id sid tid token lemma pos iajs-3048 1 1 ihjpas ihjpas PROPN iajs-3048 1 2 . . PUNCT iajs-3048 2 1 36 36 NUM iajs-3048 2 2 ( ( PUNCT iajs-3048 2 3 3 3 NUM iajs-3048 2 4 ) ) PUNCT iajs-3048 2 5 2023 2023 NUM iajs-3048 2 6 427 427 NUM iajs-3048 2 7 this this DET iajs-3048 2 8 work work NOUN iajs-3048 2 9 is be AUX iajs-3048 2 10 licensed license VERB iajs-3048 2 11 under under ADP iajs-3048 2 12 a a DET iajs-3048 2 13 creative creative ADJ iajs-3048 2 14 commons common NOUN iajs-3048 2 15 attribution attribution NOUN iajs-3048 2 16 4.0 4.0 NUM iajs-3048 2 17 international international ADJ iajs-3048 2 18 license license NOUN iajs-3048 2 19 abstract abstract NOUN iajs-3048 2 20 the the DET iajs-3048 2 21 wavelets wavelet NOUN iajs-3048 2 22 have have VERB iajs-3048 2 23 many many ADJ iajs-3048 2 24 applications application NOUN iajs-3048 2 25 in in ADP iajs-3048 2 26 engineering engineering NOUN iajs-3048 2 27 and and CCONJ iajs-3048 2 28 the the DET iajs-3048 2 29 sciences science NOUN iajs-3048 2 30 , , PUNCT iajs-3048 2 31 especially especially ADV iajs-3048 2 32 mathematics mathematic NOUN iajs-3048 2 33 . . PUNCT iajs-3048 3 1 recently recently ADV iajs-3048 3 2 , , PUNCT iajs-3048 3 3 in in ADP iajs-3048 3 4 2021 2021 NUM iajs-3048 3 5 , , PUNCT iajs-3048 3 6 the the DET iajs-3048 3 7 wavelet wavelet NOUN iajs-3048 3 8 boubaker boubaker NOUN iajs-3048 3 9 ( ( PUNCT iajs-3048 3 10 wb wb NOUN iajs-3048 3 11 ) ) PUNCT iajs-3048 3 12 polynomials polynomial NOUN iajs-3048 3 13 were be AUX iajs-3048 3 14 used use VERB iajs-3048 3 15 for for ADP iajs-3048 3 16 the the DET iajs-3048 3 17 first first ADJ iajs-3048 3 18 time time NOUN iajs-3048 3 19 to to PART iajs-3048 3 20 study study VERB iajs-3048 3 21 their their PRON iajs-3048 3 22 properties property NOUN iajs-3048 3 23 and and CCONJ iajs-3048 3 24 applications application NOUN iajs-3048 3 25 in in ADP iajs-3048 3 26 detail detail NOUN iajs-3048 3 27 . . PUNCT iajs-3048 4 1 they they PRON iajs-3048 4 2 were be AUX iajs-3048 4 3 also also ADV iajs-3048 4 4 utilized utilize VERB iajs-3048 4 5 for for ADP iajs-3048 4 6 solving solve VERB iajs-3048 4 7 the the DET iajs-3048 4 8 lane lane NOUN iajs-3048 4 9 - - PUNCT iajs-3048 4 10 emden emden NOUN iajs-3048 4 11 equation equation NOUN iajs-3048 4 12 . . PUNCT iajs-3048 5 1 the the DET iajs-3048 5 2 aim aim NOUN iajs-3048 5 3 of of ADP iajs-3048 5 4 this this DET iajs-3048 5 5 paper paper NOUN iajs-3048 5 6 is be AUX iajs-3048 5 7 to to PART iajs-3048 5 8 show show VERB iajs-3048 5 9 the the DET iajs-3048 5 10 truncated truncate VERB iajs-3048 5 11 wavelet wavelet NOUN iajs-3048 5 12 boubaker boubaker NOUN iajs-3048 5 13 polynomials polynomial NOUN iajs-3048 5 14 for for ADP iajs-3048 5 15 solving solve VERB iajs-3048 5 16 variation variation NOUN iajs-3048 5 17 problems problem NOUN iajs-3048 5 18 . . PUNCT iajs-3048 6 1 in in ADP iajs-3048 6 2 this this DET iajs-3048 6 3 research research NOUN iajs-3048 6 4 , , PUNCT iajs-3048 6 5 the the DET iajs-3048 6 6 direct direct ADJ iajs-3048 6 7 method method NOUN iajs-3048 6 8 using use VERB iajs-3048 6 9 wavelets wavelet NOUN iajs-3048 6 10 boubaker boubaker NOUN iajs-3048 6 11 was be AUX iajs-3048 6 12 presented present VERB iajs-3048 6 13 for for ADP iajs-3048 6 14 solving solve VERB iajs-3048 6 15 variational variational ADJ iajs-3048 6 16 problems problem NOUN iajs-3048 6 17 . . PUNCT iajs-3048 7 1 the the DET iajs-3048 7 2 method method NOUN iajs-3048 7 3 reduces reduce VERB iajs-3048 7 4 the the DET iajs-3048 7 5 problem problem NOUN iajs-3048 7 6 into into ADP iajs-3048 7 7 a a DET iajs-3048 7 8 set set NOUN iajs-3048 7 9 of of ADP iajs-3048 7 10 linear linear PROPN iajs-3048 7 11 algebraic algebraic ADJ iajs-3048 7 12 equations equation NOUN iajs-3048 7 13 . . PUNCT iajs-3048 8 1 the the DET iajs-3048 8 2 fundamental fundamental ADJ iajs-3048 8 3 idea idea NOUN iajs-3048 8 4 of of ADP iajs-3048 8 5 this this DET iajs-3048 8 6 method method NOUN iajs-3048 8 7 for for ADP iajs-3048 8 8 solving solve VERB iajs-3048 8 9 variation variation NOUN iajs-3048 8 10 problems problem NOUN iajs-3048 8 11 is be AUX iajs-3048 8 12 to to PART iajs-3048 8 13 convert convert VERB iajs-3048 8 14 the the DET iajs-3048 8 15 problem problem NOUN iajs-3048 8 16 of of ADP iajs-3048 8 17 a a DET iajs-3048 8 18 function function NOUN iajs-3048 8 19 into into ADP iajs-3048 8 20 one one NUM iajs-3048 8 21 that that PRON iajs-3048 8 22 involves involve VERB iajs-3048 8 23 a a DET iajs-3048 8 24 finite finite ADJ iajs-3048 8 25 number number NOUN iajs-3048 8 26 of of ADP iajs-3048 8 27 variables variable NOUN iajs-3048 8 28 . . PUNCT iajs-3048 9 1 different different ADJ iajs-3048 9 2 numerical numerical ADJ iajs-3048 9 3 examples example NOUN iajs-3048 9 4 were be AUX iajs-3048 9 5 given give VERB iajs-3048 9 6 to to PART iajs-3048 9 7 demonstrate demonstrate VERB iajs-3048 9 8 the the DET iajs-3048 9 9 applicability applicability NOUN iajs-3048 9 10 and and CCONJ iajs-3048 9 11 validity validity NOUN iajs-3048 9 12 of of ADP iajs-3048 9 13 this this DET iajs-3048 9 14 method method NOUN iajs-3048 9 15 using use VERB iajs-3048 9 16 the the DET iajs-3048 9 17 matlab matlab PROPN iajs-3048 9 18 program program NOUN iajs-3048 9 19 . . PUNCT iajs-3048 10 1 also also ADV iajs-3048 10 2 , , PUNCT iajs-3048 10 3 the the DET iajs-3048 10 4 results result NOUN iajs-3048 10 5 of of ADP iajs-3048 10 6 this this DET iajs-3048 10 7 technique technique NOUN iajs-3048 10 8 were be AUX iajs-3048 10 9 compared compare VERB iajs-3048 10 10 with with ADP iajs-3048 10 11 the the DET iajs-3048 10 12 exact exact ADJ iajs-3048 10 13 solution solution NOUN iajs-3048 10 14 , , PUNCT iajs-3048 10 15 and and CCONJ iajs-3048 10 16 graphs graph NOUN iajs-3048 10 17 were be AUX iajs-3048 10 18 added add VERB iajs-3048 10 19 to to ADP iajs-3048 10 20 these these DET iajs-3048 10 21 examples example NOUN iajs-3048 10 22 to to PART iajs-3048 10 23 test test VERB iajs-3048 10 24 the the DET iajs-3048 10 25 convergence convergence NOUN iajs-3048 10 26 of of ADP iajs-3048 10 27 wavelet wavelet NOUN iajs-3048 10 28 boubaker boubaker NOUN iajs-3048 10 29 polynomials polynomial NOUN iajs-3048 10 30 using use VERB iajs-3048 10 31 this this DET iajs-3048 10 32 method method NOUN iajs-3048 10 33 . . PUNCT iajs-3048 11 1 keywords keyword NOUN iajs-3048 11 2 : : PUNCT iajs-3048 11 3 boubaker boubaker NOUN iajs-3048 11 4 wavelets wavelet NOUN iajs-3048 11 5 , , PUNCT iajs-3048 11 6 calculus calculus NOUN iajs-3048 11 7 of of ADP iajs-3048 11 8 variation variation NOUN iajs-3048 11 9 problems problem NOUN iajs-3048 11 10 , , PUNCT iajs-3048 11 11 nonlinear nonlinear ADJ iajs-3048 11 12 programming programming NOUN iajs-3048 11 13 , , PUNCT iajs-3048 11 14 numerical numerical ADJ iajs-3048 11 15 methods method NOUN iajs-3048 11 16 . . PUNCT iajs-3048 12 1 1 1 X iajs-3048 12 2 . . X iajs-3048 12 3 introduction introduction NOUN iajs-3048 12 4 many many ADJ iajs-3048 12 5 problems problem NOUN iajs-3048 12 6 arising arise VERB iajs-3048 12 7 in in ADP iajs-3048 12 8 mathematical mathematical ADJ iajs-3048 12 9 physics physics NOUN iajs-3048 12 10 and and CCONJ iajs-3048 12 11 geometry geometry NOUN iajs-3048 12 12 are be AUX iajs-3048 12 13 connected connect VERB iajs-3048 12 14 with with ADP iajs-3048 12 15 the the DET iajs-3048 12 16 calculus calculus NOUN iajs-3048 12 17 of of ADP iajs-3048 12 18 variations variation NOUN iajs-3048 12 19 , , PUNCT iajs-3048 12 20 which which PRON iajs-3048 12 21 is be AUX iajs-3048 12 22 determined determine VERB iajs-3048 12 23 by by ADP iajs-3048 12 24 finding find VERB iajs-3048 12 25 the the DET iajs-3048 12 26 maximal maximal ADJ iajs-3048 12 27 and and CCONJ iajs-3048 12 28 minimal minimal ADJ iajs-3048 12 29 functional functional ADJ iajs-3048 12 30 functions function NOUN iajs-3048 12 31 . . PUNCT iajs-3048 13 1 the the DET iajs-3048 13 2 functionals functional NOUN iajs-3048 13 3 are be AUX iajs-3048 13 4 defined define VERB iajs-3048 13 5 by by ADP iajs-3048 13 6 definite definite ADJ iajs-3048 13 7 integrals integral NOUN iajs-3048 13 8 , , PUNCT iajs-3048 13 9 which which PRON iajs-3048 13 10 include include VERB iajs-3048 13 11 boundary boundary ADJ iajs-3048 13 12 conditions condition NOUN iajs-3048 13 13 and and CCONJ iajs-3048 13 14 appear appear VERB iajs-3048 13 15 in in ADP iajs-3048 13 16 the the DET iajs-3048 13 17 mathematical mathematical ADJ iajs-3048 13 18 formula formula NOUN iajs-3048 13 19 , , PUNCT iajs-3048 13 20 see see VERB iajs-3048 13 21 [ [ X iajs-3048 13 22 2 2 NUM iajs-3048 13 23 ] ] PUNCT iajs-3048 13 24 . . PUNCT iajs-3048 14 1 wavelet wavelet NOUN iajs-3048 14 2 theory theory NOUN iajs-3048 14 3 is be AUX iajs-3048 14 4 an an DET iajs-3048 14 5 emerging emerge VERB iajs-3048 14 6 field field NOUN iajs-3048 14 7 in in ADP iajs-3048 14 8 mathematical mathematical ADJ iajs-3048 14 9 research research NOUN iajs-3048 14 10 and and CCONJ iajs-3048 14 11 is be AUX iajs-3048 14 12 applied apply VERB iajs-3048 14 13 in in ADP iajs-3048 14 14 a a DET iajs-3048 14 15 broad broad ADJ iajs-3048 14 16 range range NOUN iajs-3048 14 17 of of ADP iajs-3048 14 18 engineering engineering NOUN iajs-3048 14 19 disciplines discipline NOUN iajs-3048 14 20 . . PUNCT iajs-3048 15 1 wavelets wavelet NOUN iajs-3048 15 2 are be AUX iajs-3048 15 3 very very ADV iajs-3048 15 4 successful successful ADJ iajs-3048 15 5 in in ADP iajs-3048 15 6 accurately accurately ADV iajs-3048 15 7 solving solve VERB iajs-3048 15 8 numerical numerical ADJ iajs-3048 15 9 problems problem NOUN iajs-3048 15 10 . . PUNCT iajs-3048 16 1 [ [ X iajs-3048 16 2 3 3 NUM iajs-3048 16 3 ] ] X iajs-3048 16 4 utilized utilize VERB iajs-3048 16 5 legendre legendre PROPN iajs-3048 16 6 wavelets wavelet NOUN iajs-3048 16 7 to to PART iajs-3048 16 8 solve solve VERB iajs-3048 16 9 variational variational ADJ iajs-3048 16 10 problems problem NOUN iajs-3048 16 11 . . PUNCT iajs-3048 17 1 [ [ X iajs-3048 17 2 4 4 X iajs-3048 17 3 ] ] PUNCT iajs-3048 17 4 used use VERB iajs-3048 17 5 the the DET iajs-3048 17 6 haar haar PROPN iajs-3048 17 7 wavelet wavelet NOUN iajs-3048 17 8 to to PART iajs-3048 17 9 solve solve VERB iajs-3048 17 10 the the DET iajs-3048 17 11 same same ADJ iajs-3048 17 12 problems problem NOUN iajs-3048 17 13 . . PUNCT iajs-3048 18 1 [ [ X iajs-3048 18 2 5 5 NUM iajs-3048 18 3 ] ] PUNCT iajs-3048 18 4 applied apply VERB iajs-3048 18 5 direct direct ADJ iajs-3048 18 6 restarted restart VERB iajs-3048 18 7 pell pell NOUN iajs-3048 18 8 to to PART iajs-3048 18 9 solve solve VERB iajs-3048 18 10 the the DET iajs-3048 18 11 problem problem NOUN iajs-3048 18 12 of of ADP iajs-3048 18 13 variation variation NOUN iajs-3048 18 14 , , PUNCT iajs-3048 18 15 then then ADV iajs-3048 18 16 used use VERB iajs-3048 18 17 the the DET iajs-3048 18 18 spectral spectral ADJ iajs-3048 18 19 method method NOUN iajs-3048 18 20 with with ADP iajs-3048 18 21 chebyshev chebyshev NOUN iajs-3048 18 22 wavelets wavelet NOUN iajs-3048 18 23 in in ADP iajs-3048 18 24 another another DET iajs-3048 18 25 paper paper NOUN iajs-3048 18 26 to to PART iajs-3048 18 27 solve solve VERB iajs-3048 18 28 the the DET iajs-3048 18 29 calculus calculus NOUN iajs-3048 18 30 of of ADP iajs-3048 18 31 variations variation NOUN iajs-3048 18 32 doi.org/10.30526/36.3.3048 doi.org/10.30526/36.3.3048 NOUN iajs-3048 18 33 article article NOUN iajs-3048 18 34 history history NOUN iajs-3048 18 35 : : PUNCT iajs-3048 18 36 received receive VERB iajs-3048 18 37 28 28 NUM iajs-3048 18 38 september september PROPN iajs-3048 18 39 2022 2022 NUM iajs-3048 18 40 , , PUNCT iajs-3048 18 41 accepted accept VERB iajs-3048 18 42 13 13 NUM iajs-3048 18 43 march march