Research Article Numerical and Analytical Study for Fourth-Order Integro-Differential Equations Using a Pseudospectral Method

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1 Matheatical Proble in Engineering Volue 213, Article ID 43473, 7 page Reearch Article Nuerical and Analytical Study for Fourth-Order Integro-Differential Equation Uing a Peudopectral Method N. H. Sweila, 1 M. M. Khader, 2 and W. Y. Kota 3 1 Departent of Matheatic, Faculty of Science, Cairo Univerity, Giza 12613, Egypt 2 Departent of Matheatic, Faculty of Science, Benha Univerity, Benha 1311, Egypt 3 Departent of Matheatic, Faculty of Science, Manoura Univerity, Daietta 316, Egypt Correpondence hould be addreed to M. M. Khader; ohaedbd@yahoo.co Received 16 July 212; Accepted 2 Deceber 212 Acadeic Editor: Pedro Ribeiro Copyright 213 N. H. Sweila et al. Thi i an open acce article ditributed under the Creative Coon Attribution Licene, which perit unretricted ue, ditribution, and reproduction in any ediu, provided the original work i properly cited. A nuerical ethod for olving fourth-order integro-differential equation i preented. Thi ethod i baed on replaceent of the unknown function by a truncated erie of well-known hifted Chebyhev epanion of function. An approiate forula of the integer derivative i introduced. The introduced ethod convert the propoed equation by ean of collocation point to yte of algebraic equation with hifted Chebyhev coefficient. Thu, by olving thi yte of equation, the hifted Chebyhev coefficient are obtained. Special attention i given to tudy the convergence analyi and derive an upper bound of the error of the preented approiate forula. Nuerical reult are perfored in order to illutrate the uefulne and how the efficiency and the accuracy of the preent work. 1. Introduction The integro-differential equation (IDE) i an equation that involve both integral and derivative of an unknown function. Matheatical odeling of real-life proble uually reult in functional equation, like ordinary or partial differential equation, and integral and integro-differential equation, tochatic equation. Many atheatical forulation of phyical phenoena contain integro-differential equation; thee equation arie in any field like phyic, atronoy, potential theory, fluid dynaic, biological odel, and cheical kinetic. Integro-differential equation; are uually difficult to olve analytically; o, it i required to obtain an efficient approiate olution [1 ]. Recently, everal nuerical ethod to olve IDE have been given uch a variational iteration ethod [6, 7], hootopy perturbation ethod [8, 9], pline function epanion [1, 11], and collocation ethod [12 1]. Chebyhev polynoial are well-known faily of orthogonal polynoial on the interval [ 1, 1] that have any application [4, 6, 8, 13]. They are widely ued becaue of their good propertie in the approiation of function. However, with our bet knowledge, very little work wa done to adapt thee polynoial to the olution of integro-differential equation. Orthogonal polynoial have a great variety and wealth of propertie. Soe of thee propertie take a very concie for in the cae of the Chebyhev polynoial, aking Chebyhev polynoial of leading iportance aong orthogonal polynoial. The Chebyhev polynoial belong to an ecluive band of orthogonal polynoial, known a Jacobi polynoial, which correpond to weight function of the for (1 ) α (1 + ) β and which are olution of Stur- Liouville equation [16]. In thi work, we derive an approiate forula of the integral derivative y (n) () and derive an upper bound of the errorofthiforula,andthenweuethiforulatoolve a cla of two-point boundary value proble (BVP) for the fourth-order integro-differential equation a y (iv) () =f()+γy()+ [p (t) y (t)+q(t) Θ(y(t))] dt,, t 1, (1)

2 2 Matheatical Proble in Engineering under the boundary and initial condition y () =α, y () =α 1, y (1) =β, y (1) =β 1, where f(), p(), and q() are known function and γ, α, α 1,β,andβ 1 areuitablecontant.severalnuericalethod to olve the fourth-order integro-differential equation have been given uch a Chebyhev cardinal function [17], variational iteration ethod [7], and other. 