Development of the Sinc Method for Nonlinear Integro-Differential Eequations

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1 Austrin Journ of Bsic nd Appied Sciences, 4(): , ISS Deveopment of the Sinc Method for oniner Integro-Differenti Eequtions K. Jei, M. Zrebni, 3 M. Mirzee Chi,3 Ismic Azd University Brnch of Astr, , Astr, Irn Deprtment of Mthemtics, University of Mohghegh Ardbii, , Ardbi, Irn Abstrct: We deveop numeric procedure for soving css of noniner integro-differenti equtions of Fredhom type, using the goby defined Sinc bsis functions. Properties of the Sinc procedure re utiized to reduce the computtion of the integro-differenti equtions to system of noniner equtions with unnown coefficients. We used three numeric exmpes to iustrte the ccurcy nd impementtion of our method. Key words: Fredhom; Integro-differenti; oniner; Sinc method. ITRODUCTIO We consider the foowing noniner Fredhom integro-differenti eqution of the form: u( x) f( x) K( x, t) g( t, u( t)) dt, x[, b], where the functions f(x) nd the erne K(x,t) re nown nd u(x) is the soution to be determined (Atinson, 997; Deves nd Mohmed, 985), nd so g(t,u(t)) is noniner in u(t). Integro-differenti equtions hve strong physic bcground nd so, hve mny prctic ppictions in scientific fieds such s popution nd poymerrheoogy (Linz, 985; Abdu Jerri, 999). oniner phenomen, tht pper in mny ppictions in scientific fieds cn be modeed by noniner integrodifferenti equtions. In recent yers, numerous methods hve been ppied for soving integr nd integrodifferenti equtions such s vrition itertion method (Wng nd He, 7), rtionized Hr functions (Ordohni nd Rzzghi, 8), CAS wveet method (Dnfu nd Xufeng, 7), Adomin method ( E-K, 8), Homotopy perturbtion method (Yusufogu, 9) nd Tyor poynomi pproch (Kurt nd Sezer, 8). Sinc methods hve incresingy been recognized s powerfu toos for ttcing probems in ppied physics nd engineering (Stenger, 993). The Sinc-Cooction overpping method is deveoped for two-point boundry-vue probems for second-order ordinry differenti equtions in ( Moret, Lybec nd Bowers, 999). In (Rshidini nd Zrebni, 5), we used Sinc-cooction procedure for numeric soution of iner Fredhom integr equtions of the second ind. In this pper gob pproximtion for the soution of the eqution () using the Sinc functions is deveoped. Our method consists of reducing the soution of () to set of gebric equtions. The properties of Sinc function re then utiized to evute the unnown coefficients. The outine of the pper is s foows. In section we review some of the min properties of Sinc function tht re necessry for the formution of the discrete system. In section 3, we iustrte how the Sinc method my be used to repce Eq. () by n expicit system of noniner gebric equtions. In section 4, we report our numeric resuts nd demonstrte the efficiency nd ccurcy of the proposed numeric scheme by considering some numeric exmpes. Sinc Function Properties nd Interpotion: Sinc function properties re discussed thoroughy in (Stenger, 993). In this section n overview of the bsic formution of the Sinc function required for our subsequent deveopment is presented. The Sinc function is defined on the whoe re ine by () Corresponding Author: M. Zrebni, Deprtment of Mthemtics, University of Mohghegh Ardbii, , Ardbi, Irn E-mi: Zrebni@um.c.ir 558

2 Aust. J. Bsic & App. Sci., 4(): , sin( x), Sinc( x) x, x x ; For ny h>, the trnsted Sinc functions with eveny spced nodes re given s foows: x h S(, h)( x) Sinc( ),,,,... h which re ced the th Sinc functions. The Sinc function form for the interpoting point by (), ; S(, h)( h), ; x h () (3) is given (4) Let ( ) sin( t) dt, t (5) ( ) ( ) ( ) then define mtrix I [ ] whose (, ) th entry is given by. If u is defined on the re ine, then for h> the series x h Cuh (, )( x) u( hsinc ) ( ), (6) h is ced the Whitter crdin expnsion of u, whenever this series converges. For further expntion of the procedure, we consider the foowing definitions nd theorems. Definition : Let D be simpy-connected domin in the compex pne ( z xiy) hving boundry D. Let nd b denote two distinct points of D nd denote conform mp of D onto D d where D d denote the region w C: Iw d, d such tht ( ) nd ( b). Let denote the inverse mp, nd et be defined by z C: z ( u), u R. Given, nd positive number h, et us set z z ( h) ( h),,,,. Definition : Let L ( D) be the set of nytic functions, for which there exists constnt, C, such tht ( z) F( z) C zd,, ( z) ( z) where ( z) e. (7) 559

