Numerical Solution of Parabolic Volterra Integro-Differential Equations via Backward-Euler Scheme

Transcription

1 America Joural of Compuaioal ad Applied Maemaics, (6): 77-8 DOI:.59/.acam.6. Numerical Soluio of Parabolic Volerra Iegro-Differeial Equaios via Bacward-Euler Sceme Ali Filiz Deparme of Maemaics, Ada Mederes Uiversiy, Aydi, 9, Turey Absrac I is sudy, we ivesigae covergece properies of ime ad space discreizaio of Parabolic Volerra iegro-differeial equaios (PVIDE) is oulied. Te recagle ad e rapezoidal rules applied for iegral erm of ese equaios ad fiie differece meod (Bacward-Euler meod) used for parial differeial par. Te iegral is approximaed i eac case by e quadraure rule wi relaively ig-order rucaio error. We cosider ime sep meods based o e bacward-euler meod ad combied wi e appropriae quadraure rules. Keywords Fiie differece meod, Bacward-Euler meod, Parabolic Volerra iegro-differeial equaios, Iegral equaios, Parial differeial equaios, Time ad space discreizaio, Mixed rule, Quadraure rule, Error. Iroducio I[5] ad[], Liz ad Baer cosidered e umerical soluio of Volerra iegral equaios of e secod id usig e recagle, e rapezium ad Simpso s rules for fidig u ( wi e quadraure rule. Wereas, i is paper we iroduce e umerical reame of parabolic Volerra iegro-differeial equaios usig e bacward-euler sceme for fidig u ( wi e fiie differece meod. I[], Douglas iroduced e umerical reame of parabolic Volerra equaios usig e bacward-euler ad Cra-Nicolso meods for fidig u ( wi e fiie differece meod of e form ( = ( g(, u( ds. + Douglas s as used quie simple equaio, bu i is impora a e used firsly ime discreizaio usig e bacward-euler ad Cra-Nicolso meods. However, I[] ad[7] e umerical examples are o give. I is paper e ime ad space discreizaio are sudied wi examples. I is paper we will sudy e umerical soluio by e ime-coiuous fiie differece meod of parabolic Volerra equaios of e form * Correspodig auor: afiliz@adu.edu.r (Ali Filiz) Publised olie a p://oural.sapub.org/acam Copyrig Scieific & Academic Publisig. All Rigs Reserved () ( = Au( + λ K(, Bu(, () i D = B L. I e ( x, plae le D = {( : < x < π, < < T}, ad le u ( ave e iiial ad boudary codiios u ( ) = f ( x), o x π, u(, π, =. Here we deoe by D e closure of D. A is a ellipic secod order parial differeial operaor ad B is a secod order parial differeial operaor respecively. We assume a bo operaors are smoo. Te equaio () is o be solved subec o e above iiial ad boudary codiios. As ey defied i[5] ad[], e erel K ( s, is a smoo, real-valued fucio of bo variables for s, ad S ( is supposed o be a smoo fucio of x ad. Te umerical meods will be a combiaio of fiie differece ad quadraure scemes for e ime seppig, wi ime sep. Iroducig a laice i D, we defie for posiive iegers m ad, = π / m, = T /, D i, = {( i, ) : i =,,,.., ; =,,,.., m}. For ay fucio U defied o D i,, we se U i, = U ( i, ). We replace e ime derivaive i () by a differece quoie, ad use a quadraure rule for e iegral erm of e form

