id sid tid token lemma pos iajs-2002 1 1 microsoft microsoft PROPN iajs-2002 1 2 word word NOUN iajs-2002 1 3 126 126 NUM iajs-2002 1 4 - - SYM iajs-2002 1 5 134 134 NUM iajs-2002 1 6 mathematics mathematic NOUN iajs-2002 1 7 | | ADV iajs-2002 1 8 126 126 NUM iajs-2002 1 9 ibn ibn PROPN iajs-2002 1 10 al al PROPN iajs-2002 1 11 - - PUNCT iajs-2002 1 12 haitham haitham PROPN iajs-2002 1 13 jour jour X iajs-2002 1 14 . . PROPN iajs-2002 2 1 for for ADP iajs-2002 2 2 pure pure ADJ iajs-2002 2 3 & & CCONJ iajs-2002 2 4 appl appl PROPN iajs-2002 2 5 . . PUNCT iajs-2002 3 1 sci sci PROPN iajs-2002 3 2 . . PROPN iajs-2002 3 3 ihjpas ihjpa VERB iajs-2002 3 4 https://doi.org/10.30526/31.3.2002 https://doi.org/10.30526/31.3.2002 PART iajs-2002 3 5 vol vol NOUN iajs-2002 3 6 . . PUNCT iajs-2002 3 7 31 31 NUM iajs-2002 3 8 ( ( PUNCT iajs-2002 3 9 3 3 NUM iajs-2002 3 10 ) ) PUNCT iajs-2002 3 11 2018 2018 NUM iajs-2002 3 12 the the DET iajs-2002 3 13 approximation approximation NOUN iajs-2002 3 14 solution solution NOUN iajs-2002 3 15 of of ADP iajs-2002 3 16 a a DET iajs-2002 3 17 nonlinear nonlinear ADJ iajs-2002 3 18 parabolic parabolic ADJ iajs-2002 3 19 boundary boundary ADJ iajs-2002 3 20 value value NOUN iajs-2002 3 21 problem problem NOUN iajs-2002 3 22 via via ADP iajs-2002 3 23 galerkin galerkin ADJ iajs-2002 3 24 finite finite PROPN iajs-2002 3 25 elements element NOUN iajs-2002 3 26 method method VERB iajs-2002 3 27 with with ADP iajs-2002 3 28 crank crank NOUN iajs-2002 3 29 - - PUNCT iajs-2002 3 30 nicolson nicolson PROPN iajs-2002 3 31 jamil jamil PROPN iajs-2002 3 32 amir amir PROPN iajs-2002 3 33 al al PROPN iajs-2002 3 34 - - PUNCT iajs-2002 3 35 hawasy hawasy PROPN iajs-2002 3 36 jhawassy17@uomustansiriyah.edu.iq jhawassy17@uomustansiriyah.edu.iq PROPN iajs-2002 3 37 marwa marwa PROPN iajs-2002 3 38 ahmed ahmed PROPN iajs-2002 3 39 jawad jawad PROPN iajs-2002 3 40 hawasy20@yahoo.com hawasy20@yahoo.com PROPN iajs-2002 3 41 department department PROPN iajs-2002 3 42 of of ADP iajs-2002 3 43 mathematics mathematics PROPN iajs-2002 3 44 , , PUNCT iajs-2002 3 45 college college NOUN iajs-2002 3 46 of of ADP iajs-2002 3 47 science science NOUN iajs-2002 3 48 , , PUNCT iajs-2002 3 49 mustansiriyah mustansiriyah NOUN iajs-2002 3 50 university university NOUN iajs-2002 3 51 , , PUNCT iajs-2002 3 52 bahgdad bahgdad NOUN iajs-2002 3 53 , , PUNCT iajs-2002 3 54 iraq iraq PROPN iajs-2002 3 55 . . PUNCT iajs-2002 4 1 article article NOUN iajs-2002 4 2 history history NOUN iajs-2002 4 3 : : PUNCT iajs-2002 4 4 received receive VERB iajs-2002 4 5 16 16 NUM iajs-2002 4 6 july july PROPN iajs-2002 4 7 2018 2018 NUM iajs-2002 4 8 , , PUNCT iajs-2002 4 9 accepted accept VERB iajs-2002 4 10 3 3 NUM iajs-2002 4 11 september september PROPN iajs-2002 4 12 2018 2018 NUM iajs-2002 4 13 , , PUNCT iajs-2002 4 14 published publish VERB iajs-2002 4 15 december december PROPN iajs-2002 4 16 2018 2018 NUM iajs-2002 4 17 abstract abstract ADV iajs-2002 4 18 this this DET iajs-2002 4 19 paper paper NOUN iajs-2002 4 20 deals deal NOUN iajs-2002 4 21 with with ADP iajs-2002 4 22 finding find VERB iajs-2002 4 23 the the DET iajs-2002 4 24 approximation approximation NOUN iajs-2002 4 25 solution solution NOUN iajs-2002 4 26 of of ADP iajs-2002 4 27 a a DET iajs-2002 4 28 nonlinear nonlinear ADJ iajs-2002 4 29 parabolic parabolic ADJ iajs-2002 4 30 boundary boundary ADJ iajs-2002 4 31 value value NOUN iajs-2002 4 32 problem problem NOUN iajs-2002 4 33 ( ( PUNCT iajs-2002 4 34 nlpbvp nlpbvp NOUN iajs-2002 4 35 ) ) PUNCT iajs-2002 4 36 by by ADP iajs-2002 4 37 using use VERB iajs-2002 4 38 the the DET iajs-2002 4 39 galekin galekin PROPN iajs-2002 4 40 finite finite PROPN iajs-2002 4 41 element element PROPN iajs-2002 4 42 method method NOUN iajs-2002 4 43 ( ( PUNCT iajs-2002 4 44 gfem gfem PROPN iajs-2002 4 45 ) ) PUNCT iajs-2002 4 46 in in ADP iajs-2002 4 47 space space NOUN iajs-2002 4 48 and and CCONJ iajs-2002 4 49 crank crank PROPN iajs-2002 4 50 nicolson nicolson PROPN iajs-2002 4 51 ( ( PUNCT iajs-2002 4 52 cn cn PROPN iajs-2002 4 53 ) ) PUNCT iajs-2002 4 54 scheme scheme NOUN iajs-2002 4 55 in in ADP iajs-2002 4 56 time time NOUN iajs-2002 4 57 , , PUNCT iajs-2002 4 58 the the DET iajs-2002 4 59 problem problem NOUN iajs-2002 4 60 then then ADV iajs-2002 4 61 reduces reduce VERB iajs-2002 4 62 to to PART iajs-2002 4 63 solve solve VERB iajs-2002 4 64 a a DET iajs-2002 4 65 galerkin galerkin ADJ iajs-2002 4 66 nonlinear nonlinear NOUN iajs-2002 4 67 algebraic algebraic ADJ iajs-2002 4 68 system(gnlas system(gnlas PROPN iajs-2002 4 69 ) ) PUNCT iajs-2002 4 70 . . PUNCT iajs-2002 5 1 the the DET iajs-2002 5 2 predictor predictor NOUN iajs-2002 5 3 and and CCONJ iajs-2002 5 4 the the DET iajs-2002 5 5 corrector corrector NOUN iajs-2002 5 6 technique technique NOUN iajs-2002 5 7 ( ( PUNCT iajs-2002 5 8 pct pct NOUN iajs-2002 5 9 ) ) PUNCT iajs-2002 5 10 is be AUX iajs-2002 5 11 applied apply VERB iajs-2002 5 12 here here ADV iajs-2002 5 13 to to PART iajs-2002 5 14 solve solve VERB iajs-2002 5 15 the the DET iajs-2002 5 16 gnlas gnla NOUN iajs-2002 5 17 , , PUNCT iajs-2002 5 18 by by ADP iajs-2002 5 19 transforms transform VERB iajs-2002 5 20 it it PRON iajs-2002 5 21 to to ADP iajs-2002 5 22 a a DET iajs-2002 5 23 galerkin galerkin ADJ iajs-2002 5 24 linear linear NOUN iajs-2002 5 25 algebraic algebraic ADJ iajs-2002 5 26 system system NOUN iajs-2002 5 27 ( ( PUNCT iajs-2002 5 28 glas glas PROPN iajs-2002 5 29 ) ) PUNCT iajs-2002 5 30 . . PUNCT iajs-2002 6 1 this this DET iajs-2002 6 2 glas glas PROPN iajs-2002 6 3 is be AUX iajs-2002 6 4 solved solve VERB iajs-2002 6 5 once once ADV iajs-2002 6 6 using use VERB iajs-2002 6 7 the the DET iajs-2002 6 8 cholesky cholesky ADJ iajs-2002 6 9 method method NOUN iajs-2002 6 10 ( ( PUNCT iajs-2002 6 11 chm chm NOUN iajs-2002 6 12 ) ) PUNCT iajs-2002 6 13 as as SCONJ iajs-2002 6 14 it it PRON iajs-2002 6 15 appears appear VERB iajs-2002 6 16 in in ADP iajs-2002 6 17 the the DET iajs-2002 6 18 matlab matlab PROPN iajs-2002 6 19 package package NOUN iajs-2002 6 20 and and CCONJ iajs-2002 6 21 once once ADV iajs-2002 6 22 again again ADV iajs-2002 6 23 using use VERB iajs-2002 6 24 the the DET iajs-2002 6 25 cholesky cholesky ADJ iajs-2002 6 26 reduction reduction NOUN iajs-2002 6 27 order order NOUN iajs-2002 6 28 technique technique NOUN iajs-2002 6 29 ( ( PUNCT iajs-2002 6 30 chrot chrot NOUN iajs-2002 6 31 ) ) PUNCT iajs-2002 6 32 which which PRON iajs-2002 6 33 we we PRON iajs-2002 6 34 employ employ VERB iajs-2002 6 35 it it PRON iajs-2002 6 36 here here ADV iajs-2002 6 37 to to PART iajs-2002 6 38 save save VERB iajs-2002 6 39 a a DET iajs-2002 6 40 massive massive ADJ iajs-2002 6 41 time time NOUN iajs-2002 6 42 . . PUNCT iajs-2002 7 1 the the DET iajs-2002 7 2 results result NOUN iajs-2002 7 3 , , PUNCT iajs-2002 7 4 for for ADP iajs-2002 7 5 chrot chrot NOUN iajs-2002 7 6 are be AUX iajs-2002 7 7 given give VERB iajs-2002 7 8 by by ADP iajs-2002 7 9 tables table NOUN iajs-2002 7 10 and and CCONJ iajs-2002 7 11 figures figure NOUN iajs-2002 7 12 and and CCONJ iajs-2002 7 13 show show VERB iajs-2002 7 14 the the DET iajs-2002 7 15 efficiency efficiency NOUN iajs-2002 7 16 of of ADP iajs-2002 7 17 this this DET iajs-2002 7 18 method method NOUN iajs-2002 7 19 , , PUNCT iajs-2002 7 20 from from ADP iajs-2002 7 21 other other ADJ iajs-2002 7 22 sides side NOUN iajs-2002 7 23 we we PRON iajs-2002 7 24 conclude conclude VERB iajs-2002 7 25 that that SCONJ iajs-2002 7 26 the the DET iajs-2002 7 27 both both DET iajs-2002 7 28 methods method NOUN iajs-2002 7 29 are be AUX iajs-2002 7 30 given give VERB iajs-2002 7 31 the the DET iajs-2002 7 32 same same ADJ iajs-2002 7 33 results result NOUN iajs-2002 7 34 , , PUNCT iajs-2002 7 35 but but CCONJ iajs-2002 7 36 the the DET iajs-2002 7 37 chrot chrot NOUN iajs-2002 7 38 is be AUX iajs-2002 7 39 very very ADV iajs-2002 7 40 fast fast ADJ iajs-2002 7 41 than than ADP iajs-2002 7 42 the the DET iajs-2002 7 43 chm chm NOUN iajs-2002 7 44 . . PUNCT iajs-2002 8 1 keywords keyword NOUN iajs-2002 8 2 : : PUNCT iajs-2002 8 3 nonlinear nonlinear ADJ iajs-2002 8 4 parabolic parabolic ADJ iajs-2002 8 5 boundary boundary ADJ iajs-2002 8 6 value value NOUN iajs-2002 8 7 problem problem NOUN iajs-2002 8 8 , , PUNCT iajs-2002 8 9 galerkin galerkin PROPN iajs-2002 8 10 finite finite PROPN iajs-2002 8 11 element element NOUN iajs-2002 8 12 methods method NOUN iajs-2002 8 13 , , PUNCT iajs-2002 8 14 crank crank NOUN iajs-2002 8 15 - - PUNCT iajs-2002 8 16 nicolson nicolson NOUN iajs-2002 8 17 . . PUNCT iajs-2002 9 1 1 1 X iajs-2002 9 2 . . X iajs-2002 9 3 introduction introduction NOUN iajs-2002 9 4 in in ADP iajs-2002 9 5 the the DET iajs-2002 9 6 last last ADJ iajs-2002 9 7 decades decade NOUN iajs-2002 9 8 many many ADJ iajs-2002 9 9 researchers researcher NOUN iajs-2002 9 10 interested interested ADJ iajs-2002 9 11 to to PART iajs-2002 9 12 study study VERB iajs-2002 9 13 the the DET iajs-2002 9 14 solution solution NOUN iajs-2002 9 15 of of ADP iajs-2002 9 16 boundary boundary ADJ iajs-2002 9 17 value value NOUN iajs-2002 9 18 problems problem NOUN iajs-2002 9 19 ( ( PUNCT iajs-2002 9 20 bvps bvps NOUN iajs-2002 9 21 ) ) PUNCT iajs-2002 9 22 in in ADP iajs-2002 9 23 general general ADJ iajs-2002 9 24 and and CCONJ iajs-2002 9 25 the the DET iajs-2002 9 26 solution solution NOUN iajs-2002 9 27 of of ADP iajs-2002 9 28 nlpbvp nlpbvp NOUN iajs-2002 9 29 in in ADP iajs-2002 9 30 particular particular ADJ iajs-2002 9 31 , , PUNCT iajs-2002 9 32 there there PRON iajs-2002 9 33 are be VERB iajs-2002 9 34 many many ADJ iajs-2002 9 35 different different ADJ iajs-2002 9 36 methods method NOUN iajs-2002 9 37 for for ADP iajs-2002 9 38 solving solve VERB iajs-2002 9 39 the the DET iajs-2002 9 40 nlpbvp nlpbvp NOUN iajs-2002 9 41 , , PUNCT iajs-2002 9 42 f.g f.g NOUN iajs-2002 9 43 . . PROPN iajs-2002 10 1 in in ADP iajs-2002 10 2 2000 2000 NUM iajs-2002 10 3 , , PUNCT iajs-2002 10 4 karlsen karlsen NOUN iajs-2002 10 5 and and CCONJ iajs-2002 10 6 riserbo riserbo NOUN iajs-2002 10 7 used use VERB iajs-2002 10 8 a a DET iajs-2002 10 9 corrected correct VERB iajs-2002 10 10 operator operator NOUN iajs-2002 10 11 splitting splitting NOUN iajs-2002 10 12 method method NOUN iajs-2002 10 13 [ [ X iajs-2002 10 14 1 1 NUM iajs-2002 10 15 ] ] PUNCT iajs-2002 10 16 , , PUNCT iajs-2002 10 17 pao pao PROPN iajs-2002 10 18 in in ADP iajs-2002 10 19 2001 2001 NUM iajs-2002 10 20 used use VERB iajs-2002 10 21 the the DET iajs-2002 10 22 time time NOUN iajs-2002 10 23 period period NOUN iajs-2002 10 24 solutions solution NOUN iajs-2002 10 25 [ [ X iajs-2002 10 26 2 2 NUM iajs-2002 10 27 ] ] PUNCT iajs-2002 10 28 , , PUNCT iajs-2002 10 29 in in ADP iajs-2002 10 30 2006 2006 NUM iajs-2002 10 31 , , PUNCT iajs-2002 10 32 alam alam PROPN iajs-2002 10 33 and and CCONJ iajs-2002 10 34 etc etc PRON iajs-2002 10 35 used use VERB iajs-2002 10 36 the the DET iajs-2002 10 37 simultaneous simultaneous ADJ iajs-2002 10 38 space space NOUN iajs-2002 10 39 – – PUNCT iajs-2002 10 40 time time NOUN iajs-2002 10 41 adaptive adaptive ADJ iajs-2002 10 42 wavelet wavelet NOUN iajs-2002 10 43 method method NOUN iajs-2002 10 44 [ [ X iajs-2002 10 45 3 3 NUM iajs-2002 10 46 ] ] PUNCT iajs-2002 10 47 . . PUNCT iajs-2002 11 1 timothy timothy PROPN iajs-2002 11 2 in in ADP iajs-2002 11 3 2010 2010 NUM iajs-2002 11 4 studied study VERB iajs-2002 11 5 the the DET iajs-2002 11 6 explicit explicit ADJ iajs-2002 11 7 and and CCONJ iajs-2002 11 8 implicit implicit ADJ iajs-2002 11 9 difference difference NOUN iajs-2002 11 10 method method NOUN iajs-2002 11 11 [ [ X iajs-2002 11 12 4 4 NUM iajs-2002 11 13 ] ] PUNCT iajs-2002 11 14 , , PUNCT iajs-2002 11 15 in in ADP iajs-2002 11 16 2011 2011 NUM iajs-2002 11 17 ghoreishi ghoreishi NOUN iajs-2002 11 18 and and CCONJ iajs-2002 11 19 ismail ismail NOUN iajs-2002 11 20 are be AUX iajs-2002 11 21 used use VERB iajs-2002 11 22 the the DET iajs-2002 11 23 homotopy homotopy NOUN iajs-2002 11 24 perturbation perturbation NOUN iajs-2002 11 25 method method NOUN iajs-2002 11 26 ( ( PUNCT iajs-2002 11 27 hpm hpm NOUN iajs-2002 11 28 ) ) PUNCT iajs-2002 11 29 [ [ PUNCT iajs-2002 11 30 5 5 NUM iajs-2002 11 31 ] ] PUNCT iajs-2002 11 32 , , PUNCT iajs-2002 11 33 and and CCONJ iajs-2002 11 34 many many ADJ iajs-2002 11 35 others other NOUN iajs-2002 11 36 . . PUNCT iajs-2002 12 1 the the DET iajs-2002 12 2 study study NOUN iajs-2002 12 3 of of ADP iajs-2002 12 4 the the DET iajs-2002 12 5 solution solution NOUN iajs-2002 12 6 for for ADP iajs-2002 12 7 the the DET iajs-2002 12 8 parabolic parabolic NOUN iajs-2002 12 9 bvp bvp NOUN iajs-2002 12 10 using use VERB iajs-2002 12 11 the the DET iajs-2002 12 12 finite finite ADJ iajs-2002 12 13 element element NOUN iajs-2002 12 14 method method NOUN iajs-2002 12 15 ( ( PUNCT iajs-2002 12 16 fem fem NOUN iajs-2002 12 17 ) ) PUNCT iajs-2002 12 18 back back ADV iajs-2002 12 19 to to ADP iajs-2002 12 20 the the DET iajs-2002 12 21 beginning beginning NOUN iajs-2002 12 22 of of ADP iajs-2002 12 23 the the DET iajs-2002 12 24 17th 17th ADJ iajs-2002 12 25 century century NOUN iajs-2002 12 26 , , PUNCT iajs-2002 12 27 and and CCONJ iajs-2002 12 28 are be AUX iajs-2002 12 29 studied study VERB iajs-2002 12 30 from from ADP iajs-2002 12 31 many many ADJ iajs-2002 12 32 researchers researcher NOUN iajs-2002 12 33 so so SCONJ iajs-2002 12 34 as as ADP iajs-2002 12 35 douglas douglas PROPN iajs-2002 12 36 and and CCONJ iajs-2002 12 37 dupont dupont PROPN iajs-2002 12 38 [ [ X iajs-2002 12 39 6 6 NUM iajs-2002 12 40 ] ] PUNCT iajs-2002 12 41 , , PUNCT iajs-2002 12 42 in in ADP iajs-2002 12 43 1993 1993 NUM iajs-2002 12 44 reddy reddy NOUN iajs-2002 12 45 introduced introduce VERB iajs-2002 12 46 in in ADP iajs-2002 12 47 his his PRON iajs-2002 12 48 book book NOUN iajs-2002 12 49 an an DET iajs-2002 12 50 introduction introduction NOUN iajs-2002 12 51 to to ADP iajs-2002 12 52 the the DET iajs-2002 12 53 fem fem NOUN iajs-2002 12 54 was be AUX iajs-2002 12 55 applied apply VERB iajs-2002 12 56 to to ADP iajs-2002 12 57 linear linear NOUN iajs-2002 12 58 , , PUNCT iajs-2002 12 59 one one NUM iajs-2002 12 60 and and CCONJ iajs-2002 12 61 two two NUM iajs-2002 12 62 - - PUNCT iajs-2002 12 63 dimensional dimensional ADJ iajs-2002 12 64 problems problem NOUN iajs-2002 12 65 of of ADP iajs-2002 12 66 engineering engineering NOUN iajs-2002 12 67 and and CCONJ iajs-2002 12 68 applied apply VERB iajs-2002 12 69 sciences science NOUN iajs-2002 13 1 [ [ X iajs-2002 13 2 7 7 NUM iajs-2002 13 3 ] ] PUNCT iajs-2002 13 4 . . PUNCT iajs-2002 14 1 in in ADP iajs-2002 14 2 1997 1997 NUM iajs-2002 14 3 - - SYM iajs-2002 14 4 2006 2006 NUM iajs-2002 14 5 thomee thomee NOUN iajs-2002 14 6 [ [ X iajs-2002 14 7 8 8 NUM iajs-2002 14 8 ] ] PUNCT iajs-2002 14 9 studied study VERB iajs-2002 14 10 