id sid tid token lemma pos iajs-1821 1 1 microsoft microsoft PROPN iajs-1821 1 2 word word NOUN iajs-1821 1 3 490 490 NUM iajs-1821 1 4 - - SYM iajs-1821 1 5 499 499 NUM iajs-1821 1 6 ihsciconf ihsciconf NOUN iajs-1821 1 7 2017 2017 NUM iajs-1821 1 8 special special ADJ iajs-1821 1 9 issue issue NOUN iajs-1821 1 10 ibn ibn PROPN iajs-1821 1 11 al al PROPN iajs-1821 1 12 - - PUNCT iajs-1821 1 13 haitham haitham PROPN iajs-1821 1 14 journal journal PROPN iajs-1821 1 15 for for ADP iajs-1821 1 16 pure pure ADJ iajs-1821 1 17 and and CCONJ iajs-1821 1 18 applied apply VERB iajs-1821 1 19 science science NOUN iajs-1821 1 20 https://doi.org/ https://doi.org/ NOUN iajs-1821 1 21 10.30526/2017.ihsciconf.1821 10.30526/2017.ihsciconf.1821 NUM iajs-1821 1 22 for for ADP iajs-1821 1 23 more more ADJ iajs-1821 1 24 information information NOUN iajs-1821 1 25 about about ADP iajs-1821 1 26 the the DET iajs-1821 1 27 conference conference NOUN iajs-1821 1 28 please please INTJ iajs-1821 1 29 visit visit VERB iajs-1821 1 30 the the DET iajs-1821 1 31 websites website NOUN iajs-1821 1 32 : : PUNCT iajs-1821 1 33 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1821 1 34     SPACE iajs-1821 1 35 www.ihsciconf.org www.ihsciconf.org ADV iajs-1821 1 36       SPACE iajs-1821 1 37 mathematics|490 mathematics|490 PROPN iajs-1821 1 38         SPACE iajs-1821 1 39 normalization normalization PROPN iajs-1821 1 40 bernstein bernstein PROPN iajs-1821 1 41 basis basis NOUN iajs-1821 1 42 for for ADP iajs-1821 1 43 solving solve VERB iajs-1821 1 44 fractional fractional ADJ iajs-1821 1 45 fredholm fredholm NOUN iajs-1821 1 46 - - PUNCT iajs-1821 1 47 integro integro NOUN iajs-1821 1 48 differential differential ADJ iajs-1821 1 49 equation equation NOUN iajs-1821 1 50 abdul abdul PROPN iajs-1821 1 51 khaleq khaleq PROPN iajs-1821 1 52 o. o. PROPN iajs-1821 1 53 al al PROPN iajs-1821 1 54 - - PROPN iajs-1821 1 55 jubory jubory PROPN iajs-1821 1 56 khaleqnoor@yahoo.com khaleqnoor@yahoo.com PROPN iajs-1821 1 57 dept dept PROPN iajs-1821 1 58 . . PROPN iajs-1821 2 1 of of ADP iajs-1821 2 2 mathematics/ mathematics/ NUM iajs-1821 2 3 college college NOUN iajs-1821 2 4 of of ADP iajs-1821 2 5 science science NOUN iajs-1821 2 6 , , PUNCT iajs-1821 2 7 university university PROPN iajs-1821 2 8 al al PROPN iajs-1821 2 9 - - PUNCT iajs-1821 2 10 mustansiriyah mustansiriyah PROPN iajs-1821 2 11 shaymaa shaymaa NOUN iajs-1821 2 12 hussain hussain PROPN iajs-1821 2 13 salih salih PROPN iajs-1821 2 14 dept dept PROPN iajs-1821 2 15 . . PROPN iajs-1821 3 1 of of ADP iajs-1821 3 2 mathematics mathematics PROPN iajs-1821 3 3 / / SYM iajs-1821 3 4 college college NOUN iajs-1821 3 5 of of ADP iajs-1821 3 6 applied apply VERB iajs-1821 3 7 science/ science/ NUM iajs-1821 3 8 university university NOUN iajs-1821 3 9 of of ADP iajs-1821 3 10 technology technology NOUN iajs-1821 3 11 abstract abstract NOUN iajs-1821 3 12 in in ADP iajs-1821 3 13 this this DET iajs-1821 3 14 work work NOUN iajs-1821 3 15 , , PUNCT iajs-1821 3 16 we we PRON iajs-1821 3 17 employ employ VERB iajs-1821 3 18 a a DET iajs-1821 3 19 new new ADJ iajs-1821 3 20 normalization normalization NOUN iajs-1821 3 21 bernstein bernstein PROPN iajs-1821 3 22 basis basis NOUN iajs-1821 3 23 for for ADP iajs-1821 3 24 solving solve VERB iajs-1821 3 25 linear linear ADJ iajs-1821 3 26 freadholm freadholm NOUN iajs-1821 3 27 of of ADP iajs-1821 3 28 fractional fractional ADJ iajs-1821 3 29 integro integro ADJ iajs-1821 3 30 - - PUNCT iajs-1821 3 31 differential differential NOUN iajs-1821 3 32 equations equation NOUN iajs-1821 3 33 nonhomogeneous nonhomogeneous ADJ iajs-1821 3 34 of of ADP iajs-1821 3 35 the the DET iajs-1821 3 36 second second ADJ iajs-1821 3 37 type type NOUN iajs-1821 3 38 ( ( PUNCT iajs-1821 3 39 lffides lffide NOUN iajs-1821 3 40 ) ) PUNCT iajs-1821 3 41 . . PUNCT iajs-1821 4 1 we we PRON iajs-1821 4 2 adopt adopt VERB iajs-1821 4 3 petrov petrov PROPN iajs-1821 4 4 - - PUNCT iajs-1821 4 5 galerkian galerkian PROPN iajs-1821 4 6 method method NOUN iajs-1821 4 7 ( ( PUNCT iajs-1821 4 8 pgm pgm PROPN iajs-1821 4 9 ) ) PUNCT iajs-1821 4 10 to to PART iajs-1821 4 11 approximate approximate ADJ iajs-1821 4 12 solution solution NOUN iajs-1821 4 13 of of ADP iajs-1821 4 14 the the DET iajs-1821 4 15 ( ( PUNCT iajs-1821 4 16 lffides lffide NOUN iajs-1821 4 17 ) ) PUNCT iajs-1821 4 18 via via ADP iajs-1821 4 19 normalization normalization NOUN iajs-1821 4 20 bernstein bernstein PROPN iajs-1821 4 21 basis basis NOUN iajs-1821 4 22 that that SCONJ iajs-1821 4 23 yields yield VERB iajs-1821 4 24 linear linear ADJ iajs-1821 4 25 system system NOUN iajs-1821 4 26 . . PUNCT iajs-1821 5 1 some some DET iajs-1821 5 2 examples example NOUN iajs-1821 5 3 are be AUX iajs-1821 5 4 given give VERB iajs-1821 5 5 and and CCONJ iajs-1821 5 6 their their PRON iajs-1821 5 7 results result NOUN iajs-1821 5 8 are be AUX iajs-1821 5 9 shown show VERB iajs-1821 5 10 in in ADP iajs-1821 5 11 tables table NOUN iajs-1821 5 12 and and CCONJ iajs-1821 5 13 figures figure NOUN iajs-1821 5 14 , , PUNCT iajs-1821 5 15 the the DET iajs-1821 5 16 petrov petrov PROPN iajs-1821 5 17 - - PUNCT iajs-1821 5 18 galerkian galerkian PROPN iajs-1821 5 19 method method NOUN iajs-1821 5 20 ( ( PUNCT iajs-1821 5 21 pgm pgm PROPN iajs-1821 5 22 ) ) PUNCT iajs-1821 5 23 is be AUX iajs-1821 5 24 very very ADV iajs-1821 5 25 effective effective ADJ iajs-1821 5 26 and and CCONJ iajs-1821 5 27 convenient convenient ADJ iajs-1821 5 28 and and CCONJ iajs-1821 5 29 overcome overcome VERB iajs-1821 5 30 the the DET iajs-1821 5 31 difficulty difficulty NOUN iajs-1821 5 32 of of ADP iajs-1821 5 33 traditional traditional ADJ iajs-1821 5 34 methods method NOUN iajs-1821 5 35 . . PUNCT iajs-1821 6 1 we we PRON iajs-1821 6 2 solve solve VERB iajs-1821 6 3 this this DET iajs-1821 6 4 problem problem NOUN iajs-1821 6 5 ( ( PUNCT iajs-1821 6 6 lffides lffides PROPN iajs-1821 6 7 ) ) PUNCT iajs-1821 6 8 by by ADP iajs-1821 6 9 the the DET iajs-1821 6 10 assistance assistance NOUN iajs-1821 6 11 of of ADP iajs-1821 6 12 matlab10 matlab10 NOUN iajs-1821 6 13 . . PUNCT iajs-1821 7 1 key key ADJ iajs-1821 7 2 words word NOUN iajs-1821 7 3 : : PUNCT iajs-1821 7 4 petrov petrov PROPN iajs-1821 7 5 - - PUNCT iajs-1821 7 6 galerkian galerkian PROPN iajs-1821 7 7 method method NOUN iajs-1821 7 8 ; ; PUNCT iajs-1821 7 9 fractional fractional ADJ iajs-1821 7 10 derivative derivative NOUN iajs-1821 7 11 ; ; PUNCT iajs-1821 7 12 caputo caputo NOUN iajs-1821 7 13 ; ; PUNCT iajs-1821 7 14 sense sense NOUN iajs-1821 7 15 . . PUNCT iajs-1821 8 1 ihsciconf ihsciconf PROPN iajs-1821 8 2 2017 2017 NUM iajs-1821 8 3 special special ADJ iajs-1821 8 4 issue issue NOUN iajs-1821 8 5 ibn ibn PROPN iajs-1821 8 6 al al PROPN iajs-1821 8 7 - - PUNCT iajs-1821 8 8 haitham haitham PROPN iajs-1821 8 9 journal journal PROPN iajs-1821 8 10 for for ADP iajs-1821 8 11 pure pure ADJ iajs-1821 8 12 and and CCONJ iajs-1821 8 13 applied apply VERB iajs-1821 8 14 science science NOUN iajs-1821 8 15 https://doi.org/ https://doi.org/ NOUN iajs-1821 8 16 10.30526/2017.ihsciconf.1821 10.30526/2017.ihsciconf.1821 NUM iajs-1821 8 17 for for ADP iajs-1821 8 18 more more ADJ iajs-1821 8 19 information information NOUN iajs-1821 8 20 about about ADP iajs-1821 8 21 the the DET iajs-1821 8 22 conference conference NOUN iajs-1821 8 23 please please INTJ iajs-1821 8 24 visit visit VERB iajs-1821 8 25 the the DET iajs-1821 8 26 websites website NOUN iajs-1821 8 27 : : PUNCT iajs-1821 8 28 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1821 8 29     SPACE iajs-1821 8 30 www.ihsciconf.org www.ihsciconf.org ADV iajs-1821 8 31       SPACE iajs-1821 8 32 mathematics|491 mathematics|491 NOUN iajs-1821 8 33         SPACE iajs-1821 8 34 1.introduction 1.introduction NUM iajs-1821 8 35 : : PUNCT iajs-1821 8 36 integro integro ADJ iajs-1821 8 37 - - PUNCT iajs-1821 8 38 differential differential NOUN iajs-1821 8 39 equations equation NOUN iajs-1821 8 40 are be AUX iajs-1821 8 41 encountered encounter VERB iajs-1821 8 42 in in ADP iajs-1821 8 43 various various ADJ iajs-1821 8 44 fields field NOUN iajs-1821 8 45 of of ADP iajs-1821 8 46 sciences science NOUN iajs-1821 8 47 . . PUNCT iajs-1821 9 1 it it PRON iajs-1821 9 2 plays play VERB iajs-1821 9 3 an an DET iajs-1821 9 4 important important ADJ iajs-1821 9 5 role role NOUN iajs-1821 9 6 in in ADP iajs-1821 9 7 many many ADJ iajs-1821 9 8 branches branch NOUN iajs-1821 9 9 of of ADP iajs-1821 9 10 linear linear PROPN iajs-1821 9 11 and and CCONJ iajs-1821 9 12 non non ADJ iajs-1821 9 13 - - ADJ iajs-1821 9 14 linear linear ADJ iajs-1821 9 15 functional functional ADJ iajs-1821 9 16 analysis analysis NOUN iajs-1821 9 17 and and CCONJ iajs-1821 9 18 their their PRON iajs-1821 9 19 applications application NOUN iajs-1821 9 20 are be AUX iajs-1821 9 21 in in ADP iajs-1821 9 22 the the DET iajs-1821 9 23 theory theory NOUN iajs-1821 9 24 of of ADP iajs-1821 9 25 science science NOUN iajs-1821 9 26 , , PUNCT iajs-1821 9 27 engineering engineering NOUN iajs-1821 9 28 and and CCONJ iajs-1821 9 29 social social ADJ iajs-1821 9 30 sciences science NOUN iajs-1821 9 31 . . PUNCT iajs-1821 10 1 many many ADJ iajs-1821 10 2 problems problem NOUN iajs-1821 10 3 can can AUX iajs-1821 10 4 be be AUX iajs-1821 10 5 modeled model VERB iajs-1821 10 6 by by ADP iajs-1821 10 7 fractional fractional ADJ iajs-1821 10 8 integro integro ADJ iajs-1821 10 9 - - PUNCT iajs-1821 10 10 differential differential NOUN iajs-1821 10 11 equations equation NOUN iajs-1821 10 12 from from ADP iajs-1821 10 13 various various ADJ iajs-1821 10 14 science science NOUN iajs-1821 10 15 and and CCONJ iajs-1821 10 16 engineering engineering NOUN iajs-1821 10 17 applications application NOUN iajs-1821 10 18 . . PUNCT iajs-1821 11 1 finding find VERB iajs-1821 11 2 the the DET iajs-1821 11 3 approximate approximate ADJ iajs-1821 11 4 or or CCONJ iajs-1821 11 5 exact exact ADJ iajs-1821 11 6 solutions solution NOUN iajs-1821 11 7 of of ADP iajs-1821 11 8 fractional fractional ADJ iajs-1821 11 9 integrodifferential integrodifferential ADJ iajs-1821 11 10 equations equation NOUN iajs-1821 11 11 is be AUX iajs-1821 11 12 an an DET iajs-1821 11 13 important important ADJ iajs-1821 11 14 task task NOUN iajs-1821 11 15 . . PUNCT iajs-1821 12 1 save save VERB iajs-1821 12 2 in in ADP iajs-1821 12 3 a a DET iajs-1821 12 4 limited limited ADJ iajs-1821 12 5 number number NOUN iajs-1821 12 6 , , PUNCT iajs-1821 12 7 there there PRON iajs-1821 12 8 is be VERB iajs-1821 12 9 difficulty difficulty NOUN iajs-1821 12 10 in in ADP iajs-1821 12 11 finding find VERB iajs-1821 12 12 the the DET iajs-1821 12 13 analytical analytical ADJ iajs-1821 12 14 solutions solution NOUN iajs-1821 12 15 of of ADP iajs-1821 12 16 fractional fractional ADJ iajs-1821 12 17 integro integro ADJ iajs-1821 12 18 - - PUNCT iajs-1821 12 19 differential differential NOUN iajs-1821 12 20 equations equation NOUN iajs-1821 