NOUN iajs-3048 18 44 2023 2023 NUM iajs-3048 18 45 , , PUNCT iajs-3048 18 46 published publish VERB iajs-3048 18 47 in in ADP iajs-3048 18 48 july july PROPN iajs-3048 18 49 2023 2023 NUM iajs-3048 18 50 . . PUNCT iajs-3048 19 1 ibn ibn PROPN iajs-3048 19 2 al al PROPN iajs-3048 19 3 - - PUNCT iajs-3048 19 4 haitham haitham PROPN iajs-3048 19 5 journal journal PROPN iajs-3048 19 6 for for ADP iajs-3048 19 7 pure pure ADJ iajs-3048 19 8 and and CCONJ iajs-3048 19 9 applied applied ADJ iajs-3048 19 10 sciences sciences PROPN iajs-3048 19 11 journal journal PROPN iajs-3048 19 12 homepage homepage NOUN iajs-3048 19 13 : : PUNCT iajs-3048 19 14 jih.uobaghdad.edu.iq jih.uobaghdad.edu.iq VERB iajs-3048 19 15 direct direct ADJ iajs-3048 19 16 method method NOUN iajs-3048 19 17 for for ADP iajs-3048 19 18 variational variational ADJ iajs-3048 19 19 problems problem NOUN iajs-3048 19 20 using use VERB iajs-3048 19 21 boubaker boubaker NOUN iajs-3048 19 22 wavelets wavelet NOUN iajs-3048 19 23 eman eman PROPN iajs-3048 19 24 hassan hassan PROPN iajs-3048 19 25 ouda ouda PROPN iajs-3048 19 26 department department PROPN iajs-3048 19 27 of of ADP iajs-3048 19 28 applied apply VERB iajs-3048 19 29 science science NOUN iajs-3048 19 30 , , PUNCT iajs-3048 19 31 university university NOUN iajs-3048 19 32 of of ADP iajs-3048 19 33 technologyiraq technologyiraq PROPN iajs-3048 19 34 . . PUNCT iajs-3048 20 1 eman.h.ouda@uotechnology.edu.iq eman.h.ouda@uotechnology.edu.iq NUM iajs-3048 20 2 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3048 20 3 mailto:eman.h.ouda@uotechnology.edu.iq mailto:eman.h.ouda@uotechnology.edu.iq PROPN iajs-3048 20 4 mailto:eman.h.ouda@uotechnology.edu.iq mailto:eman.h.ouda@uotechnology.edu.iq PROPN iajs-3048 20 5 ihjpas ihjpas PROPN iajs-3048 20 6 . . PUNCT iajs-3048 21 1 36 36 NUM iajs-3048 21 2 ( ( PUNCT iajs-3048 21 3 3 3 NUM iajs-3048 21 4 ) ) PUNCT iajs-3048 21 5 2023 2023 NUM iajs-3048 21 6 428 428 NUM iajs-3048 22 1 [ [ SYM iajs-3048 22 2 6 6 NUM iajs-3048 22 3 ] ] PUNCT iajs-3048 22 4 . . PUNCT iajs-3048 23 1 [ [ X iajs-3048 23 2 7 7 NUM iajs-3048 23 3 - - SYM iajs-3048 23 4 8 8 NUM iajs-3048 23 5 ] ] PUNCT iajs-3048 23 6 using use VERB iajs-3048 23 7 a a DET iajs-3048 23 8 spline spline ADJ iajs-3048 23 9 polynomial polynomial NOUN iajs-3048 23 10 with with ADP iajs-3048 23 11 collocation collocation NOUN iajs-3048 23 12 method method NOUN iajs-3048 23 13 and and CCONJ iajs-3048 23 14 [ [ X iajs-3048 23 15 9 9 NUM iajs-3048 23 16 ] ] PUNCT iajs-3048 23 17 used use VERB iajs-3048 23 18 a a DET iajs-3048 23 19 new new ADJ iajs-3048 23 20 technique technique NOUN iajs-3048 23 21 to to PART iajs-3048 23 22 find find VERB iajs-3048 23 23 the the DET iajs-3048 23 24 numerical numerical ADJ iajs-3048 23 25 solution solution NOUN iajs-3048 23 26 for for ADP iajs-3048 23 27 solving solve VERB iajs-3048 23 28 isoperimetric isoperimetric NOUN iajs-3048 23 29 problems problem NOUN iajs-3048 23 30 . . PUNCT iajs-3048 24 1 many many ADJ iajs-3048 24 2 researchers researcher NOUN iajs-3048 24 3 have have AUX iajs-3048 24 4 utilized utilize VERB iajs-3048 24 5 different different ADJ iajs-3048 24 6 procedures procedure NOUN iajs-3048 24 7 to to PART iajs-3048 24 8 solve solve VERB iajs-3048 24 9 calculus calculus NOUN iajs-3048 24 10 of of ADP iajs-3048 24 11 variational variational ADJ iajs-3048 24 12 problems problem NOUN iajs-3048 24 13 . . PUNCT iajs-3048 25 1 [ [ X iajs-3048 25 2 10 10 NUM iajs-3048 25 3 ] ] PUNCT iajs-3048 25 4 developed develop VERB iajs-3048 25 5 the the DET iajs-3048 25 6 new new ADJ iajs-3048 25 7 functions function NOUN iajs-3048 25 8 for for ADP iajs-3048 25 9 solving solve VERB iajs-3048 25 10 the the DET iajs-3048 25 11 problems problem NOUN iajs-3048 25 12 of of ADP iajs-3048 25 13 variational variational ADJ iajs-3048 25 14 , , PUNCT iajs-3048 25 15 then then ADV iajs-3048 25 16 [ [ X iajs-3048 25 17 11 11 NUM iajs-3048 25 18 ] ] PUNCT iajs-3048 25 19 used use VERB iajs-3048 25 20 a a DET iajs-3048 25 21 combination combination NOUN iajs-3048 25 22 of of ADP iajs-3048 25 23 many many ADJ iajs-3048 25 24 functions function NOUN iajs-3048 25 25 with with ADP iajs-3048 25 26 bernoulli bernoulli NOUN iajs-3048 25 27 polynomials polynomial NOUN iajs-3048 25 28 . . PUNCT iajs-3048 26 1 [ [ X iajs-3048 26 2 12 12 NUM iajs-3048 26 3 ] ] PUNCT iajs-3048 26 4 found find VERB iajs-3048 26 5 the the DET iajs-3048 26 6 approximate approximate ADJ iajs-3048 26 7 solution solution NOUN iajs-3048 26 8 for for ADP iajs-3048 26 9 boundary boundary ADJ iajs-3048 26 10 value value NOUN iajs-3048 26 11 problems problem NOUN iajs-3048 26 12 using use VERB iajs-3048 26 13 the the DET iajs-3048 26 14 wavelet wavelet NOUN iajs-3048 26 15 function function NOUN iajs-3048 26 16 . . PUNCT iajs-3048 27 1 [ [ X iajs-3048 27 2 13 13 NUM iajs-3048 27 3 ] ] PUNCT iajs-3048 27 4 studied study VERB iajs-3048 27 5 moving move VERB iajs-3048 27 6 or or CCONJ iajs-3048 27 7 fixed fix VERB iajs-3048 27 8 boundary boundary ADJ iajs-3048 27 9 muntz muntz PROPN iajs-3048 27 10 wavelets wavelet NOUN iajs-3048 27 11 for for ADP iajs-3048 27 12 solving solve VERB iajs-3048 27 13 variation variation NOUN iajs-3048 27 14 problems problem NOUN iajs-3048 27 15 . . PUNCT iajs-3048 28 1 this this DET iajs-3048 28 2 paper paper NOUN iajs-3048 28 3 is be AUX iajs-3048 28 4 arranged arrange VERB iajs-3048 28 5 as as SCONJ iajs-3048 28 6 follows follow VERB iajs-3048 28 7 : : PUNCT iajs-3048 28 8 in in ADP iajs-3048 28 9 section section NOUN iajs-3048 28 10 2 2 NUM iajs-3048 28 11 , , PUNCT iajs-3048 28 12 orthogonal orthogonal ADJ iajs-3048 28 13 boubaker boubaker NOUN iajs-3048 28 14 polynomials polynomial NOUN iajs-3048 28 15 and and CCONJ iajs-3048 28 16 their their PRON iajs-3048 28 17 properties property NOUN iajs-3048 28 18 with with ADP iajs-3048 28 19 recurrence recurrence NOUN iajs-3048 28 20 relations relation NOUN iajs-3048 28 21 in in ADP iajs-3048 28 22 section section NOUN iajs-3048 28 23 3 3 NUM iajs-3048 28 24 , , PUNCT iajs-3048 28 25 boubaker boubaker NOUN iajs-3048 28 26 wavelets wavelet NOUN iajs-3048 28 27 and and CCONJ iajs-3048 28 28 their their PRON iajs-3048 28 29 properties property NOUN iajs-3048 28 30 in in ADP iajs-3048 28 31 section section NOUN iajs-3048 28 32 4 4 NUM iajs-3048 28 33 , , PUNCT iajs-3048 28 34 the the DET iajs-3048 28 35 application application NOUN iajs-3048 28 36 of of ADP iajs-3048 28 37 boubaker boubaker NOUN iajs-3048 28 38 wavelet wavelet NOUN iajs-3048 28 39 polynomials polynomial NOUN iajs-3048 28 40 for for ADP iajs-3048 28 41 solving solve VERB iajs-3048 28 42 variational variational ADJ iajs-3048 28 43 problems problem NOUN iajs-3048 28 44 with with ADP iajs-3048 28 45 some some DET iajs-3048 28 46 numerical numerical ADJ iajs-3048 28 47 examples example NOUN iajs-3048 28 48 has have AUX iajs-3048 28 49 been be AUX iajs-3048 28 50 presented present VERB iajs-3048 28 51 . . PUNCT iajs-3048 29 1 in in ADP iajs-3048 29 2 section section NOUN iajs-3048 29 3 5 5 NUM iajs-3048 29 4 , , PUNCT iajs-3048 29 5 the the DET iajs-3048 29 6 convergence convergence NOUN iajs-3048 29 7 test test NOUN iajs-3048 29 8 for for ADP iajs-3048 29 9 the the DET iajs-3048 29 10 introduced introduce VERB iajs-3048 29 11 method method NOUN iajs-3048 29 12 has have AUX iajs-3048 29 13 been be AUX iajs-3048 29 14 studied study VERB iajs-3048 29 15 , , PUNCT iajs-3048 29 16 and and CCONJ iajs-3048 29 17 at at ADP iajs-3048 29 18 last last ADJ iajs-3048 29 19 the the DET iajs-3048 29 20 conclusion conclusion NOUN iajs-3048 29 21 has have AUX iajs-3048 29 22 been be AUX iajs-3048 29 23 reached reach VERB iajs-3048 29 24 . . PUNCT iajs-3048 30 1 2 2 X iajs-3048 30 2 . . X iajs-3048 30 3 orthogonal orthogonal ADJ iajs-3048 30 4 boubaker boubaker NOUN iajs-3048 30 5 polynomials polynomial NOUN iajs-3048 30 6 and and CCONJ iajs-3048 30 7 their their PRON iajs-3048 30 8 properties property NOUN iajs-3048 30 9 : : PUNCT iajs-3048 30 10 boubaker boubaker NOUN iajs-3048 30 11 polynomials polynomial NOUN iajs-3048 30 12 have have AUX iajs-3048 30 13 n't not PART iajs-3048 30 14 been be AUX iajs-3048 30 15 orthogonal orthogonal ADJ iajs-3048 30 16 , , PUNCT iajs-3048 30 17 so so CCONJ iajs-3048 30 18 by by ADP iajs-3048 30 19 applying apply VERB iajs-3048 30 20 the the DET iajs-3048 30 21 gram gram NOUN iajs-3048 30 22 - - PUNCT iajs-3048 30 23 schmit schmit PROPN iajs-3048 30 24 process process NOUN iajs-3048 30 25 to to ADP iajs-3048 30 26 boubaker boubaker NOUN iajs-3048 30 27 polynomials polynomial NOUN iajs-3048 30 28 , , PUNCT iajs-3048 30 29 one one PRON iajs-3048 30 30 can can AUX iajs-3048 30 31 obtain obtain VERB iajs-3048 30 32 orthogonal orthogonal ADJ iajs-3048 30 33 boubaker boubaker NOUN iajs-3048 30 34 polynomials polynomial NOUN iajs-3048 30 35 , , PUNCT iajs-3048 30 36 obm(t obm(t PROPN iajs-3048 30 37 ) ) PUNCT iajs-3048 30 38 . . PUNCT iajs-3048 31 1 several several ADJ iajs-3048 31 2 papers paper NOUN iajs-3048 31 3 have have AUX iajs-3048 31 4 been be AUX iajs-3048 31 5 applied apply VERB iajs-3048 31 6 with with ADP iajs-3048 31 7 different different ADJ iajs-3048 31 8 applications application NOUN iajs-3048 31 9 in in ADP iajs-3048 31 10 physics physics NOUN iajs-3048 31 11 , , PUNCT iajs-3048 31 12 applied apply VERB iajs-3048 31 13 sciences science NOUN iajs-3048 31 14 , , PUNCT iajs-3048 31 15 etc etc X iajs-3048 31 16 . . X iajs-3048 31 17 ( ( PUNCT iajs-3048 31 18 see see VERB iajs-3048 31 19 [ [ X iajs-3048 31 20 14 14 NUM iajs-3048 31 21 ] ] PUNCT iajs-3048 31 22 , , PUNCT iajs-3048 31 23 [ [ X iajs-3048 31 24 15 15 NUM iajs-3048 31 25 ] ] NUM iajs-3048 31 26 ) ) PUNCT iajs-3048 31 27 . . PUNCT iajs-3048 32 1 orthogonal orthogonal ADJ iajs-3048 32 2 boubaker boubaker NOUN iajs-3048 32 3 polynomials polynomial NOUN iajs-3048 32 4 ( ( PUNCT iajs-3048 32 5 ob ob NOUN iajs-3048 32 6 ) ) PUNCT iajs-3048 32 7 of of ADP iajs-3048 32 8 mth mth NOUN iajs-3048 32 9 degree degree NOUN iajs-3048 32 10 were be AUX iajs-3048 32 11 presented present VERB iajs-3048 32 12 on on ADP iajs-3048 32 13 the the DET iajs-3048 32 14 interval interval NOUN iajs-3048 32 15 [ [ X iajs-3048 32 16 0 0 NUM iajs-3048 32 17 , , PUNCT iajs-3048 32 18 1 1 NUM iajs-3048 32 19 ] ] PUNCT iajs-3048 32 20 by by ADP iajs-3048 32 21 the the DET iajs-3048 32 22 following follow VERB iajs-3048 32 23 equation equation NOUN iajs-3048 32 24 : : PUNCT iajs-3048 33 1 [ [ X iajs-3048 33 2 1 1 X iajs-3048 33 3 ] ] PUNCT iajs-3048 33 4 π‘‚π΅π‘š(𝑑 π‘‚π΅π‘š(𝑑 PROPN iajs-3048 33 5 ) ) PUNCT iajs-3048 33 6 = = SYM iajs-3048 33 7 ( ( PUNCT iajs-3048 33 8 π‘š!)2 π‘š!)2 INTJ iajs-3048 33 9 ( ( PUNCT iajs-3048 33 10 2π‘š 2π‘š NOUN iajs-3048 33 11 ) ) PUNCT iajs-3048 33 12 ! ! PUNCT iajs-3048 34 1 βˆ‘ βˆ‘ PUNCT iajs-3048 34 2 ( ( PUNCT iajs-3048 34 3 βˆ’1)π‘š+π‘˜ βˆ’1)π‘š+π‘˜ PROPN iajs-3048 34 4 ( ( PUNCT iajs-3048 34 5 π‘š+π‘˜ π‘š+π‘˜ NUM iajs-3048 34 6 ) ) PUNCT iajs-3048 34 7 ! ! PUNCT iajs-3048 35 1 ( ( PUNCT iajs-3048 35 2 π‘šβˆ’π‘˜)!(π‘˜!)2 π‘šβˆ’π‘˜)!