the the DET iajs-2002 14 11 gfem gfem NOUN iajs-2002 14 12 with with ADP iajs-2002 14 13 backward backward PROPN iajs-2002 14 14 euler euler NOUN iajs-2002 14 15 method method NOUN iajs-2002 14 16 for for ADP iajs-2002 14 17 nonlinear nonlinear ADJ iajs-2002 14 18 parabolic parabolic PROPN iajs-2002 14 19 bvp bvp NOUN iajs-2002 14 20 . . PUNCT iajs-2002 15 1 according accord VERB iajs-2002 15 2 to to ADP iajs-2002 15 3 these these PRON iajs-2002 15 4 studied study VERB iajs-2002 15 5 it it PRON iajs-2002 15 6 was be AUX iajs-2002 15 7 important important ADJ iajs-2002 15 8 in in ADP iajs-2002 15 9 this this DET iajs-2002 15 10 paper paper NOUN iajs-2002 15 11 to to PART iajs-2002 15 12 study study VERB iajs-2002 15 13 the the DET iajs-2002 15 14 approximate approximate ADJ iajs-2002 15 15 solution solution NOUN iajs-2002 15 16 for for ADP iajs-2002 15 17 nlpbvp nlpbvp NOUN iajs-2002 15 18 using use VERB iajs-2002 15 19 the the DET iajs-2002 15 20 gfem gfem NOUN iajs-2002 15 21 method method NOUN iajs-2002 15 22 for for ADP iajs-2002 15 23 the the DET iajs-2002 15 24 space space NOUN iajs-2002 15 25 variable variable NOUN iajs-2002 15 26 and and CCONJ iajs-2002 15 27 the the DET iajs-2002 15 28 cn cn PROPN iajs-2002 15 29 scheme scheme NOUN iajs-2002 15 30 for for ADP iajs-2002 15 31 the the DET iajs-2002 15 32 time time NOUN iajs-2002 15 33 variable variable NOUN iajs-2002 15 34 . . PUNCT iajs-2002 16 1 this this DET iajs-2002 16 2 paper paper NOUN iajs-2002 16 3 starts start VERB iajs-2002 16 4 with with ADP iajs-2002 16 5 give give VERB iajs-2002 16 6 a a DET iajs-2002 16 7 description description NOUN iajs-2002 16 8 of of ADP iajs-2002 16 9 proposed propose VERB iajs-2002 16 10 nlpbvp nlpbvp NOUN iajs-2002 16 11 and and CCONJ iajs-2002 16 12 its its PRON iajs-2002 16 13 weak weak ADJ iajs-2002 16 14 form form NOUN iajs-2002 16 15 ( ( PUNCT iajs-2002 16 16 wf wf PROPN iajs-2002 16 17 ) ) PUNCT iajs-2002 16 18 . . PUNCT iajs-2002 17 1 the the DET iajs-2002 17 2 approximation approximation NOUN iajs-2002 17 3 solution solution NOUN iajs-2002 17 4 of of ADP iajs-2002 17 5 the the DET iajs-2002 17 6 problem problem NOUN iajs-2002 17 7 is be AUX iajs-2002 17 8 obtained obtain VERB iajs-2002 17 9 by by ADP iajs-2002 17 10 discretize discretize NOUN iajs-2002 17 11 the the DET iajs-2002 17 12 wf wf PROPN iajs-2002 17 13 by by ADP iajs-2002 17 14 using use VERB iajs-2002 17 15 the the DET iajs-2002 17 16 gfem gfem NOUN iajs-2002 17 17 for for ADP iajs-2002 17 18 the the DET iajs-2002 17 19 space space NOUN iajs-2002 17 20 variable variable NOUN iajs-2002 17 21 and and CCONJ iajs-2002 17 22 the the DET iajs-2002 17 23 cn cn PROPN iajs-2002 17 24 scheme scheme NOUN iajs-2002 17 25 for for ADP iajs-2002 17 26 the the DET iajs-2002 17 27 time time NOUN iajs-2002 17 28 variable variable NOUN iajs-2002 17 29 , , PUNCT iajs-2002 17 30 the the DET iajs-2002 17 31 problem problem NOUN iajs-2002 17 32 then then ADV iajs-2002 17 33 reduces reduce VERB iajs-2002 17 34 to to PART iajs-2002 17 35 solve solve VERB iajs-2002 17 36 a a DET iajs-2002 17 37 gnlas gnla NOUN iajs-2002 17 38 which which PRON iajs-2002 17 39 transforms transform VERB iajs-2002 17 40 it it PRON iajs-2002 17 41 to to ADP iajs-2002 17 42 a a DET iajs-2002 17 43 las las NOUN iajs-2002 17 44 which which PRON iajs-2002 17 45 is be AUX iajs-2002 17 46 solved solve VERB iajs-2002 17 47 once once ADV iajs-2002 17 48 using use VERB iajs-2002 17 49 the the DET iajs-2002 17 50 chm chm NOUN iajs-2002 17 51 and and CCONJ iajs-2002 17 52 once once ADV iajs-2002 17 53 again again ADV iajs-2002 17 54 mathematics mathematic VERB iajs-2002 17 55 | | ADV iajs-2002 17 56 127 127 NUM iajs-2002 17 57 ibn ibn PROPN iajs-2002 17 58 al al PROPN iajs-2002 17 59 - - PUNCT iajs-2002 17 60 haitham haitham PROPN iajs-2002 17 61 jour jour X iajs-2002 17 62 . . PROPN iajs-2002 17 63 for for ADP iajs-2002 17 64 pure pure ADJ iajs-2002 17 65 & & CCONJ iajs-2002 17 66 appl appl PROPN iajs-2002 17 67 . . PUNCT iajs-2002 18 1 sci sci PROPN iajs-2002 18 2 . . PROPN iajs-2002 18 3 ihjpas ihjpa VERB iajs-2002 18 4 https://doi.org/10.30526/31.3.2002 https://doi.org/10.30526/31.3.2002 PART iajs-2002 18 5 vol vol NOUN iajs-2002 18 6 . . PUNCT iajs-2002 18 7 31 31 NUM iajs-2002 18 8 ( ( PUNCT iajs-2002 18 9 3 3 NUM iajs-2002 18 10 ) ) PUNCT iajs-2002 18 11 2018 2018 NUM iajs-2002 18 12 using use VERB iajs-2002 18 13 that that SCONJ iajs-2002 18 14 we we PRON iajs-2002 18 15 give give VERB iajs-2002 18 16 it it PRON iajs-2002 18 17 the the DET iajs-2002 18 18 name name NOUN iajs-2002 18 19 chrot chrot NOUN iajs-2002 18 20 to to PART iajs-2002 18 21 save save VERB iajs-2002 18 22 a a DET iajs-2002 18 23 massive massive ADJ iajs-2002 18 24 time time NOUN iajs-2002 18 25 , , PUNCT iajs-2002 18 26 which which PRON iajs-2002 18 27 is be AUX iajs-2002 18 28 explained explain VERB iajs-2002 18 29 in in ADP iajs-2002 18 30 a a DET iajs-2002 18 31 two two NUM iajs-2002 18 32 steps step NOUN iajs-2002 18 33 formula formula NOUN iajs-2002 18 34 . . PUNCT iajs-2002 19 1 finally finally ADV iajs-2002 19 2 , , PUNCT iajs-2002 19 3 illustrative illustrative ADJ iajs-2002 19 4 examples example NOUN iajs-2002 19 5 are be AUX iajs-2002 19 6 given give VERB iajs-2002 19 7 to to PART iajs-2002 19 8 solve solve VERB iajs-2002 19 9 different different ADJ iajs-2002 19 10 problems problem NOUN iajs-2002 19 11 using use VERB iajs-2002 19 12 matlab matlab PROPN iajs-2002 19 13 r2013a r2013a NOUN iajs-2002 19 14 software software NOUN iajs-2002 19 15 , , PUNCT iajs-2002 19 16 the the DET iajs-2002 19 17 results result NOUN iajs-2002 19 18 are be AUX iajs-2002 19 19 given give VERB iajs-2002 19 20 by by ADP iajs-2002 19 21 tables table NOUN iajs-2002 19 22 and and CCONJ iajs-2002 19 23 by by ADP iajs-2002 19 24 figures figure NOUN iajs-2002 19 25 and and CCONJ iajs-2002 19 26 are be AUX iajs-2002 19 27 show show VERB iajs-2002 19 28 the the DET iajs-2002 19 29 efficiency efficiency NOUN iajs-2002 19 30 of of ADP iajs-2002 19 31 this this DET iajs-2002 19 32 method method NOUN iajs-2002 19 33 , , PUNCT iajs-2002 19 34 and and CCONJ iajs-2002 19 35 are be AUX iajs-2002 19 36 show show VERB iajs-2002 19 37 that that SCONJ iajs-2002 19 38 the the DET iajs-2002 19 39 chrot chrot NOUN iajs-2002 19 40 is be AUX iajs-2002 19 41 very very ADV iajs-2002 19 42 fast fast ADJ iajs-2002 19 43 to to PART iajs-2002 19 44 solve solve VERB iajs-2002 19 45 the the DET iajs-2002 19 46 linear linear ADJ iajs-2002 19 47 algebraic algebraic ADJ iajs-2002 19 48 system system NOUN iajs-2002 19 49 than than ADP iajs-2002 19 50 the the DET iajs-2002 19 51 chm chm NOUN iajs-2002 19 52 . . PUNCT iajs-2002 20 1 definition definition NOUN iajs-2002 20 2 1 1 NUM iajs-2002 20 3 , , PUNCT iajs-2002 20 4 [ [ X iajs-2002 20 5 9 9 NUM iajs-2002 20 6 ] ] X iajs-2002 20 7 : : PUNCT iajs-2002 20 8 a a DET iajs-2002 20 9 point point NOUN iajs-2002 20 10 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2002 20 11 ∈ ∈ PROPN iajs-2002 20 12 𝑆 𝑆 PROPN iajs-2002 20 13 βŠ‚ βŠ‚ PROPN iajs-2002 20 14 ℝ ℝ PROPN iajs-2002 20 15 is be AUX iajs-2002 20 16 said say VERB iajs-2002 20 17 to to PART iajs-2002 20 18 be be AUX iajs-2002 20 19 fixed fix VERB iajs-2002 20 20 point point NOUN iajs-2002 20 21 of of ADP iajs-2002 20 22 a a DET iajs-2002 20 23 given give VERB iajs-2002 20 24 function function NOUN iajs-2002 20 25 π‘ž π‘ž NOUN iajs-2002 20 26 : : PUNCT iajs-2002 20 27 𝑆 𝑆 PROPN iajs-2002 20 28 β†’ β†’ SYM iajs-2002 20 29 ℝ ℝ PROPN iajs-2002 20 30 , , PUNCT iajs-2002 20 31 if if SCONJ iajs-2002 20 32 π‘ž π‘ž PROPN iajs-2002 20 33 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2002 20 34 π‘₯βˆ—. π‘₯βˆ—. NOUN iajs-2002 20 35 definition definition NOUN iajs-2002 20 36 2 2 NUM iajs-2002 20 37 , , PUNCT iajs-2002 20 38 [ [ X iajs-2002 20 39 9 9 NUM iajs-2002 20 40 ] ] X iajs-2002 20 41 : : PUNCT iajs-2002 20 42 a a DET iajs-2002 20 43 function function NOUN iajs-2002 20 44 π‘ž π‘ž NOUN iajs-2002 20 45 : : PUNCT iajs-2002 20 46 𝑆 𝑆 PROPN iajs-2002 20 47 βŠ‚ βŠ‚ PROPN iajs-2002 20 48 ℝ ℝ PROPN iajs-2002 20 49 β†’ β†’ SYM iajs-2002 20 50 ℝ ℝ PROPN iajs-2002 20 51 is be AUX iajs-2002 20 52 said say VERB iajs-2002 20 53 to to PART iajs-2002 20 54 be be AUX iajs-2002 20 55 contractive contractive ADJ iajs-2002 20 56 on on ADP iajs-2002 20 57 𝑆 𝑆 PROPN iajs-2002 20 58 , , PUNCT iajs-2002 20 59 if if SCONJ iajs-2002 20 60 for for ADP iajs-2002 20 61 each each DET iajs-2002 20 62 π‘₯ π‘₯ PROPN iajs-2002 20 63 , , PUNCT iajs-2002 20 64 𝑦 𝑦 PROPN iajs-2002 20 65 ∈ ∈ PROPN iajs-2002 20 66 𝑆 𝑆 PROPN iajs-2002 20 67 : : PUNCT iajs-2002 21 1 β€–π‘ž β€–π‘ž PROPN iajs-2002 21 2 π‘₯ π‘₯ PRON iajs-2002 21 3 π‘ž π‘ž X iajs-2002 21 4 𝑦 𝑦 PROPN iajs-2002 21 5 β€– β€– PROPN iajs-2002 21 6 𝛽‖π‘₯ 𝛽‖π‘₯ NOUN iajs-2002 21 7 𝑦‖ 𝑦‖ PROPN iajs-2002 21 8 , , PUNCT iajs-2002 21 9 where where SCONJ iajs-2002 21 10 0 0 NUM iajs-2002 21 11 𝛽 𝛽 NOUN iajs-2002 21 12 1 1 NUM iajs-2002 21 13 is be AUX iajs-2002 21 14 a a DET iajs-2002 21 15 constant constant ADJ iajs-2002 21 16 . . PUNCT iajs-2002 22 1 theorem theorem NOUN iajs-2002 22 2 3 3 NUM iajs-2002 22 3 , , PUNCT iajs-2002 22 4 [ [ X iajs-2002 22 5 9 9 NUM iajs-2002 22 6 ] ] X iajs-2002 22 7 : : PUNCT iajs-2002 22 8 a a DET iajs-2002 22 9 contractive contractive ADJ iajs-2002 22 10 function function NOUN iajs-2002 22 11 π‘ž π‘ž X iajs-2002 22 12 on on ADP iajs-2002 22 13 a a DET iajs-2002 22 14 complete complete ADJ iajs-2002 22 15 normed normed ADJ iajs-2002 22 16 space space NOUN iajs-2002 22 17 𝑆 𝑆 PROPN iajs-2002 22 18 has have VERB iajs-2002 22 19 a a DET iajs-2002 22 20 unique unique ADJ iajs-2002 22 21 fixed fix VERB iajs-2002 22 22 point point NOUN iajs-2002 22 23 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2002 22 24 in in ADP iajs-2002 22 25 . . PUNCT iajs-2002 23 1 theorem theorem VERB iajs-2002 23 2 4 4 NUM iajs-2002 23 3 , , PUNCT iajs-2002 23 4 [ [ X iajs-2002 23 5 9 9 NUM iajs-2002 23 6 ] ] PUNCT iajs-2002 23 7 : : PUNCT iajs-2002 23 8 let let VERB iajs-2002 23 9 β€–βˆ™β€– β€–βˆ™β€– PUNCT iajs-2002 23 10 is be AUX iajs-2002 23 11 a a DET iajs-2002 23 12 norm norm NOUN iajs-2002 23 13 in in ADP iajs-2002 23 14 ℝ ℝ PROPN iajs-2002 23 15 and and CCONJ iajs-2002 23 16 𝑆 𝑆 PROPN iajs-2002 23 17 βŠ‚ βŠ‚ PROPN iajs-2002 23 18 ℝ ℝ PROPN iajs-2002 23 19 . . PUNCT iajs-2002 24 1 if if SCONJ iajs-2002 24 2 π‘ž π‘ž X iajs-2002 24 3 : : PUNCT iajs-2002 24 4 𝑆 𝑆 PROPN iajs-2002 24 5 β†’ β†’ SYM iajs-2002 24 6 ℝ ℝ PROPN iajs-2002 24 7 is be AUX iajs-2002 24 8 contractive contractive ADJ iajs-2002 24 9 function function NOUN iajs-2002 24 10 on on ADP iajs-2002 24 11 𝑆 𝑆 PROPN iajs-2002 24 12 , , PUNCT iajs-2002 24 13 and and CCONJ iajs-2002 24 14 one one NUM iajs-2002 24 15 of of ADP iajs-2002 24 16 the the DET iajs-2002 24 17 following follow VERB iajs-2002 24 18 is be AUX iajs-2002 24 19 satisfied satisfied ADJ iajs-2002 24 20 : : PUNCT iajs-2002 24 21 ( ( PUNCT iajs-2002 24 22 1 1 X iajs-2002 24 23 ) ) PUNCT iajs-2002 24 24 π‘ž π‘ž X iajs-2002 24 25 π‘₯ π‘₯ X iajs-2002 24 26 ∈ ∈ PROPN iajs-2002 24 27 𝑆 𝑆 PROPN iajs-2002 24 28 , , PUNCT iajs-2002 24 29 βˆ€ βˆ€ VERB iajs-2002 24 30 π‘₯ π‘₯ DET iajs-2002 24 31 ∈ ∈ PROPN iajs-2002 24 32 𝑆 𝑆 PROPN iajs-2002 24 33 . . PUNCT iajs-2002 25 1 ( ( PUNCT iajs-2002 25 2 2 2 X iajs-2002 25 3 ) ) PUNCT iajs-2002 25 4 𝑆 𝑆 PROPN iajs-2002 25 5 π‘₯|β€–π‘₯ π‘₯|β€–π‘₯ PROPN iajs-2002 25 6 𝑦‖ 𝑦‖ PROPN iajs-2002 25 7 πœ‡ πœ‡ ADP iajs-2002 25 8 and and CCONJ iajs-2002 25 9 β€–π‘ž β€–π‘ž ADJ iajs-2002 25 10 𝑦 𝑦 ADJ iajs-2002 25 11 𝑦‖ 𝑦‖ ADJ iajs-2002 25 12 1 1 NUM iajs-2002 25 13 𝛽 𝛽 NOUN iajs-2002 25 14 πœ‡ πœ‡ ADP iajs-2002 25 15 . . PUNCT iajs-2002 26 1 ( ( PUNCT iajs-2002 26 2 3 3 X iajs-2002 26 3 ) ) PUNCT iajs-2002 26 4 𝑆 𝑆 PROPN iajs-2002 26 5 π‘₯|β€–π‘₯ π‘₯|β€–π‘₯ PROPN iajs-2002 26 6 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2002 26 7 β€– β€– PROPN iajs-2002 27 1 πœ‡ πœ‡ ADP iajs-2002 27 2 , , PUNCT iajs-2002 27 3 where where SCONJ iajs-2002 27 4 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2002 27 5 is be AUX iajs-2002 27 6 a a DET iajs-2002 27 7 fixed fix VERB iajs-2002 27 8 point point NOUN iajs-2002 27 9 of of ADP iajs-2002 27 10 π‘ž π‘ž PRON iajs-2002 27 11 then then ADV iajs-2002 27 12 π‘₯ π‘₯ PROPN iajs-2002 27 13 ∈ ∈ PROPN iajs-2002 27 14 𝑆 𝑆 PROPN iajs-2002 27 15 , , PUNCT iajs-2002 27 16 where where SCONJ iajs-2002 27 17 π‘₯ π‘₯ PROPN iajs-2002 27 18 is be AUX iajs-2002 27 19 the the DET iajs-2002 27 20 𝑙 𝑙 DET iajs-2002 27 21 th th NUM iajs-2002 27 22 iterative iterative NOUN iajs-2002 27 23 value value NOUN iajs-2002 27 24 of of ADP iajs-2002 27 25 π‘₯ π‘₯ PROPN iajs-2002 27 26 . . PUNCT iajs-2002 28 1 theorem theorem ADJ iajs-2002 28 2 5 5 NUM iajs-2002 28 3 , , PUNCT iajs-2002 28 4 [ [ X iajs-2002 28 5 9 9 NUM iajs-2002 28 6 ] ] PUNCT iajs-2002 28 7 : : PUNCT iajs-2002 28 8 let let VERB iajs-2002 28 9 β€–βˆ™β€– β€–βˆ™β€– PUNCT iajs-2002 28 10 is be AUX iajs-2002 28 11 a a DET iajs-2002 28 12 norm norm NOUN iajs-2002 28 13 in in ADP iajs-2002 28 14 ℝ ℝ PROPN iajs-2002 28 15 and and CCONJ iajs-2002 28 16 𝑆 𝑆 PROPN iajs-2002 28 17 be be VERB iajs-2002 28 18 a a DET iajs-2002 28 19 closed closed ADJ iajs-2002 28 20 subset subset NOUN iajs-2002 28 21 of of ADP iajs-2002 28 22 ℝ ℝ PROPN iajs-2002 28 23 . . PUNCT iajs-2002 29 1 if if SCONJ iajs-2002 29 2 π‘ž π‘ž X iajs-2002 29 3 : : PUNCT iajs-2002 29 4 𝑆 𝑆 PROPN iajs-2002 29 5 β†’ β†’ SYM iajs-2002 29 6 ℝ ℝ PROPN iajs-2002 29 7 is be AUX iajs-2002 29 8 contractive contractive ADJ iajs-2002 29 9 function function NOUN iajs-2002 29 10 on on ADP iajs-2002 29 11 , , PUNCT iajs-2002 29 12 and and CCONJ iajs-2002 29 13 π‘₯ π‘₯ DET iajs-2002 29 14 ∈ ∈ PROPN iajs-2002 29 15 𝑆 𝑆 PROPN iajs-2002 29 16 , , PUNCT iajs-2002 29 17 then then ADV iajs-2002 29 18 ( ( PUNCT iajs-2002 29 19 1 1 X iajs-2002 29 20 ) ) PUNCT iajs-2002 29 21 π‘₯ π‘₯ PROPN iajs-2002 29 22 is be AUX iajs-2002 29 23 converge converge ADJ iajs-2002 29 24 to to ADP iajs-2002 29 25 a a DET iajs-2002 29 26 fixed fix VERB iajs-2002 29 27 point point NOUN iajs-2002 29 28 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2002 29 29 ∈ ∈ PROPN iajs-2002 29 30 𝑆 𝑆 PROPN iajs-2002 29 31 ( ( PUNCT iajs-2002 29 32 2 2 NUM iajs-2002 29 33 ) ) PUNCT iajs-2002 29 34 π‘₯βˆ— π‘₯βˆ— PROPN iajs-2002 29 35 is be AUX iajs-2002 29 36 a a DET iajs-2002 29 37 unique unique ADJ iajs-2002 29 38 in in ADP iajs-2002 29 39 𝑆 𝑆 PROPN iajs-2002 29 40 3 3 NUM iajs-2002 29 41 . . PUNCT iajs-2002 30 1 description description NOUN iajs-2002 30 2 of of ADP iajs-2002 30 3 the the DET iajs-2002 30 4 ( ( PUNCT iajs-2002 30 5 nlpbvpcc nlpbvpcc PROPN iajs-2002 30 6 ) ) PUNCT iajs-2002 30 7 let let VERB iajs-2002 30 8 w w NOUN iajs-2002 30 9 = = PUNCT iajs-2002 30 10 π‘₯ π‘₯ NOUN iajs-2002 30 11 π‘₯ π‘₯ X iajs-2002 30 12 , , PUNCT iajs-2002 31 1 π‘₯ π‘₯ PROPN iajs-2002 31 2 ∈ ∈ PROPN iajs-2002 31 3 ℝ ℝ NOUN iajs-2002 