12 21 . . PUNCT iajs-1821 13 1 therefore therefore ADV iajs-1821 13 2 , , PUNCT iajs-1821 13 3 there there PRON iajs-1821 13 4 have have AUX iajs-1821 13 5 been be AUX iajs-1821 13 6 attempts attempt NOUN iajs-1821 13 7 to to PART iajs-1821 13 8 develop develop VERB iajs-1821 13 9 new new ADJ iajs-1821 13 10 methods method NOUN iajs-1821 13 11 for for ADP iajs-1821 13 12 obtaining obtain VERB iajs-1821 13 13 analytical analytical ADJ iajs-1821 13 14 solutions solution NOUN iajs-1821 13 15 which which PRON iajs-1821 13 16 reasonably reasonably ADV iajs-1821 13 17 approximate approximate VERB iajs-1821 13 18 the the DET iajs-1821 13 19 exact exact ADJ iajs-1821 13 20 solutions solution NOUN iajs-1821 13 21 . . PUNCT iajs-1821 14 1 however however ADV iajs-1821 14 2 , , PUNCT iajs-1821 14 3 several several ADJ iajs-1821 14 4 numbers number NOUN iajs-1821 14 5 of of ADP iajs-1821 14 6 algorithms algorithm NOUN iajs-1821 14 7 for for ADP iajs-1821 14 8 solving solve VERB iajs-1821 14 9 linear linear ADJ iajs-1821 14 10 fredholm fredholm NOUN iajs-1821 14 11 of of ADP iajs-1821 14 12 fractional fractional ADJ iajs-1821 14 13 integro integro ADJ iajs-1821 14 14 - - PUNCT iajs-1821 14 15 differential differential NOUN iajs-1821 14 16 equation equation NOUN iajs-1821 14 17 nonhomogeneous nonhomogeneous NOUN iajs-1821 14 18 of of ADP iajs-1821 14 19 the the DET iajs-1821 14 20 second second ADJ iajs-1821 14 21 type type NOUN iajs-1821 14 22 ( ( PUNCT iajs-1821 14 23 lffides lffide NOUN iajs-1821 14 24 ) ) PUNCT iajs-1821 14 25 have have AUX iajs-1821 14 26 been be AUX iajs-1821 14 27 investigated investigate VERB iajs-1821 14 28 . . PUNCT iajs-1821 15 1 z. z. PROPN iajs-1821 15 2 taheri taheri PROPN iajs-1821 15 3 , , PUNCT iajs-1821 15 4 sh sh PROPN iajs-1821 15 5 . . PROPN iajs-1821 15 6 javadi javadi PROPN iajs-1821 15 7 and and CCONJ iajs-1821 15 8 e. e. PROPN iajs-1821 15 9 babolian babolian PROPN iajs-1821 16 1 [ [ X iajs-1821 16 2 1 1 X iajs-1821 16 3 ] ] PUNCT iajs-1821 16 4 employed employ VERB iajs-1821 16 5 shifted shift VERB iajs-1821 16 6 legendre legendre PROPN iajs-1821 16 7 spectral spectral ADJ iajs-1821 16 8 collocation collocation NOUN iajs-1821 16 9 method method NOUN iajs-1821 16 10 to to PART iajs-1821 16 11 solve solve VERB iajs-1821 16 12 stochastic stochastic ADJ iajs-1821 16 13 integro integro ADJ iajs-1821 16 14 - - PUNCT iajs-1821 16 15 differential differential NOUN iajs-1821 16 16 equations equation NOUN iajs-1821 16 17 ( ( PUNCT iajs-1821 16 18 sfides sfide NOUN iajs-1821 16 19 ) ) PUNCT iajs-1821 16 20 . . PUNCT iajs-1821 17 1 [ [ X iajs-1821 17 2 2 2 X iajs-1821 17 3 ] ] PUNCT iajs-1821 17 4 presented present VERB iajs-1821 17 5 bernstein bernstein PROPN iajs-1821 17 6 polynomials polynomials PROPN iajs-1821 17 7 basis basis NOUN iajs-1821 17 8 for for ADP iajs-1821 17 9 solving solve VERB iajs-1821 17 10 ( ( PUNCT iajs-1821 17 11 lffides lffide NOUN iajs-1821 17 12 ) ) PUNCT iajs-1821 17 13 . . PUNCT iajs-1821 18 1 asma asma PROPN iajs-1821 18 2 a. a. PROPN iajs-1821 18 3 , , PUNCT iajs-1821 18 4 adem adem PROPN iajs-1821 18 5 kılıc kılıc PROPN iajs-1821 18 6 and and CCONJ iajs-1821 18 7 bachok bachok DET iajs-1821 18 8 m. m. NOUN iajs-1821 19 1 [ [ X iajs-1821 19 2 3 3 NUM iajs-1821 19 3 ] ] X iajs-1821 19 4 employed employ VERB iajs-1821 19 5 , , PUNCT iajs-1821 19 6 homotopy homotopy VERB iajs-1821 19 7 perturbation perturbation NOUN iajs-1821 19 8 and and CCONJ iajs-1821 19 9 the the DET iajs-1821 19 10 variational variational ADJ iajs-1821 19 11 iteration iteration NOUN iajs-1821 19 12 to to PART iajs-1821 19 13 approximate approximate VERB iajs-1821 19 14 integro integro ADJ iajs-1821 19 15 - - PUNCT iajs-1821 19 16 differential differential NOUN iajs-1821 19 17 equation equation NOUN iajs-1821 19 18 of of ADP iajs-1821 19 19 fractional fractional ADJ iajs-1821 19 20 ( ( PUNCT iajs-1821 19 21 arbitrary arbitrary ADJ iajs-1821 19 22 ) ) PUNCT iajs-1821 19 23 order order NOUN iajs-1821 19 24 . . PUNCT iajs-1821 20 1 li li PROPN iajs-1821 20 2 huanga huanga PROPN iajs-1821 20 3 , , PUNCT iajs-1821 20 4 xianfang xianfang PROPN iajs-1821 20 5 li li PROPN iajs-1821 20 6 , , PUNCT iajs-1821 20 7 yulin yulin NOUN iajs-1821 20 8 zhaoa zhaoa PROPN iajs-1821 20 9 and and CCONJ iajs-1821 20 10 xiang xiang PROPN iajs-1821 20 11 - - PROPN iajs-1821 20 12 yang yang PROPN iajs-1821 21 1 [ [ X iajs-1821 21 2 4 4 X iajs-1821 21 3 ] ] PUNCT iajs-1821 21 4 used use VERB iajs-1821 21 5 taylor taylor PROPN iajs-1821 21 6 series series PROPN iajs-1821 21 7 approach approach NOUN iajs-1821 21 8 for for ADP iajs-1821 21 9 approximately approximately ADV iajs-1821 21 10 a a DET iajs-1821 21 11 class class NOUN iajs-1821 21 12 of of ADP iajs-1821 21 13 . . PUNCT iajs-1821 22 1 peter peter PROPN iajs-1821 22 2 linzt linzt PROPN iajs-1821 23 1 [ [ X iajs-1821 23 2 5 5 X iajs-1821 23 3 ] ] PUNCT iajs-1821 23 4 used use VERB iajs-1821 23 5 nystrom nystrom PROPN iajs-1821 23 6 ’s ’s PART iajs-1821 23 7 method method NOUN iajs-1821 23 8 to to PART iajs-1821 23 9 establish establish VERB iajs-1821 23 10 numerical numerical ADJ iajs-1821 23 11 procedure procedure NOUN iajs-1821 23 12 for for ADP iajs-1821 23 13 the the DET iajs-1821 23 14 approximate approximate ADJ iajs-1821 23 15 solution solution NOUN iajs-1821 23 16 of of ADP iajs-1821 23 17 linear linear PROPN iajs-1821 23 18 integro integro ADJ iajs-1821 23 19 - - PUNCT iajs-1821 23 20 differential differential NOUN iajs-1821 23 21 equations equation NOUN iajs-1821 23 22 . . PUNCT iajs-1821 24 1 in in ADP iajs-1821 24 2 this this DET iajs-1821 24 3 wrok wrok NOUN iajs-1821 24 4 , , PUNCT iajs-1821 24 5 we we PRON iajs-1821 24 6 presented present VERB iajs-1821 24 7 the the DET iajs-1821 24 8 approximate approximate ADJ iajs-1821 24 9 solution solution NOUN iajs-1821 24 10 of of ADP iajs-1821 24 11 the the DET iajs-1821 24 12 ( ( PUNCT iajs-1821 24 13 lffides lffide NOUN iajs-1821 24 14 ) ) PUNCT iajs-1821 24 15 . . PUNCT iajs-1821 25 1 𝐷 𝐷 NOUN iajs-1821 25 2 𝑢 𝑢 PRON iajs-1821 25 3 𝑥 𝑥 X iajs-1821 25 4 𝑓 𝑓 ADP iajs-1821 25 5 𝑥 𝑥 X iajs-1821 25 6 𝑘 𝑘 PRON iajs-1821 25 7 𝑥 𝑥 NOUN iajs-1821 25 8 , , PUNCT iajs-1821 25 9 𝑡 𝑡 X iajs-1821 25 10 𝑢 𝑢 X iajs-1821 25 11 𝑡 𝑡 ADP iajs-1821 25 12 𝑎 𝑎 PROPN iajs-1821 25 13 𝑥 𝑥 NOUN iajs-1821 25 14 , , PUNCT iajs-1821 25 15 𝑡 𝑡 PROPN iajs-1821 25 16 𝑏 𝑏 NOUN iajs-1821 25 17 … … PUNCT iajs-1821 25 18 ( ( PUNCT iajs-1821 25 19 1.a 1.a NUM iajs-1821 25 20 ) ) PUNCT iajs-1821 25 21 with with ADP iajs-1821 25 22 the the DET iajs-1821 25 23 following following ADJ iajs-1821 25 24 supplementary supplementary ADJ iajs-1821 25 25 conditions condition NOUN iajs-1821 25 26 : : PUNCT iajs-1821 25 27 𝑢 𝑢 NOUN iajs-1821 25 28 0 0 NUM iajs-1821 25 29 𝛿 𝛿 NOUN iajs-1821 25 30 𝑛 𝑛 PROPN iajs-1821 25 31 1 1 NUM iajs-1821 25 32 𝛼 𝛼 NOUN iajs-1821 25 33 𝑛 𝑛 NOUN iajs-1821 25 34 , , PUNCT iajs-1821 25 35 𝑛 𝑛 DET iajs-1821 25 36 ∈ ∈ NOUN iajs-1821 25 37 𝑁 𝑁 PROPN iajs-1821 25 38 … … PUNCT iajs-1821 25 39 ( ( PUNCT iajs-1821 25 40 1.b 1.b NUM iajs-1821 25 41 ) ) PUNCT iajs-1821 25 42 where where SCONJ iajs-1821 25 43 𝐷 𝐷 NOUN iajs-1821 25 44 𝑢 𝑢 PRON iajs-1821 25 45 𝑥 𝑥 PRON iajs-1821 25 46 indicates indicate VERB iajs-1821 25 47 the the DET iajs-1821 25 48 𝛼 𝛼 PROPN iajs-1821 25 49 the the DET iajs-1821 25 50 caputo caputo PROPN iajs-1821 25 51 fractional fractional PROPN iajs-1821 25 52 derivative derivative NOUN iajs-1821 25 53 of of ADP iajs-1821 25 54 𝑢 𝑢 NOUN iajs-1821 25 55 𝑥 𝑥 X iajs-1821 25 56 ; ; PUNCT iajs-1821 25 57 𝑓(𝑥 𝑓(𝑥 NOUN iajs-1821 25 58 ) ) PUNCT iajs-1821 25 59 , , PUNCT iajs-1821 25 60 𝐾(𝑥 𝐾(𝑥 NOUN iajs-1821 25 61 , , PUNCT iajs-1821 25 62 𝑡 𝑡 X iajs-1821 25 63 ) ) PUNCT iajs-1821 25 64 are be AUX iajs-1821 25 65 given give VERB iajs-1821 25 66 functions function NOUN iajs-1821 25 67 , , PUNCT iajs-1821 25 68 𝑥 𝑥 PROPN iajs-1821 25 69 and and CCONJ iajs-1821 25 70 𝑡 𝑡 PROPN iajs-1821 25 71 are be AUX iajs-1821 25 72 real real ADJ iajs-1821 25 73 variables variable NOUN iajs-1821 25 74 varying vary VERB iajs-1821 25 75 in in ADP iajs-1821 25 76 the the DET iajs-1821 25 77 interval interval NOUN iajs-1821 25 78 [ [ X iajs-1821 25 79 a a X iajs-1821 25 80 , , PUNCT iajs-1821 25 81 b b NOUN iajs-1821 25 82 ] ] X iajs-1821 25 83 , , PUNCT iajs-1821 25 84 and and CCONJ iajs-1821 25 85 𝑢 𝑢 PRON iajs-1821 25 86 𝑥 𝑥 PROPN iajs-1821 25 87 is be AUX iajs-1821 25 88 the the DET iajs-1821 25 89 unknown unknown ADJ iajs-1821 25 90 function function NOUN iajs-1821 25 91 to to PART iajs-1821 25 92 be be AUX iajs-1821 25 93 determined determine VERB iajs-1821 25 94 . . PUNCT iajs-1821 26 1 2 2 X iajs-1821 26 2 . . X iajs-1821 26 3 basic basic ADJ iajs-1821 26 4 definition definition NOUN iajs-1821 26 5 definition1 definition1 NOUN iajs-1821 26 6 : : PUNCT iajs-1821 26 7 a a DET iajs-1821 26 8 real real ADJ iajs-1821 26 9 function function NOUN iajs-1821 26 10 f(t f(t PROPN iajs-1821 26 11 ) ) PUNCT iajs-1821 26 12 , , PUNCT iajs-1821 26 13 t t X iajs-1821 26 14 > > X iajs-1821 26 15 0 0 PROPN iajs-1821 26 16 , , PUNCT iajs-1821 26 17 is be AUX iajs-1821 26 18 said say VERB iajs-1821 26 19 to to PART iajs-1821 26 20 be be AUX iajs-1821 26 21 in in ADP iajs-1821 26 22 the the DET iajs-1821 26 23 space space NOUN iajs-1821 26 24 c c NOUN iajs-1821 26 25 , , PUNCT iajs-1821 27 1 μ μ PROPN iajs-1821 27 2 ∈ ∈ PROPN iajs-1821 27 3 r r NOUN iajs-1821 27 4 , , PUNCT iajs-1821 27 5 if if SCONJ iajs-1821 27 6 there there PRON iajs-1821 27 7 exists exist VERB iajs-1821 27 8 a a DET iajs-1821 27 9 real real ADJ iajs-1821 27 10 number number NOUN iajs-1821 27 11 p p NOUN iajs-1821 27 12 𝜇,such 𝜇,such ADV iajs-1821 27 13 that that SCONJ iajs-1821 27 14 f(t f(t NOUN iajs-1821 27 15 ) ) PUNCT iajs-1821 28 1 = = SYM iajs-1821 28 2 t t NOUN iajs-1821 28 3 h h NOUN iajs-1821 28 4 ( ( PUNCT iajs-1821 28 5 t t PROPN iajs-1821 28 6 ) ) PUNCT iajs-1821 28 7 ; ; PUNCT iajs-1821 28 8 where where SCONJ iajs-1821 28 9 f f PROPN iajs-1821 28 10 t t NOUN iajs-1821 28 11 ∈ ∈ PROPN iajs-1821 28 12 0 0 PUNCT iajs-1821 28 13 , , PUNCT iajs-1821 28 14 ∞ ∞ PROPN iajs-1821 28 15 , , PUNCT iajs-1821 28 16 and and CCONJ iajs-1821 28 17 it it PRON iajs-1821 28 18 is be AUX iajs-1821 28 19 said say VERB iajs-1821 28 20 to to PART iajs-1821 28 21 be be AUX iajs-1821 28 22 in in ADP iajs-1821 28 23 space space NOUN iajs-1821 28 24 c c NOUN iajs-1821 29 1 if if SCONJ iajs-1821 29 2 and and CCONJ iajs-1821 29 3 only only ADV iajs-1821 29 4 if if SCONJ iajs-1821 29 5 f f PROPN iajs-1821 29 6 c c PROPN iajs-1821 29 7 , , PUNCT iajs-1821 29 8 n n PROPN iajs-1821 29 9 ∈ ∈ PROPN iajs-1821 29 10 n. n. NOUN iajs-1821 29 11 definition2 definition2 NOUN iajs-1821 29 12 : : PUNCT iajs-1821 29 13 the the DET iajs-1821 29 14 riemann riemann PROPN iajs-1821 29 15 - - PUNCT iajs-1821 29 16 liouvill liouvill NOUN iajs-1821 29 17 fractional fractional ADJ iajs-1821 29 18 integral integral ADJ iajs-1821 29 19 operator operator NOUN iajs-1821 29 20 of of ADP iajs-1821 29 21 order