2. Soe Baic Propertie and Derivation of an Approiate Forula of the Derivative for Chebyhev Polynoial Epanion The Chebyhev polynoial of the firt kind i a polynoial in z of degree n, defined by the relation (2) T n (z) = co nθ, when z=co θ. (3) The Chebyhev polynoial of degree n>of the firt kind have preciely n zero and n+1 local etrea in the interval [ 1, 1].ThezeroofT n (z) are denoted by (k 1/2) π z k = co,,2,...,n. (4) n The Chebyhev polynoial can be deterined with the aid of the following recurrence forula [18]: T n+1 (z) =2zT n (z) T n 1 (z), () T (z) =1, T 1 (z) =z, n=1,2,... The analytic for of the Chebyhev polynoial T n (z) of degree n i given by [n/2] T n (z) =n i= ( 1) i 2 n 2i 1 (n i 1)! (i)! (n 2i)! zn 2i, (6) where [n/2] denote the integer part of n/2.theorthogonality condition i 1 T i (z) T j (z) d = 1 1 z 2 π, for i=j=; π 2, for i =j., for i=j =; In order to ue thee polynoial on the interval [, 1], wedefinetheocalledhiftedchebyhevpolynoialby introducing the change of variable z = 2 1.Thehifted Chebyhev polynoial are denoted by T n () and defined a T n () = T n(2 1) = T 2n ( ). The function y(), which belong to the pace of quare integrable in [, 1], ay be epreed in ter of hifted Chebyhev polynoial a y () = i= (7) c i T i (), (8) where the coefficient c i are given by c = 1 π 1 y () T () 2 d, c i = 2 π 1 y () T i () 2 d, i = 1, 2,.... (9) In practice, only the firt ( + 1) ter of hifted Chebyhev polynoial are conidered. Then, we have that y () = i= c i T i (). (1) Lea 1. The analytic for of the hifted Chebyhev polynoial T n () of degree n i given by T n n () =n ( 1) n k 2 2k (n+k 1)! (2k)! (n k)! k, k= n=1,2,... (11) Proof. Since we have T n () = T 2n( ), thenbyubtituting in (6), we can obtain that T n n () =2n ( 1) i 2 2n 2i 1 (2n i 1)! n i, (i)! (2n 2i)! i= n=1,2,... (12) Now, we put k = n iin (12) we obtain the deired reult (11). The ain approiate forula of the derivative of y (), andigiveninthefollowingtheore. Theore 2. Let y() be approiated by hifted Chebyhev polynoial a (1),and alouppoethatr iinteger; then, where λ i,k,r i given by i D r (y ()) = c i λ i,k,r k r, (13) i=rk=r λ i,k,r = ( 1) i k 2 2k i (i+k 1)!k! (i k)! (2k)! (k r)!. (14) Proof. Since the differential operator D r i linear, we can obtain that D r (y ()) = i= Since D r c=, ci a contant, and c i D r (T i ()). (1) D r n =, for n N, n<r, n! (n r)! n r, for n N, n r. Then, we have that (16) D r T i () =, i=,1,...,r 1, (17)

3 Matheatical Proble in Engineering 3 and for i=r, r+1,...,,andbyuing(16), we get that D r T i i () =i ( 1) i k 2 2k (i+k 1)! (i k)! (2k)! Dr k =i k=r i k=r ( 1) i k 2 2k (i+k 1)!k! (i k)! (2k)! (k r)! k r. (18) A cobination of (17), (18), and (14) lead to the deired reult and coplete the proof of the theore. 3. Error Analyi In thi ection, pecial attention i given to tudy the convergence analyi and evaluate the upper bound of the error of the propoed forula. Theore 3 (Chebyhev truncation theore; ee [18]). The error in approiating y() by the u of it firt ter i bounded by the u of the abolute value of all the neglected coefficient. If then y () = k= E T () y () y () for all y(),all,andall [ 1, 1]. c k T k (), (19) c k, (2) k=+1 Theore 4. The derivative of order r for the hifted Chebyhev polynoial can be epreed in ter of the hifted Chebyhev polynoial theelve in the following for: where Θ i,j,k = D r (T i k r i ()) = Θ i,j,k T j (), (21) k=rj= ( 1) i k 2i (i+k 1)!Γ (k r+1/2) h j Γ (k+1/2)(i k)!(k r j)!(k+j r)!, h =2, h j =1, j=,1,... (22) Proof. We ue the propertie of the hifted Chebyhev polynoial [18] and epand k r in (18)inthefollowingfor: k r = k r j= c kj T j (), (23) where c kj can be obtained uing (9), andy() = k r ;then, c kj = 2 h j π 1 k r T j () 2 d, h =2, h j =1, j=1,2,... (24) At j =,wefind that c k = (1/π) 1 (k r T ()/ 2 )d = (1/ π)(γ(k r + 1/2)/(k r)!); alo,atany j anduingtheforula(1), we can find that c kj = j j ( 1) j l (j+l 1)!2 2l+1 Γ (k+l r+1/2), π (j l)! (2l)! (k+l r)! l= eploying (18)and(23)give j=1,2,3,..., (2) D r (T i k r i ()) = Θ i,j,k T j (), i=r, r+1,..., (26) where Θ i,j,k = k=rj= i ( 1)i k (i+k 1)!2 2k k!γ (k r+1/2) (i k)! (2k)! π (Γ (k+1 r)) 2, j = ; ( 1) i k ij (i+k 1)!2 2k+1 k! π (k r)! (i k)! (2k)! j l= ( 1) j l (j + l 1)!2 2l Γ (k+l r+1/2), j=1,2,3,... (j l)! (2l)! (k+l r)! (27) After oe lengthy anipulation, Θ i,j,k can be put in the following for: Θ i,j,k = ( 1) i k 2i (i+k 1)!Γ (k r+1/2) h j Γ (k+1/2)(i k)!(k r j)!(k+j r)!, and thi coplete the proof of the theore. j=,1,..., (28) Theore. The error E T () = D r y() D r y () in approiating D r y() by D r y () i bounded by E T () i=+1 i k r c i ( Θ i,j,k ). (29) k=rj= Proof. A cobination of (8), (1), and (21)leadto E T () = Dr y () D r y () = i=+1 i k r but T j () 1;o,wecanobtainthat E T () c i ( i,j,k T k=rj=θ j ()), i=+1 i k r (3) c i ( Θ i,j,k ), (31) k=rj= and ubtracting the truncated erie fro the infinite erie, bounding each ter in the difference, and uing the bound coplete the proof of the theore.