3 Aust. J. Bsic & App. Sci., 4(): , Theorem : Let u L ( D), et be positive integer nd et h be seected by the formu d h ( ), then there exist positive constnt c, independent of, such tht uz ( ) ( d) uzdz ( ) h ce. ( z ) (8) (9) Theorem : u Let L ( D), with >, nd d>, et be defined s in (5), nd et. Then there exists ( ) d h ( ) constnt, c which is independent of, such tht z uz ( ) utdt () h ce. ( ) ( d) ( z ) Soution of Fredhom-Hmmerstein Integro-Differenti Eqution: We consider the noniner Fredhom-Hmmerstein integro-differenti eqution of the form u( x) f( x) K( x, t) g( t, u( t)) dt, x, b, with initi condition u ( ) u. () () In Eq. () f nd the erne K re continuous functions, nd so g(t,u(t)) is noniner in u. For convenience, consider Q(t) = g(t, u(t)). ux ( ) L( D). Let u(x) be the exct soution of the integr eqution () nd By considering Eq. () nd integrting () from to x, we get: x ux ( ) { K(, tqtdt ) ( ) f( )} d u ( ), x b,. We consider K ( ) K(, t) Q( t) dt. K L D f ow, we et ( ) nd L ( D). By setting nd ppying Theorem x x,,..., for the right-hnd side of (), we hve: () (3) x K ( x ) K ( ) d, ( ) ( ), x (4) 55

4 x Aust. J. Bsic & App. Sci., 4(): , ( ) f ( x ), ( x ) f( ) d. Thus we obtin: (5) ux ( ) h K ( x ) ( ) ( ), x h ( ), x f( x ) u. ( ) (6) For the first term on the right-hnd side of bove retion, by considering retion (3) nd using Theorem, we get: K ( x ) K( x, t ) h h Q( t ), ( ) ( ),, ( x) ( x) ( t) (7) where Qt ( ) gt (, ut ( )). Hving repced the first term on the right-hnd side of (6) with the eqution (7), we obtin: K f u h Q _ h u, ( ), ( ),, (8) where u u( x ), K K( x, t ), Q Q( t ), f f( x ), nd ( ) n( x x ), ( ), ( b), b x b be ( x)( bx) e h ( x), x ( h) ( h) h. There re (+) unnowns u,,,,, to be determined in (8). In order to determine these (+) unnowns, we rewrite these system which is the noniner system of equtions in mtrix form. Corresponding to given function u defined on we use the nottion ( ) ( ) ( ) D( u) dig( u( x,, u( x )). We set I [ ], where denotes the (, ) th eement of the mtrix ( I ). ow, we cn simpify the system (8) in the mtrix form U AQ, (9) where ( ) A h I D KD ˆ ( ( ))( ( )), Kˆ [ K( x, t )], i,,,, i T Q Q, Q,, Q, Q, f f [ h ( ) u,, h ( ) u ], ( ) ( ) T,, T U [ u,, u ]. 55

5 Aust. J. Bsic & App. Sci., 4(): , The bove noniner system consists of (+) eqution with (+) unnown { u }. Soving this noniner system by ewton`s method, we obtin unnown coefficients u,,,,. Hving used the pproximte soution u,,,,, we empoy method simir to the ystrom s ide for the Ferdhom- Hmmerstein integro-differenti eqution, i.e., we use K( ) f( ) u x h x Q h x u ( ) h, ( ) h, ( ), ( ) ( t) ( ) Where x h, ( x) Sho (, ) ( tdt ). Ech ewton itertion step invoves evution of the vector F () the Jcobin mtrix J () nd U (). Whenever the distnce between two itertion is ess thn given toernce,, then the gorithm is to stop. ( ) ( ) U U. Agorithm: initiize: U U for,,, () () () () F U AQ if () F is sm enough, stop compute J () sove () () () J U F( U ) ( ) ( ) ( ) U U U end 4 umeric exmpes: We consider the foowing exmpes to compre our computed resuts nd ustify the ccurcy nd efficiency of our method. The exmpes hve been soved by presented method with different vues of, nd d which yied h ( ). Let ux ( ) denote the exct soution of the given exmpes, nd et u ( x ) be the computed soutions by our method. Let [ b, ] nd be conform mp onto D. By expoiting of the definition, we hve z D{ zc : rg( d }, b z z ( z) n( ). b z The error is reported on set of the Sinc grid points () S { x,, x,, x }, 55