order NOUN iajs-1821 29 22 𝛼 𝛼 NOUN iajs-1821 29 23 for for ADP iajs-1821 29 24 a a DET iajs-1821 29 25 function function NOUN iajs-1821 29 26 in in ADP iajs-1821 29 27 𝐶 𝐶 PROPN iajs-1821 29 28 , , PUNCT iajs-1821 29 29 where where SCONJ iajs-1821 29 30 𝜇 𝜇 ADP iajs-1821 29 31 1 1 NUM iajs-1821 29 32 , , PUNCT iajs-1821 29 33 is be AUX iajs-1821 29 34 defined define VERB iajs-1821 29 35 as as ADP iajs-1821 29 36 𝐽 𝐽 PROPN iajs-1821 29 37 𝑓 𝑓 ADV iajs-1821 29 38 𝑥 𝑥 PROPN iajs-1821 29 39 𝑑𝑡 𝑑𝑡 PROPN iajs-1821 29 40 , , PUNCT iajs-1821 29 41 𝛼 𝛼 PROPN iajs-1821 29 42 0 0 NUM iajs-1821 29 43 𝐽 𝐽 PROPN iajs-1821 30 1 𝑓 𝑓 ADV iajs-1821 30 2 𝑥 𝑥 X iajs-1821 30 3 𝑓 𝑓 PRON iajs-1821 30 4 𝑥 𝑥 X iajs-1821 30 5 . . PUNCT iajs-1821 31 1 definition3 definition3 PROPN iajs-1821 31 2 : : PUNCT iajs-1821 31 3 let let VERB iajs-1821 31 4 𝑓 𝑓 PRON iajs-1821 31 5 ∈ ∈ PROPN iajs-1821 31 6 𝐶 𝐶 PROPN iajs-1821 31 7 1 1 NUM iajs-1821 31 8 , , PUNCT iajs-1821 31 9 𝑚 𝑚 PROPN iajs-1821 31 10 ∈ ∈ NOUN iajs-1821 31 11 𝑁 𝑁 PROPN iajs-1821 31 12 ∪ ∪ NOUN iajs-1821 31 13 0 0 NUM iajs-1821 31 14 . . PUNCT iajs-1821 32 1 then then ADV iajs-1821 32 2 the the DET iajs-1821 32 3 caputo caputo PROPN iajs-1821 32 4 fractional fractional ADJ iajs-1821 32 5 derivatives derivative NOUN iajs-1821 32 6 of of ADP iajs-1821 32 7 f(x f(x PROPN iajs-1821 32 8 ) ) PUNCT iajs-1821 32 9 is be AUX iajs-1821 32 10 defined define VERB iajs-1821 32 11 as as ADP iajs-1821 32 12 : : PUNCT iajs-1821 32 13 ihsciconf ihsciconf NOUN iajs-1821 32 14 2017 2017 NUM iajs-1821 32 15 special special ADJ iajs-1821 32 16 issue issue NOUN iajs-1821 32 17 ibn ibn PROPN iajs-1821 32 18 al al PROPN iajs-1821 32 19 - - PUNCT iajs-1821 32 20 haitham haitham PROPN iajs-1821 32 21 journal journal PROPN iajs-1821 32 22 for for ADP iajs-1821 32 23 pure pure ADJ iajs-1821 32 24 and and CCONJ iajs-1821 32 25 applied apply VERB iajs-1821 32 26 science science NOUN iajs-1821 32 27 https://doi.org/ https://doi.org/ NOUN iajs-1821 32 28 10.30526/2017.ihsciconf.1821 10.30526/2017.ihsciconf.1821 NUM iajs-1821 32 29 for for ADP iajs-1821 32 30 more more ADJ iajs-1821 32 31 information information NOUN iajs-1821 32 32 about about ADP iajs-1821 32 33 the the DET iajs-1821 32 34 conference conference NOUN iajs-1821 32 35 please please INTJ iajs-1821 32 36 visit visit VERB iajs-1821 32 37 the the DET iajs-1821 32 38 websites website NOUN iajs-1821 32 39 : : PUNCT iajs-1821 32 40 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1821 32 41     SPACE iajs-1821 32 42 www.ihsciconf.org www.ihsciconf.org ADJ iajs-1821 32 43       SPACE iajs-1821 32 44 mathematics|492 mathematics|492 NOUN iajs-1821 32 45         SPACE iajs-1821 32 46 𝐷 𝐷 PROPN iajs-1821 32 47 𝑓 𝑓 ADV iajs-1821 32 48 𝑥 𝑥 PROPN iajs-1821 32 49 𝐽 𝐽 PROPN iajs-1821 33 1 𝑓 𝑓 INTJ iajs-1821 33 2 𝑥 𝑥 X iajs-1821 33 3 , , PUNCT iajs-1821 33 4 𝑚 𝑚 PROPN iajs-1821 33 5 1 1 NUM iajs-1821 33 6 𝛼 𝛼 NOUN iajs-1821 33 7 𝑚 𝑚 NOUN iajs-1821 33 8 , , PUNCT iajs-1821 33 9 𝑚 𝑚 PROPN iajs-1821 33 10 ∈ ∈ NOUN iajs-1821 33 11 𝑁 𝑁 PROPN iajs-1821 33 12 , , PUNCT iajs-1821 33 13 𝛼 𝛼 VERB iajs-1821 33 14 𝑚 𝑚 NOUN iajs-1821 33 15 hence hence ADV iajs-1821 33 16 , , PUNCT iajs-1821 33 17 we we PRON iajs-1821 33 18 have have AUX iajs-1821 33 19 following follow VERB iajs-1821 33 20 properties property NOUN iajs-1821 33 21 1 1 NUM iajs-1821 33 22 . . PUNCT iajs-1821 34 1 𝐽 𝐽 PROPN iajs-1821 34 2 𝐽 𝐽 NOUN iajs-1821 35 1 𝑓 𝑓 PRON iajs-1821 35 2 𝐽 𝐽 PROPN iajs-1821 35 3 𝑓 𝑓 PROPN iajs-1821 35 4 , , PUNCT iajs-1821 35 5 𝛼 𝛼 PROPN iajs-1821 35 6 , , PUNCT iajs-1821 35 7 𝑣 𝑣 PRON iajs-1821 35 8 0 0 NUM iajs-1821 35 9 , , PUNCT iajs-1821 35 10 𝑓 𝑓 DET iajs-1821 35 11 ∈ ∈ PROPN iajs-1821 35 12 𝐶 𝐶 PROPN iajs-1821 35 13 , , PUNCT iajs-1821 35 14 𝜇 𝜇 ADP iajs-1821 35 15 0 0 NUM iajs-1821 35 16 2 2 NUM iajs-1821 35 17 . . PUNCT iajs-1821 36 1 𝐽 𝐽 PROPN iajs-1821 36 2 𝑥 𝑥 VERB iajs-1821 37 1 𝑥 𝑥 INTJ iajs-1821 37 2 , , PUNCT iajs-1821 37 3 𝛼 𝛼 PROPN iajs-1821 37 4 0 0 NUM iajs-1821 37 5 , , PUNCT iajs-1821 37 6 𝛾 𝛾 PROPN iajs-1821 37 7 1 1 NUM iajs-1821 37 8 , , PUNCT iajs-1821 37 9 𝑥 𝑥 PRON iajs-1821 37 10 0 0 NUM iajs-1821 37 11 3 3 X iajs-1821 37 12 . . X iajs-1821 38 1 𝐽 𝐽 PROPN iajs-1821 38 2 𝐷 𝐷 VERB iajs-1821 38 3 𝑓 𝑓 ADV iajs-1821 38 4 𝑥 𝑥 X iajs-1821 38 5 𝑓 𝑓 PRON iajs-1821 38 6 𝑥 𝑥 X iajs-1821 38 7 -∑ -∑ X iajs-1821 38 8 𝑓 𝑓 ADV iajs-1821 38 9 0 0 NUM iajs-1821 38 10 ! ! PUNCT iajs-1821 39 1 , , PUNCT iajs-1821 39 2 𝑥 𝑥 PROPN iajs-1821 39 3 0 0 NUM iajs-1821 39 4 , , PUNCT iajs-1821 39 5 𝑚 𝑚 PROPN iajs-1821 39 6 1 1 NUM iajs-1821 39 7 𝛼 𝛼 NOUN iajs-1821 39 8 𝑚 𝑚 PROPN iajs-1821 39 9 4 4 NUM iajs-1821 39 10 . . PUNCT iajs-1821 40 1 𝐽 𝐽 PROPN iajs-1821 40 2 𝐷 𝐷 PROPN iajs-1821 40 3 𝑓 𝑓 ADV iajs-1821 40 4 𝑥 𝑥 X iajs-1821 40 5 𝑓 𝑓 PRON iajs-1821 40 6 𝑥 𝑥 X iajs-1821 40 7 , , PUNCT iajs-1821 40 8 𝑥 𝑥 PROPN iajs-1821 40 9 0 0 NUM iajs-1821 40 10 , , PUNCT iajs-1821 40 11 𝑚 𝑚 PROPN iajs-1821 40 12 1 1 NUM iajs-1821 40 13 𝛼 𝛼 NOUN iajs-1821 40 14 𝑚 𝑚 PROPN iajs-1821 40 15 5 5 NUM iajs-1821 40 16 . . PUNCT iajs-1821 40 17 𝐷 𝐷 PROPN iajs-1821 40 18 𝐶 𝐶 PROPN iajs-1821 40 19 0 0 PROPN iajs-1821 40 20 , , PUNCT iajs-1821 40 21 c c PROPN iajs-1821 40 22 is be AUX iajs-1821 40 23 constant constant ADJ iajs-1821 40 24 6.𝐷 6.𝐷 NUM iajs-1821 40 25 𝑥 𝑥 NOUN iajs-1821 40 26 0 0 PUNCT iajs-1821 40 27 𝛽 𝛽 NOUN iajs-1821 40 28 ∈ ∈ NOUN iajs-1821 40 29 𝑁 𝑁 PROPN iajs-1821 40 30 , , PUNCT iajs-1821 40 31 𝛼 𝛼 VERB iajs-1821 40 32 𝑥 𝑥 X iajs-1821 40 33 𝛽 𝛽 NOUN iajs-1821 40 34 ∈ ∈ NOUN iajs-1821 40 35 𝑁 𝑁 PROPN iajs-1821 40 36 , , PUNCT iajs-1821 40 37 𝛽 𝛽 PROPN iajs-1821 40 38 𝛼 𝛼 NUM iajs-1821 40 39 where where SCONJ iajs-1821 40 40 𝛼 𝛼 PRON iajs-1821 40 41 denoted denote VERB iajs-1821 40 42 the the DET iajs-1821 40 43 smallest small ADJ iajs-1821 40 44 integer integer NOUN iajs-1821 40 45 greater great ADJ iajs-1821 40 46 than than ADP iajs-1821 40 47 or or CCONJ iajs-1821 40 48 equal equal ADJ iajs-1821 40 49 to to ADP iajs-1821 40 50 𝛼 𝛼 PRON iajs-1821 40 51 and and CCONJ iajs-1821 40 52 𝑁 𝑁 PROPN iajs-1821 40 53 0,1,2 0,1,2 NUM iajs-1821 40 54 , , PUNCT iajs-1821 40 55 … … PUNCT iajs-1821 40 56 . . PUNCT iajs-1821 41 1 2.1 2.1 NUM iajs-1821 41 2 the the DET iajs-1821 41 3 derivative derivative NOUN iajs-1821 41 4 for for ADP iajs-1821 41 5 orthonormal orthonormal ADJ iajs-1821 41 6 brnstein brnstein NOUN iajs-1821 41 7 polynomials polynomial NOUN iajs-1821 41 8 : : PUNCT iajs-1821 41 9 the the DET iajs-1821 41 10 bernstein bernstein PROPN iajs-1821 41 11 polynomials polynomial NOUN iajs-1821 41 12 of of ADP iajs-1821 41 13 𝑛th 𝑛th NOUN iajs-1821 41 14 degree degree NOUN iajs-1821 41 15 are be AUX iajs-1821 41 16 defined define VERB iajs-1821 41 17 on on ADP iajs-1821 41 18 the the DET iajs-1821 41 19 interval interval NOUN iajs-1821 41 20 [ [ X iajs-1821 41 21 0,1 0,1 NUM iajs-1821 41 22 ] ] PUNCT iajs-1821 41 23 as[6 as[6 NUM iajs-1821 41 24 ] ] PUNCT iajs-1821 41 25 . . PUNCT iajs-1821 42 1 𝐵 𝐵 NOUN iajs-1821 42 2 , , PUNCT iajs-1821 42 3 𝑥 𝑥 PROPN iajs-1821 42 4 𝑛 𝑛 VERB iajs-1821 42 5 𝑖 𝑖 SYM iajs-1821 42 6 𝑥 𝑥 PROPN iajs-1821 42 7 1 1 NUM iajs-1821 42 8 𝑥 𝑥 NOUN iajs-1821 42 9 , , PUNCT iajs-1821 42 10 𝑛 𝑛 PROPN iajs-1821 42 11 𝑖 𝑖 SYM iajs-1821 42 12 𝒏 𝒏 PROPN iajs-1821 42 13 ! ! PUNCT iajs-1821 43 1 𝒊 𝒊 X iajs-1821 43 2 ! ! PUNCT iajs-1821 43 3 𝒏 𝒏 PROPN iajs-1821 43 4 𝒊 𝒊 X iajs-1821 43 5 ! ! PUNCT iajs-1821 44 1 𝑓𝑜𝑟 𝑓𝑜𝑟 ADV iajs-1821 44 2 𝑖 𝑖 DET iajs-1821 44 3 0,1,2 0,1,2 NOUN iajs-1821 44 4 , , PUNCT iajs-1821 44 5 … … PUNCT iajs-1821 44 6 , , PUNCT iajs-1821 44 7 𝑛 𝑛 DET iajs-1821 44 8 the the DET iajs-1821 44 9 representation representation NOUN iajs-1821 44 10 of of ADP iajs-1821 44 11 the the DET iajs-1821 44 12 orthonormal orthonormal ADJ iajs-1821 44 13 bernstein bernstein PROPN iajs-1821 44 14 polynomials polynomial NOUN iajs-1821 44 15 , , PUNCT iajs-1821 44 16 denoted denote VERB iajs-1821 44 17 by by ADP iajs-1821 44 18 𝒃𝒊,𝒏 𝒃𝒊,𝒏 PUNCT iajs-1821 44 19 𝒙 𝒙 X iajs-1821 44 20 here here ADV iajs-1821 44 21 , , PUNCT iajs-1821 44 22 was be AUX iajs-1821 44 23 discovered discover VERB iajs-1821 44 24 by by ADP iajs-1821 44 25 analyzing analyze VERB iajs-1821 44 26 the the DET iajs-1821 44 27 resulting result VERB iajs-1821 44 28 orthonormal orthonormal ADJ iajs-1821 44 29 polynomials polynomial NOUN iajs-1821 44 30 after after ADP iajs-1821 44 31 applying apply VERB iajs-1821 44 32 the the DET iajs-1821 44 33 gramschmidt gramschmidt ADJ iajs-1821 44 34 process process NOUN iajs-1821 44 35 on on ADP iajs-1821 44 36 sets set NOUN iajs-1821 44 37 of of ADP iajs-1821 44 38 bernstein bernstein PROPN iajs-1821 44 39 polynomials polynomial NOUN iajs-1821 44 40 of of ADP iajs-1821 44 41 degree degree NOUN iajs-1821 44 42 𝐵 𝐵 NOUN iajs-1821 44 43 , , PUNCT iajs-1821 44 44 𝑥 𝑥 PROPN iajs-1821 44 45 . . PUNCT iajs-1821 45 1 then then ADV iajs-1821 45 2 the the DET iajs-1821 45 3 following follow VERB iajs-1821 45 4 sets set NOUN iajs-1821 45 5 of of ADP iajs-1821 45 6 orthonormal orthonormal ADJ iajs-1821 45 7 polynomials polynomial NOUN iajs-1821 45 8 𝑏 𝑏 NOUN iajs-1821 45 9 , , PUNCT iajs-1821 45 10 𝑥 𝑥 X iajs-1821 45 11 , , PUNCT iajs-1821 45 12 0 0 NUM iajs-1821 45 13 ≤ ≤ NUM iajs-1821 45 14 𝑖 𝑖 SYM iajs-1821 45 15 ≤ ≤ NOUN iajs-1821 45 16 𝑛. 𝑛. NOUN iajs-1821 45 17 for for ADP iajs-1821 45 18 𝑛 𝑛 PROPN iajs-1821 45 19 = = SYM iajs-1821 45 20 6 6 NUM iajs-1821 45 21 , , PUNCT iajs-1821 45 22 the the DET iajs-1821 45 23 four four NUM iajs-1821 45 24 orthonormal orthonormal ADJ iajs-1821 45 25 bernstein bernstein NOUN iajs-1821 45 26 polynomials polynomial NOUN iajs-1821 45 27 are be AUX iajs-1821 45 28 given give VERB iajs-1821 45 29 as as ADP iajs-1821 45 30 : : PUNCT iajs-1821 45 31 𝑏 𝑏 NOUN iajs-1821 45 32 , , PUNCT iajs-1821 45 33 𝑥 𝑥 NOUN iajs-1821 45 34 = = SYM iajs-1821 45 35 √13 √13 NUM iajs-1821 45 36 1 1 NUM iajs-1821 45 37 𝑥 𝑥 NOUN iajs-1821 45 38 , , PUNCT iajs-1821 45 39 𝑏 𝑏 PROPN iajs-1821 45 40 , , PUNCT iajs-1821 45 41 𝑥 𝑥 NOUN iajs-1821 45 42 = = SYM iajs-1821 45 43 √44 √44 PROPN iajs-1821 46 1 [ [ X iajs-1821 46 2 6x 6x NUM iajs-1821 46 3 1 1 NUM iajs-1821 46 4 𝑥 𝑥 DET iajs-1821 46 5 1 1 NUM iajs-1821 46 6 𝑥 𝑥 NOUN iajs-1821 46 7 ] ] PUNCT iajs-1821 46 8 𝑏 𝑏 NOUN iajs-1821 46 9 , , PUNCT iajs-1821 46 10 𝑥 𝑥 NOUN iajs-1821 46 11 = = SYM iajs-1821 46 12 11 11 NUM iajs-1821 46 13 15𝑥 15𝑥 NOUN iajs-1821 46 14 1 1 NUM iajs-1821 46 15 𝑥 𝑥 NOUN iajs-1821 46 16 6𝑥 6𝑥 NUM iajs-1821 46 17 1 1 NUM iajs-1821 46 18 𝑥 𝑥 NOUN iajs-1821 46 19 1 1 NUM iajs-1821 46 20 𝑥 𝑥 NOUN iajs-1821 46 21 𝑏 𝑏 NOUN iajs-1821 46 22 , , PUNCT iajs-1821 46 23 𝑥 𝑥 X iajs-1821 46 24 = = PUNCT iajs-1821 46 25 √252 √252 PROPN iajs-1821 46 26 20𝑥 20𝑥 NOUN iajs-1821 46 27 1 1 NUM iajs-1821 46 28 𝑥 𝑥 NOUN iajs-1821 46 29 𝑥 𝑥 NUM iajs-1821 46 30 1 1 NUM iajs-1821 46 31 𝑥 𝑥 NOUN iajs-1821 46 32 5𝑥 5𝑥 NUM iajs-1821 46 33 1 1 NUM iajs-1821 46 34 𝑥 𝑥 DET iajs-1821 46 35 1 1 NUM iajs-1821 46 36 𝑥 𝑥 NOUN iajs-1821 46 37 𝑏 𝑏 NOUN iajs-1821 46 38 , , PUNCT iajs-1821 46 39 𝑥 𝑥 NOUN iajs-1821 