Compact Finite Difference Schemes for Solving a Class of Weakly- Singular Partial Integro-differential Equations

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1 Ma. Sci. Le. Vol. No (0 Maemaical Scieces Leers A Ieraioal 0 NSP Naural Scieces Publisig Cor. Compac Fiie Differece Scemes for Solvig a Class of Weakly- Sigular Parial Iegro-differeial Equaios A. F. Solima, A M.A. EL-ASYED, M. S. El-Azab 3 MET Higer Isiue of Egieerig ad Tecology-Masoura-Egyp Faculy of Sciece, Alexadria Uiversiy, Egyp 3 Maemaics ad Egieerig Pysics Deparme, Faculy of Egieerig, Masoura Uiversiy, Masoura, 355, Egyp Absrac:: : I is paper, we prese a ew approac o resolve liear weakly-sigular parial iegro-differeial equaios by firs removig e sigulariy usig Taylor's approximaio ad rasform e give parial iegrodiffereial equaios io a parial differeial equaio. Afer a e four order compac fiie differece sceme ad collocaio meod is preseed o obai sysem of algebraic equaios wic solved o compue e ukow fucio. Te efficiecy ad accuracy of e meod is validaed by is applicaio o several disic es problems wic ave exac soluios.. Keywords: T weakly-sigular parial iegro-differeial equaios; Taylor's approximaio; four order compac fiie differece sceme; collocaio meod. Ay fucioal equaio i wic e ukow fucio appears uder e sig of iegraio is called a iegral equaio. I may isaces e iegral equaio origiaes from e coversio of a boudary-value problem or a iiial-value problem associaed wi a parial or a ordiary differeial equaio. Iegral equaios arise i a grea may braces of sciece; for example, i poeial eory, acousics, elasiciy, fluid mecaics, radiive rasfer eory of populaio, ec. Also iegro-differeial equaios (IDE arise widely i maemaical models of cerai biological ad pysical peomea. I is paper we sudy e liear weakly-sigular parabolic iegro-differeial equaios wi a memory erm: u u k( ds f (, x(0,, [0, T], x 0 (. I is associaed by e boudary ad iiial codiios 0, g(,, g(, 0, (. 0 u0(, 0 x k(. were Te soluio u (., ad e source erm (., L ([0,], (.3 f ake values i ([0, ] L, ad e iiial daa u 0 is a eleme of. Equaios of ype (. may be oug of as a model problem occurrig i e eory of ea coducio i maerials wi memory, populaio dyamics ad viscoelasiciy (e.g., Friedma & Sibro, 97 [5]; Heard, 98 []; Reardy e al., 987 []. Due o e wide applicaio of ese equaios, ey mus be solved successfully wi efficie umerical meods. May auors ave cosidered umerical meods for a liear problem of e form (.. Typically, e ime discreizaio is affeced by a combiaio of fiie differece ad quadraures. Fiie differece i ime ad fiie elemes i space ave bee discussed i e case of a smoo kerel (e.g., Sloa & Tomée, 98 []; Cao & Li, 988, 990 [,3]; Yaik & Fairweaer, 988 [5]; Tomée & Zag, 989 [4]; Li e al., 99 [9]; Zag, 993 []; Mclea, Tomée. & Walbi (99 [0]. For e osmoo kerel case we refer o Ce e al. (99 [4] ad Larsso e al. (998 [8]. Collocaio meod as bee discussed i (Te Riele (98 [3]; Bruer, Pedas, & Vaiikko, (00 [] Our coribuio i is paper is o develop a ew approac o resolve liear weakly-sigular parial iegrodiffereial equaios i oe dimesioal space wi o-omogeeous Diricle boudary codiios. Te suggesed umerical sceme sars wi removig e sigulariy usig Taylor's approximaio ad rasformig e give secod-order parial iegro-differeial equaios io parial differeial equaio, e we use e discreizaio i