(π‘˜!)2 PROPN iajs-3048 35 3 π‘‘π‘˜π‘š π‘‘π‘˜π‘š VERB iajs-3048 35 4 π‘˜=0 π‘˜=0 ADP iajs-3048 35 5 … … PUNCT iajs-3048 35 6 ( ( PUNCT iajs-3048 35 7 1 1 X iajs-3048 35 8 ) ) PUNCT iajs-3048 35 9 a a DET iajs-3048 35 10 recursive recursive ADJ iajs-3048 35 11 relation relation NOUN iajs-3048 35 12 of of ADP iajs-3048 35 13 the the DET iajs-3048 35 14 orthogonal orthogonal ADJ iajs-3048 35 15 boubaker boubaker NOUN iajs-3048 35 16 polynomial polynomial NOUN iajs-3048 35 17 on on ADP iajs-3048 35 18 the the DET iajs-3048 35 19 interval interval NOUN iajs-3048 35 20 [ [ X iajs-3048 35 21 0 0 NUM iajs-3048 35 22 , , PUNCT iajs-3048 35 23 1 1 NUM iajs-3048 35 24 ] ] PUNCT iajs-3048 35 25 has have AUX iajs-3048 35 26 given give VERB iajs-3048 35 27 as as SCONJ iajs-3048 35 28 follows follow VERB iajs-3048 35 29 : : PUNCT iajs-3048 35 30 π‘‚π΅π‘š+1(𝑑 π‘‚π΅π‘š+1(𝑑 X iajs-3048 35 31 ) ) PUNCT iajs-3048 35 32 = = SYM iajs-3048 35 33 ( ( PUNCT iajs-3048 35 34 ( ( PUNCT iajs-3048 35 35 π‘š+1)!)2 π‘š+1)!)2 X iajs-3048 35 36 ( ( PUNCT iajs-3048 35 37 2(π‘š+1 2(π‘š+1 NUM iajs-3048 35 38 ) ) PUNCT iajs-3048 35 39 ) ) PUNCT iajs-3048 35 40 ! ! PUNCT iajs-3048 36 1 [ [ PUNCT iajs-3048 36 2 ( ( PUNCT iajs-3048 36 3 2π‘š+1)(2π‘š 2π‘š+1)(2π‘š NUM iajs-3048 36 4 ) ) PUNCT iajs-3048 36 5 ! ! PUNCT iajs-3048 37 1 ( ( PUNCT iajs-3048 37 2 π‘š+1)(π‘š!)2 π‘š+1)(π‘š!)2 X iajs-3048 37 3 ( ( PUNCT iajs-3048 37 4 2𝑑 2𝑑 NOUN iajs-3048 37 5 βˆ’ βˆ’ PROPN iajs-3048 37 6 1)π‘‚π΅π‘š(𝑑 1)π‘‚π΅π‘š(𝑑 NUM iajs-3048 37 7 ) ) PUNCT iajs-3048 37 8 βˆ’ βˆ’ PROPN iajs-3048 37 9 ( ( PUNCT iajs-3048 37 10 π‘š π‘š NOUN iajs-3048 37 11 ) ) PUNCT iajs-3048 37 12 2(π‘šβˆ’1 2(π‘šβˆ’1 NUM iajs-3048 37 13 ) ) PUNCT iajs-3048 37 14 ! ! PUNCT iajs-3048 38 1 ( ( PUNCT iajs-3048 38 2 π‘š+1 π‘š+1 NUM iajs-3048 38 3 ) ) PUNCT iajs-3048 38 4 ( ( PUNCT iajs-3048 38 5 ( ( PUNCT iajs-3048 38 6 π‘šβˆ’1)!)2 π‘šβˆ’1)!)2 X iajs-3048 38 7 π‘‚π΅π‘šβˆ’1(𝑑 π‘‚π΅π‘šβˆ’1(𝑑 NUM iajs-3048 38 8 ) ) PUNCT iajs-3048 38 9 ] ] PUNCT iajs-3048 38 10 , , PUNCT iajs-3048 38 11 π‘š π‘š X iajs-3048 38 12 β‰₯2 β‰₯2 PROPN iajs-3048 38 13 with with ADP iajs-3048 38 14 𝑂𝐡0(𝑑 𝑂𝐡0(𝑑 NOUN iajs-3048 38 15 ) ) PUNCT iajs-3048 38 16 = = SYM iajs-3048 38 17 1 1 NUM iajs-3048 38 18 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-3048 38 19 𝑂𝐡1(𝑑 𝑂𝐡1(𝑑 NOUN iajs-3048 38 20 ) ) PUNCT iajs-3048 38 21 = = SYM iajs-3048 38 22 1 1 NUM iajs-3048 38 23 2 2 NUM iajs-3048 38 24 ( ( PUNCT iajs-3048 38 25 2𝑑 2𝑑 NOUN iajs-3048 38 26 βˆ’ βˆ’ PROPN iajs-3048 38 27 1 1 NUM iajs-3048 38 28 ) ) PUNCT iajs-3048 38 29 . . PUNCT iajs-3048 39 1 3 3 X iajs-3048 39 2 . . X iajs-3048 39 3 boubaker boubaker NOUN iajs-3048 39 4 wavelet wavelet NOUN iajs-3048 39 5 and and CCONJ iajs-3048 39 6 their their PRON iajs-3048 39 7 properties property NOUN iajs-3048 39 8 : : PUNCT iajs-3048 39 9 the the DET iajs-3048 39 10 boubaker boubaker NOUN iajs-3048 39 11 wavelets wavelet NOUN iajs-3048 39 12 can can AUX iajs-3048 39 13 be be AUX iajs-3048 39 14 defined define VERB iajs-3048 39 15 as as ADP iajs-3048 39 16 below below ADV iajs-3048 39 17 ( ( PUNCT iajs-3048 39 18 see see VERB iajs-3048 39 19 [ [ X iajs-3048 39 20 1 1 NUM iajs-3048 39 21 ] ] SYM iajs-3048 39 22 ) ) PUNCT iajs-3048 39 23 π‘Šπ΅π‘›,π‘š(𝑑 π‘Šπ΅π‘›,π‘š(𝑑 NUM iajs-3048 39 24 ) ) PUNCT iajs-3048 39 25 = = PRON iajs-3048 39 26 { { PUNCT iajs-3048 39 27 √2π‘š √2π‘š NUM iajs-3048 39 28 + + CCONJ iajs-3048 39 29 1 1 NUM iajs-3048 39 30 2 2 NUM iajs-3048 39 31 π‘˜ π‘˜ DET iajs-3048 39 32 2 2 NUM iajs-3048 39 33 0 0 NUM iajs-3048 39 34 π‘œπ‘‘β„Žπ‘’π‘Ÿ π‘œπ‘‘β„Žπ‘’π‘Ÿ NOUN iajs-3048 39 35 𝑀𝑖𝑠𝑒 𝑀𝑖𝑠𝑒 NOUN iajs-3048 39 36 ( ( PUNCT iajs-3048 39 37 2π‘š 2π‘š NOUN iajs-3048 39 38 ) ) PUNCT iajs-3048 39 39 ! ! PUNCT iajs-3048 40 1 ( ( PUNCT iajs-3048 40 2 π‘š!)2 π‘š!)2 ADV iajs-3048 40 3 π‘‚π΅π‘š(2π‘˜π‘‘ π‘‚π΅π‘š(2π‘˜π‘‘ VERB iajs-3048 40 4 βˆ’ βˆ’ PROPN iajs-3048 40 5 𝑛 𝑛 PROPN iajs-3048 40 6 ) ) PUNCT iajs-3048 40 7 , , PUNCT iajs-3048 40 8 𝑛 𝑛 DET iajs-3048 40 9 2π‘˜βˆ’1 2π‘˜βˆ’1 NUM iajs-3048 40 10 ≀ ≀ NUM iajs-3048 40 11 𝑑 𝑑 X iajs-3048 40 12 < < X iajs-3048 40 13 𝑛+1 𝑛+1 PROPN iajs-3048 40 14 2π‘˜βˆ’1 2π‘˜βˆ’1 NUM iajs-3048 40 15 … … PUNCT iajs-3048 40 16 ( ( PUNCT iajs-3048 40 17 2 2 X iajs-3048 40 18 ) ) PUNCT iajs-3048 40 19 where where SCONJ iajs-3048 40 20 π‘Šπ΅π‘›π‘š(𝑑 π‘Šπ΅π‘›π‘š(𝑑 X iajs-3048 40 21 ) ) PUNCT iajs-3048 40 22 = = SYM iajs-3048 40 23 π‘Šπ΅(π‘š π‘Šπ΅(π‘š NOUN iajs-3048 40 24 , , PUNCT iajs-3048 40 25 𝑛 𝑛 NOUN iajs-3048 40 26 , , PUNCT iajs-3048 40 27 𝑑 𝑑 NOUN iajs-3048 40 28 ) ) PUNCT iajs-3048 40 29 as as ADP iajs-3048 40 30 boubaker boubaker NOUN iajs-3048 40 31 wavelets wavelet NOUN iajs-3048 40 32 so so SCONJ iajs-3048 40 33 n n CCONJ iajs-3048 40 34 = = NOUN iajs-3048 40 35 0,1 0,1 NUM iajs-3048 40 36 , , PUNCT iajs-3048 40 37 2 2 NUM iajs-3048 40 38 , , PUNCT iajs-3048 40 39 … … PUNCT iajs-3048 40 40 , , PUNCT iajs-3048 40 41 2k-1 2k-1 NUM iajs-3048 40 42 , , PUNCT iajs-3048 40 43 k k PROPN iajs-3048 40 44 is be AUX iajs-3048 40 45 any any DET iajs-3048 40 46 positive positive ADJ iajs-3048 40 47 integer integer NOUN iajs-3048 40 48 , , PUNCT iajs-3048 40 49 m m VERB iajs-3048 40 50 = = ADJ iajs-3048 40 51 0,1,2 0,1,2 NUM iajs-3048 40 52 … … PUNCT iajs-3048 40 53 ,m ,m PUNCT iajs-3048 40 54 and and CCONJ iajs-3048 40 55 t t PROPN iajs-3048 40 56 is be AUX iajs-3048 40 57 normalized normalize VERB iajs-3048 40 58 time time NOUN iajs-3048 40 59 . . PUNCT iajs-3048 41 1 the the DET iajs-3048 41 2 first first ADJ iajs-3048 41 3 six six NUM iajs-3048 41 4 π‘Šπ΅π‘›π‘š(𝑑 π‘Šπ΅π‘›π‘š(𝑑 NOUN iajs-3048 41 5 ) ) PUNCT iajs-3048 41 6 with with ADP iajs-3048 41 7 k=1 k=1 PROPN iajs-3048 41 8 , , PUNCT iajs-3048 41 9 are be AUX iajs-3048 41 10 given give VERB iajs-3048 41 11 as as SCONJ iajs-3048 41 12 follows follow VERB iajs-3048 41 13 : : PUNCT iajs-3048 41 14 ihjpas ihjpas PROPN iajs-3048 41 15 . . PUNCT iajs-3048 42 1 36 36 NUM iajs-3048 42 2 ( ( PUNCT iajs-3048 42 3 3 3 NUM iajs-3048 42 4 ) ) PUNCT iajs-3048 42 5 2023 2023 NUM iajs-3048 42 6 429 429 NUM iajs-3048 42 7 π‘Šπ΅0(𝑑 π‘Šπ΅0(𝑑 NOUN iajs-3048 42 8 ) ) PUNCT iajs-3048 42 9 = = SYM iajs-3048 42 10 1 1 NUM iajs-3048 42 11 , , PUNCT iajs-3048 42 12 π‘Šπ΅1(𝑑 π‘Šπ΅1(𝑑 NOUN iajs-3048 42 13 ) ) PUNCT iajs-3048 43 1 = = SYM iajs-3048 43 2 √3 √3 PROPN iajs-3048 43 3 2 2 NUM iajs-3048 43 4 ( ( PUNCT iajs-3048 43 5 4𝑑 4𝑑 NOUN iajs-3048 43 6 βˆ’ βˆ’ PROPN iajs-3048 43 7 3 3 NUM iajs-3048 43 8 ) ) PUNCT iajs-3048 43 9 , , PUNCT iajs-3048 43 10 π‘Šπ΅2(𝑑 π‘Šπ΅2(𝑑 PROPN iajs-3048 43 11 ) ) PUNCT iajs-3048 43 12 = = PUNCT iajs-3048 44 1 √5 √5 ADP iajs-3048 44 2 6 6 NUM iajs-3048 44 3 ( ( PUNCT iajs-3048 44 4 24𝑑2 24𝑑2 NUM iajs-3048 44 5 βˆ’ βˆ’ NOUN iajs-3048 44 6 36𝑑 36𝑑 NUM iajs-3048 44 7 + + CCONJ iajs-3048 44 8 13 13 NUM iajs-3048 44 9 ) ) PUNCT iajs-3048 44 10 , , PUNCT iajs-3048 44 11 π‘Šπ΅3(𝑑 π‘Šπ΅3(𝑑 PROPN iajs-3048 44 12 ) ) PUNCT iajs-3048 44 13 = = PUNCT iajs-3048 45 1 √7 √7 NOUN iajs-3048 45 2 20 20 NUM iajs-3048 45 3 ( ( PUNCT iajs-3048 45 4 160𝑑3 160𝑑3 NUM iajs-3048 45 5 βˆ’ βˆ’ NOUN iajs-3048 45 6 360𝑑2 360𝑑2 NUM iajs-3048 45 7 + + NUM iajs-3048 45 8 264𝑑 264𝑑 PROPN iajs-3048 45 9 βˆ’ βˆ’ PROPN iajs-3048 45 10 63 63 NUM iajs-3048 45 11 ) ) PUNCT iajs-3048 45 12 , , PUNCT iajs-3048 45 13 π‘Šπ΅4(𝑑 π‘Šπ΅4(𝑑 X iajs-3048 45 14 ) ) PUNCT iajs-3048 45 15 = = SYM iajs-3048 45 16 √9 √9 ADP iajs-3048 45 17 70 70 NUM iajs-3048 45 18 ( ( PUNCT iajs-3048 45 19 1120𝑑4 1120𝑑4 NUM iajs-3048 45 20 βˆ’ βˆ’ NUM iajs-3048 45 21 3360𝑑3 3360𝑑3 NUM iajs-3048 45 22 + + NUM iajs-3048 45 23 3720𝑑2 3720𝑑2 NUM iajs-3048 45 24 βˆ’ βˆ’ NOUN iajs-3048 45 25 1800𝑑 1800𝑑 NOUN iajs-3048 45 26 + + CCONJ iajs-3048 45 27 321 321 NUM iajs-3048 45 28 ) ) PUNCT iajs-3048 45 29 , , PUNCT iajs-3048 45 30 π‘Šπ΅5(𝑑 π‘Šπ΅5(𝑑 X iajs-3048 45 31 ) ) PUNCT iajs-3048 45 32 = = SYM iajs-3048 45 33 √11 √11 NOUN iajs-3048 45 34 252 252 NUM iajs-3048 45 35 ( ( PUNCT iajs-3048 45 36 8064𝑑5 8064𝑑5 NUM iajs-3048 45 37 βˆ’ βˆ’ PROPN iajs-3048 45 38 30240𝑑4 30240𝑑4 NOUN iajs-3048 46 1 + + NUM iajs-3048 46 2 44800𝑑3 44800𝑑3 NOUN iajs-3048 46 3 βˆ’ βˆ’ NOUN iajs-3048 47 1 32760𝑑2 32760𝑑2 NUM iajs-3048 48 1 + + NUM iajs-3048 48 2 11820𝑑 11820𝑑 NOUN iajs-3048 48 3 βˆ’ βˆ’ PROPN iajs-3048 48 4 1683 1683 NUM iajs-3048 48 5 ) ) PUNCT iajs-3048 48 6 , , PUNCT iajs-3048 48 7 π‘Šπ΅6(𝑑 π‘Šπ΅6(𝑑 PROPN iajs-3048 48 8 ) ) PUNCT iajs-3048 49 1 = = SYM iajs-3048 49 2 √13 √13 NUM iajs-3048 49 3 924 924 NUM iajs-3048 49 4 ( ( PUNCT iajs-3048 49 5 59136𝑑6 59136𝑑6 NOUN iajs-3048 49 6 βˆ’ βˆ’ NOUN iajs-3048 49 7 266112𝑑5 266112𝑑5 PROPN iajs-3048 50 1 + + CCONJ iajs-3048 50 2 493920𝑑4 493920𝑑4 NUM iajs-3048 50 3 βˆ’ βˆ’ NUM iajs-3048 50 4 483840𝑑3 483840𝑑3 NOUN iajs-3048 51 1 + + CCONJ iajs-3048 51 2 263760𝑑2 263760𝑑2 NUM iajs-3048 51 3 βˆ’ βˆ’ NOUN iajs-3048 51 4 75852𝑑 75852𝑑 NUM iajs-3048 51 5 + + CCONJ iajs-3048 51 6 8989 8989 NUM iajs-3048 51 7 ) ) PUNCT iajs-3048 51 8 . . PUNCT iajs-3048 52 1 the the DET iajs-3048 52 2 differential differential NOUN iajs-3048 52 3 with with ADP iajs-3048 52 4 respect respect NOUN iajs-3048 52 5 to to ADP iajs-3048 52 6 t t PROPN iajs-3048 52 7 of of ADP iajs-3048 52 8 boubaker boubaker NOUN iajs-3048 52 9 wavelet wavelet NOUN iajs-3048 52 10 polynomials polynomial NOUN iajs-3048 52 11 π‘Šπ΅π‘š π‘Šπ΅π‘š NOUN iajs-3048 52 12 Μ‡ Μ‡ VERB iajs-3048 52 13 ( ( PUNCT iajs-3048 52 14 𝑑 𝑑 NOUN iajs-3048 52 15 ) ) PUNCT iajs-3048 52 16 is be AUX iajs-3048 52 17 given give VERB iajs-3048 52 18 as as SCONJ iajs-3048 52 19 follows follow VERB iajs-3048 52 20 : : PUNCT iajs-3048 52 21 π‘Šπ΅0 π‘Šπ΅0 PROPN iajs-3048 52 22 Μ‡ Μ‡ PROPN iajs-3048 52 23 ( ( PUNCT iajs-3048 52 24 𝑑 𝑑 NOUN iajs-3048 52 25 ) ) PUNCT iajs-3048 52 26 = = SYM iajs-3048 52 27 0 0 NUM iajs-3048 52 28 , , PUNCT iajs-3048 52 29 π‘Šπ΅1 π‘Šπ΅1 PROPN iajs-3048 52 30 Μ‡ Μ‡ PROPN iajs-3048 52 31 ( ( PUNCT iajs-3048 52 32 𝑑 𝑑 NOUN iajs-3048 52 33 ) ) PUNCT iajs-3048 52 34 = = SYM iajs-3048 52 35 2√3 2√3 NUM iajs-3048 52 36 , , PUNCT iajs-3048 52 37 π‘Šπ΅2 π‘Šπ΅2 PROPN iajs-3048 52 38 Μ‡ Μ‡ VERB iajs-3048 52 39 ( ( PUNCT iajs-3048 52 40 𝑑 𝑑 NOUN iajs-3048 52 41 ) ) PUNCT iajs-3048 52 42 = = SYM iajs-3048 53 1 √5 √5 ADP iajs-3048 53 2 6 6 NUM iajs-3048 53 3 ( ( PUNCT iajs-3048 53 4 48𝑑 48𝑑 NOUN iajs-3048 53 5 βˆ’ βˆ’ PROPN iajs-3048 53 6 36 36 NUM iajs-3048 53 7 ) ) PUNCT iajs-3048 53 8 , , PUNCT iajs-3048 53 9 π‘Šπ΅Μ‡ π‘Šπ΅Μ‡ NOUN iajs-3048 53 10 3(𝑑 3(𝑑 NUM iajs-3048 53 11 ) ) PUNCT iajs-3048 54 1 = = PUNCT iajs-3048 54 2 √7 √7 NOUN iajs-3048 54 3 20 20 NUM iajs-3048 54 4 ( ( PUNCT iajs-3048 54 5 480𝑑2 480𝑑2 NUM iajs-3048 54 6 βˆ’ βˆ’ NOUN iajs-3048 54 7 720𝑑 720𝑑 NOUN iajs-3048 54 8 + + CCONJ iajs-3048 54 9 264 264 NUM iajs-3048 54 10 ) ) PUNCT iajs-3048 54 11 , , PUNCT iajs-3048 54 12 π‘Šπ΅Μ‡ π‘Šπ΅Μ‡ NOUN iajs-3048 54 13 4(𝑑 4(𝑑 NUM iajs-3048 54 14 ) ) PUNCT iajs-3048 54 15 = = PUNCT iajs-3048 55 1 √9 √9 ADP iajs-3048 55 2 70 70 NUM iajs-3048 55 3 ( ( PUNCT iajs-3048 55 4 4480𝑑3 4480𝑑3 PROPN iajs-3048 55 5 βˆ’ βˆ’ NOUN iajs-3048 55 6 10080𝑑2 10080𝑑2 NOUN iajs-3048 56 1 + + CCONJ iajs-3048 56 2 7440𝑑 7440𝑑 NUM iajs-3048 56 3 βˆ’ βˆ’ PROPN iajs-3048 56 4 1800 1800 NUM iajs-3048 56 5 ) ) PUNCT iajs-3048 57 1 , , PUNCT iajs-3048 57 2 π‘Šπ΅Μ‡ π‘Šπ΅Μ‡ VERB iajs-3048 57 3 5(𝑑 5(𝑑 NUM iajs-3048 57 4 ) ) PUNCT iajs-3048 57 5 = = PUNCT iajs-3048 58 1 √11 √11 NOUN iajs-3048 58 2 252 252 NUM iajs-3048 58 3 ( ( PUNCT iajs-3048 58 4 40320𝑑4 40320𝑑4 NOUN iajs-3048 58 5 βˆ’ βˆ’ NOUN iajs-3048 58 6 120960𝑑3 120960𝑑3 NUM iajs-3048 59 1 + + CCONJ iajs-3048 60 1 134400𝑑2 134400𝑑2 NUM iajs-3048 61 1 βˆ’ βˆ’ NOUN iajs-3048 61 2 65520𝑑 65520𝑑 NUM iajs-3048 61 3 + + CCONJ iajs-3048 61 4 11820 11820 NUM iajs-3048 61 5 ) ) PUNCT iajs-3048 61 6 , , PUNCT iajs-3048 61 7 π‘Šπ΅Μ‡ π‘Šπ΅Μ‡ VERB iajs-3048 61 8 6(𝑑 6(𝑑 NUM iajs-3048 61 9 ) ) PUNCT iajs-3048 61 10 = = PUNCT iajs-3048 62 1 √13 √13 NUM iajs-3048 62 2 924 924 NUM iajs-3048 62 3 ( ( PUNCT iajs-3048 62 4 354816𝑑5 354816𝑑5 NUM iajs-3048 62 5 βˆ’ βˆ’ NOUN iajs-3048 62 6 1330560𝑑4 1330560𝑑4 X iajs-3048 63 1 + + CCONJ iajs-3048 63 2 1975680𝑑3 1975680𝑑3 NUM iajs-3048 63 3 βˆ’ βˆ’ NOUN iajs-3048 63 4 1451520𝑑2 1451520𝑑2 NUM iajs-3048 64 1 + + NUM iajs-3048 64 2 527520𝑑 527520𝑑 PROPN iajs-3048 64 3 βˆ’ βˆ’ PROPN iajs-3048 64 4 75852 75852 NUM iajs-3048 64 5 ) ) PUNCT iajs-3048 64 6 . . PUNCT iajs-3048 65 1 4 4 X iajs-3048 65 2 . . X iajs-3048 65 3 boubaker boubaker NOUN iajs-3048 65 4 wavelet wavelet NOUN iajs-3048 65 5 polynomials