4 4 Matheatical Proble in Engineering 4. Procedure Solution for the Fourth-Order Integro-Differential Equation In thi ection, we will preent the propoed ethod to olve nuerically the fourth-order integro-differential equation of the for in (1). The unknown function y() ay be epanded by finite erie of hifted Chebyhev polynoial a in the following approiation: y () = n= c n T n (), (32) and approiated forula of it derivative can be defined in Theore 2.Fro(1), (32), and Theore 2,wehavethat i i=r k=r c i λ i,k,r k r =f() +γc n T n () + n= [p (t) ( c n T n (t)) n= +q(t) Θ( c n T n (t))] dt. n= (33) We now collocate (33) at( 1 + r) point, =,1,..., ra i c i λ i,k,4 k 4 =f( )+γc n T n ( ) + n= [p (t) ( c n T n (t)) n= +q(t) Θ( c n T n (t))] dt. n= (34) For uitable collocation point, we ue root of hifted Chebyhev polynoial T +1 r (). The integral ter in (34)canbe found uing copoite trapezoidal integration technique a [p (t) ( c n T n (t))+q(t) Θ( c n T n (t))] dt n= n= h L 1 2 (Ω (t )+Ω(t L )+2Ω(t k )), (3) where Ω(t) = p(t) n= c nt n (t) + q(t)θ( n= c nt n (t)), h = /L, for an arbitrary integer L, t j+1 =t j +h, =,1,..., r, and j=,1,...,l.so,byuing(34) and(3), we obtain i c i λ i,k,r k r =f( )+γc n T n ( ) n= + h L 1 2 (Ω (t )+Ω(t L )+2Ω(t k )). (36) Alo, by ubtituting (32)intheboundarycondition(2), we can obtain r equation a follow: i=2 i= ( 1) i c i =α, c i T i () =α1, i=2 i= c i =β, c i T i (1) =β1. (37) Equation (36), together with r equation of the boundary condition (37), give ( + 1) of yte of algebraic equationwhichcanbeolved,fortheunknownc n, n =,1,...,, uing conjugate gradient ethod or Newton iteration ethod.. Nuerical Reult In thi ection, to verify the validity and the accuracy and upport our theoretical dicuion of the propoed ethod, we give oe coputation reult of nuerical eaple. Eaple 6. Conider the nonlinear fourth-order integrodifferential equation a in (1) and(2) withf() = 1, γ =, p(t) =, q(t) = e t, and Θ(y) = y 2 (); then,the integro-differential equation will be y (iv) () =1+ e t y 2 (t) dt, 1, (38) ubject to the boundary condition y () =y () =1, y(1) =y (1) =e. (39) The eact olution of thi proble i y() = e [7]. We apply the uggeted ethod with = and approiate the olution y() a follow: y () = n= Fro (38), (4), and Theore 2,wehavethat i=4 i k=4 c i λ i,k,4 k 4 =1+ c n T n (). (4) e t ( c n T n 2dt. (t)) (41) n= We now collocate (41)atpoint,, =,1a i c i λ i,k,4 k 4 =1+ e t ( c n T n 2dt. (t)) (42) n=

5 Matheatical Proble in Engineering For uitable collocation point we ue root of hifted Chebyhev polynoial T 2 (). The integral ter in (42) canbe found uing copoite trapezoidal integration technique a e t ( c n T n 2dt (t)) n= = h L 1 2 (Ω (t )+Ω(t L )+2Ω(t k )), (43) where Ω(t) = e t ( n= c nt n (t))2, h = /L, for an arbitrary integer L, t j+1 =t j +h, =,1, and j=,1,...,l.so, by uing (43)and(42), we obtain i c i λ i,k,4 k 4 =1+ h L 1 2 (Ω (t )+Ω(t L )+2Ω(t k )). (44) Alo, by ubtituting (4)intheboundarycondition(39), we can obtain four equation a follow: c c 1 +c 2 c 3 +c 4 c =1, c +c 1 +c 2 +c 3 +c 4 +c =e, l c +l 1 c 1 +l 2 c 2 +l 3 c 3 +l 4 c 4 +l c =1, c + 1 c c c c 4 + c =e, (4) where l i =T i () and i =T i (1). Equation (44), together with four equation of the boundary condition (4), repreent, a nonlinear yte of i algebraic equation in the coefficient c n ;byolvingituing the Newton iteration ethod, we obtain c = , c 1 =.839, c 2 =.1478, c 3 =.872, c 4 =.7, c =.3. (46) The behavior of the approiate olution uing the propoed ethod with =, the approiate olution uing