2 54 A. F. Solima e al : Compac Fiie Differece Scemes for Solvig. ime by e -poi Euler backward fiie differece meod. Afer a we use compac fiie differece sceme ad e we use a collocaio meod o obai sysem of algebraic equaios wic solved o compue e ukow fucio. Te proposed eciques are programmed usig Malab ver (R009a. Te paper is orgaized as follows: I Secio, we iroduce a meod of soluio for weakly-sigular parial iegro-differeial equaios wi varyig boudary codiios by usig Taylor's approximaio, e we give a brief iroducio o a ig accurae compac fiie differece formula for parial differeial equaios wi varyig boudary codiios, e we use collocaio meod. I Secio 3, e proposed meod is direcly applicable o solve several disic umerical examples o suppor e efficiecy of e suggesed umerical meod. Coclusios are draw i Secio 4.. Meod of Soluio We propose a approximae soluio for solvig weakly-sigular parial iegro-differeial equaios. Te advaage of is meod is a we remove e sigulariy of e kerel of weakly-sigular parial iegro-differeial equaios a s by udiciously applyig Taylor s approximaio ad e rasformig e give weakly-sigular parial iegro-differeial equaio io parial differeial equaio. Nex, e discreizaio i ime by e -poi Euler backward differeiaio formula is maipulaed o cover e parial differeial equaio io ordiary differeial equaio. Te we use compac fiie differece sceme ad collocaio meod o obai sysem of algebraic equaios wic solved o compue e ukow fucio.. Taylor s approximaio Cosider e followig weakly-sigular iegro-differeial equaio parial u uxx ds f (. 0 ( (.. I is associaed by e boudary ad iiial codiios u ( 0, g(,, g(, 0, (.. u ( 0 u0(, 0 x. (..3 Rewrie equaio (.. as: u uxx 0 ds f ( ( u uxx ds ds f ( 0 ( 0 ( (..5 equivalely ( u uxx ( ds f ( ( 0 ( s By usig Taylor s approximaio of u ( abou s, ( s u ( or, (..4. (.., (..7 u ( ( s From equaio (..8 io equaio (.., e ( u ( u so xx. (..8 ( u ( ( 0 ( ds f (, (..9

3 A. F. Solima e al : Compac Fiie Differece Scemes for Solvig. 55 u ( u equivalely xx ( u ( u ( ( ( ( u ( f ( ( ( xx ( ( (.3 Compac Fiie Differece Scemes f (, (..0. (.. I is secio, we give a brief iroducio o a ig accurae compac fiie differece formula for parial differeial equaios wi varyig boudary codiios..3. Formulaio of Hig-Order Compac Scemes Compac Scemes are based o a four order accurae approximaio o e derivaive calculaed from ordiary differeial equaio. To developed e sceme for oe-dimesioal uiform Caresia grids wi spacig x, le us iroduce e followig oaios [7]: If u x, e we use oaios u u u u u x, u x, (.3.. o deoe e sadard forward fiie differece ad backward fiie differece scemes for firs derivaive. Also, u u u 0 u u, (.3.. is e firs-order ceral fiie differece wi respec o x. Te sadard secod-order ceral fiie differece is deoed as x u ad is defied as u u u u. (.3..3 By usig e Taylor s series expasio, a four ad six orders accurae fiie differece for e firs ad secod derivaives ca be approximaed by 3 du d u d du du 4 0 u O( 3! 3 (.3..4 ad 4 d u d u d d u 4 x u u O(, 4 ( Compac fiie differece meod for solvig parial differeial equaios Here, we use e four order compac fiie differece meod o solve problem (.. wi boudary ad iiial codiios (..,..3. Firs e discreizaio i ime by e -poi Euler backward differeiaio formula is maipulaed o cover e parial differeial equaio io ordiary differeial equaio. To cosruc a umerical soluio, we firs cosider e odal pois x, defied i e regio [ 0,] [0, T] were 0 x x x x, x x ad 0 T,. 0 0 i i i I suc a case we ave x for 0,,,,, ad i, ( i i for i 0,,,.