polynomial NOUN iajs-3048 65 6 for for ADP iajs-3048 65 7 solving solve VERB iajs-3048 65 8 variational variational ADJ iajs-3048 65 9 problems problem NOUN iajs-3048 65 10 : : PUNCT iajs-3048 65 11 we we PRON iajs-3048 65 12 demonstrate demonstrate VERB iajs-3048 65 13 the the DET iajs-3048 65 14 application application NOUN iajs-3048 65 15 of of ADP iajs-3048 65 16 wavelet wavelet NOUN iajs-3048 65 17 boubaker boubaker NOUN iajs-3048 65 18 polynomials polynomial NOUN iajs-3048 65 19 to to PART iajs-3048 65 20 solve solve VERB iajs-3048 65 21 some some DET iajs-3048 65 22 variational variational ADJ iajs-3048 65 23 problems problem NOUN iajs-3048 65 24 . . PUNCT iajs-3048 66 1 a a DET iajs-3048 66 2 function function NOUN iajs-3048 66 3 f f X iajs-3048 66 4 ( ( PUNCT iajs-3048 66 5 t t PROPN iajs-3048 66 6 ) ) PUNCT iajs-3048 66 7 is be AUX iajs-3048 66 8 defined define VERB iajs-3048 66 9 over𝐿2[0,1 over𝐿2[0,1 ADV iajs-3048 66 10 ] ] PUNCT iajs-3048 66 11 , , PUNCT iajs-3048 66 12 the the DET iajs-3048 66 13 function function NOUN iajs-3048 66 14 approximate approximate NOUN iajs-3048 66 15 by by ADP iajs-3048 66 16 wavelet wavelet NOUN iajs-3048 66 17 boubaker boubaker NOUN iajs-3048 66 18 polynomials polynomial NOUN iajs-3048 66 19 is be AUX iajs-3048 66 20 defined define VERB iajs-3048 66 21 as as SCONJ iajs-3048 66 22 follows follow VERB iajs-3048 66 23 : : PUNCT iajs-3048 66 24 𝑓(𝑑 𝑓(𝑑 NUM iajs-3048 66 25 ) ) PUNCT iajs-3048 66 26 = = PUNCT iajs-3048 67 1 βˆ‘ βˆ‘ PUNCT iajs-3048 67 2 π‘π‘–π‘Šπ΅π‘–(𝑑 π‘π‘–π‘Šπ΅π‘–(𝑑 NUM iajs-3048 67 3 ) ) PUNCT iajs-3048 68 1 = = SYM iajs-3048 69 1 π‘π‘‡π‘Šπ΅(𝑑 π‘π‘‡π‘Šπ΅(𝑑 ADJ iajs-3048 69 2 ) ) PUNCT iajs-3048 69 3 ∞ ∞ PROPN iajs-3048 69 4 𝑖=0 𝑖=0 PROPN iajs-3048 69 5 … … PUNCT iajs-3048 69 6 ( ( PUNCT iajs-3048 69 7 3 3 X iajs-3048 69 8 ) ) PUNCT iajs-3048 69 9 where where SCONJ iajs-3048 69 10 𝑐𝑖 𝑐𝑖 NOUN iajs-3048 69 11 = = NOUN iajs-3048 69 12 βŸ¨π‘“(𝑑 βŸ¨π‘“(𝑑 NUM iajs-3048 69 13 ) ) PUNCT iajs-3048 69 14 , , PUNCT iajs-3048 69 15 π‘Šπ΅π‘–(𝑑)⟩ π‘Šπ΅π‘–(𝑑)⟩ ADJ iajs-3048 69 16 and and CCONJ iajs-3048 69 17 〈 〈 PROPN iajs-3048 69 18 , , PUNCT iajs-3048 69 19 βŒͺ βŒͺ NOUN iajs-3048 69 20 is be AUX iajs-3048 69 21 inner inner ADJ iajs-3048 69 22 product product NOUN iajs-3048 69 23 on on ADP iajs-3048 69 24 𝐿2[0,1 𝐿2[0,1 NOUN iajs-3048 69 25 ] ] PUNCT iajs-3048 69 26 . . PUNCT iajs-3048 70 1 ihjpas ihjpas PROPN iajs-3048 70 2 . . PUNCT iajs-3048 71 1 36 36 NUM iajs-3048 71 2 ( ( PUNCT iajs-3048 71 3 3 3 NUM iajs-3048 71 4 ) ) PUNCT iajs-3048 71 5 2023 2023 NUM iajs-3048 71 6 430 430 NUM iajs-3048 71 7 if if SCONJ iajs-3048 71 8 the the DET iajs-3048 71 9 series series NOUN iajs-3048 71 10 in in ADP iajs-3048 71 11 equation equation NOUN iajs-3048 71 12 ( ( PUNCT iajs-3048 71 13 3 3 X iajs-3048 71 14 ) ) PUNCT iajs-3048 71 15 is be AUX iajs-3048 71 16 truncated truncate VERB iajs-3048 71 17 , , PUNCT iajs-3048 71 18 then then ADV iajs-3048 71 19 equation equation NOUN iajs-3048 71 20 ( ( PUNCT iajs-3048 71 21 3 3 X iajs-3048 71 22 ) ) PUNCT iajs-3048 71 23 can can AUX iajs-3048 71 24 be be AUX iajs-3048 71 25 written write VERB iajs-3048 71 26 as as ADP iajs-3048 71 27 𝑓(𝑑 𝑓(𝑑 NOUN iajs-3048 71 28 ) ) PUNCT iajs-3048 71 29 = = PUNCT iajs-3048 71 30 βˆ‘ βˆ‘ PUNCT iajs-3048 71 31 π‘π‘–π‘Šπ΅π‘–(𝑑 π‘π‘–π‘Šπ΅π‘–(𝑑 NUM iajs-3048 71 32 ) ) PUNCT iajs-3048 71 33 = = PUNCT iajs-3048 72 1 π‘π‘‡π‘Šπ΅(𝑑 π‘π‘‡π‘Šπ΅(𝑑 ADJ iajs-3048 72 2 ) ) PUNCT iajs-3048 72 3 𝑁 𝑁 PROPN iajs-3048 72 4 𝑖=0 𝑖=0 PROPN iajs-3048 72 5 … … PUNCT iajs-3048 72 6 ( ( PUNCT iajs-3048 72 7 4 4 X iajs-3048 72 8 ) ) PUNCT iajs-3048 73 1 where where SCONJ iajs-3048 73 2 c c PROPN iajs-3048 73 3 =[ =[ PROPN iajs-3048 73 4 c0 c0 PROPN iajs-3048 73 5 , , PUNCT iajs-3048 73 6 c1 c1 PROPN iajs-3048 73 7 , , PUNCT iajs-3048 73 8 … … PUNCT iajs-3048 73 9 , , PUNCT iajs-3048 73 10 cn]t cn]t NOUN iajs-3048 73 11 and and CCONJ iajs-3048 73 12 wbi wbi PROPN iajs-3048 73 13 ( ( PUNCT iajs-3048 73 14 t t NOUN iajs-3048 73 15 ) ) PUNCT iajs-3048 73 16 = = PUNCT iajs-3048 74 1 [ [ X iajs-3048 74 2 wb0,wb1 wb0,wb1 NOUN iajs-3048 74 3 , , PUNCT iajs-3048 74 4 … … PUNCT iajs-3048 74 5 , , PUNCT iajs-3048 74 6 wbn]t wbn]t PROPN iajs-3048 74 7 differentiating differentiate VERB iajs-3048 74 8 equation equation NOUN iajs-3048 74 9 ( ( PUNCT iajs-3048 74 10 4 4 NUM iajs-3048 74 11 ) ) PUNCT iajs-3048 74 12 with with ADP iajs-3048 74 13 respect respect NOUN iajs-3048 74 14 t t PROPN iajs-3048 74 15 , , PUNCT iajs-3048 74 16 to to PART iajs-3048 74 17 obtain obtain VERB iajs-3048 74 18 𝑓 𝑓 DET iajs-3048 74 19 = = PUNCT iajs-3048 74 20 Μ‡ Μ‡ VERB iajs-3048 74 21 π‘π‘‡π‘Šπ΅Μ‡ π‘π‘‡π‘Šπ΅Μ‡ NOUN iajs-3048 74 22 ( ( PUNCT iajs-3048 74 23 𝑑 𝑑 NOUN iajs-3048 74 24 ) ) PUNCT iajs-3048 74 25 where where SCONJ iajs-3048 74 26 c c PROPN iajs-3048 74 27 =[ =[ PROPN iajs-3048 74 28 c0 c0 PROPN iajs-3048 74 29 , , PUNCT iajs-3048 74 30 c1 c1 PROPN iajs-3048 74 31 , , PUNCT iajs-3048 74 32 … … PUNCT iajs-3048 74 33 , , PUNCT iajs-3048 74 34 cn]t cn]t NOUN iajs-3048 74 35 and and CCONJ iajs-3048 74 36 π‘Šπ΅Μ‡ π‘Šπ΅Μ‡ PROPN iajs-3048 74 37 ( ( PUNCT iajs-3048 74 38 t t NOUN iajs-3048 74 39 ) ) PUNCT iajs-3048 74 40 = = PUNCT iajs-3048 75 1 [ [ X iajs-3048 75 2 π‘Šπ΅0 π‘Šπ΅0 NOUN iajs-3048 75 3 Μ‡ Μ‡ VERB iajs-3048 75 4 , , PUNCT iajs-3048 75 5 π‘Šπ΅Μ‡ π‘Šπ΅Μ‡ NOUN iajs-3048 75 6 1 1 NUM iajs-3048 75 7 , , PUNCT iajs-3048 75 8 … … PUNCT iajs-3048 75 9 , , PUNCT iajs-3048 75 10 π‘Šπ΅Μ‡ π‘Šπ΅Μ‡ VERB iajs-3048 75 11 n]t n]t ADV iajs-3048 75 12 the the DET iajs-3048 75 13 matrix matrix NOUN iajs-3048 75 14 of of ADP iajs-3048 75 15 derivatives derivative NOUN iajs-3048 75 16 d d NOUN iajs-3048 75 17 is be AUX iajs-3048 75 18 given give VERB iajs-3048 75 19 as as ADP iajs-3048 75 20 π‘‘π‘Šπ΅(𝑑 π‘‘π‘Šπ΅(𝑑 NUM iajs-3048 75 21 ) ) PUNCT iajs-3048 75 22 𝑑𝑑 𝑑𝑑 ADP iajs-3048 75 23 = = SYM iajs-3048 75 24 π‘Šπ΅Μ‡ π‘Šπ΅Μ‡ PROPN iajs-3048 75 25 ( ( PUNCT iajs-3048 75 26 𝑑 𝑑 NOUN iajs-3048 75 27 ) ) PUNCT iajs-3048 75 28 = = PUNCT iajs-3048 76 1 π·π‘Šπ΅(𝑑 π·π‘Šπ΅(𝑑 NOUN iajs-3048 76 2 ) ) PUNCT iajs-3048 76 3 where where SCONJ iajs-3048 76 4 π‘Šπ΅Μ‡ π‘Šπ΅Μ‡ NOUN iajs-3048 76 5 ( ( PUNCT iajs-3048 76 6 𝑑 𝑑 NOUN iajs-3048 76 7 ) ) PUNCT iajs-3048 76 8 derivative derivative NOUN iajs-3048 76 9 of of ADP iajs-3048 76 10 wavelet wavelet NOUN iajs-3048 76 11 boubaker boubaker NOUN iajs-3048 76 12 functions function NOUN iajs-3048 76 13 to to PART iajs-3048 76 14 demonstrate demonstrate VERB iajs-3048 76 15 this this DET iajs-3048 76 16 procedure procedure NOUN iajs-3048 76 17 , , PUNCT iajs-3048 76 18 we we PRON iajs-3048 76 19 consider consider VERB iajs-3048 76 20 this this DET iajs-3048 76 21 example example NOUN iajs-3048 76 22 of of ADP iajs-3048 76 23 finding find VERB iajs-3048 76 24 the the DET iajs-3048 76 25 minimum minimum NOUN iajs-3048 76 26 of of ADP iajs-3048 76 27 functional functional ADJ iajs-3048 76 28 [ [ X iajs-3048 76 29 14 14 NUM iajs-3048 76 30 ] ] PUNCT iajs-3048 76 31 example1 example1 PROPN iajs-3048 76 32 : : PUNCT iajs-3048 76 33 𝐽(π‘₯ 𝐽(π‘₯ X iajs-3048 76 34 ) ) PUNCT iajs-3048 76 35 = = SYM iajs-3048 77 1 ∫ ∫ PROPN iajs-3048 78 1 [ [ X iajs-3048 78 2 οΏ½ οΏ½ PROPN iajs-3048 78 3 Μ‡ Μ‡ PROPN iajs-3048 78 4 οΏ½ οΏ½ NOUN iajs-3048 78 5 2(𝑑 2(𝑑 NUM iajs-3048 78 6 ) ) PUNCT iajs-3048 78 7 + + CCONJ iajs-3048 78 8 𝑑 𝑑 PROPN iajs-3048 78 9 οΏ½ οΏ½ PROPN iajs-3048 78 10 Μ‡ Μ‡ NOUN iajs-3048 78 11 οΏ½ οΏ½ PROPN iajs-3048 78 12 (𝑑 (𝑑 PROPN iajs-3048 78 13 ) ) PUNCT iajs-3048 78 14 + + NUM iajs-3048 78 15 π‘₯2(𝑑)]𝑑𝑑 π‘₯2(𝑑)]𝑑𝑑 NUM iajs-3048 78 16 1 1 NUM iajs-3048 78 17 0 0 NUM iajs-3048 78 18 , , PUNCT iajs-3048 78 19 … … PUNCT iajs-3048 78 20 ( ( PUNCT iajs-3048 78 21 5 5 NUM iajs-3048 78 22 ) ) PUNCT iajs-3048 78 23 with with ADP iajs-3048 78 24 two two NUM iajs-3048 78 25 conditions condition NOUN iajs-3048 78 26 x(0 x(0 PRON iajs-3048 78 27 ) ) PUNCT iajs-3048 78 28 = = SYM iajs-3048 78 29 0 0 NUM iajs-3048 78 30 , , PUNCT iajs-3048 78 31 π‘₯(1 π‘₯(1 NOUN iajs-3048 78 32 ) ) PUNCT iajs-3048 78 33 = = SYM iajs-3048 78 34 1 1 NUM iajs-3048 78 35 4 4 NUM iajs-3048 78 36 … … PUNCT iajs-3048 78 37 ( ( PUNCT iajs-3048 78 38 6 6 NUM iajs-3048 78 39 ) ) PUNCT iajs-3048 78 40 and and CCONJ iajs-3048 78 41 the the DET iajs-3048 78 42 exact exact ADJ iajs-3048 78 43 solution solution NOUN iajs-3048 78 44 π‘₯(𝑑 π‘₯(𝑑 NOUN iajs-3048 78 45 ) ) PUNCT iajs-3048 78 46 = = SYM iajs-3048 78 47 βˆ’π‘’βˆ’π‘‘[(βˆ’1+𝑒𝑑)(π‘’βˆ’2𝑒2βˆ’2𝑒𝑑+𝑒1+𝑑 βˆ’π‘’βˆ’π‘‘[(βˆ’1+𝑒𝑑)(π‘’βˆ’2𝑒2βˆ’2𝑒𝑑+𝑒1+𝑑 X iajs-3048 78 48 ) ) PUNCT iajs-3048 78 49 ] ] PUNCT iajs-3048 79 1 4(βˆ’1+𝑒2 4(βˆ’1+𝑒2 X iajs-3048 79 2 ) ) PUNCT iajs-3048 79 3 now now ADV iajs-3048 79 4 , , PUNCT iajs-3048 79 5 we we PRON iajs-3048 79 6 use use VERB iajs-3048 79 7 wavelet wavelet NOUN iajs-3048 79 8 boubaker boubaker NOUN iajs-3048 79 9 polynomials polynomial NOUN iajs-3048 79 10 of of ADP iajs-3048 79 11 m m NOUN iajs-3048 79 12 = = NOUN iajs-3048 79 13 4 4 NUM iajs-3048 79 14 and and CCONJ iajs-3048 79 15 m=6 m=6 VERB iajs-3048 79 16 to to PART iajs-3048 79 17 approximate approximate VERB iajs-3048 79 18 the the DET iajs-3048 79 19 function function NOUN iajs-3048 79 20 x(t x(t PROPN iajs-3048 79 21 ) ) PUNCT iajs-3048 79 22 suppose suppose VERB iajs-3048 79 23 π‘₯(𝑑 π‘₯(𝑑 NOUN iajs-3048 79 24 ) ) PUNCT iajs-3048 79 25 = = PUNCT iajs-3048 79 26 βˆ‘ βˆ‘ PUNCT iajs-3048 79 27 π‘π‘–π‘Šπ΅π‘–(𝑑)5 π‘π‘–π‘Šπ΅π‘–(𝑑)5 NUM iajs-3048 79 28 0 0 NUM iajs-3048 79 29 or or CCONJ iajs-3048 79 30 π‘₯(𝑑 π‘₯(𝑑 NUM iajs-3048 79 31 ) ) PUNCT iajs-3048 79 32 = = SYM iajs-3048 80 1 π‘π‘‡π‘Šπ΅(𝑑 π‘π‘‡π‘Šπ΅(𝑑 PROPN iajs-3048 80 2 ) ) PUNCT iajs-3048 80 3 … … PUNCT iajs-3048 80 4 ( ( PUNCT iajs-3048 80 5 7 7 NUM iajs-3048 80 6 ) ) PUNCT iajs-3048 80 7 and and CCONJ iajs-3048 80 8 π‘₯2(𝑑 π‘₯2(𝑑 NOUN iajs-3048 80 9 ) ) PUNCT iajs-3048 80 10 = = SYM iajs-3048 80 11 π‘π‘‡π‘Šπ΅(π‘Šπ΅)𝑇𝑐 π‘π‘‡π‘Šπ΅(π‘Šπ΅)𝑇𝑐 NOUN iajs-3048 80 12 … … PUNCT iajs-3048 80 13 ( ( PUNCT iajs-3048 80 14 8) 8) NUM iajs-3048 80 15 where where SCONJ iajs-3048 80 16 c c NOUN iajs-3048 81 1 = = PUNCT iajs-3048 82 1 [ [ X iajs-3048 82 2 c0 c0 X iajs-3048 82 3 , , PUNCT iajs-3048 82 4 c1 c1 PROPN iajs-3048 82 5 , , PUNCT iajs-3048 82 6 … … PUNCT iajs-3048 82 7 , , PUNCT iajs-3048 82 8 c5 c5 PROPN iajs-3048 82 9 ] ] PUNCT iajs-3048 82 10 t t PROPN iajs-3048 82 11 and and CCONJ iajs-3048 82 12 wb wb PROPN iajs-3048 82 13 ( ( PUNCT iajs-3048 82 14 t t PROPN iajs-3048 82 15 ) ) PUNCT iajs-3048 82 16 = = PUNCT iajs-3048 83 1 [ [ X iajs-3048 83 2 wb0 wb0 NOUN iajs-3048 83 3 , , PUNCT iajs-3048 83 4 wb1 wb1 NOUN iajs-3048 83 5 , , PUNCT iajs-3048 83 6 … … PUNCT iajs-3048 83 7 , , PUNCT iajs-3048 83 8 wb5 wb5 X iajs-3048 83 9 ] ] PUNCT iajs-3048 83 10 , , PUNCT iajs-3048 83 11 differentiating differentiate VERB iajs-3048 83 12 equation equation NOUN iajs-3048 83 13 ( ( PUNCT iajs-3048 83 14 7 7 NUM iajs-3048 83 15 ) ) PUNCT iajs-3048 83 16 , , PUNCT iajs-3048 83 17 we we PRON iajs-3048 83 18 get get VERB iajs-3048 83 19 οΏ½ οΏ½ PROPN iajs-3048 83 20 Μ‡ Μ‡ PROPN iajs-3048 83 21 οΏ½ οΏ½ PROPN iajs-3048 83 22 (𝑑 (𝑑 NOUN iajs-3048 83 23 ) ) PUNCT iajs-3048 83 24 = = PUNCT iajs-3048 83 25 𝑐𝑇𝐷 𝑐𝑇𝐷 ADP iajs-3048 83 26 π‘Šπ΅(𝑑 π‘Šπ΅(𝑑 PROPN iajs-3048 83 27 ) ) PUNCT iajs-3048 83 28 … … PUNCT iajs-3048 83 29 ( ( PUNCT iajs-3048 83 30 9 9 NUM iajs-3048 83 31 ) ) PUNCT iajs-3048 83 32 and and CCONJ iajs-3048 83 33 οΏ½ οΏ½ PROPN iajs-3048 83 34 Μ‡ Μ‡ PROPN iajs-3048 83 35 οΏ½ οΏ½ NOUN iajs-3048 83 36 2(𝑑 2(𝑑 NUM iajs-3048 83 37 ) ) PUNCT iajs-3048 83 38 = = PUNCT iajs-3048 83 39 𝑐𝑇𝐷 𝑐𝑇𝐷 PROPN iajs-3048 83 40 ( ( PUNCT iajs-3048 83 41 π‘Šπ΅)(𝐷 π‘Šπ΅)(𝐷 PROPN iajs-3048 83 42 ( ( PUNCT iajs-3048 83 43 π‘Šπ΅))𝑇𝑐 π‘Šπ΅))𝑇𝑐 PROPN iajs-3048 83 44 … … PUNCT iajs-3048 83 45 ( ( PUNCT iajs-3048 83 