2 78 Ali Filiz: Numerical Soluio of Parabolic Volerra Iegro-Differeial Equaios via Bacward-Euler Sceme K(, u( ds α, K(, ) U ( i, ), () were α, a quadraure is weig ad U ( i, ) is e approximaio o u (.. Numerical Meods for Parabolic Volerra Iegro-differeial Equaios I is secio we will sudy e iiial boudary value problem firs cosidered i e iroducio, i.e., ( = Au( + λ K(, Bu(, (4) i D = B L. I e ( x, plae le D = {( : < x < π, < < T}, ad le u ( ave e iiial ad boudary codiios u ( ) = f ( x), o x π, u(, π, =. Oe of e difficulies ivolved i suc a meod is a if α, for, e all values of U ( i, ) ave o be reaied, causig grea demads for daa sorage. I Sloa ad Tomée[7] a quadraure rule based o fewer pois is proposed, us reducig e umber of ime levels a wic e daa eed be saved. Te quadraure rule will be of e form σ ( g) = α g( ) g() s ds, =. (5), We defie e bacward differece meod by U U = AU + λ α K(, ) U ( i, ). (6), Te simples quadraure rule of ype () wic is cosise wi e order of accuracy of e bacward-euler sceme is e recagle rule σ ( g) = g( ), =, (7) wic correspods o coosig α, = for, so a α, = = = T. (8) + Le m = [ / ] ad m Z. Se = m, ad le l be e larges ieger suc a l. I approximaig e iegral erm over [, ] we sall apply e rapezoidal rule wi sep size o, l ] ad e [ recagle rule wi sep-size o e remaiig par [ l, ]. More precisely, wi =, σ ( g) = α =, l g( [ g( ), ) + ( )] + ml g( Noe a is rule as a sorage requireme of O ( ) ). (9) / as opposed o O ( ) for e simple recagle rule. Sloa ad Tomée[7] deal wi sabiliy ad covergece resuls for wo differe ime discreizaio of (6), based o e bacward-euler ad Cra-Nicolso meods respecively. I eir paper, e ime discreizaio is sudied i e reame of e iegral erm. Le be e ime sep i e bacward-euler sceme for e equaio ( = κ ( + K(, u(. λ () or for e followig form wi ellipic operaor uder e iegral erm, ( = κ ( + K(, (. λ () If we subsiue e bacward differece for u, cered differece for u xx i (), ad a quadraure rule for e iegral erm, we obai e followig form Ui, Ui, Ui, Ui, + Ui+, = κ () + λ α v, K(, v) U ( i, v) + Si,,, ad if we subsiue e bacward differece for u, u cered differece for xx i (), ad a quadraure rule for e iegral erm, we obai e followig form Ui, Ui, Ui, Ui, + Ui+, = κ U U + U i, v iv, i+, v + λ α v, K(, v) () + S,, i, were is e mes sep for x. Te sum o e RHS is e quadraure approximaio o e iegral erm of () or () a e poi =. Te weigs i e quadraure rule

3 America Joural of Compuaioal ad Applied Maemaics, (6): correspod o (9) above. For sufficiely regular fucios g e rucaio error is of order O + ), assumig a ( / O( is bouded. If we ae = ) e accuracy is similar o a of e bacward-euler sceme. If e weigs are rapezoidal weigs for sep-size ad recagle rule weigs for sep-size, we ca wrie α v α v are α v =, we (mod m) α, v = α v =, we (mod m), v > m (4) α v =, oerwise. From (4) we ave, v v v v α max( α, α ) = α, (5) were, as easily verified (assumig ), α +. (6) v Noe a i e umerical soluio of Example., Example. ad Example.4, we did o iclude e fucio S (. Examples wi e fucio S ( ca be doe i a similar maer. Example.: Cosider e iiial-boudary value problem ( ( = κ ( + λ e u( ds, (7) for T, π, (, =, u( π, = x wi u ( ) = si( x), u ad, wic as e exac soluio ( ( ) ux (, ) e α α = cos( β + si( β si( x), (8) β were α = ( κ +) / ad β = ( κ + κ 4λ ) /. Taig κ = i e equaio (8) we ave e soluio u( = e cos( λsi( x), for λ >. (9) If we pu κ = ad λ = i (8) we obai e soluio u( = e cos( si( x). () Te las resul correspods o e simples form of our mai problem (7). We will solve e equaio (7) wi e bacward-euler meod. Firsly, we use e recagle rule rougou for e iegral erm i equaio (7). Table sows umerical resuls for is case. Uless oerwise idicaed, rougou is secio ad e ex secio we ae κ = ad λ =, ad T =.5. Table sows a e combied bacward-euler ad recagle rule as error O() for is Example. Table. Recagle rule sowig effec of ime-sep, for Example., as a error of order O () wi fixed = π / =.5 =.5 =.5 =.5 x Exac Error Error Error E E E E-.99E-.546E E-.55E-.798E E-.799E-.8456E E-.898E E-4 error.545e-.77e E-4 I is es we see a e space error as e expeced order i our umerical calculaios. Secodly, we use e rapezoidal rule rougou for e iegral erm i equaio (7). Te umerical resuls are give by Table. Tirdly, we obai e umerical soluio of equaio () usig e rapezoidal rule ad recagle rule as i (9) wic is called e mixed rule. Table illusraes a e combied bacward-euler ad rapezoidal rule also as error of order for Example. Table. Trapezoidal rule sowig effec of ime-sep, for Example., as a error of order O () wi fixed = π / =.5 =.5 =.5 x Exac Error Error E-.66E E-.7E E-.6554E E-.9464E E-.469E- error.78e e- Table sows umerical resuls for e mixed rule (bacward-euler sceme). I Table, we ae m = iegral par of [ / ], wi m = ; for e cases i Table, we ave [ / ] = ieger. Firs, we will apply e simples rule wic is e recagle rule for e iegral erm. Table illusraes a e combied rapezoidal ad recagle rule as error of order for Example.. Table. Mixed rule sowig effec of ime-sep, for Example., as a error of order O() wi fixed = π / =.5 =. =.5 =. =. =.5 =. x Exac Error Error Error E E-4.76E E-.697E-.6497E E-.7467E-.8898E E-.59E-.9E E-.597E-.957E- error.55955e-.5669e e-

