 = = SYM american_scientific_journal-4617 110 2 x2 x2 PROPN american_scientific_journal-4617 110 3 which which PRON american_scientific_journal-4617 110 4 is be AUX american_scientific_journal-4617 110 5 the the DET american_scientific_journal-4617 110 6 exact exact ADJ american_scientific_journal-4617 110 7 solution solution NOUN american_scientific_journal-4617 110 8 . . PUNCT american_scientific_journal-4617 111 1 figure figure VERB american_scientific_journal-4617 111 2 3 3 NUM american_scientific_journal-4617 111 3 : : PUNCT american_scientific_journal-4617 111 4 graphical graphical ADJ american_scientific_journal-4617 111 5 illustration illustration NOUN american_scientific_journal-4617 111 6 of of ADP american_scientific_journal-4617 111 7 example example NOUN american_scientific_journal-4617 111 8 3 3 NUM american_scientific_journal-4617 111 9 0 0 NUM american_scientific_journal-4617 111 10 0.1 0.1 NUM american_scientific_journal-4617 111 11 0.2 0.2 NUM american_scientific_journal-4617 111 12 0.3 0.3 NUM american_scientific_journal-4617 111 13 0.4 0.4 NUM american_scientific_journal-4617 111 14 0.5 0.5 NUM american_scientific_journal-4617 111 15 0.6 0.6 NUM american_scientific_journal-4617 111 16 0.7 0.7 NUM american_scientific_journal-4617 111 17 0.8 0.8 NUM american_scientific_journal-4617 111 18 0.9 0.9 NUM american_scientific_journal-4617 111 19 1 1 NUM american_scientific_journal-4617 111 20 0 0 NUM american_scientific_journal-4617 111 21 0.1 0.1 NUM american_scientific_journal-4617 111 22 0.2 0.2 NUM american_scientific_journal-4617 111 23 0.3 0.3 NUM american_scientific_journal-4617 111 24 0.4 0.4 NUM american_scientific_journal-4617 111 25 0.5 0.5 NUM american_scientific_journal-4617 111 26 0.6 0.6 NUM american_scientific_journal-4617 111 27 0.7 0.7 NUM american_scientific_journal-4617 111 28 0.8 0.8 NUM american_scientific_journal-4617 111 29 0.9 0.9 NUM american_scientific_journal-4617 111 30 1 1 NUM american_scientific_journal-4617 111 31 x x SYM american_scientific_journal-4617 111 32 y y PROPN american_scientific_journal-4617 111 33 comparison comparison NOUN american_scientific_journal-4617 111 34 with with ADP american_scientific_journal-4617 111 35 current current ADJ american_scientific_journal-4617 111 36 model model NOUN american_scientific_journal-4617 111 37 exact exact ADJ american_scientific_journal-4617 111 38 solution solution NOUN american_scientific_journal-4617 111 39 approximate approximate ADJ american_scientific_journal-4617 111 40 solution solution NOUN american_scientific_journal-4617 111 41 with with ADP american_scientific_journal-4617 111 42 current current ADJ american_scientific_journal-4617 111 43 method method NOUN american_scientific_journal-4617 111 44 error error NOUN american_scientific_journal-4617 111 45 with with ADP american_scientific_journal-4617 111 46 current current ADJ american_scientific_journal-4617 111 47 method method NOUN american_scientific_journal-4617 111 48 american american PROPN american_scientific_journal-4617 111 49 scientific scientific ADJ american_scientific_journal-4617 111 50 research research NOUN american_scientific_journal-4617 111 51 journal journal NOUN american_scientific_journal-4617 111 52 for for ADP american_scientific_journal-4617 111 53 engineering engineering NOUN american_scientific_journal-4617 111 54 , , PUNCT american_scientific_journal-4617 111 55 technology technology NOUN american_scientific_journal-4617 111 56 , , PUNCT american_scientific_journal-4617 111 57 and and CCONJ american_scientific_journal-4617 111 58 sciences science NOUN american_scientific_journal-4617 111 59 ( ( PUNCT american_scientific_journal-4617 111 60 asrjets asrjet NOUN american_scientific_journal-4617 111 61 ) ) PUNCT american_scientific_journal-4617 111 62 ( ( PUNCT american_scientific_journal-4617 111 63 2019 2019 NUM american_scientific_journal-4617 111 64 ) ) PUNCT american_scientific_journal-4617 111 65 volume volume NOUN american_scientific_journal-4617 111 66 51 51 NUM american_scientific_journal-4617 111 67 , , PUNCT american_scientific_journal-4617 111 68 no no DET american_scientific_journal-4617 111 69 1 1 NUM american_scientific_journal-4617 111 70 , , PUNCT american_scientific_journal-4617 111 71 pp pp ADV american_scientific_journal-4617 111 72 225 225 NUM american_scientific_journal-4617 111 73 - - SYM american_scientific_journal-4617 111 74 234 234 NUM american_scientific_journal-4617 111 75 233 233 NUM american_scientific_journal-4617 111 76 example example NOUN american_scientific_journal-4617 111 77 4 4 NUM american_scientific_journal-4617 111 78 : : PUNCT american_scientific_journal-4617 111 79 consider consider VERB american_scientific_journal-4617 111 80 the the DET american_scientific_journal-4617 111 81 boundary boundary ADJ american_scientific_journal-4617 111 82 value value NOUN american_scientific_journal-4617 111 83 problem problem NOUN american_scientific_journal-4617 111 84 𝑦𝑦′′ 𝑦𝑦′′ VERB american_scientific_journal-4617 112 1 + + CCONJ american_scientific_journal-4617 112 2 ln ln ADJ american_scientific_journal-4617 112 3 π‘₯π‘₯𝑦𝑦′ π‘₯π‘₯𝑦𝑦′ NOUN american_scientific_journal-4617 112 4 + + CCONJ american_scientific_journal-4617 112 5 𝑦𝑦2 𝑦𝑦2 ADJ american_scientific_journal-4617 112 6 = = SYM american_scientific_journal-4617 112 7 2 2 NUM american_scientific_journal-4617 112 8 + + NUM american_scientific_journal-4617 112 9 2π‘₯π‘₯ 2π‘₯π‘₯ ADJ american_scientific_journal-4617 112 10 ln ln ADJ american_scientific_journal-4617 112 11 π‘₯π‘₯ π‘₯π‘₯ ADP american_scientific_journal-4617 112 12 + + CCONJ american_scientific_journal-4617 112 13 π‘₯π‘₯4 π‘₯π‘₯4 ADJ american_scientific_journal-4617 112 14 , , PUNCT american_scientific_journal-4617 112 15 ( ( PUNCT american_scientific_journal-4617 112 16 4.13 4.13 NUM american_scientific_journal-4617 112 17 ) ) PUNCT american_scientific_journal-4617 112 18 with with ADP american_scientific_journal-4617 112 19 boundary boundary ADJ american_scientific_journal-4617 112 20 conditions condition NOUN american_scientific_journal-4617 112 21 𝑦𝑦(0 𝑦𝑦(0 NOUN american_scientific_journal-4617 112 22 ) ) PUNCT american_scientific_journal-4617 112 23 = = SYM american_scientific_journal-4617 112 24 0 0 NUM american_scientific_journal-4617 112 25 , , PUNCT american_scientific_journal-4617 112 26 𝑦𝑦(1 𝑦𝑦(1 NOUN american_scientific_journal-4617 112 27 ) ) PUNCT american_scientific_journal-4617 112 28 = = SYM american_scientific_journal-4617 112 29 1 1 NUM american_scientific_journal-4617 112 30 ( ( PUNCT american_scientific_journal-4617 112 31 4.14 4.14 NUM american_scientific_journal-4617 112 32 ) ) PUNCT american_scientific_journal-4617 112 33 the the DET american_scientific_journal-4617 112 34 exact exact ADJ american_scientific_journal-4617 112 35 solution solution NOUN american_scientific_journal-4617 112 36 is be AUX american_scientific_journal-4617 112 37 𝑦𝑦(π‘₯π‘₯ 𝑦𝑦(π‘₯π‘₯ PROPN american_scientific_journal-4617 112 38 ) ) PUNCT american_scientific_journal-4617 113 1 = = SYM american_scientific_journal-4617 114 1 x2 x2 ADJ american_scientific_journal-4617 114 2 substituting substitute VERB american_scientific_journal-4617 114 3 equations equation NOUN american_scientific_journal-4617 114 4 ( ( PUNCT american_scientific_journal-4617 114 5 4.14 4.14 NUM american_scientific_journal-4617 114 6 ) ) PUNCT american_scientific_journal-4617 114 7 in in ADP american_scientific_journal-4617 114 8 equation equation NOUN american_scientific_journal-4617 114 9 ( ( PUNCT american_scientific_journal-4617 114 10 2.7 2.7 NUM american_scientific_journal-4617 114 11 ) ) PUNCT american_scientific_journal-4617 114 12 we we PRON american_scientific_journal-4617 114 13 get get VERB american_scientific_journal-4617 114 14 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 114 15 βˆ’ βˆ’ NOUN american_scientific_journal-4617 114 16 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 114 17 + + CCONJ american_scientific_journal-4617 114 18 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-4617 115 1 = = SYM american_scientific_journal-4617 115 2 0 0 NUM american_scientific_journal-4617 115 3 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 115 4 + + X american_scientific_journal-4617 115 5 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 116 1 + + CCONJ american_scientific_journal-4617 116 2 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-4617 116 3 = = SYM american_scientific_journal-4617 116 4 1 1 NUM american_scientific_journal-4617 116 5 ( ( PUNCT american_scientific_journal-4617 116 6 4.15 4.15 NUM american_scientific_journal-4617 116 7 ) ) PUNCT american_scientific_journal-4617 116 8 for for ADP american_scientific_journal-4617 116 9 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 116 10 = = SYM american_scientific_journal-4617 116 11 2 2 NUM american_scientific_journal-4617 116 12 from from ADP american_scientific_journal-4617 116 13 equation equation NOUN american_scientific_journal-4617 116 14 ( ( PUNCT american_scientific_journal-4617 116 15 3.7 3.7 NUM american_scientific_journal-4617 116 16 ) ) PUNCT american_scientific_journal-4617 116 17 we we PRON american_scientific_journal-4617 116 18 get get VERB american_scientific_journal-4617 116 19 𝑐𝑐02 𝑐𝑐02 PROPN american_scientific_journal-4617 116 20 + + CCONJ american_scientific_journal-4617 116 21 1 1 NUM american_scientific_journal-4617 116 22 3 3 NUM american_scientific_journal-4617 116 23 𝑐𝑐12 𝑐𝑐12 PROPN american_scientific_journal-4617 116 24 + + CCONJ american_scientific_journal-4617 116 25 1 1 NUM american_scientific_journal-4617 116 26 5 5 NUM american_scientific_journal-4617 116 27 𝑐𝑐22 𝑐𝑐22 PROPN american_scientific_journal-4617 116 28 βˆ’ βˆ’ PROPN american_scientific_journal-4617 116 29 2𝑐𝑐1 2𝑐𝑐1 NUM american_scientific_journal-4617 117 1 + + CCONJ american_scientific_journal-4617 117 2 15𝑐𝑐2 15𝑐𝑐2 PROPN american_scientific_journal-4617 117 3 = = SYM american_scientific_journal-4617 117 4 17 17 NUM american_scientific_journal-4617 117 5 10 10 NUM american_scientific_journal-4617 117 6 ( ( PUNCT american_scientific_journal-4617 117 7 4.16 4.16 NUM american_scientific_journal-4617 117 8 ) ) PUNCT american_scientific_journal-4617 117 9 solving solve VERB american_scientific_journal-4617 117 10 a a DET american_scientific_journal-4617 117 11 3 3 NUM american_scientific_journal-4617 117 12 Γ— Γ— NOUN american_scientific_journal-4617 117 13 3 3 NUM american_scientific_journal-4617 117 14 system system NOUN american_scientific_journal-4617 117 15 of of ADP american_scientific_journal-4617 117 16 algebraic algebraic ADJ american_scientific_journal-4617 117 17 equations equation NOUN american_scientific_journal-4617 117 18 ( ( PUNCT american_scientific_journal-4617 117 19 4.15 4.15 NUM american_scientific_journal-4617 117 20 ) ) PUNCT american_scientific_journal-4617 117 21 and and CCONJ american_scientific_journal-4617 117 22 ( ( PUNCT american_scientific_journal-4617 117 23 4.16 4.16 NUM american_scientific_journal-4617 117 24 ) ) PUNCT american_scientific_journal-4617 117 25 gives give VERB american_scientific_journal-4617 117 26 𝑐𝑐0 𝑐𝑐0 NOUN american_scientific_journal-4617 117 27 = = NOUN american_scientific_journal-4617 117 28 1 1 NUM american_scientific_journal-4617 117 29 3 3 NUM american_scientific_journal-4617 117 30 , , PUNCT american_scientific_journal-4617 117 31 𝑐𝑐1 𝑐𝑐1 PROPN american_scientific_journal-4617 117 32 = = SYM american_scientific_journal-4617 117 33 1 1 NUM american_scientific_journal-4617 117 34 2 2 NUM american_scientific_journal-4617 117 35 and and CCONJ american_scientific_journal-4617 117 36 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-4617 118 1 = = NOUN american_scientific_journal-4617 118 2 1 1 NUM american_scientific_journal-4617 118 3 6 6 NUM american_scientific_journal-4617 118 4 hence hence ADV american_scientific_journal-4617 118 5 the the DET american_scientific_journal-4617 118 6 approximate approximate ADJ american_scientific_journal-4617 118 7 