31 4 : : PUNCT iajs-2002 31 5 0 0 NUM iajs-2002 31 6 π‘₯ π‘₯ NOUN iajs-2002 31 7 , , PUNCT iajs-2002 31 8 π‘₯ π‘₯ PROPN iajs-2002 31 9 1 1 NUM iajs-2002 31 10 , , PUNCT iajs-2002 31 11 with with ADP iajs-2002 31 12 lipischitz lipischitz PROPN iajs-2002 31 13 boundary boundary PROPN iajs-2002 31 14 βˆ‚w βˆ‚w PROPN iajs-2002 31 15 , , PUNCT iajs-2002 31 16 and and CCONJ iajs-2002 31 17 let let VERB iajs-2002 31 18 i i PRON iajs-2002 31 19 0 0 NUM iajs-2002 31 20 , , PUNCT iajs-2002 31 21 t t X iajs-2002 31 22 , , PUNCT iajs-2002 31 23 0 0 NUM iajs-2002 31 24 t t NOUN iajs-2002 31 25 ∞ ∞ PROPN iajs-2002 31 26 , , PUNCT iajs-2002 31 27 and and CCONJ iajs-2002 31 28 p p X iajs-2002 31 29 w w PROPN iajs-2002 31 30 i. i. PROPN iajs-2002 31 31 the the DET iajs-2002 31 32 nonlinear nonlinear ADJ iajs-2002 31 33 parabolic parabolic ADJ iajs-2002 31 34 equation equation NOUN iajs-2002 31 35 is be AUX iajs-2002 31 36 given give VERB iajs-2002 31 37 by by ADP iajs-2002 31 38 : : PUNCT iajs-2002 31 39 𝑒 𝑒 PROPN iajs-2002 31 40 βˆ†π‘’ βˆ†π‘’ PROPN iajs-2002 31 41 𝐻 𝐻 PROPN iajs-2002 31 42 οΏ½ οΏ½ PROPN iajs-2002 31 43 βƒ— βƒ— NOUN iajs-2002 31 44 οΏ½ οΏ½ PROPN iajs-2002 31 45 , , PUNCT iajs-2002 31 46 𝑑 𝑑 PROPN iajs-2002 31 47 , , PUNCT iajs-2002 31 48 𝑒 𝑒 NOUN iajs-2002 31 49 , , PUNCT iajs-2002 31 50 in in ADP iajs-2002 31 51 p p X iajs-2002 31 52 ( ( PUNCT iajs-2002 31 53 1 1 NUM iajs-2002 31 54 ) ) PUNCT iajs-2002 31 55 with with ADP iajs-2002 31 56 the the DET iajs-2002 31 57 boundary boundary ADJ iajs-2002 31 58 condition condition NOUN iajs-2002 31 59 ( ( PUNCT iajs-2002 31 60 b.c b.c NOUN iajs-2002 31 61 ) ) PUNCT iajs-2002 31 62 𝑒 𝑒 PROPN iajs-2002 31 63 οΏ½ οΏ½ PROPN iajs-2002 31 64 βƒ— βƒ— NOUN iajs-2002 31 65 οΏ½ οΏ½ PROPN iajs-2002 31 66 , , PUNCT iajs-2002 31 67 𝑑 𝑑 PROPN iajs-2002 31 68 0 0 NUM iajs-2002 31 69 , , PUNCT iajs-2002 31 70 on on ADP iajs-2002 31 71 βˆ‚π‘Š βˆ‚π‘Š ADP iajs-2002 31 72 i i PRON iajs-2002 31 73 ( ( PUNCT iajs-2002 31 74 2 2 NUM iajs-2002 31 75 ) ) PUNCT iajs-2002 31 76 and and CCONJ iajs-2002 31 77 the the DET iajs-2002 31 78 initial initial ADJ iajs-2002 31 79 condition condition NOUN iajs-2002 31 80 ( ( PUNCT iajs-2002 31 81 i.c i.c PROPN iajs-2002 31 82 ) ) PUNCT iajs-2002 31 83 𝑒 𝑒 PROPN iajs-2002 31 84 οΏ½ οΏ½ PROPN iajs-2002 31 85 βƒ— βƒ— NOUN iajs-2002 31 86 οΏ½ οΏ½ PROPN iajs-2002 31 87 , , PUNCT iajs-2002 31 88 0 0 NUM iajs-2002 31 89 𝑒 𝑒 PRON iajs-2002 31 90 οΏ½ οΏ½ PROPN iajs-2002 31 91 βƒ— βƒ— X iajs-2002 31 92 οΏ½ οΏ½ PROPN iajs-2002 31 93 , , PUNCT iajs-2002 31 94 in in ADP iajs-2002 31 95 w w PROPN iajs-2002 31 96 ( ( PUNCT iajs-2002 31 97 3 3 NUM iajs-2002 31 98 ) ) PUNCT iajs-2002 31 99 where where SCONJ iajs-2002 31 100 𝑒 𝑒 PROPN iajs-2002 31 101 𝑒 𝑒 PROPN iajs-2002 31 102 οΏ½ οΏ½ PROPN iajs-2002 31 103 βƒ— βƒ— NOUN iajs-2002 31 104 οΏ½ οΏ½ PROPN iajs-2002 31 105 , , PUNCT iajs-2002 31 106 𝑑 𝑑 X iajs-2002 31 107 , , PUNCT iajs-2002 31 108 βˆ†π‘’ βˆ†π‘’ PROPN iajs-2002 31 109 βˆ‘ βˆ‘ ADP iajs-2002 31 110 is be AUX iajs-2002 31 111 the the DET iajs-2002 31 112 laplace laplace NOUN iajs-2002 31 113 operator operator NOUN iajs-2002 31 114 and and CCONJ iajs-2002 31 115 𝐻 𝐻 PROPN iajs-2002 31 116 ∈ ∈ PROPN iajs-2002 31 117 𝐢 𝐢 PROPN iajs-2002 31 118 π‘Š π‘Š PROPN iajs-2002 31 119 . . PUNCT iajs-2002 32 1 in in ADP iajs-2002 32 2 this this DET iajs-2002 32 3 work work NOUN iajs-2002 32 4 the the DET iajs-2002 32 5 inner inner ADJ iajs-2002 32 6 product product NOUN iajs-2002 32 7 and and CCONJ iajs-2002 32 8 norm norm NOUN iajs-2002 32 9 in in ADP iajs-2002 32 10 l l PROPN iajs-2002 32 11 w w NOUN iajs-2002 32 12 will will AUX iajs-2002 32 13 be be AUX iajs-2002 32 14 denoted denote VERB iajs-2002 32 15 by by ADP iajs-2002 32 16 βˆ™ βˆ™ PROPN iajs-2002 32 17 , , PUNCT iajs-2002 32 18 βˆ™ βˆ™ PROPN iajs-2002 32 19 and and CCONJ iajs-2002 32 20 β€–βˆ™β€– β€–βˆ™β€– PROPN iajs-2002 32 21 , , PUNCT iajs-2002 32 22 the the DET iajs-2002 32 23 inner inner ADJ iajs-2002 32 24 product product NOUN iajs-2002 32 25 and and CCONJ iajs-2002 32 26 norm norm NOUN iajs-2002 32 27 in in ADP iajs-2002 32 28 sobolev sobolev PROPN iajs-2002 32 29 space space NOUN iajs-2002 32 30 v v PROPN iajs-2002 32 31 h h NOUN iajs-2002 32 32 w w NOUN iajs-2002 32 33 will will AUX iajs-2002 32 34 be be AUX iajs-2002 32 35 denoted denote VERB iajs-2002 32 36 by by ADP iajs-2002 32 37 βˆ™ βˆ™ PROPN iajs-2002 32 38 , , PUNCT iajs-2002 32 39 βˆ™ βˆ™ PROPN iajs-2002 32 40 and and CCONJ iajs-2002 32 41 β€–βˆ™β€– β€–βˆ™β€– PROPN iajs-2002 32 42 , , PUNCT iajs-2002 32 43 the the DET iajs-2002 32 44 duality duality NOUN iajs-2002 32 45 bracket bracket VERB iajs-2002 32 46 between between ADP iajs-2002 32 47 v v NOUN iajs-2002 32 48 and and CCONJ iajs-2002 32 49 its its PRON iajs-2002 32 50 dual dual ADJ iajs-2002 32 51 vβˆ— vβˆ— NOUN iajs-2002 32 52 will will AUX iajs-2002 32 53 be be AUX iajs-2002 32 54 denoted denote VERB iajs-2002 32 55 by by ADP iajs-2002 32 56 〈 〈 PROPN iajs-2002 32 57 βˆ™ βˆ™ PROPN iajs-2002 32 58 , , PUNCT iajs-2002 32 59 βˆ™ βˆ™ PROPN iajs-2002 32 60 βŒͺ βŒͺ NOUN iajs-2002 32 61 and and CCONJ iajs-2002 32 62 β€–βˆ™β€– β€–βˆ™β€– X iajs-2002 32 63 be be AUX iajs-2002 32 64 the the DET iajs-2002 32 65 norm norm NOUN iajs-2002 32 66 in in ADP iajs-2002 32 67 l l PROPN iajs-2002 32 68 p p PROPN iajs-2002 32 69 . . PUNCT iajs-2002 33 1 now now ADV iajs-2002 33 2 , , PUNCT iajs-2002 33 3 the the DET iajs-2002 33 4 wf wf PROPN iajs-2002 33 5 of of ADP iajs-2002 33 6 ( ( PUNCT iajs-2002 33 7 1 1 NUM iajs-2002 33 8 - - SYM iajs-2002 33 9 3 3 NUM iajs-2002 33 10 ) ) PUNCT iajs-2002 33 11 is be AUX iajs-2002 33 12 given give VERB iajs-2002 33 13 by by ADP iajs-2002 33 14 : : PUNCT iajs-2002 33 15 〈 〈 NOUN iajs-2002 33 16 𝑒 𝑒 NOUN iajs-2002 33 17 , , PUNCT iajs-2002 33 18 πœ‰ πœ‰ VERB iajs-2002 33 19 βŒͺ βŒͺ NOUN iajs-2002 33 20 〈 〈 NOUN iajs-2002 33 21 𝛻𝑒 𝛻𝑒 PROPN iajs-2002 33 22 , , PUNCT iajs-2002 33 23 π›»πœ‰ π›»πœ‰ NOUN iajs-2002 33 24 βŒͺ βŒͺ NOUN iajs-2002 33 25 = = SYM iajs-2002 33 26 𝐻 𝐻 PROPN iajs-2002 33 27 𝑒 𝑒 NOUN iajs-2002 33 28 , , PUNCT iajs-2002 33 29 πœ‰ πœ‰ X iajs-2002 33 30 , , PUNCT iajs-2002 33 31 βˆ€ βˆ€ X iajs-2002 33 32 πœ‰ πœ‰ ADP iajs-2002 33 33 ∈ ∈ PROPN iajs-2002 33 34 𝑉 𝑉 PROPN iajs-2002 33 35 a.e a.e NOUN iajs-2002 33 36 on on ADP iajs-2002 33 37 i i PRON iajs-2002 33 38 ( ( PUNCT iajs-2002 33 39 4 4 NUM iajs-2002 33 40 ) ) PUNCT iajs-2002 33 41 𝑒 𝑒 X iajs-2002 33 42 0 0 NUM iajs-2002 33 43 , , PUNCT iajs-2002 33 44 πœ‰ πœ‰ PROPN iajs-2002 33 45 = = PUNCT iajs-2002 33 46 𝑒 𝑒 NOUN iajs-2002 33 47 , , PUNCT iajs-2002 33 48 πœ‰ πœ‰ INTJ iajs-2002 33 49 , , PUNCT iajs-2002 33 50 in in ADP iajs-2002 33 51 w w PROPN iajs-2002 33 52 ( ( PUNCT iajs-2002 33 53 5 5 NUM iajs-2002 33 54 ) ) PUNCT iajs-2002 33 55 with with ADP iajs-2002 33 56 𝑒 𝑒 NOUN iajs-2002 33 57 belongs belong VERB iajs-2002 33 58 to to ADP iajs-2002 33 59 𝑉 𝑉 PROPN iajs-2002 33 60 and and CCONJ iajs-2002 33 61 to to ADP iajs-2002 33 62 𝐿 𝐿 PROPN iajs-2002 33 63 w w PROPN iajs-2002 33 64 since since SCONJ iajs-2002 33 65 𝑉 𝑉 PROPN iajs-2002 33 66 βŠ‚ βŠ‚ PROPN iajs-2002 33 67 𝐿 𝐿 PROPN iajs-2002 33 68 w w PROPN iajs-2002 33 69 . . PUNCT iajs-2002 34 1 assumptions assumption NOUN iajs-2002 34 2 : : PUNCT iajs-2002 34 3 ( ( PUNCT iajs-2002 34 4 1 1 X iajs-2002 34 5 ) ) PUNCT iajs-2002 34 6 for for ADP iajs-2002 34 7 some some DET iajs-2002 34 8 positive positive ADJ iajs-2002 34 9 constants constant NOUN iajs-2002 34 10 𝛾 𝛾 VERB iajs-2002 34 11 , , PUNCT iajs-2002 34 12 𝛾 𝛾 ADP iajs-2002 34 13 and and CCONJ iajs-2002 34 14 for for ADP iajs-2002 34 15 each each DET iajs-2002 34 16 πœ‚ πœ‚ NOUN iajs-2002 34 17 , , PUNCT iajs-2002 34 18 πœ‚ πœ‚ PROPN iajs-2002 34 19 ∈ ∈ PROPN iajs-2002 34 20 𝑉 𝑉 PROPN iajs-2002 34 21 & & CCONJ iajs-2002 34 22 𝑑 𝑑 PROPN iajs-2002 34 23 ∈ ∈ PROPN iajs-2002 34 24 i i PRON iajs-2002 34 25 Μ… Μ… VERB iajs-2002 34 26 , , PUNCT iajs-2002 34 27 the the DET iajs-2002 34 28 following follow VERB iajs-2002 34 29 inequality inequality NOUN iajs-2002 34 30 are be AUX iajs-2002 34 31 satisfies satisfie NOUN iajs-2002 34 32 | | ADV iajs-2002 34 33 π›»πœ‚ π›»πœ‚ NUM iajs-2002 34 34 , , PUNCT iajs-2002 34 35 π›»πœ‚ π›»πœ‚ PUNCT iajs-2002 35 1 | | ADV iajs-2002 35 2 𝛾 𝛾 ADP iajs-2002 35 3 β€–π›»πœ‚ β€–π›»πœ‚ NOUN iajs-2002 35 4 β€– β€– PROPN iajs-2002 35 5 β€–π›»πœ‚ β€–π›»πœ‚ PROPN iajs-2002 35 6 β€– β€– PROPN iajs-2002 35 7 π›»πœ‚ π›»πœ‚ PROPN iajs-2002 35 8 , , PUNCT iajs-2002 35 9 π›»πœ‚ π›»πœ‚ PUNCT iajs-2002 35 10 𝛾 𝛾 ADP iajs-2002 35 11 β€–π›»πœ‚ β€–π›»πœ‚ NOUN iajs-2002 35 12 β€– β€– PROPN iajs-2002 35 13 mathematics mathematic NOUN iajs-2002 35 14 | | ADV iajs-2002 35 15 128 128 NUM iajs-2002 35 16 ibn ibn PROPN iajs-2002 35 17 al al PROPN iajs-2002 35 18 - - PUNCT iajs-2002 35 19 haitham haitham PROPN iajs-2002 35 20 jour jour X iajs-2002 35 21 . . PROPN iajs-2002 35 22 for for ADP iajs-2002 35 23 pure pure ADJ iajs-2002 35 24 & & CCONJ iajs-2002 35 25 appl appl PROPN iajs-2002 35 26 . . PUNCT iajs-2002 36 1 sci sci PROPN iajs-2002 36 2 . . PROPN iajs-2002 36 3 ihjpas ihjpa VERB iajs-2002 36 4 https://doi.org/10.30526/31.3.2002 https://doi.org/10.30526/31.3.2002 PART iajs-2002 36 5 vol vol NOUN iajs-2002 36 6 . . PUNCT iajs-2002 36 7 31 31 NUM iajs-2002 36 8 ( ( PUNCT iajs-2002 36 9 3 3 NUM iajs-2002 36 10 ) ) PUNCT iajs-2002 36 11 2018 2018 NUM iajs-2002 36 12 ( ( PUNCT iajs-2002 36 13 2 2 X iajs-2002 36 14 ) ) PUNCT iajs-2002 36 15 the the DET iajs-2002 36 16 function function NOUN iajs-2002 36 17 h h NOUN iajs-2002 36 18 is be AUX iajs-2002 36 19 of of ADP iajs-2002 36 20 a a DET iajs-2002 36 21 carathéodory carathéodory NOUN iajs-2002 36 22 type type NOUN iajs-2002 36 23 on on ADP iajs-2002 36 24 p p PROPN iajs-2002 36 25 ℝ ℝ PROPN iajs-2002 36 26 , , PUNCT iajs-2002 36 27 satisfies satisfy VERB iajs-2002 36 28 the the DET iajs-2002 36 29 following follow VERB iajs-2002 36 30 sub sub NOUN iajs-2002 36 31 linearity linearity NOUN iajs-2002 36 32 and and CCONJ iajs-2002 36 33 lipischitz lipischitz ADJ iajs-2002 36 34 conditions condition NOUN iajs-2002 36 35 |𝐻 |𝐻 PROPN iajs-2002 36 36 οΏ½ οΏ½ PROPN iajs-2002 36 37 βƒ— βƒ— PROPN iajs-2002 36 38 οΏ½ οΏ½ PROPN iajs-2002 36 39 , , PUNCT iajs-2002 36 40 𝑑 𝑑 PROPN iajs-2002 36 41 , , PUNCT iajs-2002 36 42 𝑒 𝑒 PROPN iajs-2002 36 43 | | ADV iajs-2002 36 44 𝛿 𝛿 ADJ iajs-2002 36 45 οΏ½ οΏ½ PROPN iajs-2002 36 46 βƒ— βƒ— NOUN iajs-2002 36 47 οΏ½ οΏ½ PROPN iajs-2002 36 48 , , PUNCT iajs-2002 36 49 𝑑 𝑑 ADP iajs-2002 36 50 𝑐 𝑐 PROPN iajs-2002 36 51 |𝑒| |𝑒| PROPN iajs-2002 36 52 , , PUNCT iajs-2002 36 53 where where SCONJ iajs-2002 36 54 𝑐 𝑐 PROPN iajs-2002 36 55 0 0 NUM iajs-2002 36 56 , , PUNCT iajs-2002 36 57 ( ( PUNCT iajs-2002 36 58 οΏ½ οΏ½ PROPN iajs-2002 36 59 βƒ— βƒ— NOUN iajs-2002 36 60 οΏ½ οΏ½ PROPN iajs-2002 36 61 , , PUNCT iajs-2002 36 62 𝑑 𝑑 PROPN iajs-2002 36 63 ∈ ∈ NOUN iajs-2002 36 64 𝑃 𝑃 NOUN iajs-2002 36 65 , , PUNCT iajs-2002 36 66 , , PUNCT iajs-2002 36 67 𝑒 𝑒 PROPN iajs-2002 36 68 ∈ ∈ PROPN iajs-2002 36 69 ℝ ℝ PROPN iajs-2002 36 70 and and CCONJ iajs-2002 36 71 𝛿 𝛿 PRON iajs-2002 36 72 ∈ ∈ PROPN iajs-2002 36 73 𝐿 𝐿 NOUN iajs-2002 36 74 𝑃 𝑃 PROPN iajs-2002 36 75 , , PUNCT iajs-2002 36 76 ℝ ℝ PROPN iajs-2002 36 77 |𝐻 |𝐻 PROPN iajs-2002 36 78 οΏ½ οΏ½ PROPN iajs-2002 36 79 βƒ— βƒ— PROPN iajs-2002 36 80 οΏ½ οΏ½ PROPN iajs-2002 36 81 , , PUNCT iajs-2002 36 82 𝑑 𝑑 PROPN iajs-2002 36 83 , , PUNCT iajs-2002 36 84 𝑒 𝑒 VERB iajs-2002 36 85 𝐻 𝐻 PROPN iajs-2002 36 86 οΏ½ οΏ½ PROPN iajs-2002 36 87 βƒ— βƒ— NOUN iajs-2002 36 88 οΏ½ οΏ½ PROPN iajs-2002 36 89 , , PUNCT iajs-2002 36 90 𝑑 𝑑 PROPN iajs-2002 36 91 , , PUNCT iajs-2002 36 92 𝑒 𝑒 PRON iajs-2002 36 93 | | ADV iajs-2002 36 94 𝐿|𝑒 𝐿|𝑒 PUNCT iajs-2002 36 95 𝑒 𝑒 PROPN iajs-2002 36 96 | | ADV iajs-2002 36 97 , , PUNCT iajs-2002 36 98 where where SCONJ iajs-2002 36 99 ( ( PUNCT iajs-2002 36 100 π‘₯ π‘₯ NOUN iajs-2002 36 101 , , PUNCT iajs-2002 36 102 𝑑 𝑑 PROPN iajs-2002 36 103 ∈ ∈ PROPN iajs-2002 36 104 𝑃 𝑃 PROPN iajs-2002 36 105 , , PUNCT iajs-2002 36 106 𝑒 𝑒 X iajs-2002 36 107 , , PUNCT iajs-2002 36 108 𝑒 𝑒 PROPN iajs-2002 36 109 ∈ ∈ PROPN iajs-2002 36 110 ℝ ℝ PROPN iajs-2002 36 111 , , PUNCT iajs-2002 36 112 l l PROPN iajs-2002 36 113 is be AUX iajs-2002 36 114 a a DET iajs-2002 36 115 lipischitz lipischitz PROPN iajs-2002 36 116 constant constant NOUN iajs-2002 36 117 . . PUNCT iajs-2002 37 1 4 4 X iajs-2002 37 2 . . X iajs-2002 37 3 discretization discretization NOUN iajs-2002 37 4 of of ADP iajs-2002 37 5 the the DET iajs-2002 37 6 continuous continuous ADJ iajs-2002 37 7 equation equation NOUN iajs-2002 37 8 : : PUNCT iajs-2002 37 9 the the DET iajs-2002 37 10 wf wf PROPN iajs-2002 37 11 of of ADP iajs-2002 37 12 ( ( PUNCT iajs-2002 37 13 4 4 NUM iajs-2002 37 14 - - SYM iajs-2002 37 15 5 5 NUM iajs-2002 37 16 ) ) PUNCT iajs-2002 37 17 is be AUX iajs-2002 37 18 discretized discretize VERB iajs-2002 37 19 by by ADP iajs-2002 37 20 using use VERB iajs-2002 37 21 the the DET iajs-2002 37 22 gfem gfem NOUN iajs-2002 37 23 , , PUNCT iajs-2002 37 24 as as SCONJ iajs-2002 37 25 follows follow VERB iajs-2002 37 26 , , PUNCT iajs-2002 37 27 let let VERB iajs-2002 37 28 the the DET iajs-2002 37 29 domain domain NOUN iajs-2002 37 30 w w NOUN iajs-2002 37 31 is be AUX iajs-2002 37 32 a a DET iajs-2002 37 33 polyhedron polyhedron NOUN iajs-2002 37 34 . . PUNCT iajs-2002 38 1 for for ADP iajs-2002 38 2 every every DET iajs-2002 38 3 integer integer PROPN iajs-2002 38 4 𝑛 𝑛 NOUN iajs-2002 38 5 , , PUNCT iajs-2002 38 6 let let VERB iajs-2002 38 7 π‘Š π‘Š PRON iajs-2002 38 8 be be AUX iajs-2002 38 9 an an DET iajs-2002 38 10 admissible admissible ADJ iajs-2002 38 11 regular regular ADJ iajs-2002 38 12 traingulation traingulation NOUN iajs-2002 38 13 of of ADP iajs-2002 38 14 w w NOUN iajs-2002 38 15 into into ADP iajs-2002 38 16 closed closed ADJ iajs-2002 38 17 disimplices disimplice NOUN iajs-2002 38 18 [ [ X iajs-2002 38 19 8 8 NUM iajs-2002 38 20 ] ] PUNCT iajs-2002 38 21 , , PUNCT iajs-2002 38 22 𝐼 𝐼 PROPN iajs-2002 38 23 be be VERB iajs-2002 38 24 a a DET iajs-2002 38 25 subdivision subdivision NOUN iajs-2002 38 26 of of ADP iajs-2002 38 27 the the DET iajs-2002 38 28 interval interval NOUN iajs-2002 38 29 iΜ… iΜ… PROPN iajs-2002 38 30 into into ADP iajs-2002 38 31 𝑁𝑇 𝑁𝑇 PROPN iajs-2002 38 32 𝑛 𝑛 PRON iajs-2002 38 33 intervals interval NOUN iajs-2002 38 34 , , PUNCT iajs-2002 38 35 where where SCONJ iajs-2002 38 36 𝑑 𝑑 PROPN iajs-2002 38 37 𝑑 𝑑 PROPN iajs-2002 38 38 , , PUNCT iajs-2002 38 39 𝑑 𝑑 PROPN iajs-2002 38 40 , , PUNCT iajs-2002 38 41 of of ADP iajs-2002 38 42 equal equal ADJ iajs-2002 38 43 length length NOUN iajs-2002 38 44 βˆ†π‘‘ βˆ†π‘‘ PROPN iajs-2002 38 45 𝑇 𝑇 PROPN iajs-2002 38 46 𝑁𝑇⁄ 𝑁𝑇⁄ NOUN iajs-2002 38 47 , , PUNCT iajs-2002 38 48 let let VERB iajs-2002 38 49 𝑂 𝑂 PROPN iajs-2002 38 50 π‘Š π‘Š VERB iajs-2002 38 51 𝐼 𝐼 PROPN iajs-2002 38 52 and and CCONJ iajs-2002 38 53 𝑉 𝑉 PROPN iajs-2002 38 