46 40 = = SYM iajs-1821 46 41 √ √ NUM iajs-1821 46 42 15𝑥 15𝑥 NUM iajs-1821 46 43 1 1 NUM iajs-1821 46 44 𝑥 𝑥 NOUN iajs-1821 46 45 40 40 NUM iajs-1821 46 46 𝑥 𝑥 NOUN iajs-1821 46 47 1 1 NUM iajs-1821 46 48 𝑥 𝑥 PRON iajs-1821 46 49 𝑥 𝑥 ADP iajs-1821 46 50 1 1 NUM iajs-1821 46 51 𝑥 𝑥 NOUN iajs-1821 46 52 𝑥 𝑥 ADP iajs-1821 46 53 1 1 NUM iajs-1821 46 54 𝑥 𝑥 NOUN iajs-1821 46 55 + + NUM iajs-1821 46 56 1 1 NUM iajs-1821 46 57 𝑥 𝑥 NOUN iajs-1821 46 58 𝑏 𝑏 NOUN iajs-1821 46 59 , , PUNCT iajs-1821 46 60 𝑥 𝑥 NOUN iajs-1821 46 61 = = SYM iajs-1821 46 62 √ √ NOUN iajs-1821 46 63 6𝑥 6𝑥 NUM iajs-1821 46 64 1 1 NUM iajs-1821 46 65 𝑥 𝑥 NOUN iajs-1821 46 66 𝑥 𝑥 NUM iajs-1821 46 67 1 1 NUM iajs-1821 46 68 𝑥 𝑥 NOUN iajs-1821 46 69 60𝑥 60𝑥 NUM iajs-1821 46 70 1 1 NUM iajs-1821 46 71 𝑥 𝑥 NOUN iajs-1821 46 72 30𝑥 30𝑥 NUM iajs-1821 46 73 1 1 NUM iajs-1821 46 74 𝑥 𝑥 NOUN iajs-1821 46 75 𝑥 𝑥 NUM iajs-1821 46 76 1 1 NUM iajs-1821 46 77 𝑥 𝑥 NOUN iajs-1821 46 78 1 1 NUM iajs-1821 46 79 𝑥 𝑥 NOUN iajs-1821 46 80 𝑏 𝑏 NOUN iajs-1821 46 81 , , PUNCT iajs-1821 46 82 𝑥 𝑥 PROPN iajs-1821 46 83 = = SYM iajs-1821 46 84 7 7 NUM iajs-1821 46 85 𝑥 𝑥 PROPN iajs-1821 46 86 18𝑥 18𝑥 NUM iajs-1821 46 87 1 1 NUM iajs-1821 46 88 𝑥 𝑥 NOUN iajs-1821 46 89 75 75 NUM iajs-1821 46 90 𝑥 𝑥 NOUN iajs-1821 46 91 1 1 NUM iajs-1821 46 92 𝑥 𝑥 PROPN iajs-1821 46 93 100𝑥 100𝑥 NUM iajs-1821 46 94 1 1 NUM iajs-1821 46 95 𝑥 𝑥 PROPN iajs-1821 46 96 45𝑥 45𝑥 NOUN iajs-1821 46 97 1 1 NUM iajs-1821 46 98 𝑥 𝑥 NOUN iajs-1821 46 99 6𝑥 6𝑥 NUM iajs-1821 46 100 1 1 NUM iajs-1821 46 101 𝑥 𝑥 SYM iajs-1821 46 102 1 1 NUM iajs-1821 46 103 𝑥 𝑥 PRON iajs-1821 46 104 ihsciconf ihsciconf ADJ iajs-1821 46 105 2017 2017 NUM iajs-1821 46 106 special special ADJ iajs-1821 46 107 issue issue NOUN iajs-1821 46 108 ibn ibn PROPN iajs-1821 46 109 al al PROPN iajs-1821 46 110 - - PUNCT iajs-1821 46 111 haitham haitham PROPN iajs-1821 46 112 journal journal PROPN iajs-1821 46 113 for for ADP iajs-1821 46 114 pure pure ADJ iajs-1821 46 115 and and CCONJ iajs-1821 46 116 applied apply VERB iajs-1821 46 117 science science NOUN iajs-1821 46 118 https://doi.org/ https://doi.org/ NOUN iajs-1821 46 119 10.30526/2017.ihsciconf.1821 10.30526/2017.ihsciconf.1821 NUM iajs-1821 46 120 for for ADP iajs-1821 46 121 more more ADJ iajs-1821 46 122 information information NOUN iajs-1821 46 123 about about ADP iajs-1821 46 124 the the DET iajs-1821 46 125 conference conference NOUN iajs-1821 46 126 please please INTJ iajs-1821 46 127 visit visit VERB iajs-1821 46 128 the the DET iajs-1821 46 129 websites website NOUN iajs-1821 46 130 : : PUNCT iajs-1821 46 131 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1821 46 132     SPACE iajs-1821 46 133 www.ihsciconf.org www.ihsciconf.org ADJ iajs-1821 46 134       SPACE iajs-1821 46 135 mathematics|493 mathematics|493 NOUN iajs-1821 46 136         SPACE iajs-1821 46 137 3 3 NUM iajs-1821 46 138 . . PUNCT iajs-1821 46 139 analysis analysis NOUN iajs-1821 46 140 of of ADP iajs-1821 46 141 the the DET iajs-1821 46 142 petrov petrov PROPN iajs-1821 46 143 - - PUNCT iajs-1821 46 144 galerkin galerkin PROPN iajs-1821 46 145 method method NOUN iajs-1821 46 146 ( ( PUNCT iajs-1821 46 147 pgm pgm PROPN iajs-1821 46 148 ): ): PUNCT iajs-1821 46 149 in in ADP iajs-1821 46 150 this this DET iajs-1821 46 151 section section NOUN iajs-1821 46 152 we we PRON iajs-1821 46 153 introduce introduce VERB iajs-1821 46 154 the the DET iajs-1821 46 155 ( ( PUNCT iajs-1821 46 156 pgm pgm NOUN iajs-1821 46 157 ) ) PUNCT iajs-1821 46 158 for for ADP iajs-1821 46 159 eq eq PROPN iajs-1821 46 160 . . PUNCT iajs-1821 47 1 ( ( PUNCT iajs-1821 47 2 2 2 NUM iajs-1821 47 3 ) ) PUNCT iajs-1821 47 4 . . PUNCT iajs-1821 48 1 for for ADP iajs-1821 48 2 the the DET iajs-1821 48 3 proof proof NOUN iajs-1821 48 4 of of ADP iajs-1821 48 5 all all DET iajs-1821 48 6 results result NOUN iajs-1821 48 7 in in ADP iajs-1821 48 8 this this DET iajs-1821 48 9 section section NOUN iajs-1821 48 10 , , PUNCT iajs-1821 48 11 we we PRON iajs-1821 48 12 can can AUX iajs-1821 48 13 use use VERB iajs-1821 48 14 the the DET iajs-1821 48 15 same same ADJ iajs-1821 48 16 manner manner NOUN iajs-1821 48 17 used use VERB iajs-1821 48 18 in in ADP iajs-1821 48 19 [ [ X iajs-1821 48 20 7 7 NUM iajs-1821 48 21 ] ] PUNCT iajs-1821 48 22 , , PUNCT iajs-1821 48 23 but but CCONJ iajs-1821 48 24 for for ADP iajs-1821 48 25 eq eq NOUN iajs-1821 48 26 . . PUNCT iajs-1821 49 1 ( ( PUNCT iajs-1821 49 2 2 2 NUM iajs-1821 49 3 ) ) PUNCT iajs-1821 49 4 . . PUNCT iajs-1821 50 1 let let VERB iajs-1821 50 2 x x PRON iajs-1821 50 3 be be AUX iajs-1821 50 4 a a DET iajs-1821 50 5 banach banach NOUN iajs-1821 50 6 space space NOUN iajs-1821 50 7 with with ADP iajs-1821 50 8 the the DET iajs-1821 50 9 norm norm NOUN iajs-1821 50 10 ‖. ‖. PROPN iajs-1821 50 11 ‖ ‖ PROPN iajs-1821 50 12 and and CCONJ iajs-1821 50 13 let let VERB iajs-1821 50 14 𝑋∗ 𝑋∗ ADJ iajs-1821 50 15 denote denote VERB iajs-1821 50 16 its its PRON iajs-1821 50 17 dual dual ADJ iajs-1821 50 18 space space NOUN iajs-1821 50 19 . . PUNCT iajs-1821 51 1 assume assume VERB iajs-1821 51 2 k k X iajs-1821 51 3 : : PUNCT iajs-1821 51 4 x x X iajs-1821 51 5 → → PUNCT iajs-1821 51 6 x x X iajs-1821 51 7 is be AUX iajs-1821 51 8 a a DET iajs-1821 51 9 compact compact ADJ iajs-1821 51 10 linear linear NOUN iajs-1821 51 11 operator operator NOUN iajs-1821 51 12 . . PUNCT iajs-1821 52 1 we we PRON iajs-1821 52 2 rewrite rewrite VERB iajs-1821 52 3 this this DET iajs-1821 52 4 eq.(1 eq.(1 ADJ iajs-1821 52 5 ) ) PUNCT iajs-1821 52 6 in in ADP iajs-1821 52 7 operator operator NOUN iajs-1821 52 8 from from ADP iajs-1821 52 9 as as ADP iajs-1821 52 10 : : PUNCT iajs-1821 52 11 𝐷∗ 𝐷∗ PROPN iajs-1821 52 12 𝑢 𝑢 PROPN iajs-1821 52 13 𝐾𝑢 𝐾𝑢 PROPN iajs-1821 52 14 𝑓 𝑓 PROPN iajs-1821 52 15 , , PUNCT iajs-1821 52 16 𝑓 𝑓 DET iajs-1821 52 17 ∈ ∈ PROPN iajs-1821 52 18 𝑋 𝑋 NOUN iajs-1821 52 19 ... ... PUNCT iajs-1821 52 20 ( ( PUNCT iajs-1821 52 21 2 2 X iajs-1821 52 22 ) ) PUNCT iajs-1821 52 23 where where SCONJ iajs-1821 52 24 u u PROPN iajs-1821 52 25 ∈ ∈ PROPN iajs-1821 52 26 x x X iajs-1821 52 27 is be AUX iajs-1821 52 28 the the DET iajs-1821 52 29 unknown unknown ADJ iajs-1821 52 30 to to PART iajs-1821 52 31 be be AUX iajs-1821 52 32 determined determine VERB iajs-1821 52 33 . . PUNCT iajs-1821 53 1 the the DET iajs-1821 53 2 peterov peterov PROPN iajs-1821 53 3 - - PUNCT iajs-1821 53 4 galerkin galerkin ADJ iajs-1821 53 5 method method NOUN iajs-1821 53 6 ( ( PUNCT iajs-1821 53 7 pgm pgm NOUN iajs-1821 53 8 ) ) PUNCT iajs-1821 53 9 used use VERB iajs-1821 53 10 for for ADP iajs-1821 53 11 the the DET iajs-1821 53 12 numerical numerical ADJ iajs-1821 53 13 solutions solution NOUN iajs-1821 53 14 of of ADP iajs-1821 53 15 eq.(2 eq.(2 NOUN iajs-1821 53 16 ) ) PUNCT iajs-1821 53 17 . . PUNCT iajs-1821 54 1 the the DET iajs-1821 54 2 petrov petrov PROPN iajs-1821 54 3 - - PUNCT iajs-1821 54 4 galerkin galerkin ADJ iajs-1821 54 5 methods method NOUN iajs-1821 54 6 ( ( PUNCT iajs-1821 54 7 pgm pgm NOUN iajs-1821 54 8 ) ) PUNCT iajs-1821 54 9 interpolate interpolate NOUN iajs-1821 54 10 between between ADP iajs-1821 54 11 the the DET iajs-1821 54 12 galerkin galerkin ADJ iajs-1821 54 13 method method NOUN iajs-1821 54 14 and and CCONJ iajs-1821 54 15 the the DET iajs-1821 54 16 collocation collocation NOUN iajs-1821 54 17 method method NOUN iajs-1821 54 18 . . PUNCT iajs-1821 55 1 for for ADP iajs-1821 55 2 this this DET iajs-1821 55 3 purpose purpose NOUN iajs-1821 55 4 for for ADP iajs-1821 55 5 each each DET iajs-1821 55 6 positive positive ADJ iajs-1821 55 7 integer integer NOUN iajs-1821 55 8 n n CCONJ iajs-1821 55 9 , , PUNCT iajs-1821 55 10 we we PRON iajs-1821 55 11 assume assume VERB iajs-1821 55 12 that that SCONJ iajs-1821 55 13 𝑋 𝑋 PROPN iajs-1821 55 14 ⊂ ⊂ PROPN iajs-1821 55 15 x x X iajs-1821 55 16 , , PUNCT iajs-1821 55 17 𝑌 𝑌 PROPN iajs-1821 55 18 ⊂ ⊂ PROPN iajs-1821 55 19 𝑋∗ 𝑋∗ PROPN iajs-1821 55 20 , , PUNCT iajs-1821 55 21 and and CCONJ iajs-1821 55 22 xn xn NUM iajs-1821 55 23 , , PUNCT iajs-1821 55 24 𝑌 𝑌 PROPN iajs-1821 55 25 are be AUX iajs-1821 55 26 finite finite ADJ iajs-1821 55 27 dimensional dimensional ADJ iajs-1821 55 28 vector vector NOUN iajs-1821 55 29 spaces space NOUN iajs-1821 55 30 with with ADP iajs-1821 55 31 dim dim ADJ iajs-1821 55 32 xn xn PUNCT iajs-1821 56 1 = = SYM iajs-1821 56 2 dim dim PROPN iajs-1821 56 3 yn yn X iajs-1821 56 4 , , PUNCT iajs-1821 56 5 then then ADV iajs-1821 56 6 x x PUNCT iajs-1821 56 7 n n X iajs-1821 56 8 , , PUNCT iajs-1821 56 9 𝑌 𝑌 PROPN iajs-1821 56 10 satisfy satisfy NOUN iajs-1821 56 11 condition condition NOUN iajs-1821 56 12 ( ( PUNCT iajs-1821 56 13 h h NOUN iajs-1821 56 14 ) ) PUNCT iajs-1821 56 15 : : PUNCT iajs-1821 56 16 for for ADP iajs-1821 56 17 each each DET iajs-1821 56 18 x x SYM iajs-1821 56 19 ∈ ∈ PROPN iajs-1821 56 20 x x X iajs-1821 56 21 and and CCONJ iajs-1821 56 22 y y PROPN iajs-1821 56 23 ∈ ∈ PROPN iajs-1821 56 24 𝑋∗ 𝑋∗ ADV iajs-1821 56 25 , , PUNCT iajs-1821 56 26 there there PRON iajs-1821 56 27 exists exist VERB iajs-1821 56 28 x x PUNCT iajs-1821 56 29 n n CCONJ iajs-1821 56 30 ∈ ∈ NOUN iajs-1821 56 31 x x SYM iajs-1821 56 32 n n PROPN iajs-1821 56 33 and and CCONJ iajs-1821 56 34 y y PROPN iajs-1821 57 1 n n PRON iajs-1821 57 2 ∈ ∈ PROPN iajs-1821 57 3 yn yn INTJ iajs-1821 57 4 such such ADJ iajs-1821 57 5 that∥x that∥x PROPN iajs-1821 57 6 n n CCONJ iajs-1821 57 7 ̶ ̶ PROPN iajs-1821 57 8 x∥→ x∥→ NOUN iajs-1821 57 9 0 0 PUNCT iajs-1821 57 10 as as ADP iajs-1821 57 11 n n PROPN iajs-1821 57 12 → → SYM iajs-1821 57 13 ∞. ∞. PROPN iajs-1821 57 14 when when SCONJ iajs-1821 57 15 the the DET iajs-1821 57 16 peterov peterov PROPN iajs-1821 57 17 - - PUNCT iajs-1821 57 18 galerkin galerkin PROPN iajs-1821 57 19 method(pgm method(pgm PROPN iajs-1821 57 20 ) ) PUNCT iajs-1821 57 21 for for ADP iajs-1821 57 22 eq.(2 eq.