solution solution NOUN american_scientific_journal-4617 118 8 is be AUX american_scientific_journal-4617 118 9 now now ADV american_scientific_journal-4617 118 10 becomes become VERB american_scientific_journal-4617 118 11 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-4617 118 12 οΏ½ οΏ½ X american_scientific_journal-4617 118 13 (π‘₯π‘₯ (π‘₯π‘₯ PUNCT american_scientific_journal-4617 118 14 ) ) PUNCT american_scientific_journal-4617 119 1 = = SYM american_scientific_journal-4617 119 2 x2 x2 PROPN american_scientific_journal-4617 119 3 which which PRON american_scientific_journal-4617 119 4 is be AUX american_scientific_journal-4617 119 5 the the DET american_scientific_journal-4617 119 6 exact exact ADJ american_scientific_journal-4617 119 7 solution solution NOUN american_scientific_journal-4617 119 8 . . PUNCT american_scientific_journal-4617 120 1 figure figure VERB american_scientific_journal-4617 120 2 4 4 NUM american_scientific_journal-4617 120 3 : : PUNCT american_scientific_journal-4617 120 4 graphical graphical ADJ american_scientific_journal-4617 120 5 illustration illustration NOUN american_scientific_journal-4617 120 6 of of ADP american_scientific_journal-4617 120 7 example example NOUN american_scientific_journal-4617 120 8 4 4 NUM american_scientific_journal-4617 120 9 5 5 NUM american_scientific_journal-4617 120 10 . . PUNCT american_scientific_journal-4617 120 11 conclusion conclusion NOUN american_scientific_journal-4617 120 12 and and CCONJ american_scientific_journal-4617 120 13 recommendations recommendation NOUN american_scientific_journal-4617 120 14 for for ADP american_scientific_journal-4617 120 15 problems problem NOUN american_scientific_journal-4617 120 16 in in ADP american_scientific_journal-4617 120 17 examples example NOUN american_scientific_journal-4617 120 18 1,3 1,3 NUM american_scientific_journal-4617 120 19 & & CCONJ american_scientific_journal-4617 120 20 4 4 NUM american_scientific_journal-4617 120 21 where where SCONJ american_scientific_journal-4617 120 22 the the DET american_scientific_journal-4617 120 23 exact exact ADJ american_scientific_journal-4617 120 24 solution solution NOUN american_scientific_journal-4617 120 25 are be AUX american_scientific_journal-4617 120 26 polynomials polynomial NOUN american_scientific_journal-4617 120 27 , , PUNCT american_scientific_journal-4617 120 28 the the DET american_scientific_journal-4617 120 29 method method NOUN american_scientific_journal-4617 120 30 produces produce VERB american_scientific_journal-4617 120 31 the the DET american_scientific_journal-4617 120 32 exact exact ADJ american_scientific_journal-4617 120 33 solution solution NOUN american_scientific_journal-4617 120 34 itself itself PRON american_scientific_journal-4617 120 35 by by ADP american_scientific_journal-4617 120 36 just just ADV american_scientific_journal-4617 120 37 taking take VERB american_scientific_journal-4617 120 38 π‘šπ‘š π‘šπ‘š ADP american_scientific_journal-4617 120 39 = = SYM american_scientific_journal-4617 120 40 2 2 NUM american_scientific_journal-4617 120 41 , , PUNCT american_scientific_journal-4617 120 42 and and CCONJ american_scientific_journal-4617 120 43 using use VERB american_scientific_journal-4617 120 44 only only ADV american_scientific_journal-4617 120 45 first first ADJ american_scientific_journal-4617 120 46 few few ADJ american_scientific_journal-4617 120 47 shifted shift VERB american_scientific_journal-4617 120 48 legendre legendre NOUN american_scientific_journal-4617 120 49 polynomials polynomial NOUN american_scientific_journal-4617 120 50 . . PUNCT american_scientific_journal-4617 121 1 this this PRON american_scientific_journal-4617 121 2 makes make VERB american_scientific_journal-4617 121 3 the the DET american_scientific_journal-4617 121 4 method method NOUN american_scientific_journal-4617 121 5 more more ADV american_scientific_journal-4617 121 6 accurate accurate ADJ american_scientific_journal-4617 121 7 than than ADP american_scientific_journal-4617 121 8 others other NOUN american_scientific_journal-4617 121 9 . . PUNCT american_scientific_journal-4617 122 1 for for ADP american_scientific_journal-4617 122 2 problem problem NOUN american_scientific_journal-4617 122 3 in in ADP american_scientific_journal-4617 122 4 example example NOUN american_scientific_journal-4617 122 5 2 2 NUM american_scientific_journal-4617 122 6 , , PUNCT american_scientific_journal-4617 122 7 where where SCONJ american_scientific_journal-4617 122 8 the the DET american_scientific_journal-4617 122 9 exact exact ADJ american_scientific_journal-4617 122 10 solution solution NOUN american_scientific_journal-4617 122 11 is be AUX american_scientific_journal-4617 122 12 not not PART american_scientific_journal-4617 122 13 polynomial polynomial ADJ american_scientific_journal-4617 122 14 a a DET american_scientific_journal-4617 122 15 0 0 NUM american_scientific_journal-4617 122 16 0.1 0.1 NUM american_scientific_journal-4617 122 17 0.2 0.2 NUM american_scientific_journal-4617 122 18 0.3 0.3 NUM american_scientific_journal-4617 122 19 0.4 0.4 NUM american_scientific_journal-4617 122 20 0.5 0.5 NUM american_scientific_journal-4617 122 21 0.6 0.6 NUM american_scientific_journal-4617 122 22 0.7 0.7 NUM american_scientific_journal-4617 122 23 0.8 0.8 NUM american_scientific_journal-4617 122 24 0.9 0.9 NUM american_scientific_journal-4617 122 25 1 1 NUM american_scientific_journal-4617 122 26 0 0 NUM american_scientific_journal-4617 122 27 0.1 0.1 NUM american_scientific_journal-4617 122 28 0.2 0.2 NUM american_scientific_journal-4617 122 29 0.3 0.3 NUM american_scientific_journal-4617 122 30 0.4 0.4 NUM american_scientific_journal-4617 122 31 0.5 0.5 NUM american_scientific_journal-4617 122 32 0.6 0.6 NUM american_scientific_journal-4617 122 33 0.7 0.7 NUM american_scientific_journal-4617 122 34 0.8 0.8 NUM american_scientific_journal-4617 122 35 0.9 0.9 NUM american_scientific_journal-4617 122 36 1 1 NUM