variational iteration ethod (VIM), and the eact olution are preented in Figure1. Table 1 how the behavior of the abolute error between eact olution and approiate olution uing the preented ethod at =6and =8. Fro Figure 1 and Table 1,iticlearthatthepropoedethod can be conidered a an efficient ethod to olve the nonlinear integro-differential equation. Table 1 indicate that a increae the error decreae ore rapidly; hence, for better reult, uing nuber i recoended. Alo, we can conclude that the obtained approiated olution i in ecellent agreeent with the eact olution. Eaple 7. Conider the linear fourth-order integrodifferential equation a in (1)and(2)withf() = +(+3)e, y() Eact olution y() Chebyhev olution VIM olution Figure 1: The behavior of the eact olution, the approiate olution uing VIM, and the approiate olution uing the propoed ethod at =. γ = 1, p(t) = 1, h(t) =, and Θ(y) = y(); then,the integro-differential equation will be y (iv) () =+(+3) e +y() y (t) dt, 1, ubject to the boundary condition y () =1, y () =2, (1) =1+e, y (1) = 3e. (47) (48) The eact olution of thi proble i y() = 1 + e [17]. We apply the uggeted ethod with = and approiate the olution y() a follow: y () n= c n T n (). (49) By the ae procedure in the previou eaple, we have i c i λ i,k,4 k 4 =f( )+ c n T n ( )+ h 2 n= L 1 (Ω(t )+Ω(t L )+2 Ω(t k )), =, 1, 2, ()

6 6 Matheatical Proble in Engineering Table 1: The behavior of the abolute error between the eact olution and approiate olution at =6and =8. y e. y ap. at =6 y e. y ap. at = e e e e e e e 4.248e e e e e 1 Table 2: The behaviour of the abolute error between the eact olution and approiate olution at =7and =9. y() y e. y ap. at =7 y e. y ap. at = e e e e e e e 3 2.4e.8.24e e e e 9 where Ω(t) = n= c nt n (t), and the node t j+1 =t j +h, j =,1,...,L, t =,and h = /L. Wecanwritetheinitialboundary condition in the for c c 1 +c 2 c 3 +c 4 c +c 6 =1, c +c 1 +c 2 +c 3 +c 4 +c +c 6 =1+e, l c +l 1 c 1 +l 2 c 2 +l 3 c 3 +l 4 c 4 +l c +l 6 c 6 =2, c + 1 c c c c 4 + c + 6 c 6 =3e. (1) By uing () and(1), we obtain a linear yte of even algebraic equation in the coefficient c n ;byolvingituing the conjugate gradient ethod, we obtain c = , c 1 = , c 2 =.26461, (2) c 3 =.379, c 4 =.264, c =.1. The behavior of the approiate olution uing the propoed ethod with = 6, the approiate olution uing variational iteration ethod (VIM) and the eact olution are preented in Figure 2. Table 2 how the behaviour of the abolute error between eact olution and approiate olution uing the preented ethod at = 7 and = 9. Fro thi figure, it i clear that the propoed ethod can be conidered a an efficient ethod to olve the linear integro-differential equation. Alo, we can conclude that the obtained approiate olution i in ecellent agreeent with the eact olution. 6. Concluion and Dicuion Integro-differential equation are uually difficult to olve analytically; o, it i required to obtain the approiate olution. In thi paper, we propoed the peudopectral ethod Eact olution y() Chebyhev olution VIM olution Figure 2: The behavior of the eact olution, the approiate olution uing VIM, and the approiate olution uing the propoed ethod at =6. uing hifted Chebyhev ethod for olving the integrodifferential equation. The Chebyhev ethod i ueful for acquiring both the general olution and particular olution a deontrated in eaple. Special attention i given to tudy the convergence analyi and derive an upper bound of the error of the derived approiate forula. Fro our obtainedreult,wecanconcludethatthepropoedethod give olution in