4 5 A. F. Solima e al : Compac Fiie Differece Scemes for Solvig. Nex, e -poi Euler backward differeiaio formula is maipulaed o approximae u, give i equaio (.., a e ime-level i for i 0,,,. Terefore, we ave ( ( ui( ui ( d ui( u ( f ( ( i i ( x, (.3.. equivalely ( ( ( ui( ui ( fi( ui ( (.3.. ( ( ( were f f (, u ad u. i( i i( i Puig x,,, u i, (.3..3 le x i (.3.., e ( ( u ( ( i, f i, i( i ( u ( ( ( ( a, b,e ( ( ( u i, a ui, fi, b ui, (.3..4 were ui, u ( x, i, ui, x, i, ui, x, i, ad fi, f ( x, i. Secod e four order accurae fiie differece esimae for i,, u x is used from (.3..5 o give 4 u, u x i i, x u i, O(. (.3..5 Te, a compac (implici approximaio for u x wi four-order accuracy will be give as xui, 4 ui, O(. (.3.. x Usig is esimae ad cosiderig e discree soluio of equaio (.3..4 wic saisfies e approximaio, we ge a a a u i, ui, u i, fi, fi, f i, u i, ui, u i,. (.3..7 Te, we use collocaio meod o obai sysem of algebraic equaios wic solved o compue e ukow fucio. Le Ui( x be a fucio a approximaes i for e ime-level i i, ad is a liear combiaio of + sape fucios wic is expressed as: Ui( cmi m(, (.3..8 m0 were { cmi} m 0 are e ukow real coefficies, o be evaluaed, ad e m(x are ay kowig basis fucios e equaio (.3..7 rewrie as

5 A. F. Solima e al : Compac Fiie Differece Scemes for Solvig. 57 a a U U U f f a i, i, i, i, i, fi, u,,,, 0,,,,,,, i ui ui i. (.3..9 Replacig U i by e approximae soluio give by equaio (.3..8 yields e followig liear sysem of equaios a a a cmi m cmi m c mi m m0 m0 m0 f i, fi, fi, u i, ui, ui, (.3..0 equivalely mi d m d m d m fi, fi, d4 fi, d5 u i, ui, d ui c m0 d 3, (.3.. were a a a d d 5 d 3 d 4 b d5 d (.3.. Te sysem (.3.. cosiss of equaio i e ukows { cmi} m 0. To ge a soluio of is sysem we eed wo addiioal codiios. Tese codiios are obaied from e boudary codiios (.. a, i cmm ( a g( i, i 0, (.3..3 m0 b, i cmm ( b g( i, i 0, (.3..4 m0 Te sysem (.3.., equaios (.3..3 ad (.3..4 cosis of equaios i ukows; is sysem is of e form AC F. (.3..5 Upo solvig e sysem (.3..5, e fucio u i ( is approximaed by e sum: u ( x i c m0 mi m ( x. Numerical Experime, 0,,,,. (.3..

6 58 A. F. Solima e al : Compac Fiie Differece Scemes for Solvig. I is secio, we illusrae e procedure of solvig equaios (.. - (..3, wic deermies e soluio of secod-order liear weakly-sigular Volerra iegro-differeial equaios, by e followig examples. Example : f ( is give so a e eoreical soluio of is problem is g (, ad k ( (. u ( x. wi g ( 0 Example : f ( is give so a e eoreical soluio of is problem is g ( g (, ad k ( (. Example 3: f ( is give so a e eoreical soluio of is problem is g ( e, ad k ( (. u ( ( x x. wi u ( e. wi g ( Table. Compariso bewee errors for example ( a differe values of ad α Error = Exac Soluio Approximae Soluio x 0.5, , E E E E E E E E E-0.053E E E E E E E E E E E E E E E E E E E E-00.43E E E E E E E Table. Compariso bewee errors for example ( a differe values of ad α Error = Exac Soluio Approximae Soluio x 0., , E E E E E E E E E E E E E E E E E E E E E E E E E E E E-00