(2 NOUN iajs-1821 57 23 ) ) PUNCT iajs-1821 57 24 is be AUX iajs-1821 57 25 a a DET iajs-1821 57 26 numerical numerical ADJ iajs-1821 57 27 method method NOUN iajs-1821 57 28 for for ADP iajs-1821 57 29 finding find VERB iajs-1821 57 30 𝑢 𝑢 PRON iajs-1821 57 31 ∈ ∈ NOUN iajs-1821 57 32 𝑋 𝑋 NOUN iajs-1821 57 33 such such ADJ iajs-1821 57 34 that that SCONJ iajs-1821 57 35 〈 〈 PROPN iajs-1821 57 36 𝐷∗ 𝐷∗ NOUN iajs-1821 57 37 𝑢 𝑢 PROPN iajs-1821 57 38 ̶ ̶ PROPN iajs-1821 57 39 k k PROPN iajs-1821 57 40 un un PROPN iajs-1821 57 41 , , PUNCT iajs-1821 57 42 yn yn PROPN iajs-1821 57 43 〉 〉 NOUN iajs-1821 57 44 = = SYM iajs-1821 57 45 〈 〈 PROPN iajs-1821 57 46 f f NOUN iajs-1821 57 47 , , PUNCT iajs-1821 57 48 yn yn PROPN iajs-1821 57 49 〉 〉 NOUN iajs-1821 57 50 for for ADP iajs-1821 57 51 all all DET iajs-1821 57 52 𝑦 𝑦 NOUN iajs-1821 57 53 ∈ ∈ NOUN iajs-1821 57 54 𝑌 𝑌 PROPN iajs-1821 57 55 ... ... PUNCT iajs-1821 57 56 ( ( PUNCT iajs-1821 57 57 3 3 X iajs-1821 57 58 ) ) PUNCT iajs-1821 57 59 it it PRON iajs-1821 57 60 is be AUX iajs-1821 57 61 clear clear ADJ iajs-1821 57 62 that that SCONJ iajs-1821 57 63 the the DET iajs-1821 57 64 petrov petrov PROPN iajs-1821 57 65 - - PUNCT iajs-1821 57 66 galerkin galerkin PROPN iajs-1821 57 67 method(pgm method(pgm PROPN iajs-1821 57 68 ) ) PUNCT iajs-1821 57 69 is be AUX iajs-1821 57 70 closely closely ADV iajs-1821 57 71 related relate VERB iajs-1821 57 72 to to ADP iajs-1821 57 73 a a DET iajs-1821 57 74 generalized generalized ADJ iajs-1821 57 75 best good ADJ iajs-1821 57 76 approximation approximation NOUN iajs-1821 57 77 from from ADP iajs-1821 57 78 𝑋 𝑋 PROPN iajs-1821 57 79 to to ADP iajs-1821 57 80 x x SYM iajs-1821 57 81 ∈ ∈ PROPN iajs-1821 57 82 x x PUNCT iajs-1821 57 83 with with ADP iajs-1821 57 84 respect respect NOUN iajs-1821 57 85 to to ADP iajs-1821 57 86 𝑌 𝑌 PROPN iajs-1821 57 87 , , PUNCT iajs-1821 57 88 . . PUNCT iajs-1821 58 1 given give VERB iajs-1821 58 2 x x PUNCT iajs-1821 58 3 ∈ ∈ PROPN iajs-1821 58 4 x x NOUN iajs-1821 58 5 , , PUNCT iajs-1821 58 6 an an DET iajs-1821 58 7 element element NOUN iajs-1821 58 8 𝑃 𝑃 NOUN iajs-1821 58 9 𝑥 𝑥 PRON iajs-1821 58 10 ∈ ∈ NOUN iajs-1821 58 11 𝑋 𝑋 NOUN iajs-1821 58 12 is be AUX iajs-1821 58 13 called call VERB iajs-1821 58 14 a a DET iajs-1821 58 15 generalized generalized ADJ iajs-1821 58 16 best good ADJ iajs-1821 58 17 approximation approximation NOUN iajs-1821 58 18 from from ADP iajs-1821 58 19 𝑋 𝑋 PROPN iajs-1821 58 20 to to ADP iajs-1821 58 21 x x PUNCT iajs-1821 58 22 with with ADP iajs-1821 58 23 respect respect NOUN iajs-1821 58 24 to to ADP iajs-1821 58 25 𝑌 𝑌 PROPN iajs-1821 58 26 if if SCONJ iajs-1821 58 27 it it PRON iajs-1821 58 28 satisfies satisfy VERB iajs-1821 58 29 the the DET iajs-1821 58 30 equation equation NOUN iajs-1821 58 31 〈 〈 PROPN iajs-1821 58 32 x x PROPN iajs-1821 58 33 ⎯ ⎯ PROPN iajs-1821 58 34 𝑃 𝑃 PROPN iajs-1821 58 35 𝑥 𝑥 PROPN iajs-1821 58 36 , , PUNCT iajs-1821 58 37 yn yn NOUN iajs-1821 58 38 〉 〉 NOUN iajs-1821 58 39 = = SYM iajs-1821 58 40 0 0 NUM iajs-1821 58 41 for for ADP iajs-1821 58 42 all all PRON iajs-1821 58 43 𝑦 𝑦 NOUN iajs-1821 58 44 ∈ ∈ NOUN iajs-1821 58 45 𝑌 𝑌 PROPN iajs-1821 58 46 ... ... PUNCT iajs-1821 58 47 ( ( PUNCT iajs-1821 58 48 4 4 X iajs-1821 58 49 ) ) PUNCT iajs-1821 58 50 similarly similarly ADV iajs-1821 58 51 , , PUNCT iajs-1821 58 52 given give VERB iajs-1821 58 53 𝑦 𝑦 PRON iajs-1821 58 54 ∈ ∈ NOUN iajs-1821 58 55 𝑋∗ 𝑋∗ NOUN iajs-1821 58 56 , , PUNCT iajs-1821 58 57 an an DET iajs-1821 58 58 element element NOUN iajs-1821 58 59 𝑝 𝑝 PROPN iajs-1821 58 60 𝑦 𝑦 PROPN iajs-1821 58 61 ∈ ∈ NOUN iajs-1821 58 62 𝑌 𝑌 PROPN iajs-1821 58 63 is be AUX iajs-1821 58 64 called call VERB iajs-1821 58 65 best good ADJ iajs-1821 58 66 approxima approxima PROPN iajs-1821 58 67 tion tion NOUN iajs-1821 58 68 from from ADP iajs-1821 58 69 𝑌 𝑌 PROPN iajs-1821 58 70 𝑡𝑜 𝑡𝑜 PROPN iajs-1821 58 71 𝑦 𝑦 NOUN iajs-1821 58 72 with with ADP iajs-1821 58 73 respect respect NOUN iajs-1821 58 74 𝑌 𝑌 PROPN iajs-1821 58 75 to to ADP iajs-1821 58 76 𝑦 𝑦 PRON iajs-1821 58 77 if if SCONJ iajs-1821 58 78 it it PRON iajs-1821 58 79 satisfies satisfy VERB iajs-1821 58 80 the the DET iajs-1821 58 81 equatio equatio NOUN iajs-1821 58 82 〈 〈 NOUN iajs-1821 58 83 𝑥 𝑥 PROPN iajs-1821 58 84 , , PUNCT iajs-1821 58 85 𝑦 𝑦 NOUN iajs-1821 58 86 𝑝 𝑝 NOUN iajs-1821 58 87 𝑦〉= 𝑦〉= NOUN iajs-1821 58 88 0 0 PUNCT iajs-1821 59 1 for for ADP iajs-1821 59 2 all all PRON iajs-1821 59 3 𝑥 𝑥 DET iajs-1821 59 4 ∈ ∈ PROPN iajs-1821 59 5 𝑋 𝑋 NOUN iajs-1821 59 6 . . PUNCT iajs-1821 60 1 proposition proposition NOUN iajs-1821 60 2 : : PUNCT iajs-1821 60 3 for for ADP iajs-1821 60 4 each each DET iajs-1821 60 5 x x SYM iajs-1821 60 6 ∈ ∈ PROPN iajs-1821 60 7 x x NOUN iajs-1821 60 8 , , PUNCT iajs-1821 60 9 the the DET iajs-1821 60 10 generalized generalized ADJ iajs-1821 60 11 best good ADJ iajs-1821 60 12 approximation approximation NOUN iajs-1821 60 13 from from ADP iajs-1821 60 14 𝑋 𝑋 PROPN iajs-1821 60 15 to to ADP iajs-1821 60 16 x x PUNCT iajs-1821 60 17 with with ADP iajs-1821 60 18 respect respect NOUN iajs-1821 60 19 to to ADP iajs-1821 60 20 𝑌 𝑌 PROPN iajs-1821 60 21 exists exist VERB iajs-1821 60 22 uniquely uniquely ADV iajs-1821 60 23 if if SCONJ iajs-1821 61 1 and and CCONJ iajs-1821 61 2 only only ADV iajs-1821 61 3 if if SCONJ iajs-1821 61 4 y y PROPN iajs-1821 61 5 n n X iajs-1821 61 6 ∩ ∩ X iajs-1821 61 7 𝑋 𝑋 PROPN iajs-1821 61 8 = = PRON iajs-1821 61 9 { { PUNCT iajs-1821 61 10 0 0 NUM iajs-1821 61 11 } } PUNCT iajs-1821 61 12 ... ... PUNCT iajs-1821 61 13 ( ( PUNCT iajs-1821 61 14 5 5 X iajs-1821 61 15 ) ) PUNCT iajs-1821 61 16 under under ADP iajs-1821 61 17 this this DET iajs-1821 61 18 condition condition NOUN iajs-1821 61 19 , , PUNCT iajs-1821 61 20 𝑃 𝑃 PROPN iajs-1821 61 21 is be AUX iajs-1821 61 22 a a DET iajs-1821 61 23 projection projection NOUN iajs-1821 61 24 ; ; PUNCT iajs-1821 61 25 i.e. i.e. X iajs-1821 61 26 , , PUNCT iajs-1821 61 27 𝑃 𝑃 NOUN iajs-1821 61 28 𝑃 𝑃 NOUN iajs-1821 61 29 / / PUNCT iajs-1821 61 30 assume assume VERB iajs-1821 61 31 that that SCONJ iajs-1821 61 32 , , PUNCT iajs-1821 61 33 for for ADP iajs-1821 61 34 each each DET iajs-1821 61 35 n n CCONJ iajs-1821 61 36 , , PUNCT iajs-1821 61 37 there there PRON iajs-1821 61 38 is be VERB iajs-1821 61 39 a a DET iajs-1821 61 40 linear linear ADJ iajs-1821 61 41 operator operator NOUN iajs-1821 61 42 ∏ ∏ NOUN iajs-1821 61 43 : : PUNCT iajs-1821 61 44 𝑋 𝑋 PROPN iajs-1821 61 45 ⟶ ⟶ NOUN iajs-1821 61 46 𝑌 𝑌 PROPN iajs-1821 61 47 with with ADP iajs-1821 61 48 ∏ ∏ X iajs-1821 61 49 𝑋 𝑋 PROPN iajs-1821 61 50 = = SYM iajs-1821 61 51 𝑌 𝑌 PROPN iajs-1821 61 52 , , PUNCT iajs-1821 61 53 and and CCONJ iajs-1821 61 54 satisfying satisfy VERB iajs-1821 61 55 the the DET iajs-1821 61 56 condition condition NOUN iajs-1821 61 57 ( ( PUNCT iajs-1821 61 58 h-1 h-1 NOUN iajs-1821 61 59 ) ) PUNCT iajs-1821 62 1 ‖𝑥 ‖𝑥 PROPN iajs-1821 62 2 ‖ ‖ PROPN iajs-1821 62 3 ≤ ≤ PROPN iajs-1821 62 4 c1 c1 PROPN iajs-1821 62 5 〈 〈 PROPN iajs-1821 62 6 𝑥 𝑥 PROPN iajs-1821 62 7 , , PUNCT iajs-1821 62 8 ∏ ∏ PROPN iajs-1821 62 9 𝑥 𝑥 PRON iajs-1821 62 10 〉 〉 NOUN iajs-1821 62 11 for for ADP iajs-1821 62 12 all all DET iajs-1821 62 13 𝑥 𝑥 DET iajs-1821 62 14 ∈ ∈ PROPN iajs-1821 62 15 𝑋 𝑋 PROPN iajs-1821 62 16 ` ` PUNCT iajs-1821 62 17 , , PUNCT iajs-1821 62 18 ( ( PUNCT iajs-1821 62 19 h-2 h-2 PROPN iajs-1821 62 20 ) ) PUNCT iajs-1821 62 21 ‖∏ ‖∏ PUNCT iajs-1821 63 1 𝑥 𝑥 PRON iajs-1821 63 2 ‖ ‖ PROPN iajs-1821 63 3 c2 c2 PROPN iajs-1821 63 4 ‖𝑥 ‖𝑥 PROPN iajs-1821 63 5 ‖ ‖ PROPN iajs-1821 63 6 for for ADP iajs-1821 63 7 all all PRON iajs-1821 63 8 𝑥 𝑥 DET iajs-1821 63 9 ∈ ∈ NOUN iajs-1821 63 10 𝑋 𝑋 NOUN iajs-1821 63 11 , , PUNCT iajs-1821 63 12 where where SCONJ iajs-1821 63 13 c1 c1 PROPN iajs-1821 63 14 and and CCONJ iajs-1821 63 15 c2 c2 PROPN iajs-1821 63 16 are be AUX iajs-1821 63 17 positive positive ADJ iajs-1821 63 18 constants constant NOUN iajs-1821 63 19 independent independent ADJ iajs-1821 63 20 of of ADP iajs-1821 63 21 n. n. NOUN iajs-1821 63 22 if if SCONJ iajs-1821 63 23 a a DET iajs-1821 63 24 pair pair NOUN iajs-1821 63 25 of of ADP iajs-1821 63 26 sequence sequence NOUN iajs-1821 63 27 𝑋 𝑋 NOUN iajs-1821 63 28 and and CCONJ iajs-1821 63 29 𝑌 𝑌 PROPN iajs-1821 63 30 satisfy satisfy NOUN iajs-1821 63 31 ( ( PUNCT iajs-1821 63 32 h-1 h-1 NOUN iajs-1821 63 33 ) ) PUNCT iajs-1821 63 34 and and CCONJ iajs-1821 63 35 ( ( PUNCT iajs-1821 63 36 h-2 h-2 PROPN iajs-1821 63 37 ) ) PUNCT iajs-1821 63 38 , , PUNCT iajs-1821 63 39 we we PRON iajs-1821 63 40 call call VERB iajs-1821 63 41 { { PUNCT iajs-1821 63 42 𝑋 𝑋 NOUN iajs-1821 63 43 , , PUNCT iajs-1821 63 44 𝑌 𝑌 PROPN iajs-1821 63 45 } } PUNCT iajs-1821 63 46 a a DET iajs-1821 63 47 regular regular ADJ iajs-1821 63 48 pair pair NOUN iajs-1821 63 49 . . PUNCT iajs-1821 64 1 for for ADP iajs-1821 64 2 each each DET iajs-1821 64 3 x x SYM iajs-1821 64 4 ∈ ∈ PROPN iajs-1821 64 5 x x NOUN iajs-1821 64 6 , , PUNCT iajs-1821 64 7 let let VERB iajs-1821 64 8 𝑄 𝑄 PRON iajs-1821 64 9 𝑥 𝑥 AUX iajs-1821 64 10 be be AUX iajs-1821 64 11 a a DET iajs-1821 64 12 best good ADJ iajs-1821 64 13 approximation approximation NOUN iajs-1821 64 14 from from ADP iajs-1821 64 15 𝑋 𝑋 PROPN iajs-1821 64 16 to to ADP iajs-1821 64 17 x x PRON iajs-1821 64 18 , , PUNCT iajs-1821 64 19 that that ADV iajs-1821 64 20 is is ADV iajs-1821 64 21 , , PUNCT iajs-1821 64 22 𝑄 𝑄 PROPN iajs-1821 64 23 𝑥 𝑥 PRON iajs-1821 64 24 ∈ ∈ NOUN iajs-1821 64 25 𝑋 𝑋 NOUN iajs-1821 64 26 satisfies satisfy VERB iajs-1821 64 27 the the DET iajs-1821 64 28 equation equation NOUN iajs-1821 64 29 ‖𝑥 ‖𝑥 PRON iajs-1821 64 30 𝑄 𝑄 PRON iajs-1821 64 31 𝑥‖ 𝑥‖ VERB iajs-1821 64 32 ‖𝑥 ‖𝑥 PROPN iajs-1821 64 33 𝑥 𝑥 X iajs-1821 64 34 ‖∈ ‖∈ PUNCT iajs-1821 64 35 . . PUNCT iajs-1821 65 1 ihsciconf ihsciconf PROPN iajs-1821 65 2 2017 2017 NUM iajs-1821 65 3 special special ADJ iajs-1821 65 4 issue issue NOUN iajs-1821 65 5 ibn ibn PROPN iajs-1821 65 6 al al PROPN iajs-1821 65 7 - - PUNCT iajs-1821 65 8 haitham haitham PROPN iajs-1821 65 9 journal journal PROPN iajs-1821 65 10 for for ADP iajs-1821 65 11 pure pure ADJ iajs-1821 65 12 and and CCONJ iajs-1821 65 13 applied apply VERB iajs-1821 65 14 science science NOUN iajs-1821 65 15 https://doi.org/ https://doi.org/ NOUN iajs-1821 65 16 10.30526/2017.ihsciconf.1821 10.30526/2017.ihsciconf.1821 NUM iajs-1821 65 17 for for ADP iajs-1821 65 18 more more ADJ iajs-1821 65 19 information information NOUN iajs-1821 65 20 about about ADP iajs-1821 65 21 the the DET iajs-1821 65 22 conference conference NOUN iajs-1821 65 23 please please INTJ iajs-1821 65 24 visit visit VERB iajs-1821 65 25 the the DET iajs-1821 65 26 websites website NOUN iajs-1821 65 27 : : PUNCT iajs-1821 65 28 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1821 65 29     SPACE iajs-1821 65 30 www.ihsciconf.org www.ihsciconf.org ADJ iajs-1821 65 31       SPACE iajs-1821 65 32 mathematics|494 mathematics|494 NOUN iajs-1821 65 33         SPACE iajs-1821 65 34 if if SCONJ iajs-1821 65 35 a a DET iajs-1821 65 36 regular regular ADJ iajs-1821 65 37 pair pair NOUN iajs-1821 65 38 𝑋 𝑋 NOUN iajs-1821 65 39 , , PUNCT iajs-1821 65 40 𝑌 𝑌 PROPN iajs-1821 65 41 satisfies satisfie NOUN iajs-1821 65 42 dim dim VERB iajs-1821 65 43 xn xn PUNCT iajs-1821 66 1 = = SYM iajs-1821 66 2 dim dim VERB iajs-1821 66 3 yn yn NOUN iajs-1821 66 4 and and CCONJ iajs-1821 66 5 condition condition NOUN iajs-1821 66 6 ( ( PUNCT iajs-1821 66 7 h h NOUN iajs-1821 66 8 ) ) PUNCT iajs-1821 66 9 , , PUNCT iajs-1821 66 10 then then ADV iajs-1821 66 11 the the DET iajs-1821 66 12 corresponding corresponding ADJ iajs-1821 66 13 generalized generalized ADJ iajs-1821 66 14 projection projection NOUN iajs-1821 66 15 𝑃 𝑃 NOUN iajs-1821 66 16 satisfies satisfie NOUN iajs-1821 66 17 : : PUNCT iajs-1821 66 18 ( ( PUNCT iajs-1821 66 19 1 1 X iajs-1821 66 20 ) ) PUNCT iajs-1821 66 21 for for ADP iajs-1821 66 22 all all PRON iajs-1821 66 23 𝑥 𝑥 DET iajs-1821 66 24 ∈ ∈ PROPN iajs-1821 66 25 𝑋 𝑋 PROPN iajs-1821 66 26 , , PUNCT iajs-1821 66 27 ‖𝑃 ‖𝑃 NOUN iajs-1821 