4 8 Ali Filiz: Numerical Soluio of Parabolic Volerra Iegro-Differeial Equaios via Bacward-Euler Sceme Now cosider e equaio () were we ave a ellipic operaor uder e iegral sig. Te umerical resuls for is equaio are give i Table 4. Example.: Cosider e iiial-boudary value problem ( ( = κ ( + λ e ( ds, () for, x π, wi u ( ) = si( px), u (, =, p is a posiive ieger, p=,,,, u( π, =. Tis as e exac soluio ( a/) ( a) u( = e cos( c / ) + si( c / ) si( px), () c were a = ( κ p + ), b = p ( κ + λ) ad c = ( a 4b ). If we coose κ =, λ =- ad p = i () we obai e soluio ux (, ) = ( + e cos( )si( x). () Table 4 illusraes a e combied bacward-euler ad recagle rule as error O() for Example. Table 4. Recagle rule sowig effec of ime-sep, for Example., as a error of order O() wi fixed = π / =.5 =.5 =.5 =.5 x Exac Error Error Error E-.9495E E E-.699E-.8764E E-.5847E-.578E E E-.94E E-.689E-.94E- error.47e E-.9879E- Te las resul correspods o e simples form of our mai problem (). I Table 4 we use e recagle rule rougou for e iegral erm i equaio (). Equaio () ca be solved wi e bacward-euler ad Cra-Nicolso meod (CN) i a similar maer. Afer ese calculaios we will exed our simple examples for more geeral iiial codiios. Firs, we ae e iiial codiio u( ) π x) o x π. We ca obai e exac soluio for geeral iiial codiios by usig Fourier expasios. Example.: Solve e problem ( κ ( = for < x < π, >, (4) u( ) π x), (5) u(, π, =. (6) Te Fourier coefficie for e iiial values of u ( is b π = x( π x)si( x) d π 4 = ( cos( π )), (7) π 4 = ( ( ) ), =,,,... π 8 Tus b = if is odd or zero is eve, givig π 8 (m ) κ si((m ) x) ux (,) = e. (8) π m= (m ) I e followig example, we ave solved e equaio (7) wi a differe iiial codiio. I Example., we ave cose e iiial codiio u ( ) = si( x), owever i Example.4, we ave ae u( ) π x). Example.4: Solve e problem ( ( = κ ( + λ e u( ds, (9) u( ) π x), () u(, π, =, () for < x < π, >. If we cosider e Fourier expasio of x( π x), e we ge e soluio, for κ = ad λ = 8 si( px) ux (,) = v( p,), () π m= p were p aes odd values, v ( p, is give by ( ) v( p, ) e α α = cos( β si( β, β () Were α = ( p + ) / ad β = ( α α ). Te firs erm of (4), for p =, is 8 u( = e si( x)cos(, (4) π wic is 8 / π imes e soluio of Example., because e firs erm of e Fourier series of e iiial codiio of Example.4 is equal o 8 / π imes iiial codiio of Example.. I is eoug o ae e firs welve erms of e soluio (), o obai 9 figures i e resuls. Te errors for e recagle rule are give i Table 5. Table 5 illusraes a e combied bacward-euler ad recagle rule as error of order for Example.4.