54 βŠ‚ βŠ‚ PROPN iajs-2002 38 55 𝑉 𝑉 PROPN iajs-2002 38 56 𝐻 𝐻 PROPN iajs-2002 38 57 w w NOUN iajs-2002 38 58 be be AUX iajs-2002 38 59 the the DET iajs-2002 38 60 space space NOUN iajs-2002 38 61 of of ADP iajs-2002 38 62 continuous continuous ADJ iajs-2002 38 63 piecewise piecewise NOUN iajs-2002 38 64 affine affine NOUN iajs-2002 38 65 in in ADP iajs-2002 38 66 w. w. PROPN iajs-2002 38 67 the the DET iajs-2002 38 68 discrete discrete ADJ iajs-2002 38 69 state state NOUN iajs-2002 38 70 equation equation NOUN iajs-2002 38 71 ( ( PUNCT iajs-2002 38 72 dseq dseq NOUN iajs-2002 38 73 ) ) PUNCT iajs-2002 38 74 of of ADP iajs-2002 38 75 the the DET iajs-2002 38 76 wf wf PROPN iajs-2002 38 77 ( ( PUNCT iajs-2002 38 78 4 4 NUM iajs-2002 38 79 - - SYM iajs-2002 38 80 5 5 NUM iajs-2002 38 81 ) ) PUNCT iajs-2002 38 82 is be AUX iajs-2002 38 83 obtained obtain VERB iajs-2002 38 84 after after ADP iajs-2002 38 85 using use VERB iajs-2002 38 86 the the DET iajs-2002 38 87 cn cn PROPN iajs-2002 38 88 formula formula NOUN iajs-2002 38 89 and and CCONJ iajs-2002 38 90 is be AUX iajs-2002 38 91 given give VERB iajs-2002 38 92 by by ADP iajs-2002 38 93 𝑒 𝑒 NOUN iajs-2002 38 94 𝑒 𝑒 NOUN iajs-2002 38 95 , , PUNCT iajs-2002 38 96 πœ‰ πœ‰ PROPN iajs-2002 38 97 + + CCONJ iajs-2002 38 98 βˆ†π‘‘ βˆ†π‘‘ PROPN iajs-2002 38 99 𝛻𝑒 𝛻𝑒 PROPN iajs-2002 38 100 , , PUNCT iajs-2002 38 101 π›»πœ‰ π›»πœ‰ PROPN iajs-2002 38 102 = = SYM iajs-2002 38 103 βˆ†π‘‘ βˆ†π‘‘ PROPN iajs-2002 38 104 𝐻 𝐻 PROPN iajs-2002 38 105 𝑑 𝑑 PROPN iajs-2002 38 106 , , PUNCT iajs-2002 38 107 𝑒 𝑒 X iajs-2002 38 108 , , PUNCT iajs-2002 38 109 πœ‰ πœ‰ X iajs-2002 38 110 , , PUNCT iajs-2002 38 111 j=0,1, j=0,1, PROPN iajs-2002 38 112 ... ... PUNCT iajs-2002 38 113 ,nt-1 ,nt-1 PUNCT iajs-2002 38 114 ( ( PUNCT iajs-2002 38 115 6 6 NUM iajs-2002 38 116 ) ) PUNCT iajs-2002 38 117 𝑒 𝑒 NOUN iajs-2002 38 118 , , PUNCT iajs-2002 38 119 πœ‰ πœ‰ VERB iajs-2002 38 120 𝑒 𝑒 PROPN iajs-2002 38 121 , , PUNCT iajs-2002 38 122 πœ‰ πœ‰ X iajs-2002 38 123 ( ( PUNCT iajs-2002 38 124 7 7 NUM iajs-2002 38 125 ) ) PUNCT iajs-2002 38 126 where where SCONJ iajs-2002 38 127 πœ‰ πœ‰ PROPN iajs-2002 38 128 ∈ ∈ PROPN iajs-2002 38 129 𝑉 𝑉 PROPN iajs-2002 38 130 , , PUNCT iajs-2002 38 131 𝑒 𝑒 X iajs-2002 38 132 𝑒 𝑒 X iajs-2002 38 133 𝑒 𝑒 X iajs-2002 38 134 , , PUNCT iajs-2002 38 135 𝑑 𝑑 PROPN iajs-2002 38 136 𝑑 𝑑 PROPN iajs-2002 38 137 𝑑 𝑑 PROPN iajs-2002 38 138 , , PUNCT iajs-2002 38 139 j=0,1 j=0,1 ADV iajs-2002 38 140 ... ... PUNCT iajs-2002 38 141 , , PUNCT iajs-2002 38 142 nt not PART iajs-2002 38 143 , , PUNCT iajs-2002 38 144 𝑒 𝑒 PROPN iajs-2002 38 145 ∈ ∈ PROPN iajs-2002 38 146 𝑉 𝑉 PROPN iajs-2002 38 147 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-2002 38 148 𝑒 𝑒 NOUN iajs-2002 38 149 𝑒 𝑒 ADP iajs-2002 38 150 𝑑 𝑑 PROPN iajs-2002 38 151 ∈ ∈ PROPN iajs-2002 38 152 𝑉 𝑉 PROPN iajs-2002 38 153 , , PUNCT iajs-2002 38 154 βˆ€ βˆ€ NOUN iajs-2002 38 155 𝑗 𝑗 NOUN iajs-2002 38 156 0,1 0,1 NUM iajs-2002 38 157 , , PUNCT iajs-2002 38 158 … … PUNCT iajs-2002 38 159 , , PUNCT iajs-2002 38 160 𝑁𝑇. 𝑁𝑇. ADP iajs-2002 38 161 5 5 NUM iajs-2002 38 162 . . PUNCT iajs-2002 39 1 the the DET iajs-2002 39 2 approximation approximation NOUN iajs-2002 39 3 solution solution NOUN iajs-2002 39 4 of of ADP iajs-2002 39 5 the the DET iajs-2002 39 6 nonlinear nonlinear ADJ iajs-2002 39 7 parabolic parabolic PROPN iajs-2002 39 8 equation equation NOUN iajs-2002 39 9 : : PUNCT iajs-2002 39 10 to to PART iajs-2002 39 11 find find VERB iajs-2002 39 12 the the DET iajs-2002 39 13 approximation approximation NOUN iajs-2002 39 14 solution solution NOUN iajs-2002 39 15 ( ( PUNCT iajs-2002 39 16 app.sol app.sol NUM iajs-2002 39 17 ) ) PUNCT iajs-2002 39 18 𝑒 𝑒 PROPN iajs-2002 39 19 𝑒 𝑒 NOUN iajs-2002 39 20 , , PUNCT iajs-2002 39 21 𝑒 𝑒 X iajs-2002 39 22 , , PUNCT iajs-2002 39 23 … … PUNCT iajs-2002 39 24 , , PUNCT iajs-2002 39 25 𝑒 𝑒 PROPN iajs-2002 39 26 of of ADP iajs-2002 39 27 ( ( PUNCT iajs-2002 39 28 6 6 NUM iajs-2002 39 29 - - SYM iajs-2002 39 30 7 7 NUM iajs-2002 39 31 ) ) PUNCT iajs-2002 39 32 by by ADP iajs-2002 39 33 using use VERB iajs-2002 39 34 the the DET iajs-2002 39 35 gfem gfem NOUN iajs-2002 39 36 , , PUNCT iajs-2002 39 37 the the DET iajs-2002 39 38 following follow VERB iajs-2002 39 39 procedure procedure NOUN iajs-2002 39 40 can can AUX iajs-2002 39 41 be be AUX iajs-2002 39 42 used use VERB iajs-2002 39 43 : : PUNCT iajs-2002 39 44 ( ( PUNCT iajs-2002 39 45 1 1 X iajs-2002 39 46 ) ) PUNCT iajs-2002 39 47 for for ADP iajs-2002 39 48 fixed fix VERB iajs-2002 39 49 any any DET iajs-2002 39 50 𝑗 𝑗 NOUN iajs-2002 39 51 , , PUNCT iajs-2002 39 52 0 0 NUM iajs-2002 39 53 𝑗 𝑗 PROPN iajs-2002 39 54 𝑁𝑇 𝑁𝑇 NOUN iajs-2002 39 55 1 1 NUM iajs-2002 39 56 , , PUNCT iajs-2002 39 57 let let VERB iajs-2002 39 58 πœ‰ πœ‰ PRON iajs-2002 39 59 , , PUNCT iajs-2002 39 60 𝑖 𝑖 PROPN iajs-2002 39 61 1,2 1,2 NUM iajs-2002 39 62 , , PUNCT iajs-2002 39 63 … … PUNCT iajs-2002 39 64 , , PUNCT iajs-2002 39 65 𝑁 𝑁 PROPN iajs-2002 39 66 , , PUNCT iajs-2002 39 67 π‘€π‘–π‘‘β„Ž π‘€π‘–π‘‘β„Ž NOUN iajs-2002 39 68 πœ‰ πœ‰ ADP iajs-2002 39 69 οΏ½ οΏ½ PROPN iajs-2002 39 70 βƒ— βƒ— NOUN iajs-2002 39 71 οΏ½ οΏ½ PROPN iajs-2002 39 72 0 0 NUM iajs-2002 39 73 , , PUNCT iajs-2002 39 74 π‘œπ‘› π‘œπ‘› PROPN iajs-2002 39 75 πœ•π‘Š πœ•π‘Š NOUN iajs-2002 39 76 be be AUX iajs-2002 39 77 a a DET iajs-2002 39 78 continuous continuous ADJ iajs-2002 39 79 piecewise piecewise NOUN iajs-2002 39 80 affine affine NOUN iajs-2002 39 81 finite finite PROPN iajs-2002 39 82 basis basis NOUN iajs-2002 39 83 of of ADP iajs-2002 39 84 𝑉 𝑉 PROPN iajs-2002 39 85 in in ADP iajs-2002 39 86 w w PROPN iajs-2002 39 87 , , PUNCT iajs-2002 39 88 then then ADV iajs-2002 39 89 for for ADP iajs-2002 39 90 any any DET iajs-2002 39 91 i=1,2, i=1,2, NOUN iajs-2002 39 92 ... ... PUNCT iajs-2002 39 93 ,n ,n PUNCT iajs-2002 39 94 and and CCONJ iajs-2002 39 95 𝑒 𝑒 X iajs-2002 39 96 , , PUNCT iajs-2002 39 97 𝑒 𝑒 PROPN iajs-2002 39 98 ∈ ∈ PROPN iajs-2002 39 99 𝑉 𝑉 PROPN iajs-2002 39 100 ( ( PUNCT iajs-2002 39 101 6 6 NUM iajs-2002 39 102 - - SYM iajs-2002 39 103 7 7 NUM iajs-2002 39 104 ) ) PUNCT iajs-2002 39 105 can can AUX iajs-2002 39 106 be be AUX iajs-2002 39 107 rewritten rewrite VERB iajs-2002 39 108 as as ADP iajs-2002 39 109 : : PUNCT iajs-2002 39 110 𝑒 𝑒 NOUN iajs-2002 39 111 𝑒 𝑒 X iajs-2002 39 112 , , PUNCT iajs-2002 39 113 πœ‰ πœ‰ PROPN iajs-2002 39 114 βˆ†π‘‘ βˆ†π‘‘ NOUN iajs-2002 39 115 𝛻𝑒 𝛻𝑒 PROPN iajs-2002 39 116 , , PUNCT iajs-2002 39 117 π›»πœ‰ π›»πœ‰ PROPN iajs-2002 39 118 βˆ†π‘‘ βˆ†π‘‘ PROPN iajs-2002 39 119 𝐻 𝐻 PROPN iajs-2002 39 120 𝑑 𝑑 PROPN iajs-2002 39 121 , , PUNCT iajs-2002 39 122 𝑒 𝑒 PRON iajs-2002 39 123 𝑒 𝑒 X iajs-2002 39 124 , , PUNCT iajs-2002 39 125 πœ‰ πœ‰ X iajs-2002 39 126 ( ( PUNCT iajs-2002 39 127 8) 8) NUM iajs-2002 39 128 𝑒 𝑒 NOUN iajs-2002 39 129 , , PUNCT iajs-2002 39 130 πœ‰ πœ‰ VERB iajs-2002 39 131 𝑒 𝑒 PROPN iajs-2002 39 132 , , PUNCT iajs-2002 39 133 πœ‰ πœ‰ INTJ iajs-2002 39 134 , , PUNCT iajs-2002 39 135 πœ‰ πœ‰ PROPN iajs-2002 39 136 ∈ ∈ PROPN iajs-2002 39 137 𝑉 𝑉 PROPN iajs-2002 39 138 , , PUNCT iajs-2002 39 139 ( ( PUNCT iajs-2002 39 140 9 9 NUM iajs-2002 39 141 ) ) PUNCT iajs-2002 39 142 ( ( PUNCT iajs-2002 39 143 2 2 X iajs-2002 39 144 ) ) PUNCT iajs-2002 39 145 using use VERB iajs-2002 39 146 the the DET iajs-2002 39 147 galerkin galerkin ADJ iajs-2002 39 148 method method NOUN iajs-2002 39 149 [ [ X iajs-2002 39 150 8 8 NUM iajs-2002 39 151 ] ] PUNCT iajs-2002 39 152 , , PUNCT iajs-2002 39 153 with with ADP iajs-2002 39 154 the the DET iajs-2002 39 155 basis basis NOUN iajs-2002 39 156 πœ‰ πœ‰ NOUN iajs-2002 39 157 , , PUNCT iajs-2002 39 158 πœ‰ πœ‰ X iajs-2002 39 159 , , PUNCT iajs-2002 39 160 … … PUNCT iajs-2002 39 161 , , PUNCT iajs-2002 39 162 πœ‰ πœ‰ NOUN iajs-2002 39 163 of of ADP iajs-2002 39 164 𝑉 𝑉 PROPN iajs-2002 39 165 , , PUNCT iajs-2002 39 166 one one NUM iajs-2002 39 167 has have VERB iajs-2002 39 168 𝑒 𝑒 NOUN iajs-2002 39 169 βˆ‘ βˆ‘ PUNCT iajs-2002 39 170 𝑋 𝑋 PROPN iajs-2002 39 171 πœ‰ πœ‰ X iajs-2002 39 172 , , PUNCT iajs-2002 39 173 , , PUNCT iajs-2002 39 174 𝑒 𝑒 X iajs-2002 39 175 βˆ‘ βˆ‘ X iajs-2002 39 176 𝑋 𝑋 PROPN iajs-2002 39 177 πœ‰ πœ‰ X iajs-2002 39 178 , , PUNCT iajs-2002 39 179 and and CCONJ iajs-2002 39 180 𝑒 𝑒 PART iajs-2002 39 181 βˆ‘ βˆ‘ PUNCT iajs-2002 39 182 𝑋 𝑋 NOUN iajs-2002 39 183 πœ‰ πœ‰ ADP iajs-2002 39 184 where where SCONJ iajs-2002 39 185 , , PUNCT iajs-2002 39 186 𝑋 𝑋 PROPN iajs-2002 39 187 𝑋 𝑋 PROPN iajs-2002 39 188 𝑑 𝑑 PROPN iajs-2002 39 189 , , PUNCT iajs-2002 39 190 for for ADP iajs-2002 39 191 each each DET iajs-2002 39 192 𝑗 𝑗 PRON iajs-2002 39 193 0,1 0,1 NUM iajs-2002 39 194 , , PUNCT iajs-2002 39 195 … … PUNCT iajs-2002 39 196 , , PUNCT iajs-2002 39 197 𝑁𝑇 𝑁𝑇 PROPN iajs-2002 39 198 are be AUX iajs-2002 39 199 unknown unknown ADJ iajs-2002 39 200 constants constant NOUN iajs-2002 39 201 to to PART iajs-2002 39 202 be be AUX iajs-2002 39 203 determine determine NOUN iajs-2002 39 204 . . PUNCT iajs-2002 40 1 ( ( PUNCT iajs-2002 40 2 3 3 X iajs-2002 40 3 ) ) PUNCT iajs-2002 40 4 substitute substitute NOUN iajs-2002 40 5 𝑒 𝑒 NOUN iajs-2002 40 6 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-2002 40 7 𝑒 𝑒 PROPN iajs-2002 40 8 in in ADP iajs-2002 40 9 ( ( PUNCT iajs-2002 40 10 8) 8) NUM iajs-2002 40 11 to to PART iajs-2002 40 12 get get VERB iajs-2002 40 13 the the DET iajs-2002 40 14 following following ADJ iajs-2002 40 15 nonlinear nonlinear ADJ iajs-2002 40 16 algebraic algebraic ADJ iajs-2002 40 17 system system NOUN iajs-2002 40 18 π‘Œ π‘Œ PROPN iajs-2002 40 19 βˆ†π‘‘π‘ βˆ†π‘‘π‘ NOUN iajs-2002 40 20 𝑋 𝑋 PROPN iajs-2002 40 21 π‘Œ π‘Œ PROPN iajs-2002 40 22 βˆ†π‘‘π‘ βˆ†π‘‘π‘ NOUN iajs-2002 40 23 𝑋 𝑋 PROPN iajs-2002 40 24 𝑏 𝑏 PROPN iajs-2002 40 25 𝑑 𝑑 PROPN iajs-2002 40 26 , , PUNCT iajs-2002 40 27 𝑗 𝑗 PROPN iajs-2002 40 28 0,1 0,1 NUM iajs-2002 40 29 , , PUNCT iajs-2002 40 30 … … PUNCT iajs-2002 40 31 , , PUNCT iajs-2002 40 32 𝑁𝑇 𝑁𝑇 NOUN iajs-2002 40 33 1 1 NUM iajs-2002 40 34 ( ( PUNCT iajs-2002 40 35 10 10 NUM iajs-2002 40 36 ) ) PUNCT iajs-2002 40 37 and and CCONJ iajs-2002 40 38 substituting substitute VERB iajs-2002 40 39 𝑒 𝑒 PRON iajs-2002 40 40 in in ADP iajs-2002 40 41 ( ( PUNCT iajs-2002 40 42 9 9 NUM iajs-2002 40 43 ) ) PUNCT iajs-2002 40 44 to to PART iajs-2002 40 45 get get VERB iajs-2002 40 46 the the DET iajs-2002 40 47 following follow VERB iajs-2002 40 48 linear linear ADJ iajs-2002 40 49 algebraic algebraic ADJ iajs-2002 40 50 system system NOUN iajs-2002 40 51 π‘Œπ‘‹ π‘Œπ‘‹ PROPN iajs-2002 40 52 𝑏 𝑏 NOUN iajs-2002 40 53 ( ( PUNCT iajs-2002 40 54 11 11 NUM iajs-2002 40 55 ) ) PUNCT iajs-2002 40 56 where where SCONJ iajs-2002 40 57 π‘Œ π‘Œ PROPN iajs-2002 40 58 𝑦 𝑦 PRON iajs-2002 40 59 , , PUNCT iajs-2002 40 60 𝑦 𝑦 PRON iajs-2002 40 61 πœ‰ πœ‰ NOUN iajs-2002 40 62 , , PUNCT iajs-2002 40 63 πœ‰ πœ‰ X iajs-2002 40 64 , , PUNCT iajs-2002 40 65 𝑍 𝑍 PROPN iajs-2002 40 66 𝑧 𝑧 PRON iajs-2002 40 67 , , PUNCT iajs-2002 40 68 𝑧 𝑧 PROPN iajs-2002 40 69 π›»πœ‰ π›»πœ‰ PROPN iajs-2002 40 70 , , PUNCT iajs-2002 40 71 π›»πœ‰ π›»πœ‰ PROPN iajs-2002 40 72 , , PUNCT iajs-2002 40 73 𝑋 𝑋 PROPN iajs-2002 40 74 π‘₯ π‘₯ PROPN iajs-2002 40 75 , , PUNCT iajs-2002 40 76 π‘₯ π‘₯ X iajs-2002 40 77 , , PUNCT iajs-2002 40 78 … … PUNCT iajs-2002 40 79 , , PUNCT iajs-2002 40 80 π‘₯ π‘₯ X iajs-2002 40 81 , , PUNCT iajs-2002 40 82 𝑉 𝑉 PROPN iajs-2002 40 83 πœ‰ πœ‰ NOUN iajs-2002 40 84 , , PUNCT iajs-2002 40 85 πœ‰ πœ‰ X iajs-2002 40 86 , , PUNCT iajs-2002 40 87 … … PUNCT iajs-2002 40 88 , , PUNCT iajs-2002 40 89 πœ‰ πœ‰ X iajs-2002 40 90 , , PUNCT iajs-2002 40 91 οΏ½ οΏ½ PROPN iajs-2002 40 92 βƒ— βƒ— NOUN iajs-2002 40 93 οΏ½ οΏ½ PROPN iajs-2002 40 94 𝑏 𝑏 PROPN iajs-2002 40 95 , , PUNCT iajs-2002 40 96 𝑏 𝑏 PROPN iajs-2002 40 97 βˆ†π‘‘ βˆ†π‘‘ NOUN iajs-2002 40 98 𝐻 𝐻 NOUN iajs-2002 40 99 𝑉 𝑉 PROPN iajs-2002 40 100 𝑋 𝑋 NOUN iajs-2002 40 101 𝑉 𝑉 PROPN iajs-2002 40 102 𝑋 𝑋 NOUN iajs-2002 40 103 , , PUNCT iajs-2002 40 104 πœ‰ πœ‰ NOUN iajs-2002 40 105 and and CCONJ iajs-2002 40 106 𝑏 𝑏 ADP iajs-2002 40 107 𝑏 𝑏 PROPN iajs-2002 40 108 , , PUNCT iajs-2002 40 109 𝑏 𝑏 PROPN iajs-2002 40 110 𝑒 𝑒 PROPN iajs-2002 40 111 , , PUNCT iajs-2002 40 112 πœ‰ πœ‰ X iajs-2002 40 113 , , PUNCT iajs-2002 40 114 βˆ€ βˆ€ X iajs-2002 40 115 𝑖 𝑖 NOUN iajs-2002 40 116 , , PUNCT iajs-2002 40 117 π‘˜ π‘˜ PROPN iajs-2002 40 118 1,2 1,2 NUM iajs-2002 40 119 , , PUNCT iajs-2002 40 120 … … PUNCT iajs-2002 40 121 , , PUNCT iajs-2002 40 122 𝑁. 𝑁. NOUN iajs-2002 40 123 system system NOUN iajs-2002 40 124 ( ( PUNCT iajs-2002 40 125 10 10 NUM iajs-2002 40 126 - - SYM iajs-2002 40 127 11 11 NUM iajs-2002 40 128 ) ) PUNCT iajs-2002 40 129 has have VERB iajs-2002 40 130 a a DET iajs-2002 40 131 unique unique ADJ iajs-2002 40 132 solution solution NOUN iajs-2002 40 133 [ [ X iajs-2002 40 134 10 10 NUM iajs-2002 40 135 ] ] PUNCT iajs-2002 40 136 . . PUNCT iajs-2002 41 1 to to PART iajs-2002 41 2 solve solve VERB iajs-2002 41 3 it it PRON iajs-2002 41 4 , , PUNCT iajs-2002 41 5 the the DET iajs-2002 41 6 linear linear ADJ iajs-2002 41 7 algebraic algebraic ADJ iajs-2002 41 8 system system NOUN iajs-2002 41 9 ( ( PUNCT iajs-2002 41 10 11 11 NUM iajs-2002 41 11 ) ) PUNCT iajs-2002 41 12 is be AUX iajs-2002 41 13 solved solve VERB iajs-2002 41 14 at at ADP iajs-2002 41 15 first first ADV iajs-2002 41 16 to to PART iajs-2002 41 17 get get VERB iajs-2002 41 18 𝑋 𝑋 NOUN iajs-2002 41 19 , , PUNCT iajs-2002 41 20 then then ADV iajs-2002 41 21 to to PART iajs-2002 41 22 solve solve VERB iajs-2002 41 23 the the DET iajs-2002 41 24 nonlinear nonlinear ADJ iajs-2002 41 25 system system NOUN iajs-2002 41 26 ( ( PUNCT iajs-2002 41 27 10 10 NUM iajs-2002 41 28 ) ) PUNCT iajs-2002 41 29 the the DET iajs-2002 41 30 pct pct NOUN iajs-2002 41 31 is be AUX iajs-2002 41 32 used use VERB iajs-2002 41 33 here here ADV iajs-2002 41 34 [ [ X iajs-2002 41 35 8 8 NUM iajs-2002 41 36 ] ] PUNCT iajs-2002 41 37 , , PUNCT iajs-2002 41 38 as as SCONJ iajs-2002 41 39 follows follow VERB iajs-2002 41 40 : : PUNCT iajs-2002 41 41 for for ADP iajs-2002 41 42 each each DET iajs-2002 41 43 𝑗 𝑗 PROPN iajs-2002 41 44 0 0 NUM iajs-2002 41 45 𝑗 𝑗 NOUN iajs-2002 41 46 𝑁𝑇 𝑁𝑇 NOUN iajs-2002 41 47 1 1 NUM iajs-2002 41 48 we we PRON iajs-2002 41 49 predict predict VERB iajs-2002 41 50 at at ADP iajs-2002 41 51 first first ADV iajs-2002 41 52 the the DET iajs-2002 41 53 value value NOUN iajs-2002 41 54 𝑋 𝑋 NOUN iajs-2002 41 55 by by ADP iajs-2002 41 56 using use VERB iajs-2002 41 57 the the DET iajs-2002 41 58 mathematics mathematic NOUN iajs-2002 41 59 | | ADV iajs-2002 41 60 129 129 NUM iajs-2002 41 61 ibn ibn PROPN iajs-2002 41 62 al al PROPN iajs-2002 41 63 - - PUNCT iajs-2002 41 64 haitham haitham PROPN iajs-2002 41 65 jour jour X iajs-2002 41 66 . . PROPN iajs-2002 41 67 for for ADP iajs-2002 41 68 pure pure ADJ iajs-2002 41 69 & & CCONJ iajs-2002 41 70 appl appl PROPN iajs-2002 41 71 . . PUNCT iajs-2002 