66 28 𝑥 𝑥 PROPN iajs-1821 66 29 𝑥‖ 𝑥‖ NOUN iajs-1821 66 30 → → SYM iajs-1821 66 31 0 0 NUM iajs-1821 66 32 as as ADP iajs-1821 66 33 n→ n→ ADV iajs-1821 66 34 ∞ ∞ PROPN iajs-1821 66 35 ( ( PUNCT iajs-1821 66 36 2 2 X iajs-1821 66 37 ) ) PUNCT iajs-1821 66 38 there there PRON iajs-1821 66 39 is be VERB iajs-1821 66 40 a a DET iajs-1821 66 41 constant constant ADJ iajs-1821 66 42 𝐶 𝐶 PROPN iajs-1821 66 43 0 0 NUM iajs-1821 67 1 such such ADJ iajs-1821 67 2 that that SCONJ iajs-1821 67 3 , , PUNCT iajs-1821 67 4 ‖𝑃 ‖𝑃 NOUN iajs-1821 67 5 ‖ ‖ PROPN iajs-1821 67 6 < < X iajs-1821 67 7 c c PROPN iajs-1821 67 8 , , PUNCT iajs-1821 67 9 n n NOUN iajs-1821 67 10 = = SYM iajs-1821 67 11 1,2 1,2 NUM iajs-1821 67 12 , , PUNCT iajs-1821 67 13 .. .. PUNCT iajs-1821 67 14 ( ( PUNCT iajs-1821 67 15 3 3 X iajs-1821 67 16 ) ) PUNCT iajs-1821 67 17 for for ADP iajs-1821 67 18 some some DET iajs-1821 67 19 constant constant ADJ iajs-1821 67 20 𝐶 𝐶 PROPN iajs-1821 67 21 0 0 NUM iajs-1821 67 22 independent independent NOUN iajs-1821 67 23 of of ADP iajs-1821 67 24 n n CCONJ iajs-1821 67 25 , , PUNCT iajs-1821 67 26 ‖𝑃 ‖𝑃 PROPN iajs-1821 67 27 𝑥 𝑥 PROPN iajs-1821 67 28 𝑥‖ 𝑥‖ VERB iajs-1821 67 29 𝐶‖𝑄 𝐶‖𝑄 ADJ iajs-1821 67 30 𝑥 𝑥 NOUN iajs-1821 67 31 𝑥‖ 𝑥‖ NOUN iajs-1821 67 32 where where SCONJ iajs-1821 67 33 𝑄 𝑄 PRON iajs-1821 67 34 𝑥 𝑥 PROPN iajs-1821 67 35 is be AUX iajs-1821 67 36 the the DET iajs-1821 67 37 best good ADJ iajs-1821 67 38 approximation approximation NOUN iajs-1821 67 39 from from ADP iajs-1821 67 40 x x PROPN iajs-1821 67 41 n n ADV iajs-1821 67 42 to to PART iajs-1821 67 43 x. x. VERB iajs-1821 67 44 if if SCONJ iajs-1821 67 45 { { PUNCT iajs-1821 67 46 𝑋 𝑋 NOUN iajs-1821 67 47 , , PUNCT iajs-1821 67 48 𝑌 𝑌 PROPN iajs-1821 67 49 } } PUNCT iajs-1821 67 50 a a DET iajs-1821 67 51 regular regular ADJ iajs-1821 67 52 pair pair NOUN iajs-1821 67 53 is be AUX iajs-1821 67 54 with with ADP iajs-1821 67 55 a a DET iajs-1821 67 56 linear linear ADJ iajs-1821 67 57 operator operator NOUN iajs-1821 67 58 ∏ ∏ NOUN iajs-1821 67 59 : : PUNCT iajs-1821 67 60 𝑋 𝑋 PROPN iajs-1821 67 61 ⟶ ⟶ NOUN iajs-1821 67 62 𝑌 𝑌 PROPN iajs-1821 67 63 with with ADP iajs-1821 67 64 ∏ ∏ X iajs-1821 67 65 𝑋 𝑋 PROPN iajs-1821 67 66 = = SYM iajs-1821 67 67 𝑌 𝑌 PROPN iajs-1821 67 68 , , PUNCT iajs-1821 67 69 then then ADV iajs-1821 67 70 eq eq ADP iajs-1821 67 71 . . PUNCT iajs-1821 68 1 ( ( PUNCT iajs-1821 68 2 3 3 X iajs-1821 68 3 ) ) PUNCT iajs-1821 68 4 may may AUX iajs-1821 68 5 be be AUX iajs-1821 68 6 rewritten rewrite VERB iajs-1821 69 1 〈 〈 ADP iajs-1821 69 2 𝐷∗ 𝐷∗ NOUN iajs-1821 69 3 𝑢 𝑢 PRON iajs-1821 69 4 𝐾𝑢 𝐾𝑢 PROPN iajs-1821 69 5 , , PUNCT iajs-1821 69 6 ∏ ∏ PROPN iajs-1821 69 7 𝑥 𝑥 PRON iajs-1821 69 8 〉 〉 NOUN iajs-1821 69 9 〈 〈 NOUN iajs-1821 69 10 𝑓 𝑓 PROPN iajs-1821 69 11 , , PUNCT iajs-1821 69 12 ∏ ∏ PROPN iajs-1821 69 13 𝑥 𝑥 PRON iajs-1821 69 14 〉 〉 NOUN iajs-1821 69 15 for for ADP iajs-1821 69 16 all all DET iajs-1821 69 17 𝑥 𝑥 DET iajs-1821 69 18 ∈ ∈ NOUN iajs-1821 69 19 𝑋 𝑋 NOUN iajs-1821 69 20 ... ... PUNCT iajs-1821 69 21 ( ( PUNCT iajs-1821 69 22 6 6 X iajs-1821 69 23 ) ) PUNCT iajs-1821 69 24 using use VERB iajs-1821 69 25 the the DET iajs-1821 69 26 projection projection NOUN iajs-1821 69 27 𝑃 𝑃 NOUN iajs-1821 69 28 defined define VERB iajs-1821 69 29 earlier early ADV iajs-1821 69 30 , , PUNCT iajs-1821 69 31 eq.(3 eq.(3 NOUN iajs-1821 69 32 ) ) PUNCT iajs-1821 69 33 is be AUX iajs-1821 69 34 equivalent equivalent ADJ iajs-1821 69 35 to to ADP iajs-1821 69 36 𝐷∗ 𝐷∗ VERB iajs-1821 69 37 𝑢 𝑢 X iajs-1821 69 38 𝑃 𝑃 NOUN iajs-1821 69 39 𝐾𝑢 𝐾𝑢 PROPN iajs-1821 69 40 = = PRON iajs-1821 69 41 𝑃 𝑃 VERB iajs-1821 69 42 𝑓 𝑓 PRON iajs-1821 69 43 ... ... PUNCT iajs-1821 69 44 ( ( PUNCT iajs-1821 69 45 7 7 X iajs-1821 69 46 ) ) PUNCT iajs-1821 69 47 eq.(7 eq.(7 NOUN iajs-1821 69 48 ) ) PUNCT iajs-1821 69 49 can can AUX iajs-1821 69 50 also also ADV iajs-1821 69 51 be be AUX iajs-1821 69 52 derived derive VERB iajs-1821 69 53 from from ADP iajs-1821 69 54 the the DET iajs-1821 69 55 fact fact NOUN iajs-1821 69 56 that that SCONJ iajs-1821 69 57 𝑃 𝑃 NOUN iajs-1821 69 58 x x SYM iajs-1821 69 59 = = SYM iajs-1821 69 60 0 0 NUM iajs-1821 69 61 for for ADP iajs-1821 69 62 an an DET iajs-1821 69 63 𝑥 𝑥 DET iajs-1821 69 64 ∈ ∈ PROPN iajs-1821 69 65 𝑋 𝑋 NOUN iajs-1821 69 66 if if SCONJ iajs-1821 69 67 and and CCONJ iajs-1821 69 68 only only ADV iajs-1821 69 69 if if SCONJ iajs-1821 69 70 〈 〈 PROPN iajs-1821 69 71 𝑥 𝑥 X iajs-1821 69 72 , , PUNCT iajs-1821 69 73 𝑦 𝑦 NOUN iajs-1821 69 74 〉 〉 NOUN iajs-1821 69 75 = = SYM iajs-1821 69 76 0 0 NUM iajs-1821 69 77 for for ADP iajs-1821 69 78 all all DET iajs-1821 69 79 𝑦 𝑦 NOUN iajs-1821 69 80 ∈ ∈ NOUN iajs-1821 69 81 𝑌 𝑌 PROPN iajs-1821 69 82 . . PUNCT iajs-1821 70 1 this this DET iajs-1821 70 2 equation equation NOUN iajs-1821 70 3 indicates indicate VERB iajs-1821 70 4 that that SCONJ iajs-1821 70 5 the the DET iajs-1821 70 6 petrov petrov PROPN iajs-1821 70 7 - - PUNCT iajs-1821 70 8 galerkin galerkin PROPN iajs-1821 70 9 method method NOUN iajs-1821 70 10 is be AUX iajs-1821 70 11 a a DET iajs-1821 70 12 projection projection NOUN iajs-1821 70 13 method method NOUN iajs-1821 70 14 . . PUNCT iajs-1821 71 1 now now ADV iajs-1821 71 2 , , PUNCT iajs-1821 71 3 assume assume VERB iajs-1821 71 4 un un PROPN iajs-1821 71 5 ∈ ∈ PROPN iajs-1821 71 6 x x PUNCT iajs-1821 71 7 n n PROPN iajs-1821 71 8 and and CCONJ iajs-1821 71 9 𝑏 𝑏 PROPN iajs-1821 71 10 is be AUX iajs-1821 71 11 a a DET iajs-1821 71 12 basis basis NOUN iajs-1821 71 13 for for ADP iajs-1821 71 14 xn xn PROPN iajs-1821 71 15 ( ( PUNCT iajs-1821 71 16 trial trial NOUN iajs-1821 71 17 space space NOUN iajs-1821 71 18 ) ) PUNCT iajs-1821 71 19 and and CCONJ iajs-1821 71 20 𝑏∗ 𝑏∗ PROPN iajs-1821 71 21 ( ( PUNCT iajs-1821 71 22 test test NOUN iajs-1821 71 23 space space NOUN iajs-1821 71 24 ) ) PUNCT iajs-1821 71 25 is be AUX iajs-1821 71 26 a a DET iajs-1821 71 27 basis basis NOUN iajs-1821 71 28 for for ADP iajs-1821 71 29 yn yn PRON iajs-1821 71 30 . . PUNCT iajs-1821 72 1 therefore therefore ADV iajs-1821 72 2 the the DET iajs-1821 72 3 ( ( PUNCT iajs-1821 72 4 pgm pgm PROPN iajs-1821 72 5 ) ) PUNCT iajs-1821 72 6 on on ADP iajs-1821 72 7 𝑎 𝑎 ADP iajs-1821 72 8 , , PUNCT iajs-1821 72 9 𝑏 𝑏 NOUN iajs-1821 72 10 for for ADP iajs-1821 72 11 eq eq PROPN iajs-1821 72 12 . . PUNCT iajs-1821 73 1 ( ( PUNCT iajs-1821 73 2 2 2 X iajs-1821 73 3 ) ) PUNCT iajs-1821 73 4 is be AUX iajs-1821 73 5 : : PUNCT iajs-1821 73 6 〈 〈 ADJ iajs-1821 73 7 𝐷∗ 𝐷∗ PROPN iajs-1821 73 8 𝑢 𝑢 PROPN iajs-1821 73 9 𝐾 𝐾 PROPN iajs-1821 73 10 𝑢 𝑢 PROPN iajs-1821 73 11 , , PUNCT iajs-1821 73 12 𝑏∗ 𝑏∗ PROPN iajs-1821 73 13 〉 〉 PROPN iajs-1821 73 14 〈 〈 PROPN iajs-1821 73 15 𝑓 𝑓 PROPN iajs-1821 73 16 , , PUNCT iajs-1821 73 17 𝑏∗ 𝑏∗ PROPN iajs-1821 73 18 〉 〉 PROPN iajs-1821 73 19 , , PUNCT iajs-1821 73 20 i= i= PROPN iajs-1821 73 21 1 1 NUM iajs-1821 73 22 , , PUNCT iajs-1821 73 23 .. .. PUNCT iajs-1821 73 24 , , PUNCT iajs-1821 73 25 n n X iajs-1821 73 26 ... ... PUNCT iajs-1821 73 27 ( ( PUNCT iajs-1821 73 28 8) 8) NUM iajs-1821 73 29 4 4 NUM iajs-1821 73 30 . . PUNCT iajs-1821 73 31 application application NOUN iajs-1821 73 32 of of ADP iajs-1821 73 33 ( ( PUNCT iajs-1821 73 34 pgm pgm PROPN iajs-1821 73 35 ) ) PUNCT iajs-1821 73 36 for for ADP iajs-1821 73 37 solving solve VERB iajs-1821 73 38 ( ( PUNCT iajs-1821 73 39 lffides lffide NOUN iajs-1821 73 40 ) ) PUNCT iajs-1821 73 41 via via ADP iajs-1821 73 42 normalization normalization NOUN iajs-1821 73 43 bernstein bernstein PROPN iajs-1821 73 44 basis basis NOUN iajs-1821 73 45 : : PUNCT iajs-1821 73 46 in in ADP iajs-1821 73 47 this this DET iajs-1821 73 48 section section NOUN iajs-1821 73 49 , , PUNCT iajs-1821 73 50 the the DET iajs-1821 73 51 petrov petrov PROPN iajs-1821 73 52 - - PUNCT iajs-1821 73 53 galerkian galerkian PROPN iajs-1821 73 54 method method NOUN iajs-1821 73 55 ( ( PUNCT iajs-1821 73 56 pgm pgm PROPN iajs-1821 73 57 ) ) PUNCT iajs-1821 73 58 with with ADP iajs-1821 73 59 aid aid NOUN iajs-1821 73 60 of of ADP iajs-1821 73 61 normalization normalization NOUN iajs-1821 73 62 bernstein bernstein PROPN iajs-1821 73 63 polynomials polynomial NOUN iajs-1821 73 64 of of ADP iajs-1821 73 65 six six NUM iajs-1821 73 66 degree degree NOUN iajs-1821 73 67 are be AUX iajs-1821 73 68 interval interval NOUN iajs-1821 73 69 [ [ X iajs-1821 73 70 0 0 NUM iajs-1821 73 71 , , PUNCT iajs-1821 73 72 1 1 NUM iajs-1821 73 73 ] ] PUNCT iajs-1821 73 74 , , PUNCT iajs-1821 73 75 is be AUX iajs-1821 73 76 applied apply VERB iajs-1821 73 77 to to PART iajs-1821 73 78 study study VERB iajs-1821 73 79 the the DET iajs-1821 73 80 approximation approximation NOUN iajs-1821 73 81 solution solution NOUN iajs-1821 73 82 of of ADP iajs-1821 73 83 the the DET iajs-1821 73 84 linear linear ADJ iajs-1821 73 85 fredholm fredholm NOUN iajs-1821 73 86 fractional fractional ADJ iajs-1821 73 87 integro integro ADJ iajs-1821 73 88 - - PUNCT iajs-1821 73 89 differential differential NOUN iajs-1821 73 90 eq(1 eq(1 NOUN iajs-1821 73 91 ) ) PUNCT iajs-1821 73 92 as as ADP iajs-1821 73 93 the the DET iajs-1821 73 94 form form NOUN iajs-1821 73 95 : : PUNCT iajs-1821 73 96 𝐷∗ 𝐷∗ PROPN iajs-1821 73 97 𝑢 𝑢 PRON iajs-1821 73 98 𝑥 𝑥 X iajs-1821 73 99 𝑓 𝑓 ADP iajs-1821 73 100 𝑥 𝑥 X iajs-1821 73 101 𝑘 𝑘 PRON iajs-1821 73 102 𝑥 𝑥 NOUN iajs-1821 73 103 , , PUNCT iajs-1821 73 104 𝑡 𝑡 X iajs-1821 73 105 𝑢 𝑢 PROPN iajs-1821 73 106 𝑡 𝑡 PROPN iajs-1821 73 107 𝑑𝑡 𝑑𝑡 ADP iajs-1821 73 108 , , PUNCT iajs-1821 73 109 𝑢 𝑢 PROPN iajs-1821 73 110 0 0 PROPN iajs-1821 73 111 𝛽 𝛽 NOUN iajs-1821 73 112 , , PUNCT iajs-1821 73 113 𝑥 𝑥 PROPN iajs-1821 73 114 ∈ ∈ PROPN iajs-1821 73 115 𝑎 𝑎 NOUN iajs-1821 73 116 , , PUNCT iajs-1821 74 1 𝑏 𝑏 PROPN iajs-1821 74 2 our our PRON iajs-1821 74 3 approach approach NOUN iajs-1821 74 4 being be AUX iajs-1821 74 5 by by ADP iajs-1821 74 6 taking take VERB iajs-1821 74 7 the the DET iajs-1821 74 8 fractional fractional ADJ iajs-1821 74 9 integration integration NOUN iajs-1821 74 10 to to ADP iajs-1821 74 11 both both DET iajs-1821 74 12 sides side NOUN iajs-1821 74 13 of of ADP iajs-1821 74 14 eq eq PROPN iajs-1821 74 15 . . PUNCT iajs-1821 75 1 ( ( PUNCT iajs-1821 75 2 1 1 X iajs-1821 75 3 ) ) PUNCT iajs-1821 75 4 we we PRON iajs-1821 75 5 get get VERB iajs-1821 75 6 𝑢 𝑢 PRON iajs-1821 75 7 𝑥 𝑥 PRON iajs-1821 75 8 𝑢 𝑢 NOUN iajs-1821 75 9 0 0 NUM iajs-1821 75 10 𝐼 𝐼 ADP iajs-1821 75 11 𝑓 𝑓 PRON iajs-1821 75 12 𝑥 𝑥 NOUN iajs-1821 75 13 𝐼 𝐼 ADP iajs-1821 75 14 𝑘 𝑘 PRON iajs-1821 75 15 𝑥 𝑥 NOUN iajs-1821 75 16 , , PUNCT iajs-1821 75 17 𝑡 𝑡 X iajs-1821 75 18 𝑢 𝑢 X iajs-1821 75 19 𝑡 𝑡 NOUN iajs-1821 75 20 𝑑𝑡 𝑑𝑡 ADP iajs-1821 75 21 ... ... PUNCT iajs-1821 75 22 ( ( PUNCT iajs-1821 75 23 9 9 NUM iajs-1821 75 24 ) ) PUNCT iajs-1821 75 25 to to PART iajs-1821 75 26 approximate approximate ADJ iajs-1821 75 27 solution solution NOUN iajs-1821 75 28 of of ADP iajs-1821 75 29 eq.(1),we eq.