5 America Joural of Compuaioal ad Applied Maemaics, (6): Table 5. Recagle rule sowig effec of ime-sep, for Example.4, as a error of order O () wi fixed = π / =.5 =.5 =.5 x Exac Error Error E-.85E E-.954E E-.4546E E-.44E E-.44E- error E-.487E- I Table 6, we used e mixed rule for Example.4. I is calculaio, we ave obaied errors of O(). Table 6 illusraes a e combied bacward-euler (mixed rule) as error of order. Table 6. Mixed rule sowig effec of ime-sep, for Example.4, as a error of order O () wi fixed = π / =.5 =.5 =.77 =.5 =.5 =.5 =.5 x Exac Error Error Error E-.9697E-.8695E E-.6494E-.9964E E-.459E-.5466E E-.54E-.788E E-.5965E-.88E- error.795e-.8976e-.648e-. Error Aalysis i Time ad Space for Bacward-Euler I is secio, we iroduce e discreizaio errors for our equaio (). I applicaios e operaors A ad B will o be differeial operaors suc as = Au, ai, (5), i i x i Bu raer approximae operaors (arisig from fiie differece space discreizaio) wic deped o some mes parameer. Our resuls will be useful if e error bouds are uiform i. Te error a a mes poi is defied by ei, xi, ) U ( i, ). (6) As we defied i Secio, we abbreviae u( x i, ) ad U ( i, ) by u i, ad U i, respecively. Tus U i, = ui, ei,. I equaio () replace U ( i, ) by u i, ei, givig ui, ei, ( ui, ei, ) = ui, ei, ( ui, ei, ) + ui+, ei+, κ (7) + λ α, vk, v ( u, v e, v ) + Si,,, Te i follows from () ad () a e i, saisfy e equaio ei, ei, ei, ei, + ei+, = κ + (8) λ α, vk, ve, v + τ i,,, were τ is e rucaio error give by[]. I Table 7, i, e space error is sow for e CN ad rapezoidal rule (see[] ad[4]) for Example.. Te umerical resuls for e space error es give O ( ). Te recagle rule ca be doe similar maer. I Table 7, CN ad rapezoidal rule sowig effec of space-sep for =. / =.. Table 7. CN ad rapezoidal rule as a error of order =. = π / = π / = π / 4 x Exac Error Error Error E-4.886E E E E E E E-5.758E E-4.49E E E E-4.684E-5 error E-.8584E E- I is es we see a e space error as e expeced order i our umerical calculaios. 4. Coclusios I is paper, umerical meod as bee successfully developed for solvig parabolic Volerra iegro-differeial equaios. Numerical quadraure rules ave bee applied for iegral erm ad bacward-euler meod as bee used for parial differeial par. Errors values preseed from Table o Table 6 are calculaed based upo bacward-euler ad suiable quadraure meods wi ime sep. Values i ese ables illusrae a error of order O (). Fially, i Table 7 e space error usifies O ) i our umerical soluio. Numerical order of covergece is also calculaed: l( Error ) l( Error ) Ord =. l() (

6 8 Ali Filiz: Numerical Soluio of Parabolic Volerra Iegro-Differeial Equaios via Bacward-Euler Sceme We expec a Ord = i ime-sep ad Ord = i space-sep. Obaied eoreical resuls are cofirmed by umerical experimes. REFERENCES [] Baer, C. T. H., Te Numerical Treame of Iegral Equaios, Claredo Press, Oxford, 977. [] Douglas, JR., 96, Numerical meods for iegrodiffereial equaios of parabolic ad yperbolic ypes, Numer. Ma., 4, 96. [] Filiz, Ali, Numerical soluio of some Volerra iegral equaios, PD Tesis, Uiversiy of Maceser,. [4] Filiz, Ali,, Secod-order Meod for Parabolic Volerra Iegral Equaios wi Cra-Nicolso Meod, Maemaica Moravica, 6-, -. [5] Liz, Peer, Aalyical ad Numerical Meods for Volerra Iegral Equaios, SIAM-Piladelpia, 985. [6] Pao, C. V., Noliear Parabolic ad Ellipic Equaios, Pleum Press, New Yor, 99. [7] Sloa, H. I. ad V. Tomée, 986, Time discreizaio of a iegro-differeial equaio of parabolic ype, Siam Joural of Numerical Aalysis, 5 6. [8] Smi, G. D., Numerical Soluio of Parial Differeial Equaios: Fiie Differece Meods, Claredo Press, Oxford, 985. [9] Volerra, V., Teory of Fucioals ad of Iegro- Differeial Equaios, Dover, New Yor, 959.