42 1 sci sci PROPN iajs-2002 42 2 . . PROPN iajs-2002 42 3 ihjpas ihjpa VERB iajs-2002 42 4 https://doi.org/10.30526/31.3.2002 https://doi.org/10.30526/31.3.2002 PART iajs-2002 42 5 vol vol NOUN iajs-2002 42 6 . . PUNCT iajs-2002 42 7 31 31 NUM iajs-2002 42 8 ( ( PUNCT iajs-2002 42 9 3 3 NUM iajs-2002 42 10 ) ) SYM iajs-2002 42 11 2018 2018 NUM iajs-2002 42 12 explicit explicit ADJ iajs-2002 42 13 form form NOUN iajs-2002 42 14 ( ( PUNCT iajs-2002 42 15 just just ADV iajs-2002 42 16 the the DET iajs-2002 42 17 value value NOUN iajs-2002 42 18 of of ADP iajs-2002 42 19 𝑋 𝑋 NOUN iajs-2002 42 20 ) ) PUNCT iajs-2002 42 21 in in ADP iajs-2002 42 22 the the DET iajs-2002 42 23 component component NOUN iajs-2002 42 24 of of ADP iajs-2002 42 25 𝑏 𝑏 PROPN iajs-2002 42 26 in in ADP iajs-2002 42 27 the the DET iajs-2002 42 28 right right ADJ iajs-2002 42 29 hand hand NOUN iajs-2002 42 30 side side NOUN iajs-2002 42 31 ( ( PUNCT iajs-2002 42 32 rhs rhs PROPN iajs-2002 42 33 ) ) PUNCT iajs-2002 42 34 of of ADP iajs-2002 42 35 ( ( PUNCT iajs-2002 42 36 10 10 NUM iajs-2002 42 37 ) ) PUNCT iajs-2002 42 38 , , PUNCT iajs-2002 42 39 then then ADV iajs-2002 42 40 by by ADP iajs-2002 42 41 substitute substitute NOUN iajs-2002 42 42 𝑋 𝑋 PROPN iajs-2002 42 43 𝑋 𝑋 PROPN iajs-2002 42 44 , , PUNCT iajs-2002 42 45 in in ADP iajs-2002 42 46 the the DET iajs-2002 42 47 component component NOUN iajs-2002 42 48 of of ADP iajs-2002 42 49 𝑏 𝑏 PROPN iajs-2002 42 50 in in ADP iajs-2002 42 51 the the DET iajs-2002 42 52 rhs rhs PROPN iajs-2002 42 53 of of ADP iajs-2002 42 54 ( ( PUNCT iajs-2002 42 55 10 10 NUM iajs-2002 42 56 ) ) PUNCT iajs-2002 42 57 , , PUNCT iajs-2002 42 58 makes make VERB iajs-2002 42 59 system system NOUN iajs-2002 42 60 ( ( PUNCT iajs-2002 42 61 10 10 NUM iajs-2002 42 62 ) ) PUNCT iajs-2002 42 63 linear linear NOUN iajs-2002 42 64 , , PUNCT iajs-2002 42 65 solving solve VERB iajs-2002 42 66 it it PRON iajs-2002 42 67 w.r.t w.r.t VERB iajs-2002 42 68 𝑋 𝑋 PROPN iajs-2002 42 69 to to PART iajs-2002 42 70 get get VERB iajs-2002 42 71 the the DET iajs-2002 42 72 corrector corrector NOUN iajs-2002 42 73 solution solution NOUN iajs-2002 42 74 , , PUNCT iajs-2002 42 75 this this DET iajs-2002 42 76 procedure procedure NOUN iajs-2002 42 77 can can AUX iajs-2002 42 78 be be AUX iajs-2002 42 79 repeated repeat VERB iajs-2002 42 80 ( ( PUNCT iajs-2002 42 81 more more ADJ iajs-2002 42 82 than than ADP iajs-2002 42 83 one one NUM iajs-2002 42 84 time time NOUN iajs-2002 42 85 if if SCONJ iajs-2002 42 86 we we PRON iajs-2002 42 87 need need VERB iajs-2002 42 88 ) ) PUNCT iajs-2002 42 89 by by ADP iajs-2002 42 90 substitute substitute NOUN iajs-2002 42 91 the the DET iajs-2002 42 92 corrector corrector NOUN iajs-2002 42 93 solution solution NOUN iajs-2002 42 94 𝑋 𝑋 PROPN iajs-2002 42 95 𝑋 𝑋 PROPN iajs-2002 42 96 in in ADP iajs-2002 42 97 the the DET iajs-2002 42 98 rhs rhs PROPN iajs-2002 42 99 of of ADP iajs-2002 42 100 the the DET iajs-2002 42 101 linear linear ADJ iajs-2002 42 102 system system NOUN iajs-2002 42 103 ( ( PUNCT iajs-2002 42 104 10 10 NUM iajs-2002 42 105 ) ) PUNCT iajs-2002 42 106 and and CCONJ iajs-2002 42 107 solve solve VERB iajs-2002 42 108 it it PRON iajs-2002 42 109 again again ADV iajs-2002 42 110 w.r.t w.r.t VERB iajs-2002 42 111 𝑋 𝑋 PROPN iajs-2002 42 112 , , PUNCT iajs-2002 42 113 for for ADP iajs-2002 42 114 fixed fix VERB iajs-2002 42 115 𝑗 𝑗 NOUN iajs-2002 42 116 , , PUNCT iajs-2002 42 117 to to PART iajs-2002 42 118 get get VERB iajs-2002 42 119 a a DET iajs-2002 42 120 new new ADJ iajs-2002 42 121 corrector corrector NOUN iajs-2002 42 122 solution solution NOUN iajs-2002 42 123 ) ) PUNCT iajs-2002 42 124 . . PUNCT iajs-2002 43 1 hence hence ADV iajs-2002 43 2 the the DET iajs-2002 43 3 corrector corrector NOUN iajs-2002 43 4 equation equation NOUN iajs-2002 43 5 described describe VERB iajs-2002 43 6 as as SCONJ iajs-2002 43 7 follows follow VERB iajs-2002 43 8 : : PUNCT iajs-2002 43 9 𝑒 𝑒 PRON iajs-2002 43 10 𝑒 𝑒 X iajs-2002 43 11 , , PUNCT iajs-2002 43 12 πœ‰ πœ‰ NOUN iajs-2002 43 13 βˆ†π‘‘ βˆ†π‘‘ NOUN iajs-2002 43 14 𝛻𝑒 𝛻𝑒 X iajs-2002 43 15 𝛻𝑒 𝛻𝑒 X iajs-2002 43 16 , , PUNCT iajs-2002 43 17 π›»πœ‰ π›»πœ‰ PROPN iajs-2002 43 18 βˆ†π‘‘ βˆ†π‘‘ PROPN iajs-2002 43 19 𝐻 𝐻 PROPN iajs-2002 43 20 𝑒 𝑒 NOUN iajs-2002 43 21 𝑒 𝑒 NOUN iajs-2002 43 22 , , PUNCT iajs-2002 43 23 πœ‰ πœ‰ X iajs-2002 43 24 ( ( PUNCT iajs-2002 43 25 12 12 NUM iajs-2002 43 26 ) ) PUNCT iajs-2002 43 27 where where SCONJ iajs-2002 43 28 𝑒 𝑒 X iajs-2002 43 29 ∢ ∢ NOUN iajs-2002 43 30 𝑒 𝑒 X iajs-2002 43 31 is be AUX iajs-2002 43 32 the the DET iajs-2002 43 33 predictor predictor NOUN iajs-2002 43 34 solution solution NOUN iajs-2002 43 35 at at ADP iajs-2002 43 36 the the DET iajs-2002 43 37 iteration iteration NOUN iajs-2002 43 38 𝑙 𝑙 PROPN iajs-2002 43 39 1 1 NUM iajs-2002 43 40 , , PUNCT iajs-2002 43 41 𝑒 𝑒 PRON iajs-2002 43 42 ∢ ∢ NOUN iajs-2002 43 43 𝑒 𝑒 ADV iajs-2002 43 44 is be AUX iajs-2002 43 45 its its PRON iajs-2002 43 46 corresponding corresponding ADJ iajs-2002 43 47 corrector corrector NOUN iajs-2002 43 48 solution solution NOUN iajs-2002 43 49 at at ADP iajs-2002 43 50 the the DET iajs-2002 43 51 iteration iteration NOUN iajs-2002 43 52 𝑙 𝑙 PRON iajs-2002 43 53 and and CCONJ iajs-2002 43 54 𝑒 𝑒 PRON iajs-2002 43 55 𝑒 𝑒 NOUN iajs-2002 43 56 is be AUX iajs-2002 43 57 the the DET iajs-2002 43 58 known know VERB iajs-2002 43 59 corrector corrector NOUN iajs-2002 43 60 solution solution NOUN iajs-2002 43 61 for for ADP iajs-2002 43 62 the the DET iajs-2002 43 63 previous previous ADJ iajs-2002 43 64 step step NOUN iajs-2002 43 65 j j PROPN iajs-2002 43 66 , , PUNCT iajs-2002 43 67 i.e. i.e. X iajs-2002 43 68 ( ( PUNCT iajs-2002 43 69 12 12 NUM iajs-2002 43 70 ) ) PUNCT iajs-2002 43 71 can can AUX iajs-2002 43 72 be be AUX iajs-2002 43 73 written write VERB iajs-2002 43 74 as as ADP iajs-2002 43 75 : : PUNCT iajs-2002 43 76 𝑒 𝑒 X iajs-2002 43 77 π‘ž π‘ž X iajs-2002 43 78 𝑒 𝑒 X iajs-2002 43 79 ( ( PUNCT iajs-2002 43 80 13 13 NUM iajs-2002 43 81 ) ) PUNCT iajs-2002 43 82 theorem theorem VERB iajs-2002 43 83 6 6 NUM iajs-2002 43 84 : : PUNCT iajs-2002 43 85 the the DET iajs-2002 43 86 discrete discrete ADJ iajs-2002 43 87 state state NOUN iajs-2002 43 88 equation equation NOUN iajs-2002 43 89 ( ( PUNCT iajs-2002 43 90 6 6 NUM iajs-2002 43 91 - - SYM iajs-2002 43 92 7 7 NUM iajs-2002 43 93 ) ) PUNCT iajs-2002 43 94 with with ADP iajs-2002 43 95 fixed fix VERB iajs-2002 43 96 point point NOUN iajs-2002 43 97 and and CCONJ iajs-2002 43 98 for for ADP iajs-2002 43 99 βˆ†π‘‘ βˆ†π‘‘ PROPN iajs-2002 43 100 sufficiently sufficiently ADV iajs-2002 43 101 small small ADJ iajs-2002 43 102 has have VERB iajs-2002 43 103 a a DET iajs-2002 43 104 unique unique ADJ iajs-2002 43 105 solution solution NOUN iajs-2002 43 106 𝑒 𝑒 ADP iajs-2002 43 107 𝑒 𝑒 NOUN iajs-2002 43 108 , , PUNCT iajs-2002 43 109 𝑒 𝑒 X iajs-2002 43 110 , , PUNCT iajs-2002 43 111 … … PUNCT iajs-2002 43 112 , , PUNCT iajs-2002 43 113 𝑒 𝑒 X iajs-2002 43 114 , , PUNCT iajs-2002 43 115 and and CCONJ iajs-2002 43 116 the the DET iajs-2002 43 117 sequence sequence NOUN iajs-2002 43 118 of of ADP iajs-2002 43 119 corrector corrector NOUN iajs-2002 43 120 solutions solution NOUN iajs-2002 43 121 is be AUX iajs-2002 43 122 convergence convergence NOUN iajs-2002 43 123 in in ADP iajs-2002 43 124 ℝ. ℝ. PROPN iajs-2002 43 125 proof proof NOUN iajs-2002 43 126 : : PUNCT iajs-2002 43 127 let let VERB iajs-2002 43 128 𝑒 𝑒 PRON iajs-2002 43 129 𝑒 𝑒 X iajs-2002 43 130 , , PUNCT iajs-2002 43 131 … … PUNCT iajs-2002 43 132 , , PUNCT iajs-2002 43 133 𝑒 𝑒 X iajs-2002 43 134 , , PUNCT iajs-2002 43 135 … … PUNCT iajs-2002 43 136 , , PUNCT iajs-2002 43 137 𝑒 𝑒 NOUN iajs-2002 43 138 and and CCONJ iajs-2002 43 139 𝑣 𝑣 DET iajs-2002 43 140 𝑣 𝑣 X iajs-2002 43 141 , , PUNCT iajs-2002 43 142 … … PUNCT iajs-2002 43 143 , , PUNCT iajs-2002 43 144 𝑣 𝑣 X iajs-2002 43 145 , , PUNCT iajs-2002 43 146 … … PUNCT iajs-2002 43 147 , , PUNCT iajs-2002 43 148 𝑣 𝑣 PRON iajs-2002 43 149 are be AUX iajs-2002 43 150 two two NUM iajs-2002 43 151 solutions solution NOUN iajs-2002 43 152 of of ADP iajs-2002 43 153 ( ( PUNCT iajs-2002 43 154 12 12 NUM iajs-2002 43 155 ) ) PUNCT iajs-2002 43 156 , , PUNCT iajs-2002 43 157 i.e. i.e. X iajs-2002 43 158 ( ( PUNCT iajs-2002 43 159 𝑒 𝑒 NOUN iajs-2002 43 160 𝑒 𝑒 X iajs-2002 43 161 , , PUNCT iajs-2002 43 162 πœ‰ πœ‰ ADP iajs-2002 43 163 βˆ†π‘‘ βˆ†π‘‘ NOUN iajs-2002 43 164 βˆ‡π‘’ βˆ‡π‘’ PROPN iajs-2002 43 165 βˆ‡π‘’ βˆ‡π‘’ PROPN iajs-2002 43 166 , , PUNCT iajs-2002 43 167 βˆ‡πœ‰ βˆ‡πœ‰ PROPN iajs-2002 43 168 βˆ†π‘‘ βˆ†π‘‘ PROPN iajs-2002 43 169 𝐻 𝐻 PROPN iajs-2002 43 170 𝑒 𝑒 NOUN iajs-2002 43 171 𝑒 𝑒 NOUN iajs-2002 43 172 , , PUNCT iajs-2002 43 173 πœ‰ πœ‰ X iajs-2002 43 174 ( ( PUNCT iajs-2002 43 175 14 14 NUM iajs-2002 43 176 ) ) PUNCT iajs-2002 43 177 ( ( PUNCT iajs-2002 43 178 𝑣 𝑣 PRON iajs-2002 43 179 𝑒 𝑒 X iajs-2002 43 180 , , PUNCT iajs-2002 43 181 πœ‰ πœ‰ ADP iajs-2002 43 182 βˆ†π‘‘ βˆ†π‘‘ NOUN iajs-2002 44 1 βˆ‡π‘£ βˆ‡π‘£ PROPN iajs-2002 44 2 βˆ‡π‘’ βˆ‡π‘’ PROPN iajs-2002 44 3 , , PUNCT iajs-2002 44 4 βˆ‡πœ‰ βˆ‡πœ‰ PROPN iajs-2002 44 5 βˆ†π‘‘ βˆ†π‘‘ PROPN iajs-2002 44 6 𝐻 𝐻 PROPN iajs-2002 44 7 𝑣 𝑣 NOUN iajs-2002 44 8 𝑒 𝑒 NOUN iajs-2002 44 9 , , PUNCT iajs-2002 44 10 πœ‰ πœ‰ X iajs-2002 44 11 ( ( PUNCT iajs-2002 44 12 15 15 NUM iajs-2002 44 13 ) ) PUNCT iajs-2002 44 14 by by ADP iajs-2002 44 15 subtracting subtract VERB iajs-2002 44 16 ( ( PUNCT iajs-2002 44 17 15 15 NUM iajs-2002 44 18 ) ) PUNCT iajs-2002 44 19 from from ADP iajs-2002 44 20 ( ( PUNCT iajs-2002 44 21 14 14 NUM iajs-2002 44 22 ) ) PUNCT iajs-2002 44 23 , , PUNCT iajs-2002 44 24 setting set VERB iajs-2002 44 25 πœ‰ πœ‰ PRON iajs-2002 44 26 𝑒 𝑒 ADP iajs-2002 44 27 𝑣 𝑣 X iajs-2002 44 28 in in ADP iajs-2002 44 29 the the DET iajs-2002 44 30 obtained obtain VERB iajs-2002 44 31 equation equation NOUN iajs-2002 44 32 and and CCONJ iajs-2002 44 33 using use VERB iajs-2002 44 34 lipschitz lipschitz NOUN iajs-2002 44 35 condition condition NOUN iajs-2002 44 36 on on ADP iajs-2002 44 37 𝐻 𝐻 PROPN iajs-2002 44 38 with with ADP iajs-2002 44 39 respect respect NOUN iajs-2002 44 40 to to ADP iajs-2002 44 41 for for ADP iajs-2002 44 42 𝑒 𝑒 NOUN iajs-2002 44 43 , , PUNCT iajs-2002 44 44 once once ADV iajs-2002 44 45 get get VERB iajs-2002 44 46 that that PRON iajs-2002 44 47 𝑒 𝑒 ADP iajs-2002 44 48 𝑣 𝑣 X iajs-2002 44 49 + + NOUN iajs-2002 45 1 βˆ†π‘‘ βˆ†π‘‘ PROPN iajs-2002 45 2 βˆ‡π‘’ βˆ‡π‘’ PROPN iajs-2002 45 3 βˆ‡π‘£ βˆ‡π‘£ PROPN iajs-2002 45 4 βˆ†π‘‘πΏ βˆ†π‘‘πΏ NOUN iajs-2002 45 5 𝑒 𝑒 PART iajs-2002 45 6 𝑣 𝑣 PROPN iajs-2002 45 7 , , PUNCT iajs-2002 45 8 𝑒 𝑒 PROPN iajs-2002 45 9 𝑣 𝑣 X iajs-2002 45 10 . . PUNCT iajs-2002 46 1 keep keep VERB iajs-2002 46 2 in in ADP iajs-2002 46 3 mind mind NOUN iajs-2002 46 4 that that SCONJ iajs-2002 46 5 the the DET iajs-2002 46 6 2 2 NUM iajs-2002 46 7 term term NOUN iajs-2002 46 8 in in ADP iajs-2002 46 9 the the DET iajs-2002 46 10 left left ADJ iajs-2002 46 11 hand hand NOUN iajs-2002 46 12 side side NOUN iajs-2002 46 13 ( ( PUNCT iajs-2002 46 14 lhs lhs PROPN iajs-2002 46 15 ) ) PUNCT iajs-2002 46 16 is be AUX iajs-2002 46 17 positive positive ADJ iajs-2002 46 18 and and CCONJ iajs-2002 46 19 then then ADV iajs-2002 46 20 using use VERB iajs-2002 46 21 cauchy cauchy PROPN iajs-2002 46 22 schwarz schwarz NOUN iajs-2002 46 23 ( ( PUNCT iajs-2002 46 24 cs cs PROPN iajs-2002 46 25 ) ) PUNCT iajs-2002 46 26 inequality inequality NOUN iajs-2002 46 27 on on ADP iajs-2002 46 28 the the DET iajs-2002 46 29 rhs rhs PROPN iajs-2002 46 30 of of ADP iajs-2002 46 31 above above ADP iajs-2002 46 32 inequality inequality NOUN iajs-2002 46 33 , , PUNCT iajs-2002 46 34 once once ADV iajs-2002 46 35 get get VERB iajs-2002 46 36 that that PRON iajs-2002 46 37 𝑒 𝑒 ADP iajs-2002 46 38 𝑣 𝑣 PART iajs-2002 46 39 𝛽 𝛽 NOUN iajs-2002 46 40 𝑒 𝑒 ADP iajs-2002 46 41 𝑣 𝑣 NOUN iajs-2002 46 42 , , PUNCT iajs-2002 46 43 where where SCONJ iajs-2002 46 44 𝛽 𝛽 NOUN iajs-2002 46 45 βˆ†π‘‘πΏ βˆ†π‘‘πΏ NOUN iajs-2002 46 46 , , PUNCT iajs-2002 46 47 using use VERB iajs-2002 46 48 ( ( PUNCT iajs-2002 46 49 13 13 NUM iajs-2002 46 50 ) ) PUNCT iajs-2002 46 51 , , PUNCT iajs-2002 46 52 to to PART iajs-2002 46 53 get get VERB iajs-2002 46 54 that that PRON iajs-2002 46 55 π‘ž π‘ž PROPN iajs-2002 46 56 𝑒 𝑒 X iajs-2002 46 57 π‘ž π‘ž X iajs-2002 46 58 𝑣 𝑣 ADP iajs-2002 46 59 𝛽 𝛽 NOUN iajs-2002 46 60 𝑒 𝑒 ADP iajs-2002 46 61 𝑣 𝑣 X iajs-2002 46 62 since since SCONJ iajs-2002 46 63 βˆ†π‘‘ βˆ†π‘‘ PROPN iajs-2002 46 64 is be AUX iajs-2002 46 65 sufficiently sufficiently ADV iajs-2002 46 66 small small ADJ iajs-2002 46 67 and and CCONJ iajs-2002 46 68 𝛽 𝛽 NOUN iajs-2002 46 69 1 1 NUM iajs-2002 46 70 , , PUNCT iajs-2002 46 71 then then ADV iajs-2002 46 72 q q X iajs-2002 46 73 is be AUX iajs-2002 46 74 contractive contractive ADJ iajs-2002 46 75 , , PUNCT iajs-2002 46 76 and and CCONJ iajs-2002 46 77 by by ADP iajs-2002 46 78 theorem theorem NOUN iajs-2002 46 79 ( ( PUNCT iajs-2002 46 80 1 1 X iajs-2002 46 81 ) ) PUNCT iajs-2002 46 82 we we PRON iajs-2002 46 83 get get VERB iajs-2002 46 84 𝑒 𝑒 ADP iajs-2002 46 85 𝑣 𝑣 NOUN iajs-2002 46 86 , , PUNCT iajs-2002 46 87 hence hence ADV iajs-2002 46 88 the the DET iajs-2002 46 89 dseq dseq NOUN iajs-2002 46 90 has have VERB iajs-2002 46 91 a a DET iajs-2002 46 92 unique unique ADJ iajs-2002 46 93 solution solution NOUN iajs-2002 46 94 , , PUNCT iajs-2002 46 95 also also ADV iajs-2002 46 96 since since SCONJ iajs-2002 46 97 𝑒 𝑒 PRON iajs-2002 46 98 ∈ ∈ PROPN iajs-2002 46 99 ℝ ℝ PROPN iajs-2002 46 100 , , PUNCT iajs-2002 46 101 βˆ€ βˆ€ X iajs-2002 47 1 𝑙 𝑙 NOUN iajs-2002 47 2 then then ADV iajs-2002 47 3 mathematics mathematics PROPN iajs-2002 47 4 | | ADV iajs-2002 47 5 130 130 NUM iajs-2002 47 6 ibn ibn PROPN iajs-2002 47 7 al al PROPN iajs-2002 47 8 - - PUNCT iajs-2002 47 9 haitham haitham PROPN iajs-2002 47 10 jour jour X iajs-2002 47 11 . . PROPN iajs-2002 48 1 for for ADP iajs-2002 48 2 pure pure ADJ iajs-2002 48 3 & & CCONJ iajs-2002 48 4 appl appl PROPN iajs-2002 48 5 . . PUNCT iajs-2002 49 1 sci sci PROPN iajs-2002 49 2 . . PROPN iajs-2002 49 3 ihjpas ihjpa VERB iajs-2002 49 4 https://doi.org/10.30526/31.3.2002 https://doi.org/10.30526/31.3.2002 PART iajs-2002 49 5 vol vol NOUN iajs-2002 49 6 . . PUNCT iajs-2002 49 7 31 31 NUM iajs-2002 49 8 ( ( PUNCT iajs-2002 49 9 3 3 NUM iajs-2002 49 10 ) ) PUNCT iajs-2002 49 11 2018 2018 NUM iajs-2002 49 12 π‘ž π‘ž PART iajs-2002 49 13 𝑒 𝑒 X iajs-2002 49 14 𝑒 𝑒 X iajs-2002 49 15 ∈ ∈ PROPN iajs-2002 49 16 ℝ ℝ PROPN iajs-2002 49 17 , , PUNCT iajs-2002 49 18 βˆ€ βˆ€ X iajs-2002 50 1 𝑙 𝑙 PRON iajs-2002 50 2 implies imply VERB iajs-2002 50 3 that that SCONJ iajs-2002 50 4 π‘ž π‘ž X iajs-2002 50 5 𝑒 𝑒 PROPN iajs-2002 50 6 ∈ ∈ PROPN iajs-2002 50 7 𝑅 𝑅 PROPN iajs-2002 50 8 , , PUNCT iajs-2002 50 9 βˆ€ βˆ€ NOUN iajs-2002 50 10 𝑒 𝑒 PRON iajs-2002 50 11 ∈ ∈ PROPN iajs-2002 50 12 ℝ ℝ PROPN iajs-2002 50 13 , , PUNCT iajs-2002 50 14 and and CCONJ iajs-2002 50 15 by by ADP iajs-2002 50 16 using use VERB iajs-2002 50 17 theorem theorem NOUN iajs-2002 50 18 ( ( PUNCT iajs-2002 50 19 3 3 NUM iajs-2002 50 20 ) ) PUNCT iajs-2002 50 21 with with ADP iajs-2002 50 22 s=ℝ s=ℝ NOUN iajs-2002 50 23 , , PUNCT iajs-2002 50 24 we we PRON iajs-2002 50 25 get get VERB iajs-2002 50 26 that that SCONJ iajs-2002 50 27 𝑒 𝑒 NOUN iajs-2002 50 28 is be AUX iajs-2002 50 29 converged converge VERB iajs-2002 50 30 to to ADP iajs-2002 50 31 a a DET iajs-2002 50 32 point point NOUN iajs-2002 50 33 in in ADP iajs-2002 50 34 ℝ ℝ PROPN iajs-2002 50 35 . . PUNCT iajs-2002 51 1 6 6 NUM iajs-2002 51 2 . . PUNCT iajs-2002 51 3 cholesky cholesky ADJ iajs-2002 51 4 reduction reduction NOUN iajs-2002 51 5 order order NOUN iajs-2002 51 6 technique technique NOUN iajs-2002 51 7 this this DET iajs-2002 51 8 technique technique NOUN iajs-2002 51 9 in in ADP iajs-2002 51 10 fact fact NOUN iajs-2002 51 11 is be AUX iajs-2002 51 12 based base VERB iajs-2002 51 13 on on ADP iajs-2002 51 14 an an DET iajs-2002 51 15 idea idea NOUN iajs-2002 51 16 which which PRON iajs-2002 51 17 is be AUX iajs-2002 51 18 introduce introduce VERB iajs-2002 51 19 first first ADV iajs-2002 51 20 in in ADP iajs-2002 51 21 [ [ X iajs-2002 51 22 11 11 NUM iajs-2002 51 23 ] ] PUNCT iajs-2002 51 24 about about ADP iajs-2002 51 25 reducing reduce VERB iajs-2002 51 26 the the DET iajs-2002 51 27 diagonal diagonal ADJ iajs-2002 51 28 elements element NOUN iajs-2002 51 29 of of ADP iajs-2002 51 30 the the DET iajs-2002 51 31 galerkin galerkin ADJ iajs-2002 51 32 matrix matrix NOUN iajs-2002 51 33 into into ADP iajs-2002 51 34 columns column NOUN iajs-2002 51 35 , , PUNCT iajs-2002 51 36 we we PRON iajs-2002 51 37 formulate formulate VERB iajs-2002 51 38 it it PRON iajs-2002 51 39 by by ADP iajs-2002 51 40 the the DET iajs-2002 51 41 following following ADJ iajs-2002 51 42 steps step NOUN iajs-2002 51 43 hence hence ADV iajs-2002 51 44 we we PRON iajs-2002 51 45 called call VERB iajs-2002 51 46 it it PRON iajs-2002 51 47 by by ADP iajs-2002 51 48 the the DET iajs-2002 51 49 cholesky cholesky ADJ iajs-2002 51 50 reduction reduction NOUN iajs-2002 51 51 order order NOUN iajs-2002 51 52 technique technique NOUN iajs-2002 51 53 ( ( PUNCT iajs-2002 51 54 chrot chrot NOUN iajs-2002 51 55 ): ): PUNCT iajs-2002 51 56 first first ADV iajs-2002 51 57 , , PUNCT iajs-2002 51 58 the the DET iajs-2002 51 59 𝑁 𝑁 PROPN iajs-2002 51 60 𝑁 𝑁 PROPN iajs-2002 51 61 obtained obtain VERB iajs-2002 51 62 matrix matrix NOUN iajs-2002 51 63 is be AUX iajs-2002 51 64 reduced reduce VERB iajs-2002 51 65 to to ADP iajs-2002 51 66 𝑁 𝑁 PROPN iajs-2002 51 67 𝑀1 𝑀1 NOUN iajs-2002 51 68 matrix matrix NOUN iajs-2002 51 69 𝐴 𝐴 PROPN iajs-2002 51 70 by by ADP iajs-2002 51 71 transform transform VERB iajs-2002 51 72 the the DET iajs-2002 51 73 lower low ADJ iajs-2002 51 74 diagonals diagonal NOUN iajs-2002 51 75 ( ( PUNCT iajs-2002 51 76 𝑀1 𝑀1 NOUN iajs-2002 51 77 of of ADP iajs-2002 51 78 𝑁 𝑁 PROPN iajs-2002 51 79 𝑁 𝑁 PROPN iajs-2002 51 80 matrix matrix NOUN iajs-2002 51 81 to to ADP iajs-2002 51 82 columns column NOUN iajs-2002 51 83 , , PUNCT iajs-2002 51 84 second second ADJ iajs-2002 51 85 the the DET iajs-2002 51 86 reduction reduction NOUN iajs-2002 51 87 of of ADP iajs-2002 51 88 matrix matrix NOUN iajs-2002 51 89 𝐴 𝐴 PROPN iajs-2002 51 90 is be AUX iajs-2002 51 91 a a DET iajs-2002 51 92 new new ADJ iajs-2002 51 93 𝑁 𝑁 PROPN iajs-2002 51 94 𝑀1 𝑀1 NOUN iajs-2002 51 95 matrix matrix NOUN iajs-2002 51 96 𝑅 𝑅 PROPN iajs-2002 51 97 which which PRON iajs-2002 51 98 is be AUX iajs-2002 51 99 computed compute VERB iajs-2002 51 100 by by ADP iajs-2002 51 101 using use VERB iajs-2002 51 102 the the DET iajs-2002 51 103 following follow VERB iajs-2002 51 104 formula formula NOUN iajs-2002 51 105 : : PUNCT iajs-2002 51 106 for for ADP iajs-2002 51 107 1,2 1,2 NUM iajs-2002 51 108 , , PUNCT iajs-2002 51 109 … … PUNCT iajs-2002 51 110 , , PUNCT iajs-2002 51 111 𝑁 𝑁 PROPN iajs-2002 51 112 , , PUNCT iajs-2002 51 113 𝑗 𝑗 PROPN iajs-2002 51 114 𝑖 𝑖 ADP iajs-2002 51 115 1 1 NUM iajs-2002 51 116 , , PUNCT iajs-2002 51 117 … … PUNCT iajs-2002 51 118 , , PUNCT iajs-2002 51 119 min min PROPN iajs-2002 51 120 𝑖 𝑖 PROPN iajs-2002 51 121 𝑀 𝑀 PROPN iajs-2002 51 122 , , PUNCT iajs-2002 51 123 𝑁 𝑁 PROPN iajs-2002 51 124 β€’ β€’ NOUN iajs-2002 51 125 if if SCONJ iajs-2002 51 126 𝑖 𝑖 ADP iajs-2002 51 127 1 1 NUM iajs-2002 51 128 , , PUNCT iajs-2002 51 129 then then ADV iajs-2002 51 130 𝑅 𝑅 PROPN iajs-2002 51 131 𝐴 𝐴 PROPN iajs-2002 51 132 and and CCONJ iajs-2002 51 133 𝑅 𝑅 PROPN iajs-2002 51 134 , , PUNCT iajs-2002 51 135 𝑙 𝑙 PROPN iajs-2002 51 136 𝑖 𝑖 X iajs-2002 51 137 𝑗 𝑗 VERB iajs-2002 51 138 𝑀1 𝑀1 ADV iajs-2002 51 139 β€’ β€’ NOUN iajs-2002 51 140 if if SCONJ iajs-2002 51 141 𝑖 𝑖 ADP iajs-2002 51 142 1 1 NUM iajs-2002 51 143 , , PUNCT iajs-2002 51 144 then then ADV iajs-2002 51 145 𝑅 𝑅 PROPN iajs-2002 51 146 𝐴 𝐴 PROPN iajs-2002 51 147 βˆ‘ βˆ‘ PROPN iajs-2002 51 148 𝑅 𝑅 PROPN iajs-2002 51 149 , , PUNCT iajs-2002 51 150 𝐾 𝐾 PROPN iajs-2002 51 151 max max PROPN iajs-2002 51 152 𝑖 𝑖 PROPN iajs-2002 51 153 𝑀 𝑀 PROPN iajs-2002 51 154 , , PUNCT iajs-2002 51 155 1 1 NUM iajs-2002 51 156 : : SYM iajs-2002 51 157 𝑖 𝑖 SYM iajs-2002 51 158 1 1 NUM iajs-2002 51 159 𝑅 𝑅 PROPN iajs-2002 51 160 𝐴 𝐴 PROPN iajs-2002 51 161 βˆ‘ βˆ‘ PROPN iajs-2002 51 162 𝑅 𝑅 PROPN iajs-2002 51 163 𝑅 𝑅 PROPN iajs-2002 51 164 , , PUNCT iajs-2002 51 165 𝑠 𝑠 PROPN iajs-2002 51 166 π‘Ÿ π‘Ÿ X iajs-2002 51 167 𝑖 𝑖 SYM iajs-2002 51 168 𝑗 𝑗 VERB iajs-2002 51 169 , , PUNCT iajs-2002 51 170 with with ADP iajs-2002 51 171 𝑗 𝑗 PROPN iajs-2002 51 172 𝐾 𝐾 PROPN iajs-2002 51 173 𝑀 𝑀 PROPN iajs-2002 51 174 . . PUNCT iajs-2002 52 1 example example NOUN iajs-2002 52 2 7 7 NUM iajs-2002 52 3 : : PUNCT iajs-2002 52 4 consider consider VERB iajs-2002 52 5 the the DET iajs-2002 52 6 following follow VERB iajs-2002 52 7 nonlinear nonlinear PROPN iajs-2002 52 8 parabolic parabolic PROPN iajs-2002 52 9 b.v.p b.v.p NOUN iajs-2002 52 10 . . PUNCT iajs-2002 53 1 : : PUNCT iajs-2002 53 2 𝑒 𝑒 X iajs-2002 53 3 βˆ†π‘’ βˆ†π‘’ PROPN iajs-2002 53 4 𝐻 𝐻 PROPN iajs-2002 53 5 οΏ½ οΏ½ PROPN iajs-2002 53 6 βƒ— βƒ— NOUN iajs-2002 53 7 οΏ½ οΏ½ PROPN iajs-2002 53 8 , , PUNCT iajs-2002 53 9 𝑑 𝑑 PROPN iajs-2002 53 10 , , PUNCT iajs-2002 53 11 𝑒 𝑒 NOUN iajs-2002 53 12 , , PUNCT iajs-2002 53 13 where where SCONJ iajs-2002 53 14 οΏ½ οΏ½ NOUN iajs-2002 53 15 βƒ— βƒ— X iajs-2002 53 16 οΏ½ οΏ½ PROPN iajs-2002 53 17 π‘₯ π‘₯ PROPN iajs-2002 53 18 , , PUNCT iajs-2002 53 19 π‘₯ π‘₯ X iajs-2002 53 20 𝑒 𝑒 X iajs-2002 53 21 οΏ½ οΏ½ PROPN iajs-2002 53 22 βƒ— βƒ— NOUN iajs-2002 53 23 οΏ½ οΏ½ PROPN iajs-2002 53 24 , , PUNCT iajs-2002 53 25 𝑑 𝑑 PROPN iajs-2002 53 26 0 0 NUM iajs-2002 53 27 , , PUNCT iajs-2002 53 28 on on ADP iajs-2002 53 29 βˆ‚π‘Š βˆ‚π‘Š ADP iajs-2002 53 30 i i PRON iajs-2002 53 31 𝑒 𝑒 PROPN iajs-2002 53 32 οΏ½ οΏ½ PROPN iajs-2002 53 33 βƒ— βƒ— NOUN iajs-2002 53 34 οΏ½ οΏ½ PROPN iajs-2002 53 35 , , PUNCT iajs-2002 53 36 0 0 NUM iajs-2002 53 37 π‘₯ π‘₯ DET iajs-2002 53 38 π‘₯ π‘₯ NUM iajs-2002 53 39 1 1 NUM iajs-2002 53 40 π‘₯ π‘₯ DET iajs-2002 53 41 1 1 NUM iajs-2002 53 42 π‘₯ π‘₯ NOUN iajs-2002 53 43 , , PUNCT iajs-2002 53 44 on on ADP iajs-2002 53 45 π‘Š π‘Š PRON iajs-2002 53 46 where where SCONJ iajs-2002 53 47 , , PUNCT iajs-2002 53 48 𝐻 𝐻 PROPN iajs-2002 53 49 οΏ½ οΏ½ PROPN iajs-2002 53 50 βƒ— βƒ— NOUN iajs-2002 53 51 οΏ½ οΏ½ PROPN iajs-2002 53 52 , , PUNCT iajs-2002 53 53 𝑑 𝑑 PROPN iajs-2002 53 54 , , PUNCT iajs-2002 53 55 𝑒 𝑒 PROPN iajs-2002 53 56 𝑒 𝑒 PROPN iajs-2002 53 57 π‘₯ π‘₯ PRON iajs-2002 53 58 π‘₯ π‘₯ PRON iajs-2002 53 59 π‘₯ π‘₯ PRON iajs-2002 53 60 π‘₯ π‘₯ PART iajs-2002 53 61 π‘₯ π‘₯ PART iajs-2002 53 62 π‘₯ π‘₯ INTJ iajs-2002 53 63 π‘₯ π‘₯ INTJ iajs-2002 53 64 π‘₯ π‘₯ SYM iajs-2002 53 65 1 1 NUM iajs-2002 53 66 sin sin NOUN iajs-2002 53 67 𝑒 𝑒 PROPN iajs-2002 53 68 π‘₯ π‘₯ PRON iajs-2002 53 69 π‘₯ π‘₯ PRON iajs-2002 53 70 π‘₯ π‘₯ INTJ iajs-2002 53 71 π‘₯ π‘₯ PART iajs-2002 54 1 π‘₯ π‘₯ PART iajs-2002 54 2 π‘₯ π‘₯ INTJ iajs-2002 54 3 π‘₯ π‘₯ PART iajs-2002 54 4 π‘₯ π‘₯ X iajs-2002 54 5 𝑒 𝑒 X iajs-2002 54 6 sin sin VERB iajs-2002 54 7 𝑒 𝑒 ADP iajs-2002 54 8 the the DET iajs-2002 54 9 exact exact ADJ iajs-2002 54 10 solution solution NOUN iajs-2002 54 11 of of ADP iajs-2002 54 12 this this DET iajs-2002 54 13 problem problem NOUN iajs-2002 54 14 is be AUX iajs-2002 54 15 : : PUNCT iajs-2002 54 16 𝑒 𝑒 X iajs-2002 54 17 οΏ½ οΏ½ PROPN iajs-2002 54 18 βƒ— βƒ— NOUN iajs-2002 54 19 οΏ½ οΏ½ PROPN iajs-2002 54 20 , , PUNCT iajs-2002 54 21 𝑑 𝑑 VERB iajs-2002 54 22 π‘₯ π‘₯ VERB iajs-2002 54 23 π‘₯ π‘₯ DET iajs-2002 54 24 1 1 NUM iajs-2002 54 25 π‘₯ π‘₯ SYM iajs-2002 54 26 1 1 NUM iajs-2002 54 27 π‘₯ π‘₯ NOUN iajs-2002 54 28 𝑒 𝑒 ADP iajs-2002 54 29 this this DET iajs-2002 54 30 problem problem NOUN iajs-2002 54 31 solved solve VERB iajs-2002 54 32 using use VERB iajs-2002 54 33 the the DET iajs-2002 54 34 gfem gfem NOUN iajs-2002 54 35 for for ADP iajs-2002 54 36 m=9 m=9 PROPN iajs-2002 54 37 and and CCONJ iajs-2002 54 38 nt=20 nt=20 PROPN iajs-2002 54 39 , , PUNCT iajs-2002 54 40 the the DET iajs-2002 54 41 results result NOUN iajs-2002 54 42 are be AUX iajs-2002 54 43 shown show VERB iajs-2002 54 44 in in ADP iajs-2002 54 45 table table NOUN iajs-2002 54 46 1 1 NUM iajs-2002 54 47 . . PUNCT iajs-2002 54 48 and and CCONJ iajs-2002 54 49 figure figure VERB iajs-2002 54 50 1 1 NUM iajs-2002 54 51 . . PUNCT iajs-2002 55 1 at at ADP iajs-2002 55 2 time time NOUN iajs-2002 55 3 οΏ½ οΏ½ PROPN iajs-2002 55 4 Μ‚ Μ‚ SYM iajs-2002 55 5 οΏ½ οΏ½ PROPN iajs-2002 55 6 0.5 0.5 NUM iajs-2002 55 7 , , PUNCT iajs-2002 55 8 the the DET iajs-2002 55 9 table table NOUN iajs-2002 55 10 shows show VERB iajs-2002 55 11 the the DET iajs-2002 55 12 approximate approximate ADJ iajs-2002 55 13 solution solution NOUN iajs-2002 55 14 𝑒 𝑒 X iajs-2002 55 15 π‘₯ π‘₯ X iajs-2002 55 16 , , PUNCT iajs-2002 55 17 π‘₯ π‘₯ X iajs-2002 55 18 , , PUNCT iajs-2002 55 19 𝑑 𝑑 PROPN iajs-2002 55 20 , , PUNCT iajs-2002 55 21 the the DET iajs-2002 55 22 exact exact ADJ iajs-2002 55 23 solution solution NOUN iajs-2002 55 24 𝑒 𝑒 ADP iajs-2002 55 25 π‘₯ π‘₯ PROPN iajs-2002 55 26 , , PUNCT iajs-2002 55 27 π‘₯ π‘₯ X iajs-2002 55 28 , , PUNCT iajs-2002 55 29 𝑑 𝑑 PROPN iajs-2002 55 30 and and CCONJ iajs-2002 55 31 the the DET iajs-2002 55 32 absolute absolute ADJ iajs-2002 55 33 error error NOUN iajs-2002 55 34 at at ADP iajs-2002 55 35 π‘₯ π‘₯ PROPN iajs-2002 55 36 & & CCONJ iajs-2002 55 37 π‘₯ π‘₯ PROPN iajs-2002 55 38 . . PUNCT iajs-2002 56 1 the the DET iajs-2002 56 2 mat mat NOUN iajs-2002 56 3 lap lap NOUN iajs-2002 56 4 software software NOUN iajs-2002 56 5 is be AUX iajs-2002 56 6 used use VERB iajs-2002 56 7 to to PART iajs-2002 56 8 solve solve VERB iajs-2002 56 9 this this DET iajs-2002 56 10 problem problem NOUN iajs-2002 56 11 , , PUNCT iajs-2002 56 12 it it PRON iajs-2002 56 13 takes take VERB iajs-2002 56 14 5 5 NUM iajs-2002 56 15 - - PUNCT iajs-2002 56 16 hours hour NOUN iajs-2002 56 17 when when SCONJ iajs-2002 56 18 we we PRON iajs-2002 56 19 use use VERB iajs-2002 56 20 the the DET iajs-2002 56 21 chm chm NOUN iajs-2002 56 22 , , PUNCT iajs-2002 56 23 while while SCONJ iajs-2002 56 24 takes take VERB iajs-2002 56 25 1 1 NUM iajs-2002 56 26 - - PUNCT iajs-2002 56 27 hour hour NOUN iajs-2002 56 28 and and CCONJ iajs-2002 56 29 7 7 NUM iajs-2002 56 30 minutes minute NOUN iajs-2002 56 31 when when SCONJ iajs-2002 56 32 we we PRON iajs-2002 56 33 use use VERB iajs-2002 56 34 the the DET iajs-2002 56 35 chrot chrot NOUN iajs-2002 56 36 . . PUNCT iajs-2002 57 1 table table NOUN iajs-2002 57 2 1 1 NUM iajs-2002 57 3 . . X iajs-2002 57 4 comparison comparison NOUN iajs-2002 57 5 between between ADP iajs-2002 57 6 exact exact ADJ iajs-2002 57 7 and and CCONJ iajs-2002 57 8 approximation approximation NOUN iajs-2002 57 9 solutions solution NOUN iajs-2002 57 10 π‘₯ π‘₯ PROPN iajs-2002 57 11 Β  Β  SPACE iajs-2002 57 12 π‘₯ π‘₯ NOUN iajs-2002 57 13 Β  Β  SPACE iajs-2002 57 14 𝑒 𝑒 NOUN iajs-2002 57 15 π‘₯ π‘₯ X iajs-2002 57 16 , , PUNCT iajs-2002 57 17 π‘₯ π‘₯ X iajs-2002 57 18 , , PUNCT iajs-2002 57 19 𝑑 𝑑 VERB iajs-2002 57 20 𝑒 𝑒 X iajs-2002 57 21 π‘₯ π‘₯ X iajs-2002 57 22 , , PUNCT iajs-2002 57 23 π‘₯ π‘₯ X iajs-2002 57 24 , , PUNCT iajs-2002 57 25 𝑑 𝑑 X iajs-2002 57 26 absolute absolute ADJ iajs-2002 57 27 error error NOUN iajs-2002 57 28 π‘₯ π‘₯ ADP iajs-2002 57 29 π‘₯ π‘₯ ADP iajs-2002 57 30 𝑒 𝑒 X iajs-2002 57 31 π‘₯ π‘₯ X iajs-2002 57 32 , , PUNCT iajs-2002 57 33 π‘₯ π‘₯ X iajs-2002 57 34 , , PUNCT iajs-2002 57 35 𝑑 𝑑 VERB iajs-2002 57 36 𝑒 𝑒 X iajs-2002 57 37 π‘₯ π‘₯ X iajs-2002 57 38 , , PUNCT iajs-2002 57 39 π‘₯ π‘₯ X iajs-2002 57 40 , , PUNCT iajs-2002 57 41 𝑑 𝑑 X iajs-2002 57 42 absolute absolute ADJ iajs-2002 57 43 error error NOUN iajs-2002 57 44 0.1 0.1 NUM iajs-2002 57 45 0.1 0.1 NUM iajs-2002 57 46 0.0137 0.0137 NUM iajs-2002 57 47 0.0137 0.0137 NUM iajs-2002 57 48 0.0000 0.0000 NUM iajs-2002 57 49 0.6 0.6 NUM iajs-2002 57 50 0.5 0.5 NUM iajs-2002 57 51 0.1014 0.1014 NUM iajs-2002 57 52 0.1029 0.1029 NUM iajs-2002 57 53 0.0015 0.0015 NUM iajs-2002 57 54 0.2 0.2 NUM iajs-2002 57 55 0.1 0.1 NUM iajs-2002 57 56 0.0243 0.0243 NUM iajs-2002 57 57 0.0247 0.0247 NUM iajs-2002 57 58 0.0004 0.0004 NUM iajs-2002 57 59 0.7 0.7 NUM iajs-2002 57 60 0.5 0.5 NUM iajs-2002 57 61 0.0887 0.0887 NUM iajs-2002 57 62 0.0901 0.0901 NUM iajs-2002 58 1 0.0014 0.0014 NUM iajs-2002 58 2 0.3 0.3 NUM iajs-2002 58 3 0.1 0.1 NUM iajs-2002 58 4 Β  Β  SPACE iajs-2002 58 5 0.0319 0.0319 NUM iajs-2002 58 6 0.0323 0.0323 NUM iajs-2002 58 7 0.0004 0.0004 NUM iajs-2002 58 8 0.8 0.8 NUM iajs-2002 58 9 0.5 0.5 NUM iajs-2002 58 10 0.0676 0.0676 NUM iajs-2002 58 11 0.0687 0.0687 NUM iajs-2002 58 12 0.0011 0.0011 NUM iajs-2002 58 13 0.4 0.4 NUM iajs-2002 58 14 0.1 0.1 NUM iajs-2002 58 15 0.0365 0.0365 NUM iajs-2002 58 16 0.0370 0.0370 NUM iajs-2002 58 17 0.0005 0.0005 NUM iajs-2002 58 18 0.9 0.9 NUM iajs-2002 58 19 0.5 0.5 NUM iajs-2002 58 20 0.0380 0.0380 NUM iajs-2002 58 21 0.0386 0.0386 NUM iajs-2002 58 22 0.0006 0.0006 NUM iajs-2002 58 23 0.5 0.5 NUM iajs-2002 58 24 0.1 0.1 NUM iajs-2002 58 25 0.0380 0.0380 NUM iajs-2002 58 26 0.0386 0.0386 NUM iajs-2002 58 27 0.0006 0.0006 NUM iajs-2002 58 28 0.1 0.1 NUM iajs-2002 58 29 0.6 0.6 NUM iajs-2002 58 30 0.0365 0.0365 NUM iajs-2002 58 31 0.0370 0.0370 NUM iajs-2002 58 32 0.0005 0.0005 NUM iajs-2002 58 33 0.6 0.6 NUM iajs-2002 58 34 0.1 0.1 NUM iajs-2002 58 35 0.0365 0.0365 NUM iajs-2002 58 36 0.0370 0.0370 NUM iajs-2002 58 37 0.0005 0.0005 NUM iajs-2002 58 38 0.2 0.2 NUM iajs-2002 58 39 0.6 0.6 NUM iajs-2002 58 40 0.0649 0.0649 NUM iajs-2002 58 41 0.0659 0.0659 NUM iajs-2002 58 42 0.0010 0.0010 NUM iajs-2002 58 43 0.7 0.7 NUM iajs-2002 58 44 0.1 0.1 NUM iajs-2002 58 45 0.0319 0.0319 NUM iajs-2002 58 46 0.0324 0.0324 NUM iajs-2002 