Error Analysis of Legendre-Gauss-Radau Collocation for Linear and Non-linear Volterra Integro-differential Equations of the Second Kind
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1 Unia ona of Ina qaion wwwpapino o Anai of n-a-raa ooaion fo ina an on-ina Voa Ino-iffnia qaion of h Son in Zhao Shoo of Mahai anho i Unii ifan S Annin ii anho 737 hina hao@n MS i Shoo of Mahai anho i Unii ifan S Annin ii anho 737 hina i@n Aba In hi pap w popo an ffiin nia ho fo Voa-p ino-iffnia qaion ba on n-a-raa inpoaion whih i a o b ipn fo boh ina an nonina pob an po h pa aa o ia i on o how h pa aa of h ho W ao op a i-p ion of hi appoah ia oini w wih h hoia anai an ona h ffin of h appoah Mahai Sb aifiaion: 65M 65M5 65M7 Wo an Pha n-a-raa ooaion Mho Voa-p ina an on-ina Inoiffnia qaion aiona a Spa Aa Mi-p Vion Inoion In n a h ha bn a owin in in h Voa ina qaion Voa ina qaion ai in an phia appiaion ponia ho an iih pob oai ahaia pob of aiai qiibi h pai anpo pob of aophi an ao ho an aiai ha anf pob [39] [4] [5] [37] In hi pap w a in in nia oin Voa ino-iffnia qaion of h on in in h foowin fo: ' U wh h o fnion an h n fnion a in an i h nnown fnion o b in A na fo of a a: ' U wh i a nonina fnion wih ain oohn o h xin an niqn of nonina ina qaion on an on wih [7] [4] W wi oni h a ha h oion of o ina a a ffiin ooh - in hi a i i na o oni hih-o nia ho h a pa ho fo appoxiain h oion h a an xiin nia ho fo oin h Voa qaion an a a ai of nia ho ha bn op o api an aa obain appoxiaion o Oiw an 33
2 Zhao & MS i fn o h ia fo an xiin ho a aaiab in [] [6] [] ooaion ho [6] [3] [33] Sin ho [5] won-o ho [] Rn-a ho [6] [7] qaoaion ho [] [3] [3] a a of h an appoah ha ha pio bn oni In [3] pa ooaion ho wa ana b aho fo ina faiona ino-iffnia qaion A w now pa ho po oba ohoona ponoia a ia fnion I ofn poi xin aa nia wih ai of fo an h ha bn wi fo inifi opaion Bnai an Maa [] Bo [4] ano Haini Qaoni an Zan [7] nao [4] oib an Oa [5] o [6] Shn an an [8] an Shn an an Wan [9] h a o pbiaion abo pa fo nia oion o Voa ina qaion h a [] [8] [9] [4] Rn a n-a-raa ooaion ho i popo fo iniia a pob of oina iffnia qaion [38] Moia b h ia in [38] w popo a n-a-raa ooaion ho fo Voa ino-iffnia qaion a w a In hi pap w inia an ffiin nia ho fo Voa inoiffnia qaion of h on in a w a B h nx ion w i a nw ooaion ho whih appoxia xa oion i b n- a-raa inpoaion wih no hn w ana h nia aa in S 3 hi po ha a fainain aana i i i ai o b ipn fo nonina pob x i po h pa aa In oh wo fo an fix o h ooh h xa oion bo h o aa h nia wi b In S 4 w poi h i-p ion of n- a-raa ooaion ho B in hi aoih wih oa o w o aa nia oion p b p a i ina hi fa ipifi aa aaion an a h wo In S 5 w pn o nia an opa h popo nw ho wih a oon ho h inia h hih aa of aoih an oini w wih hoia anai h fina ion i fo o onin a n-a-raa ooaion Mho In hi ion w i h n-a-raa ooaion ho wih h nia ipnaion x b h ana n ponoia of h hif n ponoia x a fin b:! h popi of h hif n ponoia an b b in h oponin popi of h ana n ponoiaf [6] no b h no of h ana n-a-raa inpoaion on h ina [ In paia h oponin hioff nb a h no of h hif n-a-raa inpoaion on h ina [ a h iin o of no b a 34
3 Zhao & MS i h oponin hioff nb a W n n-a-raa p qaa b h of ponoia of a o h n-a-raa qaa in h hif a a a fo an P o an [ h hif n-a-raa inpoaion P i in niq b: o an noaion a an h hif n-a- Raa inpoaion on [ wih p o whih aifi ha fo an [ : P an h i h no of h hif n-a-raa inpoaion on h ina [ h oponin hioff nb a h n-a-raa ooaion ho fo oin ina a i o P h ha: an P U 3 Ba of P an h i qian o: 4 h n-a-raa ooaion ho fo oin nonina a i o P h ha: 5 an U 6 Ba of P an h 5 i qian o: B ain h nonina a fa ino h ina a So o 5 7 an 7h fa ino an 4 pi Ipnaion of h ooaion Mho W nx ib h nia ipnaion fo 5 In h of i a 35
4 Zhao & MS i op -ohoona na 8 wh i h on bo h fo an ˆ ˆ 9 W fi xp h appoxia oion a: ˆ Obio fo P hfo b ipin b an inain h o h ina w ohoonai 8 an o if ha: û a Hn w obain: ' ' 3 ' U o h nonina n ina in 5 w ha: U an i i 4 U h w : 36
5 Zhao & MS i a ; Α b B ; ; h h hn w an wi 5 a h foowin opa aix fo U ΑhU Βh - Β Α 5 In aa opaion w fi 5 o aa hn w o obain: 6 xpion abo an b in aix-o fo a: 7 w wh: w w ; Ra h nia h 5 i a nonina fo n fnion i ina In o o o i a pop o ho b h a won ho o h ina a qaion 5 i: U ΑhU Βh - B A 8 3 o Anai of h ooaion Mho In hi ion w pn o anai of h ooaion ho fo an 5 In paia w ha po h pa aa of nia oion an b h inn po an h no of pa pi W ao ino h foowin i inn po an no 3 han o fo an P 3 W ino an appoxiaion fo a 37
6 Zhao & MS i a [38] o an H wih in h ho h foowin appoxia ia: W ino h abii of n-a-raa inpoaion a o an H h ho h foowin ia: 35 Poof Ba of: hn 35 o fo a wih a 3 [34] A ha x i h h aan inpoaion ponoia aoia wih h a o a-raa o a-obao poin hn: / / ax x x B O wh B i a bon onan a 4 o an [ h ho h foowin ia: 36 / 37 Poof h ia i a i of a 3 a 5 { : } fo an an f h ho h foowin ia: f f 38 Poof Main of Hö inqai w an obain: f f f f f hi i h i 3 h ina a f A a w fi opa h nia oion wih h inpoaion of oion o o hi ppo : hn i foow fo ha fo 38
7 Zhao & MS i 39 h Sbain fo 39 i: 3 ow b ipin h h qaion of 3 b fo an pin h in qaion oh w obain: Sin i fini Bi h pio an a o: P 3 Obio P h b 3 3 hi an ha: 33 W nx ia h b i of 33 wih an w ha ha fo in h abo oh wih 34 i ha fo 34 ow w o o ia i w ha: : 3 If w a ha: } : { hn w ha: 39
8 Zhao & MS i ] [ / 3 ] [ 3 / ow w a ha: ax M hn w ha: M 35 wih: On h oh han fo an / M / / / / / / / / Hn: / / / Inain h inqai abo wih p o i: 4 / / 36 o P an w 3 an 3 i o ha: 4
9 Zhao & MS i / / / / ow w ha: 8 hn w o o h foowin ho -onn If h n aifi: } : { ax M wih 8 An i h oion o hn w ha: H wh i a poii onan pnin on M an Poof Sin w ha: h a / M / 4 onqn b a w ha: I i h i ho -onn Un h a apion a in ho w ha H 39 wh i a poii onan pnin on M an Poof o 3 w now: obinin ia 3435 wih inqai abo w ha: onqn b a on ha: I i h i 4
10 Zhao & MS i 3 h onina a W h a noaion whih i h nia oion o 5 in hi ion W ao n o opa h nia oion wih h inpoaion of oion o no ha: 3 3 hn i foow fo ha fo 3 h Sbain 5 fo 3 i: 3 3 x w ia 3 i w ha: 3 : 4 3 ow w n apion on nonina A ha: R Wih ai of h abonion apion w an : 4
11 Zhao & MS i / : ' ax M " ax ' ax 33 an b a poii onan h ha: ax 34 no hn: ' " ' M hn w ha: 3 3 / In o o ia 4 w a ha: R 35 Ba of 36 o 35 an 36 w : 43
12 Zhao & MS i 37 : ax 38 an if: 39 ax w an obain: ' " hi an: 4 4 / ow w ha: M 3 33 ho 3 Un apion of ho of nonina h ha: R R wih An i h oion o hn w ha: H 33 an: 33 wh i a poii onan pnin on M an Poof o 3 w now: 3 obinin ia of an 3 a: 44