american_scientific_journal-4617 122 37 x x SYM american_scientific_journal-4617 122 38 y y PROPN american_scientific_journal-4617 122 39 comparison comparison NOUN american_scientific_journal-4617 122 40 with with ADP american_scientific_journal-4617 122 41 current current ADJ american_scientific_journal-4617 122 42 model model NOUN american_scientific_journal-4617 122 43 exact exact ADJ american_scientific_journal-4617 122 44 solution solution NOUN american_scientific_journal-4617 122 45 approximate approximate ADJ american_scientific_journal-4617 122 46 solution solution NOUN american_scientific_journal-4617 122 47 with with ADP american_scientific_journal-4617 122 48 current current ADJ american_scientific_journal-4617 122 49 method method NOUN american_scientific_journal-4617 122 50 error error NOUN american_scientific_journal-4617 122 51 with with ADP american_scientific_journal-4617 122 52 current current ADJ american_scientific_journal-4617 122 53 method method NOUN american_scientific_journal-4617 122 54 american american PROPN american_scientific_journal-4617 122 55 scientific scientific ADJ american_scientific_journal-4617 122 56 research research NOUN american_scientific_journal-4617 122 57 journal journal NOUN american_scientific_journal-4617 122 58 for for ADP american_scientific_journal-4617 122 59 engineering engineering NOUN american_scientific_journal-4617 122 60 , , PUNCT american_scientific_journal-4617 122 61 technology technology NOUN american_scientific_journal-4617 122 62 , , PUNCT american_scientific_journal-4617 122 63 and and CCONJ american_scientific_journal-4617 122 64 sciences science NOUN american_scientific_journal-4617 122 65 ( ( PUNCT american_scientific_journal-4617 122 66 asrjets asrjet NOUN american_scientific_journal-4617 122 67 ) ) PUNCT american_scientific_journal-4617 122 68 ( ( PUNCT american_scientific_journal-4617 122 69 2019 2019 NUM american_scientific_journal-4617 122 70 ) ) PUNCT american_scientific_journal-4617 122 71 volume volume NOUN american_scientific_journal-4617 122 72 51 51 NUM american_scientific_journal-4617 122 73 , , PUNCT american_scientific_journal-4617 122 74 no no DET american_scientific_journal-4617 122 75 1 1 NUM american_scientific_journal-4617 122 76 , , PUNCT american_scientific_journal-4617 122 77 pp pp ADV american_scientific_journal-4617 122 78 225 225 NUM american_scientific_journal-4617 122 79 - - SYM american_scientific_journal-4617 122 80 234 234 NUM american_scientific_journal-4617 122 81 234 234 NUM american_scientific_journal-4617 122 82 better well ADJ american_scientific_journal-4617 122 83 approximation approximation NOUN american_scientific_journal-4617 122 84 to to ADP american_scientific_journal-4617 122 85 the the DET american_scientific_journal-4617 122 86 exact exact ADJ american_scientific_journal-4617 122 87 solution solution NOUN american_scientific_journal-4617 122 88 is be AUX american_scientific_journal-4617 122 89 found find VERB american_scientific_journal-4617 122 90 when when SCONJ american_scientific_journal-4617 122 91 compared compare VERB american_scientific_journal-4617 122 92 to to ADP american_scientific_journal-4617 122 93 the the DET american_scientific_journal-4617 122 94 finite finite ADJ american_scientific_journal-4617 122 95 difference difference NOUN american_scientific_journal-4617 122 96 method method NOUN american_scientific_journal-4617 122 97 . . PUNCT american_scientific_journal-4617 123 1 the the DET american_scientific_journal-4617 123 2 other other ADJ american_scientific_journal-4617 123 3 advantage advantage NOUN american_scientific_journal-4617 123 4 of of ADP american_scientific_journal-4617 123 5 this this DET american_scientific_journal-4617 123 6 method method NOUN american_scientific_journal-4617 123 7 over over ADP american_scientific_journal-4617 123 8 the the DET american_scientific_journal-4617 123 9 other other ADJ american_scientific_journal-4617 123 10 methods method NOUN american_scientific_journal-4617 123 11 like like ADP american_scientific_journal-4617 123 12 spline spline NOUN american_scientific_journal-4617 123 13 methods method NOUN american_scientific_journal-4617 123 14 and and CCONJ american_scientific_journal-4617 123 15 finite finite ADJ american_scientific_journal-4617 123 16 difference difference NOUN american_scientific_journal-4617 123 17 method method NOUN american_scientific_journal-4617 123 18 is be AUX american_scientific_journal-4617 123 19 that that SCONJ american_scientific_journal-4617 123 20 it it PRON american_scientific_journal-4617 123 21 needs need VERB american_scientific_journal-4617 123 22 less less ADV american_scientific_journal-4617 123 23 computational computational ADJ american_scientific_journal-4617 123 24 time time NOUN american_scientific_journal-4617 123 25 and and CCONJ american_scientific_journal-4617 123 26 effort effort NOUN american_scientific_journal-4617 123 27 . . PUNCT american_scientific_journal-4617 124 1 in in ADP american_scientific_journal-4617 124 2 the the DET american_scientific_journal-4617 124 3 feature feature NOUN american_scientific_journal-4617 124 4 , , PUNCT american_scientific_journal-4617 124 5 the the DET american_scientific_journal-4617 124 6 method method NOUN american_scientific_journal-4617 124 7 can can AUX american_scientific_journal-4617 124 8 be be AUX american_scientific_journal-4617 124 9 extended extend VERB american_scientific_journal-4617 124 10 to to PART american_scientific_journal-4617 124 11 find find VERB american_scientific_journal-4617 124 12 the the DET american_scientific_journal-4617 124 13 solutions solution NOUN american_scientific_journal-4617 124 14 of of ADP american_scientific_journal-4617 124 15 second second ADJ american_scientific_journal-4617 124 16 order order NOUN american_scientific_journal-4617 124 17 linear linear ADJ american_scientific_journal-4617 124 18 and and CCONJ american_scientific_journal-4617 124 19 nonlinear nonlinear ADJ american_scientific_journal-4617 