7 A. F. Solima e al : Compac Fiie Differece Scemes for Solvig E E E E E E E E Table 3. Compariso bewee errors for example (3 a differe values of ad α Error = Exac Soluio Approximae Soluio x, , E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Tables (-3 display, e error for differe values of ad α. I is observed a all e resuls of e proposed approximaio for e ew approac are i good agreeme wi e exac oes ad exibi e expeced covergece. 9. Coclusio We ave reduced e soluio of a class of liear weakly-sigular parial iegro-differeial equaios o e soluio of parial differeial equaios by removig e sigulariy usig a appropriae Taylor s approximaio, e e discreizaio i ime by e -poi Euler backward fiie differece meod. Afer a e four-order accurae compac fiie differece sceme for parial differeial problems was developed. Te meod reduces e uderlyig problem o liear sysem of algebraic equaios, wic ca be solved successively o obai a umerical soluio a varied ime-levels We ave cosidered several disic examples o illusrae our ew approac ad ave verified our soluios. Refereces [] Bruer, H., Pedas, A. & Vaiikko, G. (00 Piecewise polyomial collocaio for Volerra iegro-differeial equaios. SIAM J. Numer. Aal., 39, [] Cao, J. R. & Li, Y. P. (988 Noclassical H proecio ad Galerki meods for oliear parabolic iegro-differeial equaios. Calcolo, 5, [3] Cao, J. R. & Li, Y. P. (990 A priori L error esimaes for fiie-eleme meods for oliear diffusio equaios wi memory. SIAM J. Numer. Aal., 7, [4] Ce, C., Tomée, V. & Walbi, L. (99 Fiie eleme approximaio of a parabolic iegro-differeial equaio wi a weakly sigular kerel. Ma. Compu., 58,

8 0 A. F. Solima e al : Compac Fiie Differece Scemes for Solvig. [5] Friedma, A. & Sibro, M. (97 Volerra iegral equaios i Baac space. Tras. Am. Ma. Soc.,, [] Heard, M. L. (98 A absrac parabolic Volerra iegro-differeial equaio. SIAM J. Ma. Aal., 3, [7] J.C. Srikwerda, Fiie Differece Scemes ad Parial Differeial Equaios, SIAM, Piladelpia, d. Ed., 004 [8] LarssoN, S., Tomée, V. & Walbi, L. (998 Numerical soluio of parabolic iegro-differeial equaios by e discoiuous Galerki meod. Ma. Compu., 7, [9] Li, Y., Tomée, V. & Walbi, L. (99 Riz Volerra proecios o fiie eleme spaces ad applicaios o iegro-differeial ad relaed equaios. SIAM J. Numer. Aal., 8, [0] Mclea, W., Tomée, V. & Walbi, L. B. (99 Discreizaio wi variable ime seps of a evoluio equaio wi a posiive-ype memory erm. J. Compu. Appl. Ma., 9, [] Reardy, M., Hrusa, W. J. & Nobel, J. A. (987 Maemaical problems i viscoelasiciy. Pima Moograps ad Surveys i Pure ad Applied Maemaics. Logma Sciece ad Tecical, 35, New York: Wiley. pp [] Sloa, I. H. & Tomée, V. (98 Time discreizaio of a iegro-differeial equaio of parabolic ype. SIAM. J. Numer. Aal., 3, [3] Te Riele, H. J. J. (98 Collocaio meods for weakly sigular secod-kid Volerra iegral equaios wi osmoo soluio. IMA J. Numer. Aal.,, [4] Tomée, V. & Zag, N. Y. (989 Error esimaes for semidiscree fiie eleme meods for parabolic iegro-differeial equaios. Ma. Compu., 53, 39. [5] Yaik, E. G. & Fairweaer, G. (988 Fiie eleme meods for parabolic ad yperbolic parial iegro-differeial equaios. Noliear Aal.,, [] Zag, N. Y. (993 O fully discree Galerki approximaios for parial iegro-differeial equaios of parabolic ype. Ma. Compu., 0, 33.

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

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