46 10 10 NUM iajs-3048 83 47 ) ) PUNCT iajs-3048 83 48 substituting substitute VERB iajs-3048 83 49 equation equation NOUN iajs-3048 83 50 ( ( PUNCT iajs-3048 83 51 7 7 NUM iajs-3048 83 52 ) ) PUNCT iajs-3048 83 53 ( ( PUNCT iajs-3048 83 54 10 10 NUM iajs-3048 83 55 ) ) PUNCT iajs-3048 83 56 into into ADP iajs-3048 83 57 equation equation NOUN iajs-3048 83 58 ( ( PUNCT iajs-3048 83 59 5 5 NUM iajs-3048 83 60 ) ) PUNCT iajs-3048 83 61 , , PUNCT iajs-3048 83 62 we we PRON iajs-3048 83 63 obtain obtain VERB iajs-3048 83 64 𝐽(π‘₯ 𝐽(π‘₯ ADV iajs-3048 83 65 ) ) PUNCT iajs-3048 83 66 = = SYM iajs-3048 84 1 ∫ ∫ PROPN iajs-3048 85 1 [ [ X iajs-3048 85 2 𝑐𝑇𝐷 𝑐𝑇𝐷 X iajs-3048 85 3 ( ( PUNCT iajs-3048 85 4 π‘Šπ΅)(𝐷 π‘Šπ΅)(𝐷 PROPN iajs-3048 85 5 ( ( PUNCT iajs-3048 85 6 π‘Šπ΅ π‘Šπ΅ PROPN iajs-3048 85 7 ) ) PUNCT iajs-3048 85 8 ) ) PUNCT iajs-3048 86 1 𝑇1 𝑇1 NOUN iajs-3048 86 2 0 0 PUNCT iajs-3048 86 3 𝑐 𝑐 NOUN iajs-3048 86 4 + + NUM iajs-3048 86 5 𝑑𝑐𝑇𝐷 𝑑𝑐𝑇𝐷 NOUN iajs-3048 86 6 ( ( PUNCT iajs-3048 86 7 π‘Šπ΅ π‘Šπ΅ PROPN iajs-3048 86 8 ) ) PUNCT iajs-3048 86 9 + + NUM iajs-3048 86 10 π‘π‘‡π‘Šπ΅(π‘Šπ΅)𝑇𝑐]𝑑𝑑 π‘π‘‡π‘Šπ΅(π‘Šπ΅)𝑇𝑐]𝑑𝑑 NOUN iajs-3048 86 11 … … PUNCT iajs-3048 86 12 ( ( PUNCT iajs-3048 86 13 11 11 NUM iajs-3048 86 14 ) ) PUNCT iajs-3048 86 15 we we PRON iajs-3048 86 16 can can AUX iajs-3048 86 17 simplify simplify VERB iajs-3048 86 18 equation equation NOUN iajs-3048 86 19 ( ( PUNCT iajs-3048 86 20 11 11 NUM iajs-3048 86 21 ) ) PUNCT iajs-3048 86 22 to to PART iajs-3048 86 23 𝐽(π‘₯ 𝐽(π‘₯ VERB iajs-3048 86 24 ) ) PUNCT iajs-3048 86 25 = = SYM iajs-3048 86 26 1 1 NUM iajs-3048 86 27 2 2 NUM iajs-3048 86 28 𝑐𝑇𝐻𝑐 𝑐𝑇𝐻𝑐 NOUN iajs-3048 86 29 + + NUM iajs-3048 86 30 π‘žπ‘‡π‘ π‘žπ‘‡π‘ NOUN iajs-3048 86 31 … … PUNCT iajs-3048 86 32 ( ( PUNCT iajs-3048 86 33 12 12 NUM iajs-3048 86 34 ) ) PUNCT iajs-3048 86 35 where where SCONJ iajs-3048 86 36 𝐻 𝐻 PROPN iajs-3048 86 37 = = SYM iajs-3048 86 38 2 2 NUM iajs-3048 86 39 ∫ ∫ NOUN iajs-3048 87 1 [ [ X iajs-3048 87 2 𝐷 𝐷 PROPN iajs-3048 87 3 ( ( PUNCT iajs-3048 87 4 π‘Šπ΅)(𝐷 π‘Šπ΅)(𝐷 NOUN iajs-3048 87 5 ( ( PUNCT iajs-3048 87 6 π‘Šπ΅ π‘Šπ΅ PROPN iajs-3048 87 7 ) ) PUNCT iajs-3048 87 8 ) ) PUNCT iajs-3048 87 9 𝑇1 𝑇1 NOUN iajs-3048 87 10 0 0 NUM iajs-3048 87 11 + + NUM iajs-3048 87 12 π‘Šπ΅(π‘Šπ΅)𝑇]𝑑𝑑 π‘Šπ΅(π‘Šπ΅)𝑇]𝑑𝑑 PROPN iajs-3048 87 13 , , PUNCT iajs-3048 87 14 and and CCONJ iajs-3048 87 15 π‘žπ‘‡ π‘žπ‘‡ ADV iajs-3048 87 16 = = SYM iajs-3048 87 17 ∫ ∫ PROPN iajs-3048 87 18 𝑑 𝑑 PROPN iajs-3048 87 19 𝐷 𝐷 PROPN iajs-3048 87 20 ( ( PUNCT iajs-3048 87 21 π‘Šπ΅)𝑇𝑑𝑑 π‘Šπ΅)𝑇𝑑𝑑 PROPN iajs-3048 87 22 1 1 NUM iajs-3048 87 23 0 0 NUM iajs-3048 87 24 , , PUNCT iajs-3048 87 25 equation equation NOUN iajs-3048 87 26 ( ( PUNCT iajs-3048 87 27 7 7 NUM iajs-3048 87 28 ) ) PUNCT iajs-3048 87 29 with with ADP iajs-3048 87 30 boundary boundary ADJ iajs-3048 87 31 conditions condition NOUN iajs-3048 87 32 ( ( PUNCT iajs-3048 87 33 6 6 NUM iajs-3048 87 34 ) ) PUNCT iajs-3048 87 35 , , PUNCT iajs-3048 87 36 can can AUX iajs-3048 87 37 imply imply VERB iajs-3048 87 38 π‘₯(0 π‘₯(0 PROPN iajs-3048 87 39 ) ) PUNCT iajs-3048 87 40 = = PUNCT iajs-3048 88 1 π‘π‘‡π‘Šπ΅(0 π‘π‘‡π‘Šπ΅(0 ADJ iajs-3048 88 2 ) ) PUNCT iajs-3048 88 3 = = SYM iajs-3048 88 4 0 0 NUM iajs-3048 88 5 and and CCONJ iajs-3048 88 6 π‘₯(1 π‘₯(1 PROPN iajs-3048 88 7 ) ) PUNCT iajs-3048 88 8 = = SYM iajs-3048 88 9 π‘π‘‡π‘Šπ΅(1 π‘π‘‡π‘Šπ΅(1 X iajs-3048 88 10 ) ) PUNCT iajs-3048 88 11 = = SYM iajs-3048 88 12 1 1 NUM iajs-3048 88 13 4 4 NUM iajs-3048 88 14 … … PUNCT iajs-3048 88 15 ( ( PUNCT iajs-3048 88 16 13 13 NUM iajs-3048 88 17 ) ) PUNCT iajs-3048 88 18 ihjpas ihjpa NOUN iajs-3048 88 19 . . PUNCT iajs-3048 89 1 36 36 NUM iajs-3048 89 2 ( ( PUNCT iajs-3048 89 3 3 3 NUM iajs-3048 89 4 ) ) PUNCT iajs-3048 89 5 2023 2023 NUM iajs-3048 89 6 431 431 NUM iajs-3048 89 7 we we PRON iajs-3048 89 8 can can AUX iajs-3048 89 9 rewrite rewrite VERB iajs-3048 89 10 equations equation NOUN iajs-3048 89 11 ( ( PUNCT iajs-3048 89 12 12 12 NUM iajs-3048 89 13 ) ) PUNCT iajs-3048 89 14 & & CCONJ iajs-3048 89 15 ( ( PUNCT iajs-3048 89 16 13 13 NUM iajs-3048 89 17 ) ) PUNCT iajs-3048 89 18 , , PUNCT iajs-3048 89 19 as as SCONJ iajs-3048 89 20 follows follow VERB iajs-3048 89 21 min min NOUN iajs-3048 89 22 𝐽(π‘₯ 𝐽(π‘₯ PROPN iajs-3048 89 23 ) ) PUNCT iajs-3048 90 1 = = SYM iajs-3048 90 2 1 1 NUM iajs-3048 90 3 2 2 NUM iajs-3048 90 4 𝑐𝑇𝐻𝑐 𝑐𝑇𝐻𝑐 NOUN iajs-3048 90 5 + + NUM iajs-3048 90 6 π‘žπ‘‡π‘ π‘žπ‘‡π‘ NOUN iajs-3048 90 7 subject subject ADJ iajs-3048 90 8 to to ADP iajs-3048 90 9 fc fc NOUN iajs-3048 90 10 - - PROPN iajs-3048 90 11 b=0 b=0 PROPN iajs-3048 90 12 where where SCONJ iajs-3048 90 13 𝐹 𝐹 PROPN iajs-3048 90 14 = = PUNCT iajs-3048 91 1 [ [ X iajs-3048 91 2 π‘Šπ΅π‘‡(0 π‘Šπ΅π‘‡(0 NOUN iajs-3048 91 3 ) ) PUNCT iajs-3048 91 4 π‘Šπ΅π‘‡(1 π‘Šπ΅π‘‡(1 NUM iajs-3048 91 5 ) ) PUNCT iajs-3048 91 6 ] ] PUNCT iajs-3048 91 7 , , PUNCT iajs-3048 92 1 𝑏 𝑏 NOUN iajs-3048 92 2 = = PUNCT iajs-3048 93 1 [ [ X iajs-3048 93 2 0 0 NUM iajs-3048 93 3 1 1 NUM iajs-3048 93 4 4 4 NUM iajs-3048 93 5 ] ] PUNCT iajs-3048 93 6 we we PRON iajs-3048 93 7 can can AUX iajs-3048 93 8 find find VERB iajs-3048 93 9 the the DET iajs-3048 93 10 parameter parameter NOUN iajs-3048 93 11 c c NOUN iajs-3048 93 12 using use VERB iajs-3048 93 13 lagrange lagrange NOUN iajs-3048 93 14 equation equation NOUN iajs-3048 93 15 as as ADP iajs-3048 93 16 π‘βˆ— π‘βˆ— NOUN iajs-3048 93 17 = = PUNCT iajs-3048 94 1 βˆ’π»βˆ’1𝑐 βˆ’π»βˆ’1𝑐 PROPN iajs-3048 94 2 + + CCONJ iajs-3048 94 3 π»βˆ’1𝐹𝑇(πΉπ»βˆ’1𝐹𝑇)βˆ’1(πΉπ»βˆ’1𝑐 π»βˆ’1𝐹𝑇(πΉπ»βˆ’1𝐹𝑇)βˆ’1(πΉπ»βˆ’1𝑐 ADJ iajs-3048 94 4 + + CCONJ iajs-3048 94 5 𝑏 𝑏 NOUN iajs-3048 94 6 ) ) PUNCT iajs-3048 94 7 when when SCONJ iajs-3048 94 8 m=4 m=4 AUX iajs-3048 94 9 the the DET iajs-3048 94 10 approximate approximate ADJ iajs-3048 94 11 value value NOUN iajs-3048 94 12 of of ADP iajs-3048 94 13 x x PUNCT iajs-3048 94 14 is be AUX iajs-3048 94 15 x(t)= x(t)= PUNCT iajs-3048 95 1 [ [ X iajs-3048 95 2 0.21463728 0.21463728 NUM iajs-3048 95 3 , , PUNCT iajs-3048 95 4 0.04727207 0.04727207 NUM iajs-3048 95 5 , , PUNCT iajs-3048 95 6 -0.01564479 -0.01564479 PROPN iajs-3048 95 7 , , PUNCT iajs-3048 95 8 0.00192278 0.00192278 NUM iajs-3048 95 9 ] ] X iajs-3048 95 10 wb wb PROPN iajs-3048 95 11 ( ( PUNCT iajs-3048 95 12 t t PROPN iajs-3048 95 13 ) ) PUNCT iajs-3048 95 14 and and CCONJ iajs-3048 95 15 when when SCONJ iajs-3048 95 16 m=6 m=6 ADP iajs-3048 95 17 an an DET iajs-3048 95 18 approximate approximate ADJ iajs-3048 95 19 value value NOUN iajs-3048 95 20 of of ADP iajs-3048 95 21 x x PUNCT iajs-3048 95 22 is be AUX iajs-3048 95 23 x(t)= x(t)= PUNCT iajs-3048 96 1 [ [ X iajs-3048 96 2 0.21464181 0.21464181 NUM iajs-3048 96 3 , , PUNCT iajs-3048 96 4 0.04746694,-0.01585894 0.04746694,-0.01585894 NUM iajs-3048 96 5 , , PUNCT iajs-3048 96 6 0.00129175 0.00129175 NUM iajs-3048 96 7 , , PUNCT iajs-3048 96 8 -0.00023926 -0.00023926 NOUN iajs-3048 96 9 , , PUNCT iajs-3048 96 10 0.00001933 0.00001933 NUM iajs-3048 96 11 ] ] X iajs-3048 96 12 wb wb PROPN iajs-3048 96 13 ( ( PUNCT iajs-3048 96 14 t t NOUN iajs-3048 96 15 ) ) PUNCT iajs-3048 96 16 table table NOUN iajs-3048 96 17 ( ( PUNCT iajs-3048 96 18 1 1 X iajs-3048 96 19 ) ) PUNCT iajs-3048 96 20 shows show VERB iajs-3048 96 21 the the DET iajs-3048 96 22 numerical numerical ADJ iajs-3048 96 23 results result NOUN iajs-3048 96 24 for for ADP iajs-3048 96 25 example example NOUN iajs-3048 96 26 ( ( PUNCT iajs-3048 96 27 1 1 X iajs-3048 96 28 ) ) PUNCT iajs-3048 96 29 with with ADP iajs-3048 96 30 m=4 m=4 ADV iajs-3048 96 31 and and CCONJ iajs-3048 96 32 m=6 m=6 PROPN iajs-3048 96 33 compared compare VERB iajs-3048 96 34 with with ADP iajs-3048 96 35 exact exact ADJ iajs-3048 96 36 solution solution NOUN iajs-3048 96 37 , , PUNCT iajs-3048 96 38 and and CCONJ iajs-3048 96 39 graphically graphically ADV iajs-3048 96 40 in in ADP iajs-3048 96 41 figure figure NOUN iajs-3048 96 42 ( ( PUNCT iajs-3048 96 43 1 1 NUM iajs-3048 96 44 ) ) PUNCT iajs-3048 96 45 . . PUNCT iajs-3048 97 1 table table NOUN iajs-3048 97 2 1 1 NUM iajs-3048 97 3 . . PUNCT iajs-3048 97 4 results result NOUN iajs-3048 97 5 for for ADP iajs-3048 97 6 example1 example1 PROPN iajs-3048 97 7 t t PROPN iajs-3048 97 8 xexact xexact NOUN iajs-3048 97 9 xapp.(t xapp.(t PROPN iajs-3048 97 10 ) ) PUNCT iajs-3048 98 1 m=4 m=4 ADP iajs-3048 98 2 m=4 m=4 ADP iajs-3048 98 3 absolute absolute ADJ iajs-3048 98 4 error error NOUN iajs-3048 98 5 xapp.(t xapp.(t PUNCT iajs-3048 98 6 ) ) PUNCT iajs-3048 99 1 m=6 m=6 PUNCT iajs-3048 99 2 m=6 m=6 X iajs-3048 100 1 absolute absolute ADJ iajs-3048 100 2 error error NOUN iajs-3048 100 3 0 0 NUM iajs-3048 100 4 0.00000000 0.00000000 NUM iajs-3048 100 5 0.00000000 0.00000000 NUM iajs-3048 100 6 0.00000000 0.00000000 NUM iajs-3048 100 7 0.00000000 0.00000000 NUM iajs-3048 100 8 0.0000000 0.0000000 NUM iajs-3048 100 9 0.1 0.1 NUM iajs-3048 100 10 0.04195072 0.04195072 NUM iajs-3048 100 11 0.04180602 0.04180602 NUM iajs-3048 100 12 0.00014470 0.00014470 NUM iajs-3048 100 13 0.04195070 0.04195070 NUM iajs-3048 100 14 0.00000002 0.00000002 NUM iajs-3048 100 15 0.2 0.2 NUM iajs-3048 100 16 0.07931714 0.07931714 NUM iajs-3048 100 17 0.07922621 0.07922621 NUM iajs-3048 100 18 0.00009093 0.00009093 NUM iajs-3048 100 19 0.07931742 0.07931742 NUM iajs-3048 100 20 0.00000027 0.00000027 NUM iajs-3048 100 21 0.3 0.3 NUM iajs-3048 100 22 0.11247322 0.11247322 NUM iajs-3048 100 23 0.11250475 0.11250475 NUM iajs-3048 100 24 0.00003152 0.00003152 NUM iajs-3048 100 25 0.11247334 0.11247334 NUM iajs-3048 100 26 0.00000011 0.00000011 NUM iajs-3048 100 27 0.4 0.4 NUM iajs-3048 100 28 0.14175081 0.14175081 NUM iajs-3048 100 29 0.14188583 0.14188583 NUM iajs-3048 101 1 0.00013502 0.00013502 NUM iajs-3048 101 2 0.14175057 0.14175057 NUM iajs-3048 101 3 0.00000023 0.00000023 NUM iajs-3048 101 4 0.5 0.5 NUM iajs-3048 101 5 0.16744291 0.16744291 NUM iajs-3048 101 6 0.16761363 0.16761363 NUM iajs-3048 101 7 0.00017071 0.00017071 NUM iajs-3048 101 8 0.16744257 0.16744257 NUM iajs-3048 101 9 0.00000034 0.00000034 NUM iajs-3048 101 10 0.6 0.6 NUM iajs-3048 101 11 0.18980668 0.18980668 NUM iajs-3048 101 12 0.18993234 0.18993234 NUM iajs-3048 101 13 0.00012566 0.00012566 NUM iajs-3048 101 14 0.18980660 0.18980660 NUM iajs-3048 101 15 0.00000007 0.00000007 NUM iajs-3048 101 16 0.7 0.7 NUM iajs-3048 101 17 0.20906592 0.20906592 NUM iajs-3048 101 18 0.20908615 0.20908615 NUM iajs-3048 101 19 0.00002022 0.00002022 NUM iajs-3048 101 20 0.20906621 0.20906621 NUM iajs-3048 101 21 0.00000029 0.00000029 NUM iajs-3048 101 22 0.8 0.8 NUM iajs-3048 101 23 0.22541340 0.22541340 NUM iajs-3048 101 24 0.22531923 0.22531923 NUM iajs-3048 101 25 0.00009416 0.00009416 NUM iajs-3048 101 26 0.22541370 0.22541370 NUM iajs-3048 101 27 0.00000030 0.00000030 NUM iajs-3048 101 28 0.9 0.9 NUM iajs-3048 101 29 0.23901272 0.23901272 NUM iajs-3048 101 30 0.23887579 0.23887579 NUM iajs-3048 101 31 0.00013693 0.00013693 NUM iajs-3048 101 32 0.23901258 0.23901258 NUM iajs-3048 101 33 0.00000014 0.00000014 NUM iajs-3048 101 34 1.0 1.0 NUM iajs-3048 101 35 0.2500000000 0.2500000000 NUM iajs-3048 101 36 0.25000000 0.25000000 NUM iajs-3048 101 37 0.00000000 0.00000000 NUM iajs-3048 101 38 0.25000000 0.25000000 NUM iajs-3048 101 39 0.00000000 0.00000000 NUM iajs-3048 101 40 figure figure NOUN iajs-3048 101 41 1 1 NUM iajs-3048 101 42 . . PUNCT iajs-3048 101 43 shows show VERB iajs-3048 101 44 that that SCONJ iajs-3048 101 45 the the DET iajs-3048 101 46 absolute absolute ADJ iajs-3048 101 47 error error NOUN iajs-3048 101 48 when when SCONJ iajs-3048 101 49 m=6 m=6 PRON iajs-3048 101 50 is be AUX iajs-3048 101 51 very very ADV iajs-3048 101 52 less less ADJ iajs-3048 101 53 than than ADP iajs-3048 101 54 m=4 m=4 