124 20 two two NUM american_scientific_journal-4617 124 21 point point NOUN american_scientific_journal-4617 124 22 boundary boundary ADJ american_scientific_journal-4617 124 23 value value NOUN american_scientific_journal-4617 124 24 problems problem NOUN american_scientific_journal-4617 124 25 of of ADP american_scientific_journal-4617 124 26 partial partial ADJ american_scientific_journal-4617 124 27 differential differential ADJ american_scientific_journal-4617 124 28 equations equation NOUN american_scientific_journal-4617 124 29 . . PUNCT american_scientific_journal-4617 125 1 acknowledgements acknowledgement NOUN american_scientific_journal-4617 125 2 the the DET american_scientific_journal-4617 125 3 author author NOUN american_scientific_journal-4617 125 4 would would AUX american_scientific_journal-4617 125 5 like like VERB american_scientific_journal-4617 125 6 to to PART american_scientific_journal-4617 125 7 express express VERB american_scientific_journal-4617 125 8 his his PRON american_scientific_journal-4617 125 9 sincere sincere ADJ american_scientific_journal-4617 125 10 gratitude gratitude NOUN american_scientific_journal-4617 125 11 to to ADP american_scientific_journal-4617 125 12 the the DET american_scientific_journal-4617 125 13 anonymous anonymous ADJ american_scientific_journal-4617 125 14 referees referee NOUN american_scientific_journal-4617 125 15 for for ADP american_scientific_journal-4617 125 16 their their PRON american_scientific_journal-4617 125 17 exciting exciting ADJ american_scientific_journal-4617 125 18 comments comment NOUN american_scientific_journal-4617 125 19 which which PRON american_scientific_journal-4617 125 20 helped help VERB american_scientific_journal-4617 125 21 him he PRON american_scientific_journal-4617 125 22 in in ADP american_scientific_journal-4617 125 23 improving improve VERB american_scientific_journal-4617 125 24 this this DET american_scientific_journal-4617 125 25 manuscript manuscript NOUN american_scientific_journal-4617 125 26 significantly significantly ADV american_scientific_journal-4617 125 27 . . PUNCT american_scientific_journal-4617 126 1 he he PRON american_scientific_journal-4617 126 2 is be AUX american_scientific_journal-4617 126 3 also also ADV american_scientific_journal-4617 126 4 thankful thankful ADJ american_scientific_journal-4617 126 5 to to ADP american_scientific_journal-4617 126 6 the the DET american_scientific_journal-4617 126 7 previous previous ADJ american_scientific_journal-4617 126 8 authors author NOUN american_scientific_journal-4617 126 9 since since SCONJ american_scientific_journal-4617 126 10 their their PRON american_scientific_journal-4617 126 11 efforts effort NOUN american_scientific_journal-4617 126 12 have have AUX american_scientific_journal-4617 126 13 been be AUX american_scientific_journal-4617 126 14 used use VERB american_scientific_journal-4617 126 15 as as ADP american_scientific_journal-4617 126 16 references reference NOUN american_scientific_journal-4617 126 17 in in ADP american_scientific_journal-4617 126 18 the the DET american_scientific_journal-4617 126 19 study study NOUN american_scientific_journal-4617 126 20 . . PUNCT american_scientific_journal-4617 127 1 references reference NOUN american_scientific_journal-4617 127 2 [ [ X american_scientific_journal-4617 127 3 1 1 X american_scientific_journal-4617 127 4 ] ] PUNCT american_scientific_journal-4617 127 5 j. j. PROPN american_scientific_journal-4617 127 6 douglas douglas PROPN american_scientific_journal-4617 127 7 faires faires PROPN american_scientific_journal-4617 127 8 & & CCONJ american_scientific_journal-4617 127 9 richard richard PROPN american_scientific_journal-4617 127 10 l.burden l.burden PROPN american_scientific_journal-4617 127 11 . . PUNCT american_scientific_journal-4617 128 1 β€œnumerical β€œnumerical ADJ american_scientific_journal-4617 128 2 methods method NOUN american_scientific_journal-4617 128 3 . . PUNCT american_scientific_journal-4617 128 4 ” " PUNCT american_scientific_journal-4617 129 1 brooks brooks PROPN american_scientific_journal-4617 129 2 cole cole PROPN american_scientific_journal-4617 129 3 , , PUNCT american_scientific_journal-4617 129 4 third third ADJ american_scientific_journal-4617 129 5 edition edition NOUN american_scientific_journal-4617 129 6 , , PUNCT american_scientific_journal-4617 129 7 june june PROPN american_scientific_journal-4617 129 8 18 18 NUM american_scientific_journal-4617 129 9 , , PUNCT american_scientific_journal-4617 129 10 2002 2002 NUM american_scientific_journal-4617 129 11 . . PUNCT american_scientific_journal-4617 130 1 [ [ X american_scientific_journal-4617 130 2 2 2 NUM american_scientific_journal-4617 130 3 ] ] X american_scientific_journal-4617 130 4 chahn chahn PROPN american_scientific_journal-4617 130 5 yong yong PROPN american_scientific_journal-4617 130 6 jung jung PROPN american_scientific_journal-4617 130 7 , , PUNCT american_scientific_journal-4617 130 8 zeqing zeqing PROPN american_scientific_journal-4617 130 9 liu liu PROPN american_scientific_journal-4617 130 10 , , PUNCT american_scientific_journal-4617 130 11 arif arif PROPN american_scientific_journal-4617 130 12 rafiq rafiq PROPN american_scientific_journal-4617 130 13 , , PUNCT american_scientific_journal-4617 130 14 faisal faisal PROPN american_scientific_journal-4617 130 15 ali ali PROPN american_scientific_journal-4617 130 16 , , PUNCT american_scientific_journal-4617 130 17 & & CCONJ american_scientific_journal-4617 130 18 shin shin PROPN american_scientific_journal-4617 130 19 min min PROPN american_scientific_journal-4617 130 20 kang kang PROPN american_scientific_journal-4617 130 21 . . PUNCT american_scientific_journal-4617 131 1 β€œ " PUNCT american_scientific_journal-4617 131 2 solution solution NOUN american_scientific_journal-4617 131 3 of of ADP american_scientific_journal-4617 131 4 second second ADJ american_scientific_journal-4617 131 5 order order NOUN