58 47 0.0005 0.0005 NUM iajs-2002 58 48 0.3 0.3 NUM iajs-2002 58 49 0.6 0.6 NUM iajs-2002 58 50 0.0852 0.0852 NUM iajs-2002 58 51 0.0865 0.0865 NUM iajs-2002 58 52 0.0013 0.0013 NUM iajs-2002 58 53 0.8 0.8 NUM iajs-2002 58 54 0.1 0.1 NUM iajs-2002 58 55 0.0243 0.0243 NUM iajs-2002 58 56 0.0247 0.0247 NUM iajs-2002 58 57 0.0004 0.0004 NUM iajs-2002 58 58 0.4 0.4 NUM iajs-2002 58 59 0.6 0.6 NUM iajs-2002 58 60 0.0974 0.0974 NUM iajs-2002 58 61 0.0988 0.0988 NUM iajs-2002 58 62 0.0014 0.0014 NUM iajs-2002 58 63 0.9 0.9 NUM iajs-2002 58 64 0.1 0.1 NUM iajs-2002 58 65 0.0137 0.0137 NUM iajs-2002 58 66 0.0140 0.0140 NUM iajs-2002 58 67 0.0003 0.0003 NUM iajs-2002 58 68 0.5 0.5 NUM iajs-2002 58 69 0.6 0.6 NUM iajs-2002 58 70 0.1014 0.1014 NUM iajs-2002 58 71 0.1029 0.1029 NUM 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iajs-2002 61 241 0.0015 0.0015 NUM iajs-2002 61 242 0.9 0.9 NUM iajs-2002 61 243 0.9 0.9 NUM iajs-2002 61 244 0.0137 0.0137 NUM iajs-2002 61 245 0.0137 0.0137 NUM iajs-2002 61 246 0.0000 0.0000 NUM iajs-2002 61 247 0.5 0.5 NUM iajs-2002 61 248 0.5 0.5 NUM iajs-2002 61 249 0.1057 0.1057 NUM iajs-2002 61 250 0.1072 0.1072 NUM iajs-2002 61 251 0.0015 0.0015 NUM iajs-2002 61 252 figure figure NOUN iajs-2002 61 253 1 1 NUM iajs-2002 61 254 . . PUNCT iajs-2002 62 1 ( ( PUNCT iajs-2002 62 2 a a X iajs-2002 62 3 ) ) PUNCT iajs-2002 62 4 shows show VERB iajs-2002 62 5 the the DET iajs-2002 62 6 approximation approximation NOUN iajs-2002 62 7 solution solution NOUN iajs-2002 62 8 and and CCONJ iajs-2002 62 9 ( ( PUNCT iajs-2002 62 10 b b X iajs-2002 62 11 ) ) PUNCT iajs-2002 62 12 shows show VERB iajs-2002 62 13 the the DET iajs-2002 62 14 exact exact ADJ iajs-2002 62 15 solution solution NOUN iajs-2002 62 16 example example NOUN iajs-2002 62 17 8 8 NUM iajs-2002 62 18 : : PUNCT iajs-2002 62 19 consider consider VERB iajs-2002 62 20 the the DET iajs-2002 62 21 following follow VERB iajs-2002 62 22 nonlinear nonlinear ADJ iajs-2002 62 23 b.v.p b.v.p NOUN iajs-2002 62 24 . . PUNCT iajs-2002 63 1 : : PUNCT iajs-2002 64 1 𝑒 𝑒 X iajs-2002 64 2 βˆ†π‘’ βˆ†π‘’ PROPN iajs-2002 64 3 𝐻 𝐻 PROPN iajs-2002 64 4 οΏ½ οΏ½ PROPN iajs-2002 64 5 βƒ— βƒ— NOUN iajs-2002 64 6 οΏ½ οΏ½ PROPN iajs-2002 64 7 , , PUNCT iajs-2002 64 8 𝑑 𝑑 PROPN iajs-2002 64 9 , , PUNCT iajs-2002 64 10 𝑒 𝑒 NOUN iajs-2002 64 11 , , PUNCT iajs-2002 64 12 where where SCONJ iajs-2002 64 13 οΏ½ οΏ½ NOUN iajs-2002 64 14 βƒ— βƒ— X iajs-2002 64 15 οΏ½ οΏ½ PROPN iajs-2002 64 16 π‘₯ π‘₯ PROPN iajs-2002 64 17 , , PUNCT iajs-2002 64 18 π‘₯ π‘₯ PROPN iajs-2002 64 19 associated associate VERB iajs-2002 64 20 with with ADP iajs-2002 64 21 the the DET iajs-2002 64 22 i.c i.c PROPN iajs-2002 64 23 and and CCONJ iajs-2002 64 24 b.c b.c PROPN iajs-2002 64 25 𝑒 𝑒 PROPN iajs-2002 64 26 οΏ½ οΏ½ PROPN iajs-2002 64 27 βƒ— βƒ— NOUN iajs-2002 64 28 οΏ½ οΏ½ PROPN iajs-2002 64 29 , , PUNCT iajs-2002 64 30 𝑑 𝑑 PROPN iajs-2002 64 31 0 0 NUM iajs-2002 64 32 , , PUNCT iajs-2002 64 33 on on ADP iajs-2002 64 34 βˆ‚π‘Š βˆ‚π‘Š ADP iajs-2002 64 35 i i PRON iajs-2002 64 36 𝑒 𝑒 PROPN iajs-2002 64 37 οΏ½ οΏ½ PROPN iajs-2002 64 38 βƒ— βƒ— NOUN iajs-2002 64 39 οΏ½ οΏ½ PROPN iajs-2002 64 40 , , PUNCT iajs-2002 64 41 0 0 NUM iajs-2002 64 42 0 0 NUM iajs-2002 64 43 , , PUNCT iajs-2002 64 44 on on ADP iajs-2002 64 45 π‘Š π‘Š PRON iajs-2002 64 46 where where SCONJ iajs-2002 64 47 𝐻 𝐻 PROPN iajs-2002 64 48 οΏ½ οΏ½ NOUN iajs-2002 64 49 βƒ— βƒ— NOUN iajs-2002 64 50 οΏ½ οΏ½ PROPN iajs-2002 64 51 , , PUNCT iajs-2002 64 52 𝑑 𝑑 PROPN iajs-2002 64 53 , , PUNCT iajs-2002 64 54 𝑒 𝑒 VERB iajs-2002 64 55 π‘₯ π‘₯ X iajs-2002 64 56 π‘₯ π‘₯ VERB iajs-2002 64 57 π‘₯ π‘₯ X iajs-2002 64 58 π‘₯ π‘₯ PART iajs-2002 64 59 π‘₯ π‘₯ PART iajs-2002 64 60 π‘₯ π‘₯ INTJ iajs-2002 64 61 π‘₯ π‘₯ X iajs-2002 64 62 π‘₯ π‘₯ SYM iajs-2002 64 63 1 1 NUM iajs-2002 64 64 𝑑 𝑑 PROPN iajs-2002 64 65 sin sin NOUN iajs-2002 64 66 𝑑 𝑑 NOUN iajs-2002 64 67 π‘₯ π‘₯ VERB iajs-2002 64 68 π‘₯ π‘₯ PRON iajs-2002 64 69 π‘₯ π‘₯ X iajs-2002 64 70 π‘₯ π‘₯ PART iajs-2002 64 71 π‘₯ π‘₯ PART iajs-2002 64 72 π‘₯ π‘₯ PART iajs-2002 64 73 π‘₯ π‘₯ NOUN iajs-2002 64 74 π‘₯ π‘₯ NOUN iajs-2002 64 75 2𝑑 2𝑑 NOUN iajs-2002 65 1 π‘₯ π‘₯ INTJ iajs-2002 65 2 π‘₯ π‘₯ INTJ iajs-2002 65 3 π‘₯ π‘₯ X iajs-2002 65 4 π‘₯ π‘₯ X iajs-2002 65 5 𝑒 𝑒 X iajs-2002 65 6 sin sin VERB iajs-2002 65 7 𝑒 𝑒 ADP iajs-2002 65 8 the the DET iajs-2002 65 9 exact exact ADJ iajs-2002 65 10 solution solution NOUN iajs-2002 65 11 of of ADP iajs-2002 65 12 this this DET iajs-2002 65 13 problem problem NOUN iajs-2002 65 14 is be AUX iajs-2002 65 15 : : PUNCT iajs-2002 65 16 𝑒 𝑒 X iajs-2002 65 17 οΏ½ οΏ½ PROPN iajs-2002 65 18 βƒ— βƒ— NOUN iajs-2002 65 19 οΏ½ οΏ½ PROPN iajs-2002 65 20 , , PUNCT iajs-2002 65 21 𝑑 𝑑 X iajs-2002 65 22 π‘₯ π‘₯ VERB iajs-2002 65 23 π‘₯ π‘₯ PUNCT iajs-2002 65 24 𝑑 𝑑 PROPN iajs-2002 65 25 1 1 NUM iajs-2002 65 26 π‘₯ π‘₯ DET iajs-2002 65 27 1 1 NUM iajs-2002 65 28 π‘₯ π‘₯ NOUN iajs-2002 65 29 0 0 NUM iajs-2002 65 30 0.5 0.5 NUM iajs-2002 65 31 1 1 NUM iajs-2002 65 32 0 0 NUM iajs-2002 65 33 0.5 0.5 NUM iajs-2002 65 34 1 1 NUM iajs-2002 65 35 0 0 NUM iajs-2002 65 36 0.02 0.02 NUM iajs-2002 65 37 0.04 0.04 NUM iajs-2002 65 38 0.06 0.06 NUM iajs-2002 65 39 0.08 0.08 NUM iajs-2002 65 40 0.1 0.1 NUM iajs-2002 65 41 0.12 0.12 NUM iajs-2002 65 42 a a DET iajs-2002 65 43 0 0 NUM iajs-2002 65 44 0.5 0.5 NUM iajs-2002 65 45 1 1 NUM iajs-2002 65 46 0 0 NUM iajs-2002 65 47 0.5 0.5 NUM iajs-2002 65 48 1 1 NUM iajs-2002 65 49 0 0 NUM iajs-2002 65 50 0.02 0.02 NUM iajs-2002 65 51 0.04 0.04 NUM iajs-2002 65 52 0.06 0.06 NUM iajs-2002 65 53 0.08 0.08 NUM iajs-2002 65 54 0.1 0.1 NUM iajs-2002 65 55 0.12 0.12 NUM iajs-2002 65 56 b b NOUN iajs-2002 65 57 mathematics mathematic NOUN iajs-2002 65 58 | | ADV iajs-2002 65 59 132 132 NUM iajs-2002 65 60 ibn ibn PROPN iajs-2002 65 61 al al PROPN iajs-2002 65 62 - - PUNCT iajs-2002 65 63 haitham haitham PROPN iajs-2002 65 64 jour jour X iajs-2002 65 65 . . PROPN iajs-2002 66 1 for for ADP iajs-2002 66 2 pure pure ADJ iajs-2002 66 3 & & CCONJ iajs-2002 66 4 appl appl PROPN iajs-2002 66 5 . . PUNCT iajs-2002 67 1 sci sci PROPN iajs-2002 67 2 . . PROPN iajs-2002 67 3 ihjpas ihjpa VERB iajs-2002 67 4 https://doi.org/10.30526/31.3.2002 https://doi.org/10.30526/31.3.2002 PART iajs-2002 67 5 vol vol NOUN iajs-2002 67 6 . . 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PUNCT iajs-2002 68 1 at at ADP iajs-2002 68 2 οΏ½ οΏ½ PROPN iajs-2002 68 3 Μ‚ Μ‚ SYM iajs-2002 68 4 οΏ½ οΏ½ PROPN iajs-2002 68 5 0.5 0.5 NUM iajs-2002 68 6 , , PUNCT iajs-2002 68 7 the the DET iajs-2002 68 8 table table NOUN iajs-2002 68 9 shows show VERB iajs-2002 68 10 the the DET iajs-2002 68 11 approximate approximate ADJ iajs-2002 68 12 solution solution NOUN iajs-2002 68 13 𝑒 𝑒 X iajs-2002 68 14 π‘₯ π‘₯ X iajs-2002 68 15 , , PUNCT iajs-2002 68 16 π‘₯ π‘₯ X iajs-2002 68 17 , , PUNCT iajs-2002 68 18 𝑑 𝑑 PROPN iajs-2002 68 19 , , PUNCT iajs-2002 68 20 the the DET iajs-2002 68 21 exact exact ADJ iajs-2002 68 22 solution solution NOUN iajs-2002 68 23 𝑒 𝑒 ADP iajs-2002 68 24 π‘₯ π‘₯ PROPN iajs-2002 68 25 , , PUNCT iajs-2002 68 26 π‘₯ π‘₯ X iajs-2002 68 27 , , PUNCT iajs-2002 68 28 𝑑 𝑑 PROPN iajs-2002 68 29 and and CCONJ iajs-2002 68 30 the the DET iajs-2002 68 31 absolute absolute ADJ iajs-2002 68 32 error error NOUN iajs-2002 68 33 at at ADP iajs-2002 68 34 π‘₯ π‘₯ PROPN iajs-2002 68 35 & & CCONJ iajs-2002 68 36 π‘₯ π‘₯ PROPN iajs-2002 68 37 . . PUNCT iajs-2002 69 1 this this DET iajs-2002 69 2 problem problem NOUN iajs-2002 69 3 is be AUX iajs-2002 69 4 take take VERB iajs-2002 69 5 13 13 NUM iajs-2002 69 6 hours hour NOUN iajs-2002 69 7 when when SCONJ iajs-2002 69 8 we we PRON iajs-2002 69 9 use use VERB iajs-2002 69 10 the the DET iajs-2002 69 11 chm chm NOUN iajs-2002 69 12 to to PART iajs-2002 69 13 solve solve VERB iajs-2002 69 14 the the DET iajs-2002 69 15 pct pct NOUN iajs-2002 69 16 , , PUNCT iajs-2002 69 17 while while SCONJ iajs-2002 69 18 takes take VERB iajs-2002 69 19 3 3 NUM iajs-2002 69 20 hours hour NOUN iajs-2002 69 21 and and CCONJ iajs-2002 69 22 27 27 NUM iajs-2002 69 23 minute minute NOUN iajs-2002 69 24 when when SCONJ iajs-2002 69 25 we we PRON iajs-2002 69 26 use use VERB iajs-2002 69 27 the the DET iajs-2002 69 28 chrot chrot NOUN iajs-2002 69 29 . . PUNCT iajs-2002 69 30 table table NOUN iajs-2002 69 31 2 2 NUM iajs-2002 69 32 . . X iajs-2002 69 33 comparison comparison NOUN iajs-2002 69 34 between between ADP iajs-2002 69 35 exact exact ADJ iajs-2002 69 36 and and CCONJ iajs-2002 69 37 approximation approximation NOUN iajs-2002 69 38 solutions solution NOUN iajs-2002 69 39 π‘₯ π‘₯ NOUN iajs-2002 69 40 π‘₯ π‘₯ X iajs-2002 69 41 𝑒 𝑒 X iajs-2002 69 42 π‘₯ π‘₯ X iajs-2002 69 43 , , PUNCT iajs-2002 69 44 π‘₯ π‘₯ X iajs-2002 69 45 , , PUNCT iajs-2002 69 46 𝑑 𝑑 VERB iajs-2002 69 47 𝑒 𝑒 X iajs-2002 69 48 π‘₯ π‘₯ X iajs-2002 69 49 , , PUNCT iajs-2002 69 50 π‘₯ π‘₯ X iajs-2002 69 51 , , PUNCT iajs-2002 69 52 𝑑 𝑑 X iajs-2002 69 53 absolute absolute ADJ iajs-2002 69 54 error error NOUN iajs-2002 69 55 π‘₯ π‘₯ ADP iajs-2002 69 56 π‘₯ π‘₯ ADP iajs-2002 69 57 𝑒 𝑒 X iajs-2002 69 58 π‘₯ π‘₯ X iajs-2002 69 59 , , PUNCT iajs-2002 69 60 π‘₯ π‘₯ X iajs-2002 69 61 , , PUNCT iajs-2002 69 62 𝑑 𝑑 VERB iajs-2002 69 63 𝑒 𝑒 X iajs-2002 69 64 π‘₯ π‘₯ X iajs-2002 69 65 , , PUNCT iajs-2002 69 66 π‘₯ π‘₯ X iajs-2002 69 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107 1 0.0009 0.0009 NUM iajs-2002 107 2 0.3 0.3 NUM iajs-2002 107 3 0.8 0.8 NUM iajs-2002 107 4 -0.0176 -0.0176 NOUN iajs-2002 107 5 -0.0184 -0.0184 VERB iajs-2002 108 1 0.0008 0.0008 NUM iajs-2002 108 2 0.8 0.8 NUM iajs-2002 108 3 0.3 0.3 NUM iajs-2002 108 4 -0.0176 -0.0176 NOUN iajs-2002 108 5 -0.0184 -0.0184 PUNCT iajs-2002 109 1 0.0008 0.0008 NUM iajs-2002 109 2 0.4 0.4 NUM iajs-2002 109 3 0.8 0.8 NUM iajs-2002 109 4 -0.0202 -0.0202 NOUN iajs-2002 109 5 -0.0210 -0.0210 PROPN iajs-2002 110 1 0.0008 0.0008 NUM iajs-2002 110 2 0.9 0.9 NUM iajs-2002 110 3 0.3 0.3 NUM iajs-2002 110 4 -0.0099 -0.0099 NOUN iajs-2002 110 5 -0.0103 -0.0103 PROPN iajs-2002 111 1 0.0004 0.0004 NUM iajs-2002 111 2 0.5 0.5 NUM iajs-2002 111 3 0.8 0.8 NUM iajs-2002 111 4 -0.0210 -0.0210 NOUN iajs-2002 111 5 -0.0219 -0.0219 NOUN iajs-2002 112 1 0.0009 0.0009 NUM iajs-2002 112 2 0.1 0.1 NUM iajs-2002 112 3 0.4 0.4 NUM iajs-2002 112 4 -0.0113 -0.0113 PROPN iajs-2002 112 5 -0.0118 -0.0118 PUNCT iajs-2002 113 1 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0.0013 NUM iajs-2002 119 2 0.9 0.9 NUM iajs-2002 119 3 0.8 0.8 NUM iajs-2002 119 4 -0.0076 -0.0076 NOUN iajs-2002 119 5 -0.0079 -0.0079 NOUN iajs-2002 119 6 0.0003 0.0003 NUM iajs-2002 119 7 0.5 0.5 NUM iajs-2002 119 8 0.4 0.4 NUM iajs-2002 119 9 -0.0315 -0.0315 NOUN iajs-2002 119 10 -0.0328 -0.0328 PROPN iajs-2002 120 1 0.0013 0.0013 NUM iajs-2002 120 2 0.1 0.1 NUM iajs-2002 120 3 0.9 0.9 NUM iajs-2002 120 4 -0.0043 -0.0043 NOUN iajs-2002 120 5 -0.0044 -0.0044 PUNCT iajs-2002 121 1 0.0001 0.0001 NUM iajs-2002 121 2 0.6 0.6 NUM iajs-2002 121 3 0.4 0.4 NUM iajs-2002 121 4 -0.0302 -0.0302 NOUN iajs-2002 121 5 -0.0315 -0.0315 PROPN iajs-2002 122 1 0.0013 0.0013 NUM iajs-2002 122 2 0.2 0.2 NUM iajs-2002 122 3 0.9 0.9 NUM iajs-2002 122 4 -0.0076 -0.0076 NOUN iajs-2002 122 5 -0.0079 -0.0079 NOUN iajs-2002 122 6 0.0003 0.0003 NUM iajs-2002 122 7 0.7 0.7 NUM iajs-2002 122 8 0.4 0.4 NUM iajs-2002 122 9 -0.0265 -0.0265 PUNCT iajs-2002 122 10 -0.0276 -0.0276 PROPN iajs-2002 123 1 0.0011 0.0011 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iajs-2002 129 2 0.6 0.6 NUM iajs-2002 129 3 0.9 0.9 NUM iajs-2002 129 4 -0.0113 -0.0113 NOUN iajs-2002 129 5 -0.0118 -0.0118 PUNCT iajs-2002 130 1 0.0005 0.0005 NUM iajs-2002 130 2 0.2 0.2 NUM iajs-2002 130 3 0.5 0.5 NUM iajs-2002 130 4 -0.0210 -0.0210 NOUN iajs-2002 130 5 -0.0219 -0.0219 NOUN iajs-2002 131 1 0.0009 0.0009 NUM iajs-2002 131 2 0.7 0.7 NUM iajs-2002 131 3 0.9 0.9 NUM iajs-2002 131 4 -0.0099 -0.0099 NOUN iajs-2002 131 5 -0.0103 -0.0103 PROPN iajs-2002 132 1 0.0004 0.0004 NUM iajs-2002 132 2 0.3 0.3 NUM iajs-2002 132 3 0.5 0.5 NUM iajs-2002 132 4 -0.0276 -0.0276 NOUN iajs-2002 132 5 -0.0287 -0.0287 PROPN iajs-2002 133 1 0.0011 0.0011 NUM iajs-2002 133 2 0.8 0.8 NUM iajs-2002 133 3 0.9 0.9 NUM iajs-2002 133 4 -0.0076 -0.0076 NOUN iajs-2002 133 5 -0.0079 -0.0079 NOUN iajs-2002 133 6 0.0003 0.0003 NUM iajs-2002 133 7 0.4 0.4 NUM iajs-2002 133 8 0.5 0.5 NUM iajs-2002 133 9 -0.0315 -0.0315 NOUN iajs-2002 133 10 -0.0328 -0.0328 PROPN iajs-2002 134 1 0.0013 0.0013 NUM iajs-2002 134 2 0.9 0.9 NUM iajs-2002 134 3 0.9 0.9 NUM iajs-2002 134 4 -0.0043 -0.0043 NOUN iajs-2002 134 5 -0.0044 -0.0044 PUNCT iajs-2002 135 1 0.0001 0.0001 NUM iajs-2002 135 2 0.5 0.5 NUM iajs-2002 135 3 0.5 0.5 NUM iajs-2002 135 4 -0.0328 -0.0328 NOUN iajs-2002 135 5 -0.0342 -0.0342 PUNCT iajs-2002 136 1 0.0014 0.0014 NUM iajs-2002 136 2 mathematics mathematic NOUN iajs-2002 136 3 | | ADV iajs-2002 136 4 133 133 NUM iajs-2002 136 5 ibn ibn PROPN iajs-2002 136 6 al al PROPN iajs-2002 136 7 - - PUNCT iajs-2002 136 8 haitham haitham PROPN iajs-2002 136 9 jour jour X iajs-2002 136 10 . . PROPN iajs-2002 136 11 for for ADP iajs-2002 136 12 pure pure ADJ iajs-2002 136 13 & & CCONJ iajs-2002 136 14 appl appl PROPN iajs-2002 136 15 . . PUNCT iajs-2002 137 1 sci sci PROPN iajs-2002 137 2 . . PROPN iajs-2002 137 3 ihjpas ihjpa VERB iajs-2002 137 4 https://doi.org/10.30526/31.3.2002 https://doi.org/10.30526/31.3.2002 PART iajs-2002 137 5 vol vol NOUN iajs-2002 137 6 . . PUNCT iajs-2002 137 7 31 31 NUM iajs-2002 137 8 ( ( PUNCT iajs-2002 137 9 3 3 NUM iajs-2002 137 10 ) ) SYM iajs-2002 137 11 2018 2018 NUM iajs-2002 137 12 figure figure NOUN iajs-2002 137 13 2 2 NUM iajs-2002 137 14 . . PUNCT iajs-2002 138 1 ( ( PUNCT iajs-2002 138 2 a a X iajs-2002 138 3 ) ) PUNCT iajs-2002 138 4 shows show VERB iajs-2002 138 5 the the DET iajs-2002 138 6 approximation approximation NOUN iajs-2002 138 7 solution solution NOUN iajs-2002 138 8 and and CCONJ iajs-2002 138 9 ( ( PUNCT iajs-2002 138 10 b b X iajs-2002 138 11 ) ) PUNCT iajs-2002 138 12 shows show VERB iajs-2002 138 13 the the DET iajs-2002 138 14 exact exact ADJ iajs-2002 138 15 solution solution NOUN iajs-2002 138 16 7 7 NUM iajs-2002 138 17 . . PUNCT iajs-2002 138 18 conclusion conclusion NOUN iajs-2002 138 19 β€’ β€’ ADP iajs-2002 138 20 the the DET iajs-2002 138 21 gfem gfem NOUN iajs-2002 138 22 associated associate VERB iajs-2002 138 23 with with ADP iajs-2002 138 24 the the DET iajs-2002 138 25 pct pct NOUN iajs-2002 138 26 is be AUX iajs-2002 138 27 suitable suitable ADJ iajs-2002 138 28 , , PUNCT iajs-2002 138 29 efficient efficient ADJ iajs-2002 138 30 and and CCONJ iajs-2002 138 31 very very ADV iajs-2002 138 32 fast fast ADV iajs-2002 138 33 to to PART iajs-2002 138 34 solve solve VERB iajs-2002 138 35 the the DET iajs-2002 138 36 nonlinear nonlinear ADJ iajs-2002 138 37 parabolic parabolic PROPN iajs-2002 138 38 boundary boundary ADJ iajs-2002 