ADV iajs-3048 101 55 for for ADP iajs-3048 101 56 example1 example1 PROPN iajs-3048 101 57 ihjpas ihjpas PROPN iajs-3048 101 58 . . PUNCT iajs-3048 102 1 36 36 NUM iajs-3048 102 2 ( ( PUNCT iajs-3048 102 3 3 3 NUM iajs-3048 102 4 ) ) PUNCT iajs-3048 102 5 2023 2023 NUM iajs-3048 102 6 432 432 NUM iajs-3048 102 7 example example NOUN iajs-3048 102 8 2 2 NUM iajs-3048 102 9 : : PUNCT iajs-3048 102 10 min min NOUN iajs-3048 102 11 j= j= PROPN iajs-3048 102 12 ∫ ∫ PROPN iajs-3048 102 13 ( ( PUNCT iajs-3048 102 14 οΏ½ οΏ½ PROPN iajs-3048 102 15 Μ‡ Μ‡ PROPN iajs-3048 102 16 οΏ½ οΏ½ PROPN iajs-3048 102 17 2 2 NUM iajs-3048 102 18 + + NUM iajs-3048 102 19 π‘₯2 π‘₯2 NOUN iajs-3048 102 20 ) ) PUNCT iajs-3048 102 21 1 1 NUM iajs-3048 102 22 0 0 NUM iajs-3048 102 23 𝑑𝑑 𝑑𝑑 ADP iajs-3048 102 24 with with ADP iajs-3048 102 25 boundary boundary ADJ iajs-3048 102 26 conditions condition NOUN iajs-3048 102 27 π‘₯(0 π‘₯(0 PROPN iajs-3048 102 28 ) ) PUNCT iajs-3048 103 1 = = SYM iajs-3048 103 2 0 0 NUM iajs-3048 103 3 , , PUNCT iajs-3048 103 4 π‘₯(1 π‘₯(1 PROPN iajs-3048 103 5 ) ) PUNCT iajs-3048 103 6 = = SYM iajs-3048 103 7 1 1 NUM iajs-3048 103 8 , , PUNCT iajs-3048 103 9 and and CCONJ iajs-3048 103 10 exact exact ADJ iajs-3048 103 11 solution solution NOUN iajs-3048 103 12 is be AUX iajs-3048 103 13 π‘₯(𝑑 π‘₯(𝑑 NOUN iajs-3048 103 14 ) ) PUNCT iajs-3048 103 15 = = PUNCT iajs-3048 103 16 π‘’π‘‘βˆ’π‘’βˆ’π‘‘ π‘’π‘‘βˆ’π‘’βˆ’π‘‘ VERB iajs-3048 103 17 𝑒1βˆ’π‘’βˆ’1 𝑒1βˆ’π‘’βˆ’1 PROPN iajs-3048 103 18 in in ADP iajs-3048 103 19 the the DET iajs-3048 103 20 same same ADJ iajs-3048 103 21 procedure procedure NOUN iajs-3048 103 22 , , PUNCT iajs-3048 103 23 we we PRON iajs-3048 103 24 can can AUX iajs-3048 103 25 solve solve VERB iajs-3048 103 26 example2 example2 PROPN iajs-3048 103 27 . . PUNCT iajs-3048 104 1 table table NOUN iajs-3048 104 2 ( ( PUNCT iajs-3048 104 3 2 2 X iajs-3048 104 4 ) ) PUNCT iajs-3048 104 5 shows show VERB iajs-3048 104 6 the the DET iajs-3048 104 7 numerical numerical ADJ iajs-3048 104 8 results result NOUN iajs-3048 104 9 for for ADP iajs-3048 104 10 example example NOUN iajs-3048 104 11 ( ( PUNCT iajs-3048 104 12 2 2 NUM iajs-3048 104 13 ) ) PUNCT iajs-3048 104 14 with with ADP iajs-3048 104 15 m=4 m=4 ADV iajs-3048 104 16 and and CCONJ iajs-3048 104 17 m=6 m=6 PROPN iajs-3048 104 18 compared compare VERB iajs-3048 104 19 with with ADP iajs-3048 104 20 exact exact ADJ iajs-3048 104 21 solution solution NOUN iajs-3048 104 22 , , PUNCT iajs-3048 104 23 and and CCONJ iajs-3048 104 24 graphically graphically ADV iajs-3048 104 25 in in ADP iajs-3048 104 26 figure figure NOUN iajs-3048 104 27 ( ( PUNCT iajs-3048 104 28 2 2 NUM iajs-3048 104 29 ) ) PUNCT iajs-3048 104 30 . . PUNCT iajs-3048 105 1 when when SCONJ iajs-3048 105 2 m=4 m=4 ADV iajs-3048 105 3 the the DET iajs-3048 105 4 approximate approximate ADJ iajs-3048 105 5 value value NOUN iajs-3048 105 6 of of ADP iajs-3048 105 7 x x PUNCT iajs-3048 105 8 is be AUX iajs-3048 105 9 x(t)= x(t)= PUNCT iajs-3048 106 1 [ [ X iajs-3048 106 2 0.70703400,0.32030428,0.03906027 0.70703400,0.32030428,0.03906027 X iajs-3048 106 3 , , PUNCT iajs-3048 106 4 0.00769113 0.00769113 NUM iajs-3048 106 5 ] ] X iajs-3048 106 6 wb wb PROPN iajs-3048 106 7 ( ( PUNCT iajs-3048 106 8 t t PROPN iajs-3048 106 9 ) ) PUNCT iajs-3048 106 10 and and CCONJ iajs-3048 106 11 when when SCONJ iajs-3048 106 12 m=6 m=6 ADP iajs-3048 106 13 the the DET iajs-3048 106 14 approximate approximate ADJ iajs-3048 106 15 value value NOUN iajs-3048 106 16 of of ADP iajs-3048 106 17 x x PUNCT iajs-3048 106 18 is be AUX iajs-3048 106 19 x(t)= x(t)= PROPN iajs-3048 107 1 [ [ X iajs-3048 107 2 0.70703595,0.32001609,0.03929145,0.00870795,0.00060523,0.00007777 0.70703595,0.32001609,0.03929145,0.00870795,0.00060523,0.00007777 X iajs-3048 107 3 ] ] X iajs-3048 107 4 wb wb PROPN iajs-3048 107 5 ( ( PUNCT iajs-3048 107 6 t t NOUN iajs-3048 107 7 ) ) PUNCT iajs-3048 107 8 table table NOUN iajs-3048 107 9 2 2 NUM iajs-3048 107 10 . . X iajs-3048 107 11 results result NOUN iajs-3048 107 12 for for ADP iajs-3048 107 13 example2 example2 PROPN iajs-3048 107 14 t t PROPN iajs-3048 107 15 xexact xexact NOUN iajs-3048 107 16 m=4 m=4 PROPN iajs-3048 107 17 xapp.(t xapp.(t PROPN iajs-3048 107 18 ) ) PUNCT iajs-3048 108 1 m=4 m=4 ADP iajs-3048 108 2 absolute absolute ADJ iajs-3048 108 3 error error NOUN iajs-3048 108 4 m=6 m=6 X iajs-3048 108 5 xapp.(t xapp.(t PROPN iajs-3048 108 6 ) ) PUNCT iajs-3048 109 1 m=6 m=6 X iajs-3048 109 2 absolute absolute ADJ iajs-3048 109 3 error error NOUN iajs-3048 109 4 0 0 NUM iajs-3048 109 5 0 0 NUM iajs-3048 109 6 0.00000000 0.00000000 NUM iajs-3048 109 7 0.00000000 0.00000000 NUM iajs-3048 109 8 0.00000000 0.00000000 NUM iajs-3048 109 9 0.00000000 0.00000000 NUM iajs-3048 109 10 0.1 0.1 NUM iajs-3048 109 11 0.08523370 0.08523370 NUM iajs-3048 109 12 0.08540591 0.08540591 NUM iajs-3048 110 1 0.00017221 0.00017221 NUM iajs-3048 110 2 0.08523414 0.08523414 NUM iajs-3048 110 3 0.00000043 0.00000043 NUM iajs-3048 110 4 0.2 0.2 NUM iajs-3048 110 5 0.17132045 0.17132045 NUM iajs-3048 110 6 0.17145031 0.17145031 NUM iajs-3048 110 7 0.00012986 0.00012986 NUM iajs-3048 110 8 0.17131995 0.17131995 NUM iajs-3048 110 9 0.00000049 0.00000049 NUM iajs-3048 110 10 0.3 0.3 NUM iajs-3048 110 11 0.25912183 0.25912183 NUM iajs-3048 110 12 0.25910993 0.25910993 NUM iajs-3048 110 13 0.00001190 0.00001190 NUM iajs-3048 110 14 0.25912108 0.25912108 NUM iajs-3048 110 15 0.00000075 0.00000075 NUM iajs-3048 110 16 0.4 0.4 NUM iajs-3048 110 17 0.34951660 0.34951660 NUM iajs-3048 110 18 0.34936152 0.34936152 NUM iajs-3048 110 19 0.00015507 0.00015507 NUM iajs-3048 110 20 0.34951638 0.34951638 NUM iajs-3048 110 21 0.00000021 0.00000021 NUM iajs-3048 110 22 0.5 0.5 NUM iajs-3048 110 23 0.44340944 0.44340944 NUM iajs-3048 110 24 0.44318181 0.44318181 NUM iajs-3048 110 25 0.00022762 0.00022762 NUM iajs-3048 111 1 0.44340990 0.44340990 NUM iajs-3048 111 2 0.00000046 0.00000046 NUM iajs-3048 111 3 0.6 0.6 NUM iajs-3048 111 4 0.54174007 0.54174007 NUM iajs-3048 111 5 0.54154756 0.54154756 NUM iajs-3048 111 6 0.00019250 0.00019250 NUM iajs-3048 111 7 0.54174071 0.54174071 NUM iajs-3048 111 8 0.00000063 0.00000063 NUM iajs-3048 111 9 0.7 0.7 NUM iajs-3048 111 10 0.64549262 0.64549262 NUM iajs-3048 111 11 0.64543551 0.64543551 NUM iajs-3048 111 12 0.00005710 0.00005710 NUM iajs-3048 111 13 0.64549284 0.64549284 NUM iajs-3048 111 14 0.00000021 0.00000021 NUM iajs-3048 111 15 0.8 0.8 NUM iajs-3048 111 16 0.75570548 0.75570548 NUM iajs-3048 111 17 0.75582241 0.75582241 NUM iajs-3048 111 18 0.00011693 0.00011693 NUM iajs-3048 111 19 0.75570520 0.75570520 NUM iajs-3048 111 20 0.00000027 0.00000027 NUM iajs-3048 111 21 0.9 0.9 NUM iajs-3048 111 22 0.87348169 0.87348169 NUM iajs-3048 112 1 0.87368498 0.87368498 NUM iajs-3048 112 2 0.00020329 0.00020329 NUM iajs-3048 112 3 0.87348147 0.87348147 NUM iajs-3048 112 4 0.00000021 0.00000021 NUM iajs-3048 112 5 1.0 1.0 NUM iajs-3048 112 6 1.00000000 1.00000000 NUM iajs-3048 112 7 1.00000000 1.00000000 NUM iajs-3048 112 8 0.00000000 0.00000000 NUM iajs-3048 112 9 1.00000000 1.00000000 NUM iajs-3048 112 10 0.00000000 0.00000000 NUM iajs-3048 112 11 figure2 figure2 X iajs-3048 112 12 . . PUNCT iajs-3048 113 1 shows show VERB iajs-3048 113 2 that that SCONJ iajs-3048 113 3 the the DET iajs-3048 113 4 absolute absolute ADJ iajs-3048 113 5 error error NOUN iajs-3048 113 6 when when SCONJ iajs-3048 113 7 m=6 m=6 PRON iajs-3048 113 8 is be AUX iajs-3048 113 9 very very ADV iajs-3048 113 10 less less ADJ iajs-3048 113 11 than than ADP iajs-3048 113 12 m=4 m=4 ADV iajs-3048 113 13 for for ADP iajs-3048 113 14 example2 example2 PROPN iajs-3048 113 15 . . PUNCT iajs-3048 114 1 ihjpas ihjpas PROPN iajs-3048 114 2 . . PUNCT iajs-3048 115 1 36 36 NUM iajs-3048 115 2 ( ( PUNCT iajs-3048 115 3 3 3 NUM iajs-3048 115 4 ) ) PUNCT iajs-3048 115 5 2023 2023 NUM iajs-3048 115 6 433 433 NUM iajs-3048 115 7 example example NOUN iajs-3048 115 8 3 3 NUM iajs-3048 115 9 : : PUNCT iajs-3048 115 10 min min NOUN iajs-3048 115 11 j= j= PROPN iajs-3048 115 12 ∫ ∫ PROPN iajs-3048 115 13 ( ( PUNCT iajs-3048 115 14 οΏ½ οΏ½ PROPN iajs-3048 115 15 Μ‡ Μ‡ PROPN iajs-3048 115 16 οΏ½ οΏ½ PROPN iajs-3048 115 17 2 2 NUM iajs-3048 115 18 βˆ’ βˆ’ NOUN iajs-3048 115 19 π‘₯2 π‘₯2 NOUN iajs-3048 115 20 ) ) PUNCT iajs-3048 115 21 1 1 NUM iajs-3048 115 22 0 0 NUM iajs-3048 115 23 𝑑𝑑 𝑑𝑑 ADP iajs-3048 115 24 with with ADP iajs-3048 115 25 boundary boundary ADJ iajs-3048 115 26 conditions condition NOUN iajs-3048 115 27 π‘₯(0 π‘₯(0 PROPN iajs-3048 115 28 ) ) PUNCT iajs-3048 116 1 = = SYM iajs-3048 116 2 0 0 NUM iajs-3048 116 3 , , PUNCT iajs-3048 116 4 π‘₯(1 π‘₯(1 NOUN iajs-3048 116 5 ) ) PUNCT iajs-3048 116 6 = = SYM iajs-3048 116 7 1 1 NUM iajs-3048 116 8 , , PUNCT iajs-3048 116 9 and and CCONJ iajs-3048 116 10 exact exact ADJ iajs-3048 116 11 solution solution NOUN iajs-3048 116 12 is be AUX iajs-3048 116 13 π‘₯(𝑑 π‘₯(𝑑 NOUN iajs-3048 116 14 ) ) PUNCT iajs-3048 117 1 = = SYM iajs-3048 117 2 𝑠𝑖𝑛𝑑 𝑠𝑖𝑛𝑑 ADJ iajs-3048 117 3 𝑠𝑖𝑛1 𝑠𝑖𝑛1 PROPN iajs-3048 117 4 table table NOUN iajs-3048 117 5 ( ( PUNCT iajs-3048 117 6 3 3 NUM iajs-3048 117 7 ) ) PUNCT iajs-3048 117 8 and and CCONJ iajs-3048 117 9 figure figure NOUN iajs-3048 117 10 ( ( PUNCT iajs-3048 117 11 3 3 X iajs-3048 117 12 ) ) PUNCT iajs-3048 117 13 illustrate illustrate VERB iajs-3048 117 14 the the DET iajs-3048 117 15 results result NOUN iajs-3048 117 16 of of ADP iajs-3048 117 17 example example NOUN iajs-3048 117 18 ( ( PUNCT iajs-3048 117 19 3 3 NUM iajs-3048 117 20 ) ) PUNCT iajs-3048 117 21 . . PUNCT iajs-3048 118 1 when when SCONJ iajs-3048 118 2 m=4 m=4 ADV iajs-3048 118 3 the the DET iajs-3048 118 4 approximate approximate ADJ iajs-3048 118 5 value value NOUN iajs-3048 118 6 of of ADP iajs-3048 118 7 x x PUNCT iajs-3048 118 8 is be AUX iajs-3048 118 9 x(t)= x(t)= PROPN iajs-3048 118 10 [ [ PUNCT iajs-3048 118 11 0.80163166 0.80163166 NUM iajs-3048 118 12 , , PUNCT iajs-3048 118 13 0.24981351 0.24981351 NUM iajs-3048 118 14 , , PUNCT iajs-3048 118 15 0.04537278 0.04537278 NUM iajs-3048 118 16 , , PUNCT iajs-3048 118 17 0.00806631 0.00806631 NUM iajs-3048 118 18 ] ] X iajs-3048 118 19 wb wb PROPN iajs-3048 118 20 ( ( PUNCT iajs-3048 118 21 t t PROPN iajs-3048 118 22 ) ) PUNCT iajs-3048 118 23 and and CCONJ iajs-3048 118 24 when when SCONJ iajs-3048 118 25 m m VERB iajs-3048 118 26 = = NOUN iajs-3048 118 27 6 6 NUM iajs-3048 118 28 the the DET iajs-3048 118 29 approximate approximate ADJ iajs-3048 118 30 value value NOUN iajs-3048 118 31 of of ADP iajs-3048 118 32 x x PUNCT iajs-3048 118 33 is be AUX iajs-3048 118 34 x(t)= x(t)= PUNCT iajs-3048 119 1 [ [ X iajs-3048 119 2 0.80164433 0.80164433 NUM iajs-3048 119 3 , , PUNCT iajs-3048 119 4 0.24944837 0.24944837 NUM iajs-3048 119 5 , , PUNCT iajs-3048 119 6 0.04508324 0.04508324 NUM iajs-3048 119 7 , , PUNCT iajs-3048 119 8 0.00682505 0.00682505 NUM iajs-3048 119 9 , , PUNCT iajs-3048 119 10 0.00070902 0.00070902 NUM iajs-3048 119 11 , , PUNCT iajs-3048 119 12 0.00008006 0.00008006 NUM iajs-3048 119 13 ] ] X iajs-3048 119 14 wb wb PROPN iajs-3048 119 15 ( ( PUNCT iajs-3048 119 16 t t NOUN iajs-3048 119 17 ) ) PUNCT iajs-3048 119 18 table table NOUN iajs-3048 119 19 3 3 NUM iajs-3048 119 20 . . NOUN iajs-3048 119 21 results result NOUN iajs-3048 119 22 for for ADP iajs-3048 119 23 example3 example3 PROPN iajs-3048 119 24 t t PROPN iajs-3048 119 25 uexact uexact ADJ iajs-3048 119 26 m= m= NUM iajs-3048 119 27 4 4 NUM iajs-3048 119 28 uappr uappr NOUN iajs-3048 119 29 ( ( PUNCT iajs-3048 119 30 t t NOUN iajs-3048 119 31 ) ) PUNCT iajs-3048 119 32 m=4 m=4 ADP iajs-3048 120 1 absolute absolute ADJ iajs-3048 120 2 error error NOUN iajs-3048 120 3 m= m= X iajs-3048 120 4 6 6 NUM iajs-3048 120 5 uappr.