13 Zhao & MS i hn w ha: ho b in 36 w an obain: Ao w ha: h poof i op Ra o ha h apion 39 whih i abo bonn of an b aifi in onn in ho 3 4 Mi-Sp Vion of h ooaion Mho In aa opaion i i no onnin o o h i 5 wih a o On h oh han fo nin h onn of h 5 h nh of i ii o h fiini w now poi h i-p ion of n-a-raa ooaion ho whih ipifi opaion a wo an i p h pa aa M b a poii in an M b poii in W ii h ina a: [ ] M W Rpain an b an in 5-6 w obain h oa nia oion wih h iniia a P x w aa h oa nia oion b in h foowin po 3 M P 4 M 4 Obio h oa nia oion i a pop appoxiaion o h oa xa oion wih h appoxia iniia a Spifia w no h hif n ponoia of o on na: ] [ ] [ h oponin hif n-a-raa ooaion poin an wih a: an pi : 45
14 Zhao & MS i ; a A ; b B ; h h b b ; ; n n n n n n n ˆ f o oin h oa nia oion w n o o h a : P U U h B h A B A an: b x p b p h oa nia oion an b obain b oin: 3 M P h A B A ˆ h B b oin ha h po abo an b own wiho an ap Obio h oa nia oion i a pop appoxiaion o h oa xa oion wih h appoxia iniia a U 5 ia xpin B hi ion w wi pn o nia xpin o onfi h ffin an obn of h h 5 o 5 h fi i in pfoin h popo ho i how o o h nonina 5 Sin h aix B in h ih-han i of h 5 pn on n fnion w an no ai an f on i Anwa a ip ia ho 46
15 Zhao & MS i i: fo A B AhU Bh U 5 If h qn on hn w an obain a oo appoxiaion oion of h 5 xap O fi xap i onn wih ina pob in on-inion oni Voa ino-iffnia qaion wih: xp xp xp h xa oion i xp Ba h xa oion i nown w an op h o bwn nia oion an xa oion h o in an a fin b: inf ax wh an a xa an nia oion pi Anoh poinwi o a i In ab w pn h o wih iffn h how "pa aa" ab o wih iffn fo xap inf xap O on xap i onn wih nonina pob in on-inion oni Voa ino-iffnia qaion wih: o in o xp in o 8 xp in o 6 xp h xa oion i in o oin h nonina 5 ip ia ho 5 i po wih oa 6 ha i In ab w pn h o wih iffn Ao h iaion nb of ip ia ho 5 i in I i how ha a fw iaion an ahi oo piion h how "pa aa" ab o wih iffn fo xap Iaion inf
16 Zhao & MS i xap 3 oni Voa ino-iffnia qaion wih: xp xp o6 6 in6 36 xp in3 3 xp o3 xp 3 h xa oion i xp in3 h o in an a fin b h a a in xap o oin h nonina 5 ip ia ho 5 i po wih oa 6 ha i In ab 3 w pn h o an ia nb wih iffn h ao how "pa aa" ab 3 o wih iffn fo xap Iaion - 9 inf xap 4 oni h i-p ho 4 o o ina Voa inoiffnia qaion wih: o in in h xa oion i o W oni ha h opaiona ina [ ] i ii qa a i i M / o ipii nb i of ooaion poin in bina [ ] a a h nia a i in ab 4-7 ab 4 o wih iffn fo xap 4 wih ab 5 o wih iffn fo xap 4 wih ab 6 o wih iffn fo xap 4 wih ab 7 o wih iffn fo xap 4 wih onin Ra In hi pap w popo a n-a-raa ooaion ho fo oin Voa-p ino-iffnia qaion of on in hi ho i a o b 48
17 Zhao & MS i ipn fo nonina pob In paia bnfiin fo h api onn of h n-a-raa inpoaion hi ho po pa aa I an ha fo an fix o h ooh h xa oion bo h o aa h nia wi b W ao poi a i-p ion of n-a-raa ooaion how o hi po wih oa o o aa h nia oion p b p hi ipifi aa opaion an a wo nia ho fo an fix o h nia oion ha hih onn a Manwhi fo an fix p i in i h nia o a api a h oohn of oion an h o ina h nia ona h pa aa of popo ho an oini w wih h hoia anai I wa ao inia ha o nw ho poi o aa nia On h oh han in h iaion of h xiin ooaion ho on o h aan inpoaion whih i nab fo a Wha w h a-p inpoaion a in [8] [9] [] whih a o ho h o ab fo a hi i ao onfi