(1),we PROPN iajs-1821 75 30 use use VERB iajs-1821 75 31 the the DET iajs-1821 75 32 normalization normalization NOUN iajs-1821 75 33 polynomial polynomial ADJ iajs-1821 75 34 basis basis NOUN iajs-1821 75 35 on on ADP iajs-1821 75 36 𝑎 𝑎 PROPN iajs-1821 75 37 , , PUNCT iajs-1821 75 38 𝑏 𝑏 NOUN iajs-1821 75 39 as as ADP iajs-1821 75 40 : : PUNCT iajs-1821 75 41 𝑢 𝑢 PART iajs-1821 75 42 𝑥 𝑥 X iajs-1821 75 43 ∑ ∑ PUNCT iajs-1821 75 44 𝑎 𝑎 PROPN iajs-1821 75 45 𝑏 𝑏 NOUN iajs-1821 75 46 , , PUNCT iajs-1821 75 47 𝑥 𝑥 X iajs-1821 75 48 ... ... PUNCT iajs-1821 75 49 ( ( PUNCT iajs-1821 75 50 10 10 NUM iajs-1821 75 51 ) ) PUNCT iajs-1821 75 52 where where SCONJ iajs-1821 75 53 ( ( PUNCT iajs-1821 75 54 𝑎 𝑎 X iajs-1821 75 55 , , PUNCT iajs-1821 75 56 𝑖 𝑖 NOUN iajs-1821 75 57 0,1 0,1 NUM iajs-1821 75 58 , , PUNCT iajs-1821 75 59 … … PUNCT iajs-1821 75 60 . . PUNCT iajs-1821 76 1 . . PUNCT iajs-1821 77 1 , , PUNCT iajs-1821 77 2 𝑛 𝑛 PROPN iajs-1821 77 3 are be AUX iajs-1821 77 4 unknown unknown ADJ iajs-1821 77 5 constants constant NOUN iajs-1821 77 6 to to PART iajs-1821 77 7 be be AUX iajs-1821 77 8 determined determine VERB iajs-1821 77 9 substituting substitute VERB iajs-1821 77 10 eq.(10 eq.(10 NOUN iajs-1821 77 11 ) ) PUNCT iajs-1821 77 12 in in ADP iajs-1821 77 13 to to ADP iajs-1821 77 14 eq.(9),we eq.(9),we PROPN iajs-1821 77 15 get get VERB iajs-1821 77 16 ∑ ∑ PUNCT iajs-1821 77 17 𝑎 𝑎 NOUN iajs-1821 77 18 𝑏 𝑏 NOUN iajs-1821 77 19 , , PUNCT iajs-1821 77 20 𝑥 𝑥 PRON iajs-1821 77 21 𝑢 𝑢 NOUN iajs-1821 77 22 0 0 NUM iajs-1821 77 23 𝐼 𝐼 ADP iajs-1821 77 24 𝑓 𝑓 PRON iajs-1821 77 25 𝑥 𝑥 NOUN iajs-1821 77 26 𝐼 𝐼 ADP iajs-1821 77 27 𝑘 𝑘 PROPN iajs-1821 77 28 𝑥 𝑥 NOUN iajs-1821 77 29 , , PUNCT iajs-1821 77 30 𝑡 𝑡 VERB iajs-1821 77 31 ∑ ∑ PROPN iajs-1821 77 32 𝑎 𝑎 PROPN iajs-1821 77 33 𝑏 𝑏 NOUN iajs-1821 77 34 , , PUNCT iajs-1821 77 35 𝑡 𝑡 PROPN iajs-1821 77 36 𝑑𝑡 𝑑𝑡 ADP iajs-1821 77 37 ... ... PUNCT iajs-1821 77 38 ( ( PUNCT iajs-1821 77 39 11 11 NUM iajs-1821 77 40 ) ) PUNCT iajs-1821 77 41 hence hence ADV iajs-1821 77 42 ihsciconf ihsciconf VERB iajs-1821 77 43 2017 2017 NUM iajs-1821 77 44 special special ADJ iajs-1821 77 45 issue issue NOUN iajs-1821 77 46 ibn ibn PROPN iajs-1821 77 47 al al PROPN iajs-1821 77 48 - - PUNCT iajs-1821 77 49 haitham haitham PROPN iajs-1821 77 50 journal journal PROPN iajs-1821 77 51 for for ADP iajs-1821 77 52 pure pure ADJ iajs-1821 77 53 and and CCONJ iajs-1821 77 54 applied apply VERB iajs-1821 77 55 science science NOUN iajs-1821 77 56 https://doi.org/ https://doi.org/ NOUN iajs-1821 77 57 10.30526/2017.ihsciconf.1821 10.30526/2017.ihsciconf.1821 NUM iajs-1821 77 58 for for ADP iajs-1821 77 59 more more ADJ iajs-1821 77 60 information information NOUN iajs-1821 77 61 about about ADP iajs-1821 77 62 the the DET iajs-1821 77 63 conference conference NOUN iajs-1821 77 64 please please INTJ iajs-1821 77 65 visit visit VERB iajs-1821 77 66 the the DET iajs-1821 77 67 websites website NOUN iajs-1821 77 68 : : PUNCT iajs-1821 77 69 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1821 77 70     SPACE iajs-1821 77 71 www.ihsciconf.org www.ihsciconf.org ADJ iajs-1821 77 72       SPACE iajs-1821 77 73 mathematics|495 mathematics|495 NOUN iajs-1821 77 74         SPACE iajs-1821 77 75 ∑ ∑ PROPN iajs-1821 77 76 𝑎 𝑎 PROPN iajs-1821 77 77 𝑏 𝑏 NOUN iajs-1821 77 78 , , PUNCT iajs-1821 78 1 𝑥 𝑥 PROPN iajs-1821 78 2 𝐼 𝐼 ADP iajs-1821 78 3 𝑘 𝑘 PRON iajs-1821 78 4 𝑥 𝑥 NOUN iajs-1821 78 5 , , PUNCT iajs-1821 78 6 𝑡 𝑡 VERB iajs-1821 78 7 ∑ ∑ PROPN iajs-1821 78 8 𝑎 𝑎 PROPN iajs-1821 78 9 𝑏 𝑏 NOUN iajs-1821 78 10 , , PUNCT iajs-1821 78 11 𝑡 𝑡 PROPN iajs-1821 78 12 𝑑𝑡 𝑑𝑡 ADP iajs-1821 78 13 𝑢 𝑢 NOUN iajs-1821 78 14 0 0 NUM iajs-1821 78 15 𝐼 𝐼 ADP iajs-1821 78 16 𝑓 𝑓 PRON iajs-1821 78 17 𝑥 𝑥 X iajs-1821 78 18 ... ... PUNCT iajs-1821 78 19 ( ( PUNCT iajs-1821 78 20 12 12 NUM iajs-1821 78 21 ) ) PUNCT iajs-1821 78 22 in in ADP iajs-1821 78 23 the the DET iajs-1821 78 24 next next ADJ iajs-1821 78 25 step step NOUN iajs-1821 78 26 , , PUNCT iajs-1821 78 27 apply apply VERB iajs-1821 78 28 petrov petrov PROPN iajs-1821 78 29 - - PUNCT iajs-1821 78 30 galerkin galerkin ADJ iajs-1821 78 31 method method NOUN iajs-1821 78 32 ( ( PUNCT iajs-1821 78 33 pgm pgm PROPN iajs-1821 78 34 ) ) PUNCT iajs-1821 78 35 for for ADP iajs-1821 78 36 eq.(1 eq.(1 ADJ iajs-1821 78 37 ) ) PUNCT iajs-1821 78 38 is be AUX iajs-1821 78 39 a a DET iajs-1821 78 40 numerical numerical ADJ iajs-1821 78 41 method method NOUN iajs-1821 78 42 for for ADP iajs-1821 78 43 finding find VERB iajs-1821 78 44 𝑢 𝑢 PRON iajs-1821 78 45 𝑥 𝑥 X iajs-1821 78 46 ∑ ∑ PUNCT iajs-1821 78 47 𝑎 𝑎 PROPN iajs-1821 78 48 𝑏 𝑏 NOUN iajs-1821 78 49 , , PUNCT iajs-1821 78 50 𝑥 𝑥 DET iajs-1821 78 51 ∊ ∊ NOUN iajs-1821 78 52 x x SYM iajs-1821 78 53 n n CCONJ iajs-1821 78 54 , , PUNCT iajs-1821 78 55 such such ADJ iajs-1821 78 56 that that SCONJ iajs-1821 78 57 𝒂𝒊 𝒂𝒊 PROPN iajs-1821 78 58 is be AUX iajs-1821 78 59 unknown unknown ADJ iajs-1821 78 60 and and CCONJ iajs-1821 78 61 must must AUX iajs-1821 78 62 be be AUX iajs-1821 78 63 determined determine VERB iajs-1821 78 64 from from ADP iajs-1821 78 65 eq.(12 eq.(12 PROPN iajs-1821 78 66 ) ) PUNCT iajs-1821 78 67 . . PUNCT iajs-1821 79 1 from from ADP iajs-1821 79 2 eq.(8 eq.(8 ADJ iajs-1821 79 3 ) ) PUNCT iajs-1821 80 1 it it PRON iajs-1821 80 2 is be AUX iajs-1821 80 3 clear clear ADJ iajs-1821 80 4 that that SCONJ iajs-1821 80 5 the the DET iajs-1821 80 6 eq.(12 eq.(12 NOUN iajs-1821 80 7 ) ) PUNCT iajs-1821 80 8 can can AUX iajs-1821 80 9 be be AUX iajs-1821 80 10 written write VERB iajs-1821 80 11 as as ADP iajs-1821 80 12 : : PUNCT iajs-1821 80 13 < < X iajs-1821 80 14 ∑ ∑ X iajs-1821 80 15 a a DET iajs-1821 80 16 b b NOUN iajs-1821 80 17 , , PUNCT iajs-1821 80 18 x x PROPN iajs-1821 80 19 i i PRON iajs-1821 80 20 k k PROPN iajs-1821 80 21 x x PROPN iajs-1821 80 22 , , PUNCT iajs-1821 80 23 t t PROPN iajs-1821 80 24 ∑ ∑ PROPN iajs-1821 80 25 a a DET iajs-1821 80 26 b b PROPN iajs-1821 80 27 , , PUNCT iajs-1821 80 28 t t PROPN iajs-1821 80 29 dt dt X iajs-1821 80 30 , , PUNCT iajs-1821 80 31 b b PROPN iajs-1821 80 32 , , PUNCT iajs-1821 80 33 ∗ ∗ NOUN iajs-1821 80 34 > > PUNCT iajs-1821 80 35 = = SYM iajs-1821 80 36 < < X iajs-1821 80 37 u u X iajs-1821 80 38 0 0 PUNCT iajs-1821 81 1 i i PRON iajs-1821 81 2 f f PROPN iajs-1821 81 3 x x X iajs-1821 81 4 , , PUNCT iajs-1821 81 5 b b PROPN iajs-1821 81 6 , , PUNCT iajs-1821 81 7 ∗ ∗ NOUN iajs-1821 81 8 > > X iajs-1821 81 9 ... ... PUNCT iajs-1821 81 10 ( ( PUNCT iajs-1821 81 11 13 13 NUM iajs-1821 81 12 ) ) PUNCT iajs-1821 82 1 thus thus ADV iajs-1821 82 2 ∑ ∑ DET iajs-1821 82 3 a a DET iajs-1821 82 4 b b NOUN iajs-1821 82 5 , , PUNCT iajs-1821 82 6 x x PROPN iajs-1821 83 1 i i PRON iajs-1821 83 2 k k PROPN iajs-1821 83 3 x x PROPN iajs-1821 83 4 , , PUNCT iajs-1821 83 5 t t PROPN iajs-1821 83 6 ∑ ∑ PROPN iajs-1821 83 7 a a DET iajs-1821 83 8 b b PROPN iajs-1821 83 9 , , PUNCT iajs-1821 83 10 t t PROPN iajs-1821 83 11 dt dt X iajs-1821 83 12 b b PROPN iajs-1821 83 13 , , PUNCT iajs-1821 83 14 ∗ ∗ NOUN iajs-1821 83 15 u u NOUN iajs-1821 83 16 0 0 PUNCT iajs-1821 84 1 i i PRON iajs-1821 84 2 f f PROPN iajs-1821 85 1 x x X iajs-1821 85 2 dx dx PROPN iajs-1821 85 3 b b PROPN iajs-1821 85 4 , , PUNCT iajs-1821 85 5 ∗ ∗ NOUN iajs-1821 85 6 … … PUNCT iajs-1821 85 7 ( ( PUNCT iajs-1821 85 8 14 14 NUM iajs-1821 85 9 ) ) PUNCT iajs-1821 85 10 then then ADV iajs-1821 85 11 , , PUNCT iajs-1821 85 12 eq.(14 eq.(14 NOUN iajs-1821 85 13 ) ) PUNCT iajs-1821 85 14 is be AUX iajs-1821 85 15 equivalent equivalent ADJ iajs-1821 85 16 to to AUX iajs-1821 85 17 linear linear ADJ iajs-1821 85 18 system system NOUN iajs-1821 85 19 can can AUX iajs-1821 85 20 be be AUX iajs-1821 85 21 formed form VERB iajs-1821 85 22 as as SCONJ iajs-1821 85 23 follows follow VERB iajs-1821 85 24 : : PUNCT iajs-1821 85 25 l l NOUN iajs-1821 85 26 x x X iajs-1821 85 27 , , PUNCT iajs-1821 85 28 𝑎 𝑎 NOUN iajs-1821 85 29 ∑ ∑ PUNCT iajs-1821 85 30 𝑎 𝑎 PROPN iajs-1821 85 31 𝑏 𝑏 NOUN iajs-1821 85 32 , , PUNCT iajs-1821 85 33 𝑥 𝑥 PROPN iajs-1821 86 1 𝐼 𝐼 ADP iajs-1821 86 2 𝑘 𝑘 PRON iajs-1821 86 3 𝑥 𝑥 NOUN iajs-1821 86 4 , , PUNCT iajs-1821 86 5 𝑡 𝑡 VERB iajs-1821 86 6 ∑ ∑ PROPN iajs-1821 86 7 𝑎 𝑎 PROPN iajs-1821 86 8 𝑏 𝑏 NOUN iajs-1821 86 9 , , PUNCT iajs-1821 86 10 𝑡 𝑡 PROPN iajs-1821 86 11 𝑑𝑡 𝑑𝑡 ADP iajs-1821 86 12 𝑚 𝑚 X iajs-1821 86 13 𝑢 𝑢 NOUN iajs-1821 86 14 0 0 NUM iajs-1821 86 15 𝐼 𝐼 ADP iajs-1821 86 16 𝑓 𝑓 PROPN iajs-1821 86 17 𝑥 𝑥 X iajs-1821 86 18 𝑏 𝑏 PROPN iajs-1821 86 19 , , PUNCT iajs-1821 86 20 ∗ ∗ NOUN iajs-1821 86 21 … … PUNCT iajs-1821 86 22 15 15 NUM iajs-1821 86 23 we we PRON iajs-1821 86 24 can can AUX iajs-1821 86 25 represent represent VERB iajs-1821 86 26 the the DET iajs-1821 86 27 system system NOUN iajs-1821 86 28 eq.(15 eq.(15 NOUN iajs-1821 86 29 ) ) PUNCT iajs-1821 86 30 as as ADP iajs-1821 86 31 a a DET iajs-1821 86 32 matrix matrix NOUN iajs-1821 86 33 form form NOUN iajs-1821 86 34 : : PUNCT iajs-1821 86 35 la la X iajs-1821 86 36 = = NOUN iajs-1821 86 37 m m VERB iajs-1821 86 38 ... ... PUNCT iajs-1821 86 39 ( ( PUNCT iajs-1821 86 40 16 16 NUM iajs-1821 86 41 ) ) PUNCT iajs-1821 87 1 where where SCONJ iajs-1821 87 2 𝐿 𝐿 PROPN iajs-1821 87 3 𝐿 𝐿 PROPN iajs-1821 87 4 𝑥 𝑥 PROPN iajs-1821 87 5 , , PUNCT iajs-1821 87 6 𝑎 𝑎 NOUN iajs-1821 87 7 𝑏 𝑏 NOUN iajs-1821 87 8 , , PUNCT iajs-1821 87 9 ∗ ∗ NOUN iajs-1821 87 10 𝑑𝑡 𝑑𝑡 ADP iajs-1821 87 11 ⋯ ⋯ PROPN iajs-1821 87 12 𝐿 𝐿 PROPN iajs-1821 87 13 𝑥 𝑥 PROPN iajs-1821 87 14 , , PUNCT iajs-1821 87 15 𝑎 𝑎 NOUN iajs-1821 87 16 𝑏 𝑏 NOUN iajs-1821 87 17 , , PUNCT iajs-1821 87 18 ∗ ∗ NOUN iajs-1821 87 19 𝑑𝑡 𝑑𝑡 ADP iajs-1821 87 20 ⋮ ⋮ NOUN iajs-1821 87 21 ⋱ ⋱ PUNCT iajs-1821 87 22 ⋮ ⋮ NOUN iajs-1821 87 23 𝐿 𝐿 PROPN iajs-1821 87 24 𝑥 𝑥 PROPN iajs-1821 87 25 , , PUNCT iajs-1821 87 26 𝑎 𝑎 NOUN iajs-1821 87 27 𝑏 𝑏 NOUN iajs-1821 87 28 , , PUNCT iajs-1821 87 29 ∗ ∗ NOUN iajs-1821 87 30 𝑑𝑡 𝑑𝑡 ADP iajs-1821 87 31 ⋯ ⋯ PROPN iajs-1821 87 32 𝐿 𝐿 PROPN iajs-1821 87 33 𝑥 𝑥 PROPN iajs-1821 87 34 , , PUNCT iajs-1821 87 35 𝑎 𝑎 NOUN iajs-1821 87 36 𝑏 𝑏 NOUN iajs-1821 87 37 , , PUNCT iajs-1821 87 38 ∗ ∗ NOUN iajs-1821 87 39 𝑑𝑡 𝑑𝑡 ADP iajs-1821 87 40 , , PUNCT iajs-1821 87 41 𝐴 𝐴 PROPN iajs-1821 87 42 𝑎 𝑎 NOUN iajs-1821 87 43 𝑎 𝑎 DET iajs-1821 87 44 ⋮ ⋮ NOUN iajs-1821 87 45 𝑎 𝑎 NOUN iajs-1821 87 46 , , PUNCT iajs-1821 87 47 𝑀 𝑀 PROPN iajs-1821 87 48 𝑚 𝑚 NOUN iajs-1821 87 49 𝑚 𝑚 ADP iajs-1821 87 50 ⋮ ⋮ NOUN iajs-1821 87 51 𝑚 𝑚 PROPN iajs-1821 87 52 then then ADV iajs-1821 87 53 we we PRON iajs-1821 87 54 are be AUX iajs-1821 87 55 solving solve VERB iajs-1821 87 56 the the DET iajs-1821 87 57 system system NOUN iajs-1821 87 58 to to PART iajs-1821 87 59 calculate calculate VERB iajs-1821 87 60 the the DET iajs-1821 87 61 value value NOUN iajs-1821 87 62 𝑎 𝑎 PROPN iajs-1821 87 63 5 5 NUM iajs-1821 87 64 . . PUNCT iajs-1821 87 65 numerical numerical ADJ iajs-1821 87 66 examples example NOUN iajs-1821 87 67 : : PUNCT iajs-1821 87 68 example example NOUN iajs-1821 87 69 1 1 NUM iajs-1821 87 70 : : PUNCT iajs-1821 87 71 consider consider VERB iajs-1821 87 72 the the DET iajs-1821 87 73 following follow VERB iajs-1821 87 74 linear linear PROPN iajs-1821 87 75 fredholm fredholm NOUN iajs-1821 87 76 fractional fractional ADJ iajs-1821 87 77 integro integro ADJ iajs-1821 87 78 - - PUNCT iajs-1821 87 79 differential differential NOUN iajs-1821 87 80 equation equation NOUN iajs-1821 87 81 : : PUNCT iajs-1821 87 82 𝐷 𝐷 NOUN iajs-1821 87 83 𝑢 𝑢 ADV iajs-1821 87 84 𝑥 𝑥 X iajs-1821 87 85 3𝑥 3𝑥 NUM iajs-1821 87 86 𝑢 𝑢 X iajs-1821 87 87 𝑡 𝑡 PROPN iajs-1821 87 88 𝑑𝑡 𝑑𝑡 ADP iajs-1821 87 89 , , PUNCT iajs-1821 87 90 u(0)= u(0)= PROPN iajs-1821 87 91 0 0 NUM iajs-1821 87 92 , , PUNCT iajs-1821 87 93 0 0 NUM iajs-1821 87 94 𝛼 𝛼 SYM iajs-1821 87 95 1 1 NUM iajs-1821 87 96 where where SCONJ iajs-1821 87 97 the the DET iajs-1821 87 98 exact exact ADJ iajs-1821 87 99 solution solution NOUN iajs-1821 87 100 u(x u(x VERB iajs-1821 87 101 ) ) PUNCT iajs-1821 87 102 = = SYM iajs-1821 88 1 𝑥 𝑥 DET iajs-1821 88 2 ihsciconf ihsciconf NOUN iajs-1821 88 3 2017 2017 NUM iajs-1821 88 4 special special ADJ iajs-1821 88 5 issue issue NOUN iajs-1821 88 6 ibn ibn PROPN iajs-1821 88 7 al al PROPN iajs-1821 88 8 - - PUNCT iajs-1821 88 9 haitham haitham PROPN iajs-1821 88 10 journal journal PROPN iajs-1821 88 11 for for ADP iajs-1821 88 12 pure pure ADJ iajs-1821 88 13 and and CCONJ iajs-1821 88 14 applied apply VERB iajs-1821 88 15 science science NOUN iajs-1821 88 16 https://doi.org/ https://doi.org/ NOUN iajs-1821 88 17 10.30526/2017.ihsciconf.1821 10.30526/2017.ihsciconf.1821 NUM iajs-1821 88 18 for for ADP iajs-1821 88 19 more more ADJ iajs-1821 88 20 information information NOUN iajs-1821 88 21 about about ADP iajs-1821 88 22 the the DET iajs-1821 88 23 conference conference NOUN iajs-1821 88 24 please please INTJ iajs-1821 88 25 visit visit VERB iajs-1821 88 26 the the DET iajs-1821 88 27 websites website NOUN iajs-1821 88 28 : : PUNCT iajs-1821 88 29 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1821 88 30     SPACE iajs-1821 88 31 www.ihsciconf.org www.ihsciconf.org ADJ iajs-1821 88 32       SPACE iajs-1821 88 33 mathematics|496 mathematics|496 NOUN iajs-1821 88 34         SPACE iajs-1821 88 35 table table NOUN iajs-1821 88 36 ( ( PUNCT iajs-1821 88 37 1 1 NUM iajs-1821 88 38 ): ): PUNCT iajs-1821 88 39 represents represent VERB iajs-1821 88 40 a a DET iajs-1821 88 41 comparison comparison NOUN iajs-1821 88 42 between between ADP iajs-1821 88 43 the the DET iajs-1821 88 44 exact exact ADJ iajs-1821 88 45 solution solution NOUN iajs-1821 88 46 and and CCONJ iajs-1821 88 47 approximate approximate ADJ iajs-1821 88 48 solution solution NOUN iajs-1821 88 49 with with ADP iajs-1821 88 50 different different ADJ iajs-1821 88 51 value value NOUN iajs-1821 89 1 𝛼 𝛼 NOUN iajs-1821 89 2 1,0.5,0.25 1,0.5,0.25 NOUN iajs-1821 89 3 x x SYM iajs-1821 89 4 exact exact ADJ iajs-1821 89 5 solution solution NOUN iajs-1821 89 6 approximate approximate ADJ iajs-1821 89 7 solution solution NOUN iajs-1821 89 8 𝛼 𝛼 PROPN iajs-1821 89 9 1 1 NUM iajs-1821 89 10 𝛼 𝛼 PROPN iajs-1821 89 11 0.5 0.5 NUM iajs-1821 89 12 𝛼 𝛼 NUM iajs-1821 89 13 0.25 0.25 NUM iajs-1821 89 14 0 0 NUM iajs-1821 89 15 0 0 NUM iajs-1821 89 16 0 0 NUM iajs-1821 89 17 0.000606 0.000606 NUM iajs-1821 89 18 0.002462 0.002462 NUM iajs-1821 89 19 0.1 0.1 NUM iajs-1821 89 20 0.001 0.001 NUM iajs-1821 89 21 0.001 0.001 NUM iajs-1821 89 22 0.015119 0.015119 NUM iajs-1821 89 23 0.052247 0.052247 NUM iajs-1821 89 24 0.2 0.2 NUM iajs-1821 89 25 0.008 0.008 NUM iajs-1821 89 26 0.008 0.008 NUM iajs-1821 89 27 0.05891 0.05891 NUM iajs-1821 89 28 0.15574 0.15574 NUM iajs-1821 89 29 0.3 0.3 NUM iajs-1821 89 30 0.027 0.027 NUM iajs-1821 89 31 0.027 0.027 NUM iajs-1821 89 32 0.13789 0.13789 NUM iajs-1821 89 33 0.31079 0.31079 NUM iajs-1821 89 34 0.4 0.4 NUM iajs-1821 89 35 0.064 0.064 NUM iajs-1821 89 36 0.064 0.064 NUM iajs-1821 89 37 0.25797 0.25797 NUM iajs-1821 89 38 0.52016 0.52016 NUM iajs-1821 89 39 0.5 0.5 NUM iajs-1821 89 40 0.125 0.125 NUM iajs-1821 89 41 0.125 0.125 NUM iajs-1821 89 42 0.42506 0.42506 NUM iajs-1821 89 43 0.78662 0.78662 NUM iajs-1821 89 44 0.6 0.6 NUM iajs-1821 89 45 0.216 0.216 NUM iajs-1821 89 46 0.216 0.216 NUM iajs-1821 89 47 0.64507 0.64507 NUM iajs-1821 89 48 1.1129 1.1129 NUM iajs-1821 89 49 0.7 0.7 NUM iajs-1821 89 50 0.343 0.343 NUM iajs-1821 89 51 0.343 0.343 NUM iajs-1821 89 52 0.9239 0.9239 NUM iajs-1821 89 53 1.5018 1.5018 NUM iajs-1821 89 54 0.8 0.8 NUM iajs-1821 89 55 0.512 0.512 NUM iajs-1821 89 56 0.512 0.512 NUM iajs-1821 89 57 1.2675 1.2675 NUM iajs-1821 89 58 1.9562 1.9562 NUM iajs-1821 89 59 0.9 0.9 NUM iajs-1821 89 60 0.729 0.729 NUM iajs-1821 89 61 0.729 0.729 NUM iajs-1821 89 62 1.6817 1.6817 NUM iajs-1821 89 63 2.4786 2.4786 NUM iajs-1821 89 64 1 1 NUM iajs-1821 89 65 1.0 1.0 NUM iajs-1821 89 66 1.0 1.0 NUM iajs-1821 89 67 2.1725 2.1725 NUM iajs-1821 89 68 3.072 3.072 NUM iajs-1821 89 69 figure figure NOUN iajs-1821 89 70 ( ( PUNCT iajs-1821 89 71 1 1 NUM iajs-1821 89 72 ): ): PUNCT iajs-1821 89 73 comparison comparison NOUN iajs-1821 89 74 between between ADP iajs-1821 89 75 the the DET iajs-1821 89 76 approximate approximate ADJ iajs-1821 89 77 solution solution NOUN iajs-1821 89 78 and and CCONJ iajs-1821 89 79 exact exact ADJ iajs-1821 89 80 solution solution NOUN iajs-1821 89 81 example example NOUN iajs-1821 89 82 2 2 NUM iajs-1821 89 83 : : PUNCT iajs-1821 89 84 consider consider VERB iajs-1821 89 85 the the DET iajs-1821 89 86 following follow VERB iajs-1821 89 87 linear linear PROPN iajs-1821 89 88 fredholm fredholm NOUN iajs-1821 89 89 fractional fractional ADJ iajs-1821 89 90 integro integro ADJ iajs-1821 89 91 - - PUNCT iajs-1821 89 92 differential differential NOUN iajs-1821 89 93 equation equation NOUN iajs-1821 89 94 : : PUNCT iajs-1821 89 95 𝐷 𝐷 PROPN iajs-1821 89 96 𝑦 𝑦 NOUN iajs-1821 89 97 𝑥 𝑥 NOUN iajs-1821 89 98 𝑥𝑒 𝑥𝑒 INTJ iajs-1821 89 99 𝑒 𝑒 PART iajs-1821 89 100 𝑥 𝑥 PROPN iajs-1821 89 101 𝑥𝑦 𝑥𝑦 ADP iajs-1821 89 102 𝑡 𝑡 PROPN iajs-1821 89 103 𝑑𝑡 𝑑𝑡 ADP iajs-1821 89 104 , , PUNCT iajs-1821 89 105 y(0)=0 y(0)=0 PROPN iajs-1821 89 106 , , PUNCT iajs-1821 89 107 0 0 NUM iajs-1821 89 108 𝛼 𝛼 SYM iajs-1821 89 109 1 1 NUM iajs-1821 89 110 the the DET iajs-1821 89 111 exact exact ADJ iajs-1821 89 112 solution solution NOUN iajs-1821 89 113 y(x)=𝑥𝑒 y(x)=𝑥𝑒 NOUN iajs-1821 89 114 0 0 NUM iajs-1821 89 115 0.1 0.1 NUM iajs-1821 89 116 0.2 0.2 NUM iajs-1821 89 117 0.3 0.3 NUM iajs-1821 89 118 0.4 0.4 NUM iajs-1821 89 119 0.5 0.5 NUM iajs-1821 89 120 0.6 0.6 NUM iajs-1821 89 121 0.7 0.7 NUM iajs-1821 89 122 0.8 0.8 NUM iajs-1821 89 123 0.9 0.9 NUM iajs-1821 89 124 1 1 NUM iajs-1821 89 125 -0.5 -0.5 X iajs-1821 89 126 0 0 NUM iajs-1821 89 127 0.5 0.5 NUM iajs-1821 89 128 1 1 NUM iajs-1821 89 129 1.5 1.5 NUM iajs-1821 89 130 2 2 NUM iajs-1821 89 131 2.5 2.5 NUM iajs-1821 89 132 3 3 NUM iajs-1821 89 133 3.5 3.5 NUM iajs-1821 89 134 exact exact ADJ iajs-1821 89 135 solution solution NOUN iajs-1821 89 136 alpha alpha NOUN iajs-1821 89 137 = = NOUN iajs-1821 89 138 1 1 NUM iajs-1821 89 139 alpha alpha NOUN iajs-1821 89 140 = = NOUN iajs-1821 89 141 0.5 0.5 NUM iajs-1821 89 142 alpha alpha NOUN iajs-1821 89 143 = = NUM iajs-1821 89 144 0.25 0.25 NUM iajs-1821 89 145 ihsciconf ihsciconf NOUN iajs-1821 89 146 2017 2017 NUM iajs-1821 89 147 special special ADJ iajs-1821 89 148 issue issue NOUN iajs-1821 89 149 ibn ibn PROPN iajs-1821 89 150 al al PROPN iajs-1821 89 151 - 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PUNCT iajs-1821 93 1 in in ADP iajs-1821 93 2 this this DET iajs-1821 93 3 paper paper NOUN iajs-1821 93 4 , , PUNCT iajs-1821 93 5 petrov petrov PROPN iajs-1821 93 6 - - PUNCT iajs-1821 93 7 galerkin galerkin PROPN iajs-1821 93 8 method(pgm method(pgm PROPN iajs-1821 93 9 ) ) PUNCT iajs-1821 93 10 has have AUX iajs-1821 93 11 been be AUX iajs-1821 93 12 successfully successfully ADV iajs-1821 93 13 applied apply VERB iajs-1821 93 14 to to PART iajs-1821 93 15 find find VERB iajs-1821 93 16 0 0 NUM iajs-1821 93 17 0.1 0.1 NUM iajs-1821 93 18 0.2 0.2 NUM iajs-1821 93 19 0.3 0.3 NUM iajs-1821 93 20 0.4 0.4 NUM iajs-1821 93 21 0.5 0.5 NUM iajs-1821 93 22 0.6 0.6 NUM iajs-1821 93 23 0.7 0.7 NUM iajs-1821 93 24 0.8 0.8 NUM iajs-1821 93 25 0.9 0.9 NUM iajs-1821 93 26 1 1 NUM iajs-1821 93 27 0 0 NUM iajs-1821 93 28 0.2 0.2 NUM iajs-1821 93 29 0.4 0.4 NUM iajs-1821 93 30 0.6 0.6 NUM iajs-1821 93 31 0.8 0.8 NUM iajs-1821 93 32 1 1 NUM iajs-1821 93 33 1.2 1.2 NUM iajs-1821 93 34 1.4 1.4 NUM iajs-1821 93 35 1.6 1.6 NUM iajs-1821 93 36 1.8 1.8 NUM iajs-1821 93 37 exact exact ADJ iajs-1821 93 38 solution solution NOUN iajs-1821 93 39 alpha alpha NOUN iajs-1821 93 40 = = NOUN iajs-1821 93 41 1 1 NUM iajs-1821 93 42 alpha alpha NOUN iajs-1821 93 43 = = NOUN iajs-1821 93 44 0.25 0.25 NUM iajs-1821 93 45 alpha alpha NOUN iajs-1821 93 46 = = NOUN iajs-1821 93 47 0.5 0.5 NUM iajs-1821 93 48 ihsciconf ihsciconf NOUN iajs-1821 93 49 2017 2017 NUM iajs-1821 93 50 special special ADJ iajs-1821 93 51 issue issue NOUN iajs-1821 93 52 ibn ibn PROPN iajs-1821 93 53 al al PROPN iajs-1821 93 54 - - PUNCT iajs-1821 93 55 haitham haitham PROPN iajs-1821 93 56 journal journal PROPN iajs-1821 93 57 for for ADP iajs-1821 93 58 pure pure ADJ iajs-1821 93 59 and and CCONJ iajs-1821 93 60 applied apply VERB iajs-1821 93 61 science science NOUN iajs-1821 93 62 https://doi.org/ https://doi.org/ NOUN iajs-1821 93 63 10.30526/2017.ihsciconf.1821 10.30526/2017.ihsciconf.1821 NUM iajs-1821 93 64 for for ADP iajs-1821 93 65 more more ADJ iajs-1821 93 66 information information NOUN iajs-1821 93 67 about about ADP iajs-1821 93 68 the the DET iajs-1821 93 69 conference conference NOUN iajs-1821 93 70 please please INTJ iajs-1821 93 71 visit visit VERB iajs-1821 93 72 the the DET iajs-1821 93 73 websites website NOUN iajs-1821 93 74 : : PUNCT iajs-1821 93 75 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1821 93 76     SPACE iajs-1821 93 77 www.ihsciconf.org www.ihsciconf.org NOUN iajs-1821 93 78       SPACE iajs-1821 93 79 mathematics|499 mathematics|499 NOUN iajs-1821 93 80         SPACE iajs-1821 93 81 the the DET iajs-1821 93 82 approximate approximate ADJ iajs-1821 93 83 solution solution NOUN iajs-1821 93 84 of of ADP iajs-1821 93 85 linear linear PROPN iajs-1821 93 86 fractional fractional PROPN iajs-1821 93 87 volterra volterra PROPN iajs-1821 93 88 integro integro PROPN iajs-1821 93 89 - - PUNCT iajs-1821 93 90 differential differential NOUN iajs-1821 93 91 equation equation NOUN iajs-1821 93 92 of of ADP iajs-1821 93 93 the the DET iajs-1821 93 94 second second ADJ iajs-1821 93 95 type type NOUN iajs-1821 93 96 ( ( PUNCT iajs-1821 93 97 lffides lffide NOUN iajs-1821 93 98 ) ) PUNCT iajs-1821 93 99 via via ADP iajs-1821 93 100 the the DET iajs-1821 93 101 normalization normalization NOUN iajs-1821 93 102 bernstein bernstein PROPN iajs-1821 93 103 basis basis NOUN iajs-1821 93 104 . . 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