(t uappr.(t NUM iajs-3048 120 6 ) ) PUNCT iajs-3048 120 7 m=6 m=6 ADP iajs-3048 121 1 absolute absolute ADJ iajs-3048 121 2 error error NOUN iajs-3048 121 3 0 0 NUM iajs-3048 121 4 0.00000000 0.00000000 NUM iajs-3048 121 5 0.00000000 0.00000000 NUM iajs-3048 121 6 0.00000000 0.00000000 NUM iajs-3048 121 7 0.00000000 0.00000000 NUM iajs-3048 121 8 0.0000000 0.0000000 NUM iajs-3048 121 9 0.1 0.1 NUM iajs-3048 121 10 0.11864154 0.11864154 NUM iajs-3048 121 11 0.11885365 0.11885365 NUM iajs-3048 121 12 0.00021211 0.00021211 NUM iajs-3048 121 13 0.11862523 0.11862523 NUM iajs-3048 121 14 0.00001631 0.00001631 NUM iajs-3048 121 15 0.2 0.2 NUM iajs-3048 121 16 0.23609766 0.23609766 NUM iajs-3048 121 17 0.23624932 0.23624932 NUM iajs-3048 121 18 0.00015166 0.00015166 NUM iajs-3048 121 19 0.23610543 0.23610543 NUM iajs-3048 121 20 0.00000777 0.00000777 NUM iajs-3048 121 21 0.3 0.3 NUM iajs-3048 121 22 0.35119476 0.35119476 NUM iajs-3048 121 23 0.35116260 0.35116260 NUM iajs-3048 121 24 0.00003216 0.00003216 NUM iajs-3048 121 25 0.35121956 0.35121956 NUM iajs-3048 121 26 0.00002480 0.00002480 NUM iajs-3048 121 27 0.4 0.4 NUM iajs-3048 121 28 0.46278285 0.46278285 NUM iajs-3048 121 29 0.46256910 0.46256910 NUM iajs-3048 122 1 0.00021374 0.00021374 NUM iajs-3048 122 2 0.46280277 0.46280277 NUM iajs-3048 122 3 0.00001992 0.00001992 NUM iajs-3048 122 4 0.5 0.5 NUM iajs-3048 122 5 0.56974696 0.56974696 NUM iajs-3048 122 6 0.56944444 0.56944444 NUM iajs-3048 122 7 0.00030251 0.00030251 NUM iajs-3048 122 8 0.56974637 0.56974637 NUM iajs-3048 122 9 0.00000058 0.00000058 NUM iajs-3048 122 10 0.6 0.6 NUM iajs-3048 122 11 0.67101835 0.67101835 NUM iajs-3048 122 12 0.67076422 0.67076422 NUM iajs-3048 122 13 0.00025412 0.00025412 NUM iajs-3048 122 14 0.67099789 0.67099789 NUM iajs-3048 122 15 0.00002045 0.00002045 NUM iajs-3048 122 16 0.7 0.7 NUM iajs-3048 122 17 0.76558514 0.76558514 NUM iajs-3048 122 18 0.76550406 0.76550406 NUM iajs-3048 122 19 0.00008108 0.00008108 NUM iajs-3048 122 20 0.76556103 0.76556103 NUM iajs-3048 122 21 0.00002411 0.00002411 NUM iajs-3048 122 22 0.8 0.8 NUM iajs-3048 122 23 0.85250246 0.85250246 NUM iajs-3048 122 24 0.85263956 0.85263956 NUM iajs-3048 122 25 0.00013709 0.00013709 NUM iajs-3048 122 26 0.85249567 0.85249567 NUM iajs-3048 122 27 0.00000678 0.00000678 NUM iajs-3048 122 28 0.9 0.9 NUM iajs-3048 122 29 0.93090186 0.93090186 NUM iajs-3048 122 30 0.93114634 0.93114634 NUM iajs-3048 122 31 0.00024447 0.00024447 NUM iajs-3048 122 32 0.93091791 0.93091791 NUM iajs-3048 122 33 0.00001604 0.00001604 NUM iajs-3048 122 34 1.0 1.0 NUM iajs-3048 122 35 1.00000000 1.00000000 NUM iajs-3048 122 36 1.00000000 1.00000000 NUM iajs-3048 122 37 0.00000000 0.00000000 NUM iajs-3048 122 38 1.00000000 1.00000000 NUM iajs-3048 122 39 0.00000000 0.00000000 NUM iajs-3048 122 40 figure figure NOUN iajs-3048 122 41 3 3 NUM iajs-3048 122 42 . . PUNCT iajs-3048 122 43 shows show VERB iajs-3048 122 44 that that SCONJ iajs-3048 122 45 the the DET iajs-3048 122 46 absolute absolute ADJ iajs-3048 122 47 error error NOUN iajs-3048 122 48 when when SCONJ iajs-3048 122 49 m=6 m=6 PRON iajs-3048 122 50 is be AUX iajs-3048 122 51 very very ADV iajs-3048 122 52 less less ADJ iajs-3048 122 53 than than ADP iajs-3048 122 54 m=4 m=4 ADV iajs-3048 122 55 for for ADP iajs-3048 122 56 example3 example3 PROPN iajs-3048 122 57 ihjpas ihjpas PROPN iajs-3048 122 58 . . PUNCT iajs-3048 123 1 36 36 NUM iajs-3048 123 2 ( ( PUNCT iajs-3048 123 3 3 3 NUM iajs-3048 123 4 ) ) PUNCT iajs-3048 123 5 2023 2023 NUM iajs-3048 123 6 434 434 NUM iajs-3048 123 7 5 5 NUM iajs-3048 123 8 . . PUNCT iajs-3048 124 1 the the DET iajs-3048 124 2 convergence convergence NOUN iajs-3048 124 3 test test NOUN iajs-3048 124 4 of of ADP iajs-3048 124 5 wavelet wavelet NOUN iajs-3048 124 6 boubaker boubaker NOUN iajs-3048 124 7 polynomials polynomial NOUN iajs-3048 124 8 : : PUNCT iajs-3048 124 9 by by ADP iajs-3048 124 10 ( ( PUNCT iajs-3048 124 11 theorem theorem NOUN iajs-3048 124 12 ( ( PUNCT iajs-3048 124 13 1 1 NUM iajs-3048 124 14 ) ) PUNCT iajs-3048 124 15 , , PUNCT iajs-3048 124 16 see see VERB iajs-3048 124 17 [ [ X iajs-3048 124 18 1 1 NUM iajs-3048 124 19 ] ] PUNCT iajs-3048 124 20 ) ) PUNCT iajs-3048 124 21 if if SCONJ iajs-3048 124 22 x(t x(t PROPN iajs-3048 124 23 ) ) PUNCT iajs-3048 124 24 is be AUX iajs-3048 124 25 continually continually ADV iajs-3048 124 26 defined define VERB iajs-3048 124 27 on on ADP iajs-3048 124 28 [ [ X iajs-3048 124 29 0,1 0,1 NUM iajs-3048 124 30 ] ] PUNCT iajs-3048 124 31 and and CCONJ iajs-3048 124 32 Ξ±(t Ξ±(t PROPN iajs-3048 124 33 ) ) PUNCT iajs-3048 124 34 is be AUX iajs-3048 124 35 the the DET iajs-3048 124 36 approximate approximate NOUN iajs-3048 124 37 of of ADP iajs-3048 124 38 π›Όβˆ—(t π›Όβˆ—(t PROPN iajs-3048 124 39 ) ) PUNCT iajs-3048 124 40 by by ADP iajs-3048 124 41 applying apply VERB iajs-3048 124 42 boubaker boubaker NOUN iajs-3048 124 43 wavelet wavelet NOUN iajs-3048 124 44 . . PUNCT iajs-3048 125 1 also also ADV iajs-3048 125 2 , , PUNCT iajs-3048 125 3 suppose suppose VERB iajs-3048 125 4 that that SCONJ iajs-3048 125 5 x x SYM iajs-3048 125 6 ( ( PUNCT iajs-3048 125 7 t t NOUN iajs-3048 125 8 ) ) PUNCT iajs-3048 125 9 is be AUX iajs-3048 125 10 bounded bound VERB iajs-3048 125 11 by by ADP iajs-3048 125 12 a a DET iajs-3048 125 13 positive positive ADJ iajs-3048 125 14 constant constant NOUN iajs-3048 125 15 that that PRON iajs-3048 125 16 is be AUX iajs-3048 125 17 |π‘₯(𝑑)| |π‘₯(𝑑)| PROPN iajs-3048 125 18 < < X iajs-3048 125 19 πœ–. πœ–. NOUN iajs-3048 125 20 then then ADV iajs-3048 125 21 , , PUNCT iajs-3048 125 22 the the DET iajs-3048 125 23 coefficients coefficient NOUN iajs-3048 125 24 of of ADP iajs-3048 125 25 x x X iajs-3048 125 26 ( ( PUNCT iajs-3048 125 27 t t NOUN iajs-3048 125 28 ) ) PUNCT iajs-3048 125 29 are be AUX iajs-3048 125 30 bounded bound VERB iajs-3048 125 31 . . PUNCT iajs-3048 126 1 the the DET iajs-3048 126 2 state state NOUN iajs-3048 126 3 can can AUX iajs-3048 126 4 be be AUX iajs-3048 126 5 expanded expand VERB iajs-3048 126 6 using use VERB iajs-3048 126 7 wavelet wavelet NOUN iajs-3048 126 8 boubaker boubaker NOUN iajs-3048 126 9 polynomials polynomial NOUN iajs-3048 126 10 , , PUNCT iajs-3048 126 11 as as ADP iajs-3048 126 12 : : PUNCT iajs-3048 126 13 π‘₯𝑖𝑁(𝑑 π‘₯𝑖𝑁(𝑑 PROPN iajs-3048 126 14 ) ) PUNCT iajs-3048 126 15 = = PUNCT iajs-3048 127 1 βˆ‘ βˆ‘ PUNCT iajs-3048 127 2 π‘π‘–π‘˜π‘Šπ΅π‘˜(𝑑)𝑁 π‘π‘–π‘˜π‘Šπ΅π‘˜(𝑑)𝑁 PROPN iajs-3048 127 3 π‘˜=1 π‘˜=1 NOUN iajs-3048 127 4 π‘₯𝑖(𝑑 π‘₯𝑖(𝑑 NUM iajs-3048 127 5 ) ) PUNCT iajs-3048 127 6 = = SYM iajs-3048 127 7 π‘₯𝑖𝑁(𝑑 π‘₯𝑖𝑁(𝑑 PROPN iajs-3048 127 8 ) ) PUNCT iajs-3048 128 1 + + CCONJ iajs-3048 128 2 βˆ‘ βˆ‘ PUNCT iajs-3048 128 3 π‘π‘–π‘˜π‘Šπ΅π‘˜(𝑑)∞ π‘π‘–π‘˜π‘Šπ΅π‘˜(𝑑)∞ ADJ iajs-3048 128 4 π‘˜=𝑁+1 π‘˜=𝑁+1 ADP iajs-3048 128 5 or or CCONJ iajs-3048 128 6 π‘₯𝑖(𝑑 π‘₯𝑖(𝑑 NUM iajs-3048 128 7 ) ) PUNCT iajs-3048 128 8 = = SYM iajs-3048 128 9 π‘₯𝑖𝑁(𝑑 π‘₯𝑖𝑁(𝑑 PROPN iajs-3048 128 10 ) ) PUNCT iajs-3048 128 11 + + CCONJ iajs-3048 128 12 𝑒𝑖(𝑑 𝑒𝑖(𝑑 NOUN iajs-3048 128 13 ) ) PUNCT iajs-3048 128 14 … … PUNCT iajs-3048 128 15 ( ( PUNCT iajs-3048 128 16 14 14 NUM iajs-3048 128 17 ) ) PUNCT iajs-3048 128 18 the the DET iajs-3048 128 19 coefficients coefficient NOUN iajs-3048 128 20 in in ADP iajs-3048 128 21 equation equation NOUN iajs-3048 128 22 ( ( PUNCT iajs-3048 128 23 14 14 NUM iajs-3048 128 24 ) ) PUNCT iajs-3048 128 25 is be AUX iajs-3048 128 26 limited limit VERB iajs-3048 128 27 by by ADP iajs-3048 128 28 [ [ X iajs-3048 128 29 1 1 NUM iajs-3048 128 30 ] ] PUNCT iajs-3048 128 31 such such ADJ iajs-3048 128 32 that that SCONJ iajs-3048 128 33 the the DET iajs-3048 128 34 residual residual ADJ iajs-3048 128 35 ‖𝑒(𝑑)β€– ‖𝑒(𝑑)β€– NOUN iajs-3048 128 36 is be AUX iajs-3048 128 37 less less ADJ iajs-3048 128 38 than than ADP iajs-3048 128 39 some some DET iajs-3048 128 40 πœ€ πœ€ NOUN iajs-3048 128 41 where where SCONJ iajs-3048 128 42 𝑒(𝑑 𝑒(𝑑 NOUN iajs-3048 128 43 ) ) PUNCT iajs-3048 128 44 = = PUNCT iajs-3048 128 45 π‘šπ‘Žπ‘₯ π‘šπ‘Žπ‘₯ ADJ iajs-3048 128 46 { { PUNCT iajs-3048 128 47 𝑒1(𝑑 𝑒1(𝑑 NOUN iajs-3048 128 48 ) ) PUNCT iajs-3048 128 49 , , PUNCT iajs-3048 128 50 𝑒2(𝑑 𝑒2(𝑑 PROPN iajs-3048 128 51 ) ) PUNCT iajs-3048 128 52 , , PUNCT iajs-3048 128 53 … … PUNCT iajs-3048 128 54 , , PUNCT iajs-3048 128 55 𝑒𝑁(𝑑 𝑒𝑁(𝑑 NUM iajs-3048 128 56 ) ) PUNCT iajs-3048 128 57 } } PUNCT iajs-3048 128 58 , , PUNCT iajs-3048 128 59 then then ADV iajs-3048 128 60 , , PUNCT iajs-3048 128 61 using use VERB iajs-3048 128 62 the the DET iajs-3048 128 63 convergence convergence NOUN iajs-3048 128 64 test test NOUN iajs-3048 128 65 for for ADP iajs-3048 128 66 the the DET iajs-3048 128 67 technique technique NOUN iajs-3048 128 68 to to ADP iajs-3048 128 69 the the DET iajs-3048 128 70 variable variable NOUN iajs-3048 128 71 x x PUNCT iajs-3048 128 72 in in ADP iajs-3048 128 73 terms term NOUN iajs-3048 128 74 of of ADP iajs-3048 128 75 n n PRON iajs-3048 128 76 proposed propose VERB iajs-3048 128 77 𝐿2norm 𝐿2norm NOUN iajs-3048 128 78 of of ADP iajs-3048 128 79 π‘₯𝑖 π‘₯𝑖 PRON iajs-3048 128 80 , , PUNCT iajs-3048 128 81 i=1,2, i=1,2, NOUN iajs-3048 128 82 … … X iajs-3048 128 83 ,n ,n NOUN iajs-3048 129 1 [ [ X iajs-3048 129 2 ∫ ∫ X iajs-3048 129 3 ( ( PUNCT iajs-3048 129 4 π‘₯(𝑑 π‘₯(𝑑 PROPN iajs-3048 129 5 ) ) PUNCT iajs-3048 129 6 βˆ’ βˆ’ PUNCT iajs-3048 129 7 π‘₯𝑖𝑁(𝑑))2𝑑𝑑 π‘₯𝑖𝑁(𝑑))2𝑑𝑑 PUNCT iajs-3048 129 8 1 1 NUM iajs-3048 129 9 0 0 NUM iajs-3048 129 10 ] ] SYM iajs-3048 129 11 1 1 NUM iajs-3048 129 12 2 2 NUM iajs-3048 129 13 < < X iajs-3048 129 14 πœ€π‘– πœ€π‘– NOUN iajs-3048 129 15 , , PUNCT iajs-3048 129 16 𝑖=1,2, 𝑖=1,2, PROPN iajs-3048 129 17 … … SYM iajs-3048 129 18 ,𝑛 ,𝑛 PUNCT iajs-3048 129 19 πœ€ πœ€ X iajs-3048 129 20 = = SYM iajs-3048 129 21 π‘šπ‘Žπ‘₯{πœ€1(𝑑 π‘šπ‘Žπ‘₯{πœ€1(𝑑 PROPN iajs-3048 129 22 ) ) PUNCT iajs-3048 129 23 , , PUNCT iajs-3048 129 24 πœ€2(𝑑 πœ€2(𝑑 NOUN iajs-3048 129 25 ) ) PUNCT iajs-3048 129 26 , , PUNCT iajs-3048 129 27 … … PUNCT iajs-3048 129 28 , , PUNCT iajs-3048 129 29 πœ€π‘›(𝑑 πœ€π‘›(𝑑 NOUN iajs-3048 129 30 ) ) PUNCT iajs-3048 129 31 } } PUNCT iajs-3048 129 32 , , PUNCT iajs-3048 129 33 [ [ X iajs-3048 129 34 ∫ ∫ X iajs-3048 129 35 ( ( PUNCT iajs-3048 129 36 π‘₯(𝑑 π‘₯(𝑑 PROPN iajs-3048 129 37 ) ) PUNCT iajs-3048 129 38 βˆ’ βˆ’ NOUN iajs-3048 130 1 π‘₯𝑁(𝑑))2𝑑𝑑 π‘₯𝑁(𝑑))2𝑑𝑑 PROPN iajs-3048 130 2 1 1 NUM iajs-3048 130 3 0 0 NUM iajs-3048 130 4 ] ] SYM iajs-3048 130 5 1 1 NUM iajs-3048 130 6 2 2 NUM iajs-3048 130 7 < < X iajs-3048 130 8 πœ€ πœ€ PROPN iajs-3048 130 9 , , PUNCT iajs-3048 130 10 using use VERB iajs-3048 130 11 the the DET iajs-3048 130 12 boubaker boubaker NOUN iajs-3048 130 13 wavelet wavelet NOUN iajs-3048 130 14 polynomials polynomial NOUN iajs-3048 130 15 for for ADP iajs-3048 130 16 approximating approximate VERB iajs-3048 130 17 x x PROPN iajs-3048 130 18 variables variable NOUN iajs-3048 130 19 , , PUNCT iajs-3048 130 20 we we PRON iajs-3048 130 21 get get VERB iajs-3048 130 22 [ [ X iajs-3048 130 23 ∫ ∫ X iajs-3048 130 24 ( ( PUNCT iajs-3048 130 25 βˆ‘ βˆ‘ ADP iajs-3048 130 26 π‘π‘–π‘Šπ΅π‘–(𝑑 π‘π‘–π‘Šπ΅π‘–(𝑑 NUM iajs-3048 130 27 ) ) PUNCT iajs-3048 130 28 𝑁+𝑀 𝑁+𝑀 X iajs-3048 130 29 𝑖=0 𝑖=0 PROPN iajs-3048 130 30 βˆ’ βˆ’ PROPN iajs-3048 130 31 βˆ‘ βˆ‘ PUNCT iajs-3048 130 32 π‘π‘–π‘Šπ΅π‘–(𝑑 π‘π‘–π‘Šπ΅π‘–(𝑑 NUM iajs-3048 130 33 ) ) PUNCT iajs-3048 131 1 𝑁 𝑁 PROPN iajs-3048 131 2 𝑖=0 𝑖=0 PROPN iajs-3048 131 3 ) ) PUNCT iajs-3048 131 4 2𝑑𝑑 2𝑑𝑑 NOUN iajs-3048 131 5 1 1 NUM iajs-3048 131 6 0 0 NUM iajs-3048 131 7 ] ] SYM iajs-3048 131 8 1 1 NUM iajs-3048 131 9 2 2 NUM iajs-3048 131 10 < < X iajs-3048 131 11 πœ€ πœ€ X iajs-3048 131 12 = = PRON