138 39 value value NOUN iajs-2002 138 40 problems problem NOUN iajs-2002 138 41 . . PUNCT iajs-2002 139 1 β€’ β€’ NOUN iajs-2002 139 2 the the DET iajs-2002 139 3 chrot chrot NOUN iajs-2002 139 4 is be AUX iajs-2002 139 5 very very ADV iajs-2002 139 6 fast fast ADJ iajs-2002 139 7 than than ADP iajs-2002 139 8 the the DET iajs-2002 139 9 chm chm NOUN iajs-2002 139 10 with with ADP iajs-2002 139 11 same same ADJ iajs-2002 139 12 results result NOUN iajs-2002 139 13 and and CCONJ iajs-2002 139 14 this this PRON iajs-2002 139 15 is be AUX iajs-2002 139 16 important important ADJ iajs-2002 139 17 when when SCONJ iajs-2002 139 18 we we PRON iajs-2002 139 19 have have VERB iajs-2002 139 20 problems problem NOUN iajs-2002 139 21 gives give VERB iajs-2002 139 22 very very ADV iajs-2002 139 23 large large ADJ iajs-2002 139 24 algebraic algebraic ADJ iajs-2002 139 25 systems system NOUN iajs-2002 139 26 which which PRON iajs-2002 139 27 take take VERB iajs-2002 139 28 a a DET iajs-2002 139 29 long long ADJ iajs-2002 139 30 time time NOUN iajs-2002 139 31 in in ADP iajs-2002 139 32 the the DET iajs-2002 139 33 classical classical ADJ iajs-2002 139 34 chm chm NOUN iajs-2002 139 35 . . PROPN iajs-2002 140 1 β€’ β€’ NUM iajs-2002 140 2 the the DET iajs-2002 140 3 value value NOUN iajs-2002 140 4 of of ADP iajs-2002 140 5 οΏ½ οΏ½ PROPN iajs-2002 140 6 Μ‚ Μ‚ SYM iajs-2002 140 7 οΏ½ οΏ½ NOUN iajs-2002 140 8 is be AUX iajs-2002 140 9 chose choose VERB iajs-2002 140 10 arbitral arbitral ADJ iajs-2002 140 11 in in ADP iajs-2002 140 12 the the DET iajs-2002 140 13 interval interval NOUN iajs-2002 140 14 i i PRON iajs-2002 140 15 , , PUNCT iajs-2002 140 16 same same ADJ iajs-2002 140 17 results result NOUN iajs-2002 140 18 with with ADP iajs-2002 140 19 same same ADJ iajs-2002 140 20 accuracy accuracy NOUN iajs-2002 140 21 will will AUX iajs-2002 140 22 obtained obtain VERB iajs-2002 140 23 if if SCONJ iajs-2002 140 24 we we PRON iajs-2002 140 25 can can AUX iajs-2002 140 26 take take VERB iajs-2002 140 27 any any DET iajs-2002 140 28 other other ADJ iajs-2002 140 29 value value NOUN iajs-2002 140 30 of of ADP iajs-2002 140 31 οΏ½ οΏ½ PROPN iajs-2002 140 32 Μ‚ Μ‚ VERB iajs-2002 140 33 οΏ½ οΏ½ NOUN iajs-2002 140 34 provided provide VERB iajs-2002 140 35 this this DET iajs-2002 140 36 value value NOUN iajs-2002 140 37 belong belong VERB iajs-2002 140 38 to to ADP iajs-2002 140 39 i. i. PROPN iajs-2002 140 40 acknowledgement acknowledgement PROPN iajs-2002 140 41 : : PUNCT iajs-2002 140 42 the the DET iajs-2002 140 43 authors author NOUN iajs-2002 140 44 thank thank VERB iajs-2002 140 45 prof prof PROPN iajs-2002 140 46 . . PUNCT iajs-2002 141 1 dr dr PROPN iajs-2002 141 2 . . PROPN iajs-2002 141 3 i. i. PROPN iajs-2002 141 4 chryssoverghi chryssoverghi PROPN iajs-2002 141 5 for for ADP iajs-2002 141 6 fruitful fruitful ADJ iajs-2002 141 7 discussion discussion NOUN iajs-2002 141 8 . . PUNCT iajs-2002 142 1 references reference NOUN iajs-2002 142 2 1 1 NUM iajs-2002 142 3 . . PUNCT iajs-2002 142 4 karlsen karlsen PROPN iajs-2002 142 5 k. k. PROPN iajs-2002 142 6 h. h. PROPN iajs-2002 142 7 ; ; PUNCT iajs-2002 142 8 riserbo riserbo NOUN iajs-2002 142 9 , , PUNCT iajs-2002 142 10 n. n. PROPN iajs-2002 142 11 h. h. PROPN iajs-2002 142 12 corrected correct VERB iajs-2002 142 13 operator operator NOUN iajs-2002 142 14 splitting splitting NOUN iajs-2002 142 15 for for ADP iajs-2002 142 16 nonlinear nonlinear ADJ iajs-2002 142 17 parabolic parabolic ADJ iajs-2002 142 18 equations equation NOUN iajs-2002 142 19 . . PUNCT iajs-2002 143 1 siam siam PROPN iajs-2002 143 2 j. j. PROPN iajs-2002 143 3 numr numr PROPN iajs-2002 143 4 . . PUNCT iajs-2002 144 1 anal anal ADJ iajs-2002 144 2 . . PUNCT iajs-2002 145 1 2000 2000 NUM iajs-2002 145 2 , , PUNCT iajs-2002 145 3 37 37 NUM iajs-2002 145 4 , , PUNCT iajs-2002 145 5 3 3 NUM iajs-2002 145 6 , , PUNCT iajs-2002 145 7 980–1003 980–1003 NUM iajs-2002 145 8 . . NOUN iajs-2002 145 9 2 2 NUM iajs-2002 145 10 . . X iajs-2002 145 11 pao pao PROPN iajs-2002 145 12 , , PUNCT iajs-2002 145 13 c. c. PROPN iajs-2002 145 14 v. v. ADP iajs-2002 145 15 numerical numerical ADJ iajs-2002 145 16 methods method NOUN iajs-2002 145 17 for for ADP iajs-2002 145 18 time time NOUN iajs-2002 145 19 - - PUNCT iajs-2002 145 20 periodic periodic ADJ iajs-2002 145 21 solutions solution NOUN iajs-2002 145 22 of of ADP iajs-2002 145 23 nonlinear nonlinear ADJ iajs-2002 145 24 parabolic parabolic PROPN iajs-2002 145 25 boundary boundary ADJ iajs-2002 145 26 value value NOUN iajs-2002 145 27 problems problem NOUN iajs-2002 145 28 . . PUNCT iajs-2002 146 1 siam siam PROPN iajs-2002 146 2 journal journal PROPN iajs-2002 146 3 on on ADP iajs-2002 146 4 numerical numerical ADJ iajs-2002 146 5 analysis analysis NOUN iajs-2002 146 6 . . PUNCT iajs-2002 147 1 2001 2001 NUM iajs-2002 147 2 , , PUNCT iajs-2002 147 3 39 39 NUM iajs-2002 147 4 , , PUNCT iajs-2002 147 5 2 2 NUM iajs-2002 147 6 , , PUNCT iajs-2002 147 7 647667 647667 NUM iajs-2002 147 8 . . PUNCT iajs-2002 148 1 3 3 X iajs-2002 148 2 . . X iajs-2002 148 3 alam alam PROPN iajs-2002 148 4 , , PUNCT iajs-2002 148 5 j. j. PROPN iajs-2002 148 6 m. m. PROPN iajs-2002 148 7 ; ; PUNCT iajs-2002 148 8 kevlahan kevlahan PROPN iajs-2002 148 9 , , PUNCT iajs-2002 148 10 n. n. PROPN iajs-2002 148 11 k.-r k.-r PROPN iajs-2002 148 12 . . PUNCT iajs-2002 148 13 ; ; PUNCT iajs-2002 149 1 vasilyev vasilyev VERB iajs-2002 149 2 , , PUNCT iajs-2002 149 3 o. o. PROPN iajs-2002 149 4 v. v. ADP iajs-2002 149 5 simultaneous simultaneous ADJ iajs-2002 149 6 space space NOUN iajs-2002 149 7 – – PUNCT iajs-2002 149 8 time time NOUN iajs-2002 149 9 adaptive adaptive ADJ iajs-2002 149 10 wavelet wavelet NOUN iajs-2002 149 11 solution solution NOUN iajs-2002 149 12 of of ADP iajs-2002 149 13 nonlinear nonlinear ADJ iajs-2002 149 14 parabolic parabolic PROPN iajs-2002 149 15 differential differential PROPN iajs-2002 149 16 equations equation NOUN iajs-2002 149 17 . . PUNCT iajs-2002 150 1 journal journal NOUN iajs-2002 150 2 of of ADP iajs-2002 150 3 computational computational ADJ iajs-2002 150 4 physics physic NOUN iajs-2002 150 5 . . PUNCT iajs-2002 151 1 2006 2006 NUM iajs-2002 151 2 , , PUNCT iajs-2002 151 3 214 214 NUM iajs-2002 151 4 , , PUNCT iajs-2002 151 5 829–857 829–857 NUM iajs-2002 151 6 . . NOUN iajs-2002 152 1 4 4 NUM iajs-2002 152 2 . . X iajs-2002 152 3 timothy timothy PROPN iajs-2002 152 4 , , PUNCT iajs-2002 152 5 b. b. PROPN iajs-2002 152 6 explicit explicit ADJ iajs-2002 152 7 and and CCONJ iajs-2002 152 8 implicit implicit ADJ iajs-2002 152 9 methods method NOUN iajs-2002 152 10 in in ADP iajs-2002 152 11 solving solve VERB iajs-2002 152 12 differential differential ADJ iajs-2002 152 13 equations equation NOUN iajs-2002 152 14 . . PUNCT iajs-2002 153 1 honors honor NOUN iajs-2002 153 2 scholarheses scholarhese NOUN iajs-2002 153 3 . . PUNCT iajs-2002 154 1 university university NOUN iajs-2002 154 2 of of ADP iajs-2002 154 3 connecticut connecticut PROPN iajs-2002 154 4 – – PUNCT iajs-2002 154 5 storrs storrs PROPN iajs-2002 154 6 . . PROPN iajs-2002 154 7 2010 2010 NUM iajs-2002 154 8 . . PUNCT iajs-2002 155 1 5 5 NUM iajs-2002 155 2 . . X iajs-2002 155 3 ghoreishi ghoreishi PROPN iajs-2002 155 4 , , PUNCT iajs-2002 155 5 m. m. NOUN iajs-2002 155 6 ismail ismail PROPN iajs-2002 155 7 , , PUNCT iajs-2002 155 8 a. a. PROPN iajs-2002 155 9 i. i. PROPN iajs-2002 155 10 b b PROPN iajs-2002 155 11 , , PUNCT iajs-2002 155 12 the the DET iajs-2002 155 13 homotopy homotopy NOUN iajs-2002 155 14 perturbation perturbation NOUN iajs-2002 155 15 method method NOUN iajs-2002 155 16 ( ( PUNCT iajs-2002 155 17 hpm hpm NOUN iajs-2002 155 18 ) ) PUNCT iajs-2002 155 19 for for ADP iajs-2002 155 20 nonlinear nonlinear ADJ iajs-2002 155 21 parabolic parabolic ADJ iajs-2002 155 22 equation equation NOUN iajs-2002 155 23 with with ADP iajs-2002 155 24 nonlocal nonlocal ADJ iajs-2002 155 25 boundary boundary ADJ iajs-2002 155 26 conditions condition NOUN iajs-2002 155 27 . . PUNCT iajs-2002 156 1 applied apply VERB iajs-2002 156 2 mathematical mathematical ADJ iajs-2002 156 3 sciences science NOUN iajs-2002 156 4 , , PUNCT iajs-2002 156 5 2011 2011 NUM iajs-2002 156 6 , , PUNCT iajs-2002 156 7 5 5 NUM iajs-2002 156 8 , , PUNCT iajs-2002 156 9 3 3 NUM iajs-2002 156 10 , , PUNCT iajs-2002 156 11 113 113 NUM iajs-2002 156 12 - - SYM iajs-2002 156 13 123 123 NUM iajs-2002 156 14 . . PUNCT iajs-2002 157 1 6 6 NUM iajs-2002 157 2 . . X iajs-2002 158 1 douglas douglas PROPN iajs-2002 158 2 , , PUNCT iajs-2002 158 3 j. j. PROPN iajs-2002 158 4 ; ; PUNCT iajs-2002 158 5 dupont dupont PROPN iajs-2002 158 6 , , PUNCT iajs-2002 158 7 t. t. NOUN iajs-2002 158 8 galerkin galerkin ADJ iajs-2002 158 9 methods method NOUN iajs-2002 158 10 for for ADP iajs-2002 158 11 parabolic parabolic NOUN iajs-2002 158 12 problems problem NOUN iajs-2002 158 13 , , PUNCT iajs-2002 158 14 siam siam PROPN iajs-2002 158 15 j. j. PROPN iajs-2002 158 16 numerical numerical PROPN iajs-2002 158 17 analysis analysis PROPN iajs-2002 158 18 . . PUNCT iajs-2002 159 1 1970 1970 NUM iajs-2002 159 2 , , PUNCT iajs-2002 159 3 7 7 NUM iajs-2002 159 4 , , PUNCT iajs-2002 159 5 4 4 NUM iajs-2002 159 6 , , PUNCT iajs-2002 159 7 575 575 NUM iajs-2002 159 8 - - SYM iajs-2002 159 9 626 626 NUM iajs-2002 159 10 . . PUNCT iajs-2002 160 1 7 7 X iajs-2002 160 2 . . X iajs-2002 160 3 reddy reddy PROPN iajs-2002 160 4 , , PUNCT iajs-2002 160 5 j.n j.n PROPN iajs-2002 160 6 . . PROPN iajs-2002 161 1 an an DET iajs-2002 161 2 introduction introduction NOUN iajs-2002 161 3 to to ADP iajs-2002 161 4 the the DET iajs-2002 161 5 finite finite ADJ iajs-2002 161 6 element element NOUN iajs-2002 161 7 method method NOUN iajs-2002 161 8 , , PUNCT iajs-2002 161 9 Β  Β  SPACE iajs-2002 161 10 and and CCONJ iajs-2002 161 11 ed ed NOUN iajs-2002 161 12 . . PROPN iajs-2002 161 13 , , PUNCT iajs-2002 161 14 mcgraw mcgraw PROPN iajs-2002 161 15 - - PUNCT iajs-2002 161 16 hill hill NOUN iajs-2002 161 17 , , PUNCT iajs-2002 161 18 1993 1993 NUM iajs-2002 161 19 . . PUNCT iajs-2002 161 20 Β  Β  SPACE iajs-2002 162 1 0 0 NUM iajs-2002 162 2 0.5 0.5 NUM iajs-2002 162 3 1 1 NUM iajs-2002 162 4 0 0 NUM iajs-2002 162 5 0.5 0.5 NUM iajs-2002 162 6 1 1 NUM iajs-2002 162 7 -0.04 -0.04 NUM iajs-2002 162 8 -0.03 -0.03 NUM iajs-2002 162 9 -0.02 -0.02 NUM iajs-2002 162 10 -0.01 -0.01 NUM iajs-2002 162 11 0 0 NUM iajs-2002 162 12 a a DET iajs-2002 162 13 0 0 NUM iajs-2002 162 14 0.5 0.5 NUM iajs-2002 162 15 1 1 NUM iajs-2002 162 16 0 0 NUM iajs-2002 162 17 0.5 0.5 NUM iajs-2002 162 18 1 1 NUM iajs-2002 162 19 -0.04 -0.04 NUM iajs-2002 162 20 -0.03 -0.03 NUM iajs-2002 162 21 -0.02 -0.02 NUM iajs-2002 162 22 -0.01 -0.01 NUM iajs-2002 162 23 0 0 NUM iajs-2002 162 24 b b NOUN iajs-2002 162 25 mathematics mathematic NOUN iajs-2002 162 26 | | ADV iajs-2002 162 27 134 134 NUM iajs-2002 162 28 ibn ibn PROPN iajs-2002 162 29 al al PROPN iajs-2002 162 30 - - PUNCT iajs-2002 162 31 haitham haitham PROPN iajs-2002 162 32 jour jour X iajs-2002 162 33 . . PROPN iajs-2002 163 1 for for ADP iajs-2002 163 2 pure pure ADJ iajs-2002 163 3 & & CCONJ iajs-2002 163 4 appl appl PROPN iajs-2002 163 5 . . PUNCT iajs-2002 164 1 sci sci PROPN iajs-2002 164 2 . . PROPN iajs-2002 164 3 ihjpas ihjpa VERB iajs-2002 164 4 https://doi.org/10.30526/31.3.2002 https://doi.org/10.30526/31.3.2002 PART iajs-2002 164 5 vol vol NOUN iajs-2002 164 6 . . PUNCT iajs-2002 164 7 31 31 NUM iajs-2002 164 8 ( ( PUNCT iajs-2002 164 9 3 3 NUM iajs-2002 164 10 ) ) PUNCT iajs-2002 164 11 2018 2018 NUM iajs-2002 164 12 8 8 NUM iajs-2002 164 13 . . PUNCT iajs-2002 165 1 thomee thomee PROPN iajs-2002 165 2 , , PUNCT iajs-2002 165 3 v. v. ADP iajs-2002 165 4 galerkin galerkin PROPN iajs-2002 165 5 finite finite PROPN iajs-2002 165 6 element element NOUN iajs-2002 165 7 methods method NOUN iajs-2002 165 8 for for ADP iajs-2002 165 9 parabolic parabolic NOUN iajs-2002 165 10 problems problem NOUN iajs-2002 165 11 , , PUNCT iajs-2002 165 12 springer springer NOUN iajs-2002 165 13 verlag verlag PROPN iajs-2002 165 14 berlin berlin PROPN iajs-2002 165 15 heidelberg heidelberg PROPN iajs-2002 165 16 , , PUNCT iajs-2002 165 17 2006 2006 NUM iajs-2002 165 18 . . PUNCT iajs-2002 166 1 9 9 X iajs-2002 166 2 . . X iajs-2002 166 3 suhas suha NOUN iajs-2002 166 4 , , PUNCT iajs-2002 166 5 p. p. NOUN iajs-2002 166 6 ; ; PUNCT iajs-2002 166 7 uttam uttam PROPN iajs-2002 166 8 , , PUNCT iajs-2002 166 9 d. d. PROPN iajs-2002 166 10 random random ADJ iajs-2002 166 11 fixed fix VERB iajs-2002 166 12 point point NOUN iajs-2002 166 13 theorems theorem NOUN iajs-2002 166 14 for for ADP iajs-2002 166 15 contraction contraction NOUN iajs-2002 166 16 mappings mapping NOUN iajs-2002 166 17 in in ADP iajs-2002 166 18 metric metric ADJ iajs-2002 166 19 space space NOUN iajs-2002 166 20 . . PUNCT iajs-2002 167 1 international international ADJ iajs-2002 167 2 journal journal PROPN iajs-2002 167 3 of of ADP iajs-2002 167 4 science science NOUN iajs-2002 167 5 and and CCONJ iajs-2002 167 6 research research NOUN iajs-2002 167 7 ( ( PUNCT iajs-2002 167 8 ijsr ijsr NOUN iajs-2002 167 9 ) ) PUNCT iajs-2002 167 10 . . PUNCT iajs-2002 168 1 2015 2015 NUM iajs-2002 168 2 , , PUNCT iajs-2002 168 3 23,197064 23,197064 NOUN iajs-2002 168 4 . . PUNCT iajs-2002 169 1 10 10 NUM iajs-2002 169 2 . . PUNCT iajs-2002 169 3 burden burden NOUN iajs-2002 169 4 , , PUNCT iajs-2002 169 5 r. r. PROPN iajs-2002 169 6 l. l. PROPN iajs-2002 169 7 ; ; PUNCT iajs-2002 169 8 douglas douglas PROPN iajs-2002 169 9 faires faires PROPN iajs-2002 169 10 , , PUNCT iajs-2002 169 11 j. j. PROPN iajs-2002 169 12 numerical numerical PROPN iajs-2002 169 13 analysis analysis PROPN iajs-2002 169 14 , , PUNCT iajs-2002 169 15 seven seven NUM iajs-2002 169 16 ed ed NOUN iajs-2002 169 17 . . PUNCT iajs-2002 170 1 unite unite VERB iajs-2002 170 2 state state NOUN iajs-2002 170 3 , , PUNCT iajs-2002 170 4 2001 2001 NUM iajs-2002 170 5 . . PUNCT iajs-2002 171 1 11 11 NUM iajs-2002 171 2 . . PUNCT iajs-2002 172 1 bacopoulos bacopoulo NOUN iajs-2002 172 2 , , PUNCT iajs-2002 172 3 a. a. NOUN iajs-2002 172 4 ; ; PUNCT iajs-2002 172 5 chryssoverghi chryssoverghi PROPN iajs-2002 172 6 , , PUNCT iajs-2002 172 7 i. i. PROPN iajs-2002 172 8 numerical numerical PROPN iajs-2002 172 9 solutions solution NOUN iajs-2002 172 10 of of ADP iajs-2002 172 11 partial partial ADJ iajs-2002 172 12 differential differential ADJ iajs-2002 172 13 equations equation NOUN iajs-2002 172 14 by by ADP iajs-2002 172 15 finite finite ADJ iajs-2002 172 16 elements element NOUN iajs-2002 172 17 methods method NOUN iajs-2002 172 18 . . PUNCT iajs-2002 173 1 symeon symeon PROPN iajs-2002 173 2 publishing publishing PROPN iajs-2002 173 3 co. co. PROPN iajs-2002 173 4 athens athens PROPN iajs-2002 173 5 . . PROPN iajs-2002 173 6 2003 2003 NUM iajs-2002 173 7 . . PUNCT