american_scientific_journal-4617 131 6 linear linear ADJ american_scientific_journal-4617 131 7 and and CCONJ american_scientific_journal-4617 131 8 nonlinear nonlinear ADJ american_scientific_journal-4617 131 9 ordinary ordinary ADJ american_scientific_journal-4617 131 10 differential differential ADJ american_scientific_journal-4617 131 11 equations equation NOUN american_scientific_journal-4617 131 12 using use VERB american_scientific_journal-4617 131 13 legendre legendre PROPN american_scientific_journal-4617 131 14 operational operational ADJ american_scientific_journal-4617 131 15 matrix matrix NOUN american_scientific_journal-4617 131 16 of of ADP american_scientific_journal-4617 131 17 differentiation differentiation NOUN american_scientific_journal-4617 131 18 . . PUNCT american_scientific_journal-4617 131 19 ” " PUNCT american_scientific_journal-4617 132 1 international international ADJ american_scientific_journal-4617 132 2 journal journal NOUN american_scientific_journal-4617 132 3 of of ADP american_scientific_journal-4617 132 4 pure pure ADJ american_scientific_journal-4617 132 5 and and CCONJ american_scientific_journal-4617 132 6 applied applied ADJ american_scientific_journal-4617 132 7 mathematics mathematic NOUN american_scientific_journal-4617 132 8 . . PUNCT american_scientific_journal-4617 133 1 volume volume NOUN american_scientific_journal-4617 133 2 93 93 NUM american_scientific_journal-4617 134 1 no no NOUN american_scientific_journal-4617 134 2 . . NOUN american_scientific_journal-4617 134 3 2 2 NUM american_scientific_journal-4617 134 4 2014 2014 NUM american_scientific_journal-4617 134 5 . . PUNCT american_scientific_journal-4617 135 1 [ [ X american_scientific_journal-4617 135 2 3 3 NUM american_scientific_journal-4617 135 3 ] ] X american_scientific_journal-4617 135 4 md md PROPN american_scientific_journal-4617 135 5 . . PROPN american_scientific_journal-4617 135 6 shafiqul shafiqul PROPN american_scientific_journal-4617 135 7 islam islam PROPN american_scientific_journal-4617 135 8 , , PUNCT american_scientific_journal-4617 135 9 afroza afroza PROPN american_scientific_journal-4617 135 10 shirin shirin PROPN american_scientific_journal-4617 135 11 . . PUNCT american_scientific_journal-4617 136 1 β€œnumerical β€œnumerical ADJ american_scientific_journal-4617 136 2 solutions solution NOUN american_scientific_journal-4617 136 3 of of ADP american_scientific_journal-4617 136 4 a a DET american_scientific_journal-4617 136 5 class class NOUN american_scientific_journal-4617 136 6 of of ADP american_scientific_journal-4617 136 7 second second ADJ american_scientific_journal-4617 136 8 order order NOUN american_scientific_journal-4617 136 9 boundary boundary ADJ american_scientific_journal-4617 136 10 value value NOUN american_scientific_journal-4617 136 11 problems problem NOUN american_scientific_journal-4617 136 12 on on ADP american_scientific_journal-4617 136 13 using use VERB american_scientific_journal-4617 136 14 bernoulli bernoulli NOUN american_scientific_journal-4617 136 15 polynomials polynomial NOUN american_scientific_journal-4617 136 16 . . PUNCT american_scientific_journal-4617 136 17 ” " PUNCT american_scientific_journal-4617 137 1 applied apply VERB american_scientific_journal-4617 137 2 mathematics mathematic NOUN american_scientific_journal-4617 137 3 , , PUNCT american_scientific_journal-4617 137 4 2 2 NUM american_scientific_journal-4617 137 5 , , PUNCT american_scientific_journal-4617 137 6 pp.1059 pp.1059 PROPN american_scientific_journal-4617 137 7 - - PUNCT american_scientific_journal-4617 137 8 1067,2011 1067,2011 NUM american_scientific_journal-4617 137 9 . . PUNCT american_scientific_journal-4617 138 1 [ [ X american_scientific_journal-4617 138 2 4 4 NUM american_scientific_journal-4617 138 3 ] ] X american_scientific_journal-4617 138 4 yogesh yogesh PROPN american_scientific_journal-4617 138 5 gupta gupta PROPN american_scientific_journal-4617 138 6 . . PUNCT american_scientific_journal-4617 139 1 β€œa β€œa PROPN american_scientific_journal-4617 139 2 numerical numerical ADJ american_scientific_journal-4617 139 3 algorithm algorithm NOUN american_scientific_journal-4617 139 4 for for ADP american_scientific_journal-4617 139 5 solution solution NOUN american_scientific_journal-4617 139 6 of of ADP american_scientific_journal-4617 139 7 boundary boundary ADJ american_scientific_journal-4617 139 8 value value NOUN american_scientific_journal-4617 139 9 problems problem NOUN american_scientific_journal-4617 139 10 with with ADP american_scientific_journal-4617 139 11 applications application NOUN american_scientific_journal-4617 139 12 . . PUNCT american_scientific_journal-4617 139 13 ” " PUNCT american_scientific_journal-4617 140 1 international international ADJ american_scientific_journal-4617 140 2 journal journal NOUN american_scientific_journal-4617 140 3 of of ADP american_scientific_journal-4617 140 4 computer computer NOUN american_scientific_journal-4617 140 5 applications application NOUN american_scientific_journal-4617 140 6 . . PUNCT american_scientific_journal-4617 141 1 volume volume NOUN american_scientific_journal-4617 141 2 40 40 NUM american_scientific_journal-4617 141 3 , , PUNCT american_scientific_journal-4617 141 4 no.8 no.8 PROPN american_scientific_journal-4617 141 5 , , PUNCT american_scientific_journal-4617 141 6 february february PROPN american_scientific_journal-4617 141 7 2012 2012 NUM american_scientific_journal-4617 141 8 . . PUNCT american_scientific_journal-4617 142 1 [ [ X american_scientific_journal-4617 142 2 5 5 NUM american_scientific_journal-4617 142 3 ] ] X american_scientific_journal-4617 142 4 abbas abbas PROPN american_scientific_journal-4617 142 5 saadatmandi saadatmandi PROPN american_scientific_journal-4617 142 6 , , PUNCT american_scientific_journal-4617 142 7 mehdi mehdi ADJ american_scientific_journal-4617 142 8 dehghan dehghan PROPN american_scientific_journal-4617 142 9 . . PUNCT american_scientific_journal-4617 