iajs-3048 132 1 [ [ X iajs-3048 132 2 ∫ ∫ X iajs-3048 132 3 ( ( PUNCT iajs-3048 132 4 βˆ‘ βˆ‘ ADP iajs-3048 132 5 π‘π‘–π‘Šπ΅π‘–(𝑑 π‘π‘–π‘Šπ΅π‘–(𝑑 NUM iajs-3048 132 6 ) ) PUNCT iajs-3048 132 7 𝑁+𝑀 𝑁+𝑀 NOUN iajs-3048 132 8 𝑖=𝑁+1 𝑖=𝑁+1 X iajs-3048 132 9 ) ) PUNCT iajs-3048 133 1 2𝑑𝑑 2𝑑𝑑 NOUN iajs-3048 133 2 1 1 NUM iajs-3048 133 3 0 0 NUM iajs-3048 133 4 ] ] SYM iajs-3048 133 5 1 1 NUM iajs-3048 133 6 2 2 NUM iajs-3048 133 7 < < X iajs-3048 133 8 πœ€ πœ€ X iajs-3048 133 9 = = PRON iajs-3048 134 1 [ [ X iajs-3048 134 2 ∫ ∫ X iajs-3048 134 3 ( ( PUNCT iajs-3048 134 4 βˆ‘ βˆ‘ ADP iajs-3048 134 5 π‘π‘–π‘Šπ΅π‘–(𝑑 π‘π‘–π‘Šπ΅π‘–(𝑑 NUM iajs-3048 134 6 ) ) PUNCT iajs-3048 134 7 𝑁+𝑀 𝑁+𝑀 NOUN iajs-3048 134 8 𝑖=𝑁+1 𝑖=𝑁+1 X iajs-3048 134 9 ) ) PUNCT iajs-3048 134 10 ( ( PUNCT iajs-3048 134 11 βˆ‘ βˆ‘ ADP iajs-3048 134 12 π‘π‘–π‘Šπ΅π‘–(𝑑 π‘π‘–π‘Šπ΅π‘–(𝑑 NUM iajs-3048 134 13 ) ) PUNCT iajs-3048 134 14 𝑁+𝑀 𝑁+𝑀 NOUN iajs-3048 134 15 𝑖=𝑁+1 𝑖=𝑁+1 X iajs-3048 134 16 ) ) PUNCT iajs-3048 134 17 𝑑𝑑 𝑑𝑑 ADP iajs-3048 134 18 1 1 NUM iajs-3048 134 19 0 0 NUM iajs-3048 134 20 ] ] SYM iajs-3048 134 21 1 1 NUM iajs-3048 134 22 2 2 NUM iajs-3048 134 23 < < X iajs-3048 134 24 πœ€ πœ€ X iajs-3048 134 25 = = SYM iajs-3048 134 26 βˆ‘ βˆ‘ PROPN iajs-3048 134 27 βˆ‘ βˆ‘ PUNCT iajs-3048 134 28 𝑐𝑖𝑐𝑗 𝑐𝑖𝑐𝑗 PROPN iajs-3048 134 29 𝑁+𝑀 𝑁+𝑀 PROPN iajs-3048 134 30 𝑗=𝑁+1 𝑗=𝑁+1 X iajs-3048 134 31 ∫ ∫ PROPN iajs-3048 135 1 π‘Šπ΅π‘–(𝑑 π‘Šπ΅π‘–(𝑑 NUM iajs-3048 135 2 ) ) PUNCT iajs-3048 135 3 1 1 NUM iajs-3048 135 4 0 0 NUM iajs-3048 135 5 π‘Šπ΅π‘—(𝑑 π‘Šπ΅π‘—(𝑑 NOUN iajs-3048 135 6 ) ) PUNCT iajs-3048 135 7 𝑑𝑑 𝑑𝑑 ADP iajs-3048 135 8 < < X iajs-3048 135 9 πœ€ πœ€ X iajs-3048 135 10 𝑁+𝑀 𝑁+𝑀 X iajs-3048 135 11 𝑖=𝑁+1 𝑖=𝑁+1 X iajs-3048 135 12 hence hence ADV iajs-3048 135 13 , , PUNCT iajs-3048 135 14 the the DET iajs-3048 135 15 wavelet wavelet NOUN iajs-3048 135 16 boubaker boubaker NOUN iajs-3048 135 17 polynomials polynomial NOUN iajs-3048 135 18 can can AUX iajs-3048 135 19 be be AUX iajs-3048 135 20 reduced reduce VERB iajs-3048 135 21 to to ADP iajs-3048 135 22 the the DET iajs-3048 135 23 form form NOUN iajs-3048 135 24 ihjpas ihjpa VERB iajs-3048 135 25 . . PUNCT iajs-3048 136 1 36 36 NUM iajs-3048 136 2 ( ( PUNCT iajs-3048 136 3 3 3 NUM iajs-3048 136 4 ) ) PUNCT iajs-3048 136 5 2023 2023 NUM iajs-3048 136 6 435 435 NUM iajs-3048 136 7 βˆ‘ βˆ‘ SYM iajs-3048 136 8 𝑐𝑖 𝑐𝑖 NOUN iajs-3048 136 9 2 2 NUM iajs-3048 136 10 𝑁+𝑀 𝑁+𝑀 NOUN iajs-3048 136 11 𝑖=𝑁+1 𝑖=𝑁+1 X iajs-3048 136 12 < < X iajs-3048 136 13 πœ€ πœ€ X iajs-3048 136 14 when when SCONJ iajs-3048 136 15 the the DET iajs-3048 136 16 squares square NOUN iajs-3048 136 17 of of ADP iajs-3048 136 18 the the DET iajs-3048 136 19 remaining remain VERB iajs-3048 136 20 coefficients coefficient NOUN iajs-3048 136 21 becomes becomes AUX iajs-3048 136 22 neglected neglect VERB iajs-3048 136 23 , , PUNCT iajs-3048 136 24 a a DET iajs-3048 136 25 favorable favorable ADJ iajs-3048 136 26 approximation approximation NOUN iajs-3048 136 27 to to ADP iajs-3048 136 28 the the DET iajs-3048 136 29 solution solution NOUN iajs-3048 136 30 is be AUX iajs-3048 136 31 achieved achieve VERB iajs-3048 136 32 . . PUNCT iajs-3048 137 1 6 6 X iajs-3048 137 2 . . X iajs-3048 137 3 conclusion conclusion NOUN iajs-3048 137 4 in in ADP iajs-3048 137 5 this this DET iajs-3048 137 6 paper paper NOUN iajs-3048 137 7 , , PUNCT iajs-3048 137 8 a a DET iajs-3048 137 9 direct direct ADJ iajs-3048 137 10 method method NOUN iajs-3048 137 11 with with ADP iajs-3048 137 12 wavelet wavelet NOUN iajs-3048 137 13 boubaker boubaker NOUN iajs-3048 137 14 polynomials polynomial NOUN iajs-3048 137 15 was be AUX iajs-3048 137 16 achieved achieve VERB iajs-3048 137 17 as as ADP iajs-3048 137 18 an an DET iajs-3048 137 19 evaluation evaluation NOUN iajs-3048 137 20 solution solution NOUN iajs-3048 137 21 for for ADP iajs-3048 137 22 reducing reduce VERB iajs-3048 137 23 the the DET iajs-3048 137 24 variational variational ADJ iajs-3048 137 25 problems problem NOUN iajs-3048 137 26 into into ADP iajs-3048 137 27 quadratic quadratic ADJ iajs-3048 137 28 programming programming NOUN iajs-3048 137 29 problems problem NOUN iajs-3048 137 30 . . PUNCT iajs-3048 138 1 only only ADV iajs-3048 138 2 a a DET iajs-3048 138 3 few few ADJ iajs-3048 138 4 terms term NOUN iajs-3048 138 5 of of ADP iajs-3048 138 6 wavelet wavelet NOUN iajs-3048 138 7 boubaker boubaker NOUN iajs-3048 138 8 polynomials polynomial NOUN iajs-3048 138 9 were be AUX iajs-3048 138 10 needed need VERB iajs-3048 138 11 to to PART iajs-3048 138 12 obtain obtain VERB iajs-3048 138 13 an an DET iajs-3048 138 14 accurate accurate ADJ iajs-3048 138 15 solution solution NOUN iajs-3048 138 16 . . PUNCT iajs-3048 139 1 the the DET iajs-3048 139 2 numerical numerical ADJ iajs-3048 139 3 results result NOUN iajs-3048 139 4 were be AUX iajs-3048 139 5 compared compare VERB iajs-3048 139 6 with with ADP iajs-3048 139 7 the the DET iajs-3048 139 8 exact exact ADJ iajs-3048 139 9 solution solution NOUN iajs-3048 139 10 , , PUNCT iajs-3048 139 11 which which PRON iajs-3048 139 12 proved prove VERB iajs-3048 139 13 the the DET iajs-3048 139 14 capability capability NOUN iajs-3048 139 15 and and CCONJ iajs-3048 139 16 validity validity NOUN iajs-3048 139 17 of of ADP iajs-3048 139 18 such such ADJ iajs-3048 139 19 problems problem NOUN iajs-3048 139 20 with with ADP iajs-3048 139 21 easy easy ADJ iajs-3048 139 22 steps step NOUN iajs-3048 139 23 . . PUNCT iajs-3048 140 1 matlab matlab PROPN iajs-3048 140 2 plotting plotting NOUN iajs-3048 140 3 was be AUX iajs-3048 140 4 also also ADV iajs-3048 140 5 used use VERB iajs-3048 140 6 to to PART iajs-3048 140 7 demonstrate demonstrate VERB iajs-3048 140 8 the the DET iajs-3048 140 9 results result NOUN iajs-3048 140 10 . . PUNCT iajs-3048 141 1 references reference NOUN iajs-3048 141 2 1 1 NUM iajs-3048 141 3 . . PUNCT iajs-3048 142 1 eman eman PROPN iajs-3048 142 2 , , PUNCT iajs-3048 142 3 h. h. PROPN iajs-3048 142 4 ; ; PUNCT iajs-3048 142 5 najeeb najeeb PROPN iajs-3048 142 6 , , PUNCT iajs-3048 142 7 s. s. PROPN iajs-3048 142 8 ; ; PUNCT iajs-3048 142 9 samaa samaa PROPN iajs-3048 142 10 , , PUNCT iajs-3048 142 11 f. f. PROPN iajs-3048 142 12 , , PUNCT iajs-3048 142 13 boubaker boubaker NOUN iajs-3048 142 14 wavelets wavelet NOUN iajs-3048 142 15 functions function NOUN iajs-3048 142 16 : : PUNCT iajs-3048 142 17 properties property NOUN iajs-3048 142 18 and and CCONJ iajs-3048 142 19 applications application NOUN iajs-3048 142 20 , , PUNCT iajs-3048 142 21 baghdad baghdad PROPN iajs-3048 142 22 science science PROPN iajs-3048 142 23 journal journal PROPN iajs-3048 142 24 , , PUNCT iajs-3048 142 25 2021 2021 NUM iajs-3048 142 26 , , PUNCT iajs-3048 142 27 18 18 NUM iajs-3048 142 28 , , PUNCT iajs-3048 142 29 4 4 NUM iajs-3048 142 30 , , PUNCT iajs-3048 142 31 1226 1226 NUM iajs-3048 142 32 - - SYM iajs-3048 142 33 1233 1233 NUM iajs-3048 142 34 . . PUNCT iajs-3048 143 1 2 2 X iajs-3048 143 2 . . X iajs-3048 143 3 mohammadi mohammadi NOUN iajs-3048 143 4 , , PUNCT iajs-3048 143 5 r. r. PROPN iajs-3048 143 6 ; ; PUNCT iajs-3048 143 7 alavi alavi NOUN iajs-3048 143 8 , , PUNCT iajs-3048 143 9 a. a. NOUN iajs-3048 143 10 , , PUNCT iajs-3048 143 11 quadratic quadratic ADJ iajs-3048 143 12 spline spline NOUN iajs-3048 143 13 solution solution NOUN iajs-3048 143 14 of of ADP iajs-3048 143 15 calculus calculus NOUN iajs-3048 143 16 of of ADP iajs-3048 143 17 variation variation NOUN iajs-3048 143 18 problems problem NOUN iajs-3048 143 19 , , PUNCT iajs-3048 143 20 j. j. PROPN iajs-3048 143 21 app app PROPN iajs-3048 143 22 . . PUNCT iajs-3048 144 1 eng eng PROPN iajs-3048 144 2 . . PROPN iajs-3048 144 3 math math PROPN iajs-3048 144 4 . . PUNCT iajs-3048 144 5 , , PUNCT iajs-3048 144 6 2015 2015 NUM iajs-3048 144 7 , , PUNCT iajs-3048 144 8 5 5 NUM iajs-3048 144 9 , , PUNCT iajs-3048 144 10 2 2 NUM iajs-3048 144 11 , , PUNCT iajs-3048 144 12 276 276 NUM iajs-3048 144 13 - - SYM iajs-3048 144 14 285 285 NUM iajs-3048 144 15 . . PUNCT iajs-3048 145 1 3 3 X iajs-3048 145 2 . . X iajs-3048 145 3 razzaghi razzaghi PROPN iajs-3048 145 4 , , PUNCT iajs-3048 145 5 m. m. NOUN iajs-3048 145 6 ; ; PUNCT iajs-3048 145 7 yousefi yousefi PROPN iajs-3048 145 8 , , PUNCT iajs-3048 145 9 s. s. PROPN iajs-3048 145 10 , , PUNCT iajs-3048 145 11 legendre legendre PROPN iajs-3048 145 12 wavelets wavelet VERB iajs-3048 145 13 direct direct ADJ iajs-3048 145 14 method method NOUN iajs-3048 145 15 for for ADP iajs-3048 145 16 variational variational ADJ iajs-3048 145 17 problems problem NOUN iajs-3048 145 18 , , PUNCT iajs-3048 145 19 elsevier elsevier NOUN iajs-3048 145 20 , , PUNCT iajs-3048 145 21 mathematics mathematic NOUN iajs-3048 145 22 and and CCONJ iajs-3048 145 23 computers computer NOUN iajs-3048 145 24 in in ADP iajs-3048 145 25 simulation simulation NOUN iajs-3048 145 26 , , PUNCT iajs-3048 145 27 2000 2000 NUM iajs-3048 145 28 , , PUNCT iajs-3048 145 29 53 53 NUM iajs-3048 145 30 , , PUNCT iajs-3048 145 31 185 185 NUM iajs-3048 145 32 - - SYM iajs-3048 145 33 192 192 NUM iajs-3048 145 34 . . PUNCT iajs-3048 146 1 4 4 NUM iajs-3048 146 2 . . X iajs-3048 147 1 ravindra ravindra PROPN iajs-3048 147 2 , , PUNCT iajs-3048 147 3 m. m. NOUN iajs-3048 147 4 , , PUNCT iajs-3048 147 5 direct direct ADJ iajs-3048 147 6 method method NOUN iajs-3048 147 7 for for ADP iajs-3048 147 8 solving solve VERB iajs-3048 147 9 variational variational ADJ iajs-3048 147 10 problems problem NOUN iajs-3048 147 11 using use VERB iajs-3048 147 12 haar haar PROPN iajs-3048 147 13 wavelet wavelet PROPN iajs-3048 147 14 , , PUNCT iajs-3048 147 15 international international ADJ iajs-3048 147 16 journal journal PROPN iajs-3048 147 17 research research NOUN iajs-3048 147 18 & & CCONJ iajs-3048 147 19 advance advance NOUN iajs-3048 147 20 in in ADP iajs-3048 147 21 engineer engineer NOUN iajs-3048 147 22 ( ( PUNCT iajs-3048 147 23 ijmrae ijmrae PROPN iajs-3048 147 24 ) ) PUNCT iajs-3048 147 25 , , PUNCT iajs-3048 147 26 2010 2010 NUM iajs-3048 147 27 , , PUNCT iajs-3048 147 28 2 2 NUM iajs-3048 147 29 , , PUNCT iajs-3048 147 30 iii iii NOUN iajs-3048 147 31 , , PUNCT iajs-3048 147 32 85 85 NUM iajs-3048 147 33 - - SYM iajs-3048 147 34 90 90 NUM iajs-3048 147 35 . . PUNCT iajs-3048 147 36 5 5 NUM iajs-3048 147 37 . . X iajs-3048 147 38 suha suha PROPN iajs-3048 147 39 , , PUNCT iajs-3048 147 40 najeeb najeeb PROPN iajs-3048 147 41 ; ; PUNCT iajs-3048 147 42 mohammed mohammed PROPN iajs-3048 147 43 , , PUNCT iajs-3048 147 44 r. r. PROPN iajs-3048 147 45 ; ; PUNCT iajs-3048 147 46 ahmed ahmed PROPN iajs-3048 147 47 r. r. PROPN iajs-3048 147 48 , , PUNCT iajs-3048 147 49 direct direct PROPN iajs-3048 147 50 restarted restart VERB iajs-3048 147 51 pell pell NOUN iajs-3048 147 52 algorithm algorithm NOUN iajs-3048 147 53 for for ADP iajs-3048 147 54 solving solve VERB iajs-3048 147 55 calculus calculus NOUN iajs-3048 147 56 of of ADP iajs-3048 147 57 variation variation NOUN iajs-3048 147 58 problems problem NOUN iajs-3048 147 59 , , PUNCT iajs-3048 147 60 journal journal NOUN iajs-3048 147 61 of of ADP iajs-3048 147 62 al al PROPN iajs-3048 147 63 - - PUNCT iajs-3048 147 64 qadisiyah qadisiyah NOUN iajs-3048 147 65 for for ADP iajs-3048 147 66 computer computer NOUN iajs-3048 147 67 science science NOUN iajs-3048 147 68 and and CCONJ iajs-3048 147 69 mathematics mathematic NOUN iajs-3048 147 70 , , PUNCT iajs-3048 147 71 2021 2021 NUM iajs-3048 147 72 , , PUNCT iajs-3048 147 73 13 13 NUM iajs-3048 147 74 , , PUNCT iajs-3048 147 75 2 2 NUM iajs-3048 147 76 , , PUNCT iajs-3048 147 77 169 169 NUM iajs-3048 147 78 - - SYM iajs-3048 147 79 180 180 NUM iajs-3048 147 80 . . 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