ecellent agreeent with the eact olution and better than the other ethod. An intereting feature of thi ethod i that when an integral yte ha linearly independent polynoial olution of degree or le than, the ethod can be ued for finding the analytical olution. All coputationaredoneuingmatlab8. Reference [1] R. P. Agarwal, Boundary value proble for higher order integro-differential equation, Nonlinear Analyi: Theory, Method & Application,vol.7,no.3,pp.29 27,1983. [2] E. Babolian, F. Fattahzadeh, and E. G. Raboky, A Chebyhev approiation for olving nonlinear integral equation of Haertein type, Applied Matheatic and Coputation, vol. 189, no. 1, pp , 27. [3] A.H.Borzabadi,A.V.Kayad,andH.H.Mehne, Adifferent approach for olving the nonlinear Fredhol integral equation of the econd kind, Applied Matheatic and Coputation, vol. 173, no. 2, pp , 26. [4] M. M. Khader, On the nuerical olution for the fractional diffuion equation, Counication in Nonlinear Science and Nuerical Siulation,vol.16,no.6,pp ,211. []N.H.Sweila,M.M.Khader,andA.M.Nagy, Nuerical olution of two-ided pace-fractional wave equation uing finite difference ethod, Coputational and Applied Matheatic,vol.23,no.8,pp ,211.

7 Matheatical Proble in Engineering 7 [6] M. M. Khader, Introducing an efficient odification of the variational iteration ethod by uing Chebyhev polynoial, Application and Applied Matheatic,vol.7,no.1,pp , 212. [7] N. H. Sweila, Fourth order integro-differential equation uing variational iteration ethod, Coputer & Matheatic with Application,vol.4,no.7-8,pp ,27. [8] M. M. Khader, Introducing an efficient odification of the hootopy perturbation ethod by uing Chebyhev polynoial, Arab Matheatical Science,vol.18,no.1,pp , 212. [9]N.H.Sweila,M.M.Khader,andR.F.Al-Bar, Hootopy perturbation ethod for linear and nonlinear yte of fractional integro-differential equation, International Coputational Matheatic and Nuerical Siulation, vol.1, no. 1, pp , 28. [1] M. M. Khader and S. T. Mohaed, Nuerical treatent for firt order neutral delay differential equation uing pline function, Engineering Matheatic Letter,vol.1,no.1,pp.32 43, 212. [11] S. T. Mohaed and M. M. Khader, Nuerical olution to the econd order Fredhol integro-differential equation uing the pline function epanion, Global Pure and Applied Matheatic,vol.34,pp.21 29,211. [12] M. M. Khader and A. S. Hendy, The approiate and eact olution of the fractional-order delay differential equation uing Legendre peudo-pectral ethod, International Journal of Pure and Applied Matheatic, vol.74,no.3,pp , 212. [13] M. M. Khader, N. H. Sweila, and A. M. S. Mahdy, An efficient nuerical ethod for olving the fractional difuion equation, Applied Matheatic and Bioinforatic,vol.1,no.2, pp. 1 12, 211. [14] N. H. Sweila, M. M. Khader, and W. Y. Kota, On the nuerical olution of Haertein integral equation uing Legendre approiation, International Applied Matheatical Reearch,vol.1,no.1,pp.6 76,212. [1] S. Youefi and M. Razzaghi, Legendre wavelet ethod for the nonlinear Volterra-Fredhol integral equation, Matheatic and Coputer in Siulation, vol. 7, no. 1, pp. 1 8, 2. [16] J. C. Maon and D. C. Handcob, Chebyhev Polynoial, Chapan & Hall/CRC, Wahington, DC, USA, 23. [17] M. Laketani and M. Dehghan, Nuerical olution of fourthorder integro-differential equation uing Chebyhev cardinal function, International Coputer Matheatic,vol. 87,no.6,pp ,21. [18] M. A. Snyder, Chebyhev Method in Nuerical Approiation, Prentice-Hall Inc., Englewood Cliff, NJ, USA, 1966.

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