143 1 β€œ " PUNCT american_scientific_journal-4617 143 2 a a DET american_scientific_journal-4617 143 3 new new ADJ american_scientific_journal-4617 143 4 operational operational ADJ american_scientific_journal-4617 143 5 matrix matrix NOUN american_scientific_journal-4617 143 6 for for ADP american_scientific_journal-4617 143 7 solving solve VERB american_scientific_journal-4617 143 8 fractional fractional ADJ american_scientific_journal-4617 143 9 - - PUNCT american_scientific_journal-4617 143 10 order order NOUN american_scientific_journal-4617 143 11 differential differential ADJ american_scientific_journal-4617 143 12 equations equation NOUN american_scientific_journal-4617 143 13 . . PUNCT american_scientific_journal-4617 143 14 ” " PUNCT american_scientific_journal-4617 144 1 computers computer NOUN american_scientific_journal-4617 144 2 and and CCONJ american_scientific_journal-4617 144 3 mathematics mathematic NOUN american_scientific_journal-4617 144 4 with with ADP american_scientific_journal-4617 144 5 applications application NOUN american_scientific_journal-4617 144 6 59 59 NUM american_scientific_journal-4617 144 7 , , PUNCT american_scientific_journal-4617 144 8 pp pp ADJ american_scientific_journal-4617 144 9 . . PUNCT american_scientific_journal-4617 145 1 1326 1326 NUM american_scientific_journal-4617 145 2 - - SYM american_scientific_journal-4617 145 3 1336 1336 NUM american_scientific_journal-4617 145 4 , , PUNCT american_scientific_journal-4617 145 5 2010 2010 NUM american_scientific_journal-4617 145 6 . . PUNCT american_scientific_journal-4617 146 1 [ [ X american_scientific_journal-4617 146 2 6 6 NUM american_scientific_journal-4617 146 3 ] ] PUNCT american_scientific_journal-4617 146 4 nur nur VERB american_scientific_journal-4617 146 5 nadiah nadiah PROPN american_scientific_journal-4617 146 6 abd abd PROPN american_scientific_journal-4617 146 7 hamid hamid PROPN american_scientific_journal-4617 146 8 , , PUNCT american_scientific_journal-4617 146 9 ahmad ahmad PROPN american_scientific_journal-4617 146 10 abd abd PROPN american_scientific_journal-4617 146 11 . . PUNCT american_scientific_journal-4617 147 1 majid majid PROPN american_scientific_journal-4617 147 2 & & CCONJ american_scientific_journal-4617 147 3 ahmad ahmad PROPN american_scientific_journal-4617 147 4 izani izani PROPN american_scientific_journal-4617 147 5 md md PROPN american_scientific_journal-4617 147 6 . . PUNCT american_scientific_journal-4617 147 7 ismail ismail PROPN american_scientific_journal-4617 147 8 . . PUNCT american_scientific_journal-4617 148 1 β€œ " PUNCT american_scientific_journal-4617 148 2 extended extend VERB american_scientific_journal-4617 148 3 cubic cubic ADJ american_scientific_journal-4617 148 4 b b NOUN american_scientific_journal-4617 148 5 - - PUNCT american_scientific_journal-4617 148 6 spline spline NOUN american_scientific_journal-4617 148 7 method method NOUN american_scientific_journal-4617 148 8 for for ADP american_scientific_journal-4617 148 9 linear linear ADJ american_scientific_journal-4617 148 10 two two NUM american_scientific_journal-4617 148 11 - - PUNCT american_scientific_journal-4617 148 12 point point NOUN american_scientific_journal-4617 148 13 boundary boundary ADJ american_scientific_journal-4617 148 14 value value NOUN american_scientific_journal-4617 148 15 problems problem NOUN american_scientific_journal-4617 148 16 . . PUNCT american_scientific_journal-4617 148 17 ” " PUNCT american_scientific_journal-4617 149 1 sains sain VERB american_scientific_journal-4617 149 2 malaysiana malaysiana PROPN american_scientific_journal-4617 149 3 40(11 40(11 PROPN american_scientific_journal-4617 149 4 ) ) PUNCT american_scientific_journal-4617 149 5 , , PUNCT american_scientific_journal-4617 149 6 pp.1285–1290 pp.1285–1290 PROPN american_scientific_journal-4617 149 7 , , PUNCT american_scientific_journal-4617 149 8 2011 2011 NUM american_scientific_journal-4617 149 9 . . PUNCT american_scientific_journal-4617 150 1 [ [ X american_scientific_journal-4617 150 2 7 7 X american_scientific_journal-4617 150 3 ] ] X american_scientific_journal-4617 150 4 j. j. PROPN american_scientific_journal-4617 150 5 rashidinia rashidinia PROPN american_scientific_journal-4617 150 6 and and CCONJ american_scientific_journal-4617 150 7 sh sh PROPN american_scientific_journal-4617 150 8 . . PROPN american_scientific_journal-4617 150 9 sharif sharif PROPN american_scientific_journal-4617 150 10 . . PUNCT american_scientific_journal-4617 151 1 β€œ " PUNCT american_scientific_journal-4617 151 2 b b X american_scientific_journal-4617 151 3 - - PUNCT american_scientific_journal-4617 151 4 spline spline NOUN american_scientific_journal-4617 151 5 method method NOUN american_scientific_journal-4617 151 6 for for ADP american_scientific_journal-4617 151 7 two two NUM american_scientific_journal-4617 151 8 - - PUNCT american_scientific_journal-4617 151 9 point point NOUN american_scientific_journal-4617 151 10 boundary boundary ADJ american_scientific_journal-4617 151 11 value value NOUN american_scientific_journal-4617 151 12 problems problem NOUN american_scientific_journal-4617 151 13 . . PUNCT american_scientific_journal-4617 151 14 ” " PUNCT american_scientific_journal-4617 152 1 international international ADJ american_scientific_journal-4617 152 2 journal journal NOUN american_scientific_journal-4617 152 3 of of ADP american_scientific_journal-4617 152 4 mathematical mathematical ADJ american_scientific_journal-4617 152 5 modelling modelling NOUN american_scientific_journal-4617 152 6 & & CCONJ american_scientific_journal-4617 152 7 computations computation NOUN american_scientific_journal-4617 152 8 . . PUNCT american_scientific_journal-4617 153 1 vol vol NOUN american_scientific_journal-4617 153 2 . . PROPN american_scientific_journal-4617 154 1 05 05 NUM american_scientific_journal-4617 154 2 , , PUNCT american_scientific_journal-4617 154 3 no no INTJ american_scientific_journal-4617 154 4 . . NOUN american_scientific_journal-4617 154 5 02 02 NUM american_scientific_journal-4617 154 6 , , PUNCT american_scientific_journal-4617 154 7 pp pp ADJ american_scientific_journal-4617 154 8 . . PUNCT american_scientific_journal-4617 155 1 111125 111125 NUM american_scientific_journal-4617 155 2 , , PUNCT american_scientific_journal-4617 155 3 spring spring NOUN american_scientific_journal-4617 155 4 2015 2015 NUM american_scientific_journal-4617 155 5 . . PUNCT american_scientific_journal-4617 156 1 [ [ X american_scientific_journal-4617 156 2 8 8 NUM american_scientific_journal-4617 156 3 ] ] PUNCT american_scientific_journal-4617 156 4 m. m. NOUN american_scientific_journal-4617 156 5 m. m. PROPN american_scientific_journal-4617 156 6 rahman rahman PROPN american_scientific_journal-4617 156 7 , , PUNCT american_scientific_journal-4617 156 8 m.a m.a PROPN american_scientific_journal-4617 156 9 . . PROPN american_scientific_journal-4617 156 10 hossen hossen PROPN american_scientific_journal-4617 156 11 , , PUNCT american_scientific_journal-4617 156 12 m. m. NOUN american_scientific_journal-4617 156 13 nurul nurul PROPN american_scientific_journal-4617 156 14 islam islam PROPN american_scientific_journal-4617 156 15 and and CCONJ american_scientific_journal-4617 156 16 md md PROPN american_scientific_journal-4617 156 17 . . PROPN american_scientific_journal-4617 156 18 shajib shajib PROPN american_scientific_journal-4617 156 19 ali ali PROPN american_scientific_journal-4617 156 20 . . PUNCT american_scientific_journal-4617 157 1 β€œ " PUNCT american_scientific_journal-4617 157 2 numerical numerical ADJ american_scientific_journal-4617 157 3 solutions solution NOUN american_scientific_journal-4617 157 4 of of ADP american_scientific_journal-4617 157 5 second second ADJ american_scientific_journal-4617 157 6 order order NOUN american_scientific_journal-4617 157 7 boundary boundary ADJ american_scientific_journal-4617 157 8 value value NOUN american_scientific_journal-4617 157 9 problems problem NOUN american_scientific_journal-4617 157 10 by by ADP american_scientific_journal-4617 157 11 galerkin galerkin PROPN american_scientific_journal-4617 157 12 method method NOUN american_scientific_journal-4617 157 13 with with ADP american_scientific_journal-4617 157 14 hermite hermite ADJ american_scientific_journal-4617 157 15 polynomials polynomial NOUN american_scientific_journal-4617 157 16 . . PUNCT american_scientific_journal-4617 157 17 ” " PUNCT american_scientific_journal-4617 158 1 annals annal NOUN american_scientific_journal-4617 158 2 of of ADP american_scientific_journal-4617 158 3 pure pure ADJ american_scientific_journal-4617 158 4 and and CCONJ american_scientific_journal-4617 158 5 applied applied ADJ american_scientific_journal-4617 158 6 mathematics mathematic NOUN american_scientific_journal-4617 158 7 . . PUNCT american_scientific_journal-4617 159 1 vol vol NOUN american_scientific_journal-4617 159 2 . . PROPN american_scientific_journal-4617 160 1 1 1 NUM american_scientific_journal-4617 160 2 , , PUNCT american_scientific_journal-4617 160 3 no no INTJ american_scientific_journal-4617 160 4 . . NOUN american_scientific_journal-4617 160 5 2 2 NUM american_scientific_journal-4617 160 6 , , PUNCT american_scientific_journal-4617 160 7 pp.138 pp.138 PROPN american_scientific_journal-4617 160 8 - - PUNCT american_scientific_journal-4617 160 9 148 148 NUM american_scientific_journal-4617 160 10 , , PUNCT american_scientific_journal-4617 160 11 2012 2012 NUM american_scientific_journal-4617 160 12 . . PUNCT american_scientific_journal-4617 161 1 [ [ X american_scientific_journal-4617 161 2 9 9 NUM american_scientific_journal-4617 161 3 ] ] SYM american_scientific_journal-4617 161 4 li li PROPN american_scientific_journal-4617 161 5 - - PROPN american_scientific_journal-4617 161 6 bin bin PROPN american_scientific_journal-4617 161 7 liu liu PROPN american_scientific_journal-4617 161 8 , , PUNCT american_scientific_journal-4617 161 9 huan huan PROPN american_scientific_journal-4617 161 10 - - PUNCT american_scientific_journal-4617 161 11 wen wen PROPN american_scientific_journal-4617 161 12 liu liu PROPN american_scientific_journal-4617 161 13 , , PUNCT american_scientific_journal-4617 161 14 yanping yanpe VERB american_scientific_journal-4617 161 15 chen chen PROPN american_scientific_journal-4617 161 16 . . PUNCT american_scientific_journal-4617 162 1 β€œ " PUNCT american_scientific_journal-4617 162 2 polynomial polynomial ADJ american_scientific_journal-4617 162 3 spline spline NOUN american_scientific_journal-4617 162 4 approach approach NOUN american_scientific_journal-4617 162 5 for for ADP american_scientific_journal-4617 162 6 solving solve VERB american_scientific_journal-4617 162 7 second second ADJ american_scientific_journal-4617 162 8 - - PUNCT american_scientific_journal-4617 162 9 order order NOUN american_scientific_journal-4617 162 10 boundary boundary ADJ american_scientific_journal-4617 162 11 - - PUNCT american_scientific_journal-4617 162 12 value value NOUN american_scientific_journal-4617 162 13 problems problem NOUN american_scientific_journal-4617 162 14 with with ADP american_scientific_journal-4617 162 15 neumann neumann PROPN american_scientific_journal-4617 162 16 conditions condition NOUN american_scientific_journal-4617 162 17 . . PUNCT american_scientific_journal-4617 162 18 ” " PUNCT american_scientific_journal-4617 162 19 applied apply VERB american_scientific_journal-4617 162 20 mathematics mathematic NOUN american_scientific_journal-4617 162 21 and and CCONJ american_scientific_journal-4617 162 22 computation computation NOUN american_scientific_journal-4617 162 23 217 217 NUM american_scientific_journal-4617 162 24 , , PUNCT american_scientific_journal-4617 162 25 pp.6872–6882 pp.6872–6882 NUM american_scientific_journal-4617 162 26 , , PUNCT american_scientific_journal-4617 162 27 2011 2011 NUM american_scientific_journal-4617 162 28 . . PUNCT