b h nia Anown h aho han h anono f fo hi aab ion an hpf on h wo of h fi aho wa pa ppo b aiona aa Sin onaion of hina n an o 66 Rfn Ba H 'A ppi on h nia an of Voa qaion' op App Mah Bnai an Maa Y' Spa ho in Hanboo of ia Anai' VoV hniq of Sinifi opin Pa P ia an ion oh-hoan Aa Bo Y Bnn H 'Aoih 689: ii ooaion an ia ooaion fo nonina Voa ina qaion of h on in' AM an Mah Sofwa Bo P 'hbh an oi pa ho' n iion o Pbiaion In Minoa w Yo 5 Bnn H Maoo A Mi R 'Mix inpoaion ooaion ho fo fi an on Voa ino-iffnia qaion wih pioi oion' App Mah Bnn H 'ooaion Mho fo Voa Ina an Ra niona qaion' abi Unii P w Yo 4 7 ano Haini M Y Qaoni A Zan A 'Spa Mho: nana in Sin oain' Spin Bin 6 8 hn Y P an an 'Spa ho fo wa ina Voa ina qaion wih ooh oion' op App Mah hn Y P an an 'onn anai of h aobi pa-ooaion ho fo Voa ina qaion wih a wa ina n' Mah op hhan M Sahi R 'h nia oion of h non-ina ino-iffnia qaion ba on h h ho' op App Mah M Moha 'opaiona Mho fo Ina qaion' abi Unii P abi 985 naa an ai M 'hbh pa oion of nonina Voa-Hain ina qaion' op App Mah iwaa H 'Hih-aa nia ho fo ina qaion of h fi in n ippiion aihi' Ppin RIMS oo Unii 6 4 nao 'Ponoia Appoxiaion of iffnia qaion' Spin Bin99 5 oib Oa S A 'ia Anai of Spa Mho: ho an Appiaion' SIAM- BMS Phiaphia
18 Zhao & MS i 6 o B Y 'h pa ho an i appiaion' Wo Sinifi Sinapo o an Sn X 'onina Ina qaion' in hin Shanon Sin an hnoo P P R hina o B Y Wan Z Q 'ia inaion ba on a-a inpoaion' op Mho App Mh n o B Y Wan Z Q 'n-a ooaion ho fo oina iffnia qaion' A op Mah o B Y Wan Z Q 'A pa ooaion ho fo oin iniia a pob of fi o oina iffnia qaion' i onin n S S B Han R Sibann B'Opao qaion an nia anai Sina Anai' a- Wia-Ini f Mahai Bin987 h On abii of h qaoaion ho 43-5 ian Y 'On pa ho fo Voa-p ino-iffnia qaion' op App Mah Ma X H Han M 'Spa ooaion ho fo ina faiona ino-iffnia qaion' App Mah Moin Mana Ahaah 'ia oion of Voa ina qaion of h on in wih onoion n b in ao-i xpanion ho' App Mah op Mhaa M haa A Moi M Sihaa M 'ia oion of ina qaion b an of h in ooaion ho ba on h ob xponnia anfoaion' op App Mah Shap P W Vn H 'xn xpii B o pai of o 4 an 5 fo Voa ina qaion of h on in' App Mah Shap P W Vn H 'So xn xpii B o pai fo Voa ina qaion of h on in' SIAM Ana Shn an an 'Spa an hih-o ho wih appiaion' Sin P Biin 6 9 Shn an an Wan 'Spa ho: Aoih Anai an Appiaion' Spin Si in opaiona Mahai 4 Spin-Va Bin 3 Soan I H'A qaa-ba appoah o ipoin h ooaion ho' Mah Soan I H 'Qaoaion' op App Mah Soan I H Wnan W 'A qaa-ba appoah o ipoin h ooa-ion ho fo pin of n ' Zi f Ana ih Anw an 'A no on ooaion ho fo Voa ino-iffnia qaion wih wa ina n' IMA Ana an X X an hn 'On pa ho fo Voa ina qaion an h onn anai' op Mah ao X Xi Z Q an Zho X 'Spa Po-ain ho fo h on in Voa p ino-iffnia qaion' Mah ho Mh App fhn 'Spa ho in Maab' SIAM Phiaphia PA 37 Wan W 'A hania aoih fo oin h Voa ina qaion' App Mah op Wan Z Q an o B Y 'n-a-raa ooaion ho fo oin iniia a pob of fi o oina iffnia qaion' Si op Wawa AM 'wo ho fo oin ina qaion' App Mah op Wawa AM 'ina an onina Ina qaion: Mho an Appiaion' Spin 4 Wi Y X an hn Y P 'A pa ho fo na Voa ino-iffnia qaion wih wa ina n' Mah ho Mh App Xi Z Q i X an an 'onn anai of pa ain ho fo Voa p ina qaion' Si op
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