6 Aust. J. Bsic & App. Sci., 4(): , h be x,,,. h e The mximum bsoute error on the grid points Sinc is defined s E ( h) mx u( x ) u ( x ). s The mximum bsoute errors in numeric resuts re tbuted in tbes 3. () () Exmpe : Consider the foowing noniner Fredhom integro-differenti eqution with exct soution ux ( ). x ( ) ( ),, 3 u x x xu t dt x u(), We ppied the Sinc function pproch nd soved exmpe. Mximum bsoute errors in numeric soution of exmpe re tbuted in Tbe. These resuts show the efficiency nd ppicbiity of our method. The exct nd pproximte soutions of exmpe re shown in Fig. for = nd =4. For rge vues of the pproximte soution is indistinguishbe(for the given sce) from the exct soution. Tbe : Resuts for Exmpe. h E ( h) s Fig. : Exct nd pproximte soutions for Exmpe, (=,4). Exmpe : We consider the integro-differenti eqution x x 5 tx 3 u( x) e e ( e ) e u ( t) dt, 5 u(), x with exct soution ux ( ) e. 553

7 Aust. J. Bsic & App. Sci., 4(): , We soved the exmpe by our method for different vues of. The mximum bsoute errors on the Sinc grids S re tbuted in Tbe. The exct nd pproximte soutions of exmpe re shown in Fig. for = nd =4. Tbe : Resuts for Exmpe. h E ( h) Exmpe 3: Consider the noniner Fredhom integro-differenti eqution x t ut () ( ) ( ( ) ),, u x e x x e dt x u(), with exct soution ux ( ) x. s Fig. : Exct nd pproximte soutions for Exmpe, (=,4). The mximum bsoute errors in computed soutions re tbuted in Tbe 3. These resuts show the ccurcy nd efficiency of our Sinc method. The exct nd pproximte soutions for exmpe 3 re shown in Fig.3, incuding the pproximtions for = nd =4, which re indistinguishbe (on this sce) from the exct soution. Tbe 3: Resuts for Exmpe 3. h E ( h) s 554

8 Aust. J. Bsic & App. Sci., 4(): , Fig. 3: Exct nd pproximte soutions for Exmpe 3, (=,4). Concusion: The Sinc method is used to sove the first-order Fredhom type integro-differenti eqution with initi condition. The numeric resuts show tht the ccurcy improves with incresing the. The Tbes 3 nd Figures 3 indicte tht s increses the errors re decrese more rpidy, then for better resuts, using the rger is recommended. REFERECES Atinson, E., 997. The umeric Soution of Integr Equtions of the Second Kind, Cmbridge University Press, Cmbridge. Abdu Jerri, J., 999. Introduction to Integr Equtions with Appictions, John Wiey nd Sons, ew Yor. Dnfu, H., S. Xufeng, 7. umeric Soution of Integro-Differenti Equtions by Using CAS Wveet Opertion Mtrix of Integrtion, Appied Mthemtics nd Computtion, 94: Deves, L. M., J. L. Mohmed, 985. Computtion Methods for Integr Equtions, Cmbridge University Press, Cmbridge. E-K, I.L., 8. Convergence of the Adomin Method Appied to Css of oniner Integr Equtions, Appied Mthemtics Letters, : Kurt,., M. Sezer, 8. Poynomi Soution of High-Order Liner Fredhom Integro-Differenti Equtions with Constnt Coefficients, Journ of the Frnin Institute, 345: Linz, P., 985. Anytic nd umeric Methods for Voterr Equtions, SIAM, Phidephi, PA. Ordohni, Y., M. Rzzghi, 8. Soution of oniner Voterr-Fredhom-Hmmerstein Integr equtions vi Cooction Method nd Rtionized Hr Functions, Appied Mthemtics Letters, : 4-9. Moret, A.C.,.J. Lybec, K.L. Bowers, 999. Convergence of the Sinc overpping domin decomposition method, App. Mth. Comput., 98: 9-7. Rshidini, J., M. Zrebni, 5. umeric soution of iner integr equtions by using Sinc-cooction method, App. Mth. Comput., 68: Stenger, F., 993. umeric Methods Bsed on Sinc nd Anytic Functions, Springer-Verg, ew Yor. Wng, Shu-Qing, Ji-Hun He, 7. Vrition Itertion Method for Soving Integro-Differenti Equtions, Physics Letters A, 367: Yusufogu, E., 9. Improved homotopy perturbtion method for soving Fredhom type integrodifferenti equtions, Chos, Soitons nd Frcts, 4: