Jacobi Operational Matrix Approach for Solving Systems of Linear and Nonlinear Integro Differential Equations

Transcription

1 Internatonal Journal of Mathematcal Modellng & Computatons Vol. 7, No. 1, Sprng 217, 1-25 Jacob Operatonal Matrx Approach for Solvng Systems of Lnear and Nonlnear Integro Dfferental Equatons K. Sadr a, and Z. Ayat b a Young Researchers and Elte Club, Rasht Branch, Islamc Azad Unversty, Rasht, Iran, b Department of Engneerg Scences, Faculty of Technology and Engneerng East of Gulan, P.C , Rudsar Vaargah, Iran. Abstract. Ths paper ams to construct a general formulaton for the shfted Jacob operatonal matrces of ntegraton and product. The man am s to generalze the Jacob ntegral and product operatonal matrces to the solvng system of Fredholm and Volterra ntegro dfferental equatons whch appear n varous felds of scence such as physcs and engneerng. The Operatonal matrces together wth the collocaton method are appled to reduce the soluton of these problems to the soluton of a system of algebrac equatons. Indeed, to solve the system of ntegro dfferental equatons, a fast algorthm s used for smplfyng the problem under study. The method s appled to solve system of lnear and nonlnear Fredholm and Volterra ntegro dfferental equatons. Illustratve examples are ncluded to demonstrate the valdty and effcency of the presented method. It s further found that the absolute errors are almost constant n the studed nterval. Also, several theorems related to the convergence of the proposed method, wll be presented. Receved: 7 October 216, Revsed: 31 December 216, Accepted: 25 January 217. Keywords: Collocaton method, Shfted Jacob polynomals, System of Fredholm and Volterra ntegro dfferental equatons, Operatonal matrces of ntegraton and product, Convergence. Index to nformaton contaned n ths paper 1 Introducton 2 Jacob Polynomals and Ther Propertes 3 Convergence Analyss 4 The Jacob Operatonal Matrces 5 Applcatons of Operatonal Matrces of Integraton and Product 6 Illustratve Examples 7 Concluson 1. Introducton Fndng the analytcal solutons of functonal equatons has been devoted the attenton of mathematcans nterest n recent years. Several methods are proposed to Correspondng author. Emal: kh.sadr@uma.ac.r. c 217 IAUCTB

2 2 K. Sadr & Z. Ayat/ IJM 2 C, acheve ths purpose, such as [3, 5 12, 16 19, 26, 29, 3, 33, 34]. Systems of ntegro dfferental equatons arse n the mathematcal modelng of many phenomena. Varous technques have been used for solvng these systems such as, Adoman decomposton method [13], Homotopy perturbaton method [2, 14], Varatonal teraton method [15], the Tau method [1, 32], dfferental transform method [4], and others. Between of present methods, spectral methods have been used to solve dfferent functonal equatons, because of ther hgh accuracy and easy applyng. Specfc types of spectral methods that more applcable and wdely used, are the Galerkn, collocaton, and Tau methods [2 25, 27, 31]. The mportance of Sturm Louvlle problems for spectral methods les n the fact that the spectral approxmaton of the soluton of a functonal equaton s usually regarded as a fnte expanson of egenfunctons of a sutable Sturm Louvlle problem. In recent years, many dfferent orthogonal functons and polynomals have been used to approxmate the soluton of varous functonal equatons. The man goal of usng orthogonal bass s that the equaton under study reduces to a system of lnear or nonlnear algebrac equatons. Ths can be done by truncatng seres of functons wth orthogonal bass for the soluton of equatons and usng the operatonal matrces. In ths paper, Jacob polynomals, on nterval [, 1], have been consdered for solvng systems of ntegro dfferental equatons. The Jacob polynomals P α,β x, α, β > 1 play mportant roles n mathematcal analyss and ts applcatons [2]. It s proven that Jacob polynomals are precsely the only polynomals arsng as egenfunctons of a sngular Sturm Louvlle problem [22, 31]. Ths class of polynomals comprses all the polynomal soluton to sngular Sturm Louvlle problems on [ 1, 1]. Chebyshev, Legendre, and ultrasphercal polynomals are partcular cases of the Jacob polynomals. In ths paper, the shfted Jacob operatonal matrces of ntegraton and product s ntroduced, whch s based on Jacob collocaton method for solvng numercally the systems of the lnear and nonlnear Fredholm and Volterra ntegrodfferental equatons on the nterval [, 1], to fnd the approxmate solutons u N x, 1,..., n. The resultant systems are collocated at nn + 1 nodes of the shfted Jacob Gauss nterpolaton on the nterval, 1. These equatons generate nn + 1 lnear or nonlnear algebrac equatons. The nonlnear systems can be solved usng Newton teratve method. The remander of ths paper s organzed as follows: The Jacob polynomals and some ther propertes are ntrodued n Secton 2. In Secton 3, The Jacob operatonal matrces of ntegraton and product are derved. In Secton 4, the convergence of the method s studed. Secton 5 s devoted to applyng the Jacob operatonal matrces for solvng system of ntegro dfferental equatons. In Secton 6, the proposed method s appled to several examples. A concluson s presented n Secton Jacob Polynomals and Ther Propertes The Jacob polynomals, assocated wth the real parameters α, β > 1, are a sequence of polynomals P α,β t, 1, 2,..., each of degree, are orthogonal wth Jacob weghted functon wx 1 x α 1 + x β over I [ 1, 1], and wth the followng orthogonalty condton: 1 P n α,β tp m α,β twt dt h n δ nm,

3 where δ nm s the Delta functon and K. Sadr & Z. Ayat/ IJM 2 C, h n 2α+β+1 Γn + α + 1Γn + β + 1 2n + α + β + 1n! Γn + α + β + 1. These polynomals can be generated wth the followng recurrence formula: α + β α 2 β 2 + tα + β + 2α + β P α,β t P α,β 1 2α + β + α + β t where α + 1β + 1α + β + 2 α + β + α + β P α,β t 1, P α,β 1 t α + β P α,β 2 t, 2, 3,..., t + α β. 2 In order to use these polynomals on the nterval [, 1], shfted Jacob polynomals are defned by ntroducng the change of varable t 2x 1. Let the shfted Jacob polynomals P α,β 2x 1 be denoted by x, then x can be generated from followng formula: α + β α 2 β 2 + 2x 1α + β + 2α + β x 2α + β + α + β where 1 α + 1β + 1α + β + 2 x α + β + α + β x 1, 1 x α + β x, 2, 3,..., 2x 1 + α β. 2 Remark 1 Of ths polynomals, the most commonly used are the shfted Gegenbauer polynomals, CS, α x, the shfted Chebyshev polynomals of the frst knd, T S, x, the shfted Legendre polynomals, P S, x, the shfted Chebyshev polynomals of the second knd, U S, x. These orthogonal polynomals are related to the shfted Jacob polynomals by the followng relatons. 1 CS,x α!γα ,β 1 Γ + α Rα x, T S, x!γ 1 2 Γ R P S, x R, x, U S, x + 1!Γ 1 2 Γ R 1, , The analytc form of the shfted Jacob polynomals, x, s gven by: x k x, x. 1 k Γ + β + 1Γ + k + α + β + 1x k, 2 Γk + β + 1Γ + α + β + 1 k! k!

4 4 K. Sadr & Z. Ayat/ IJM 2 C, Some propertes of the shfted Jacob polynomals are presented as: α, β,,, 3 d Γn + α + β dx Rα,β n x Γn + α + β + 1 The orthogonalty condton of shfted Jacob polynomals s: R α+,β+ n x. L x k xw α,β x dx θ k δ k, 3 where W α,β x, the shfted weghted functon, and θ k are as follows: W α,β x x β 1 x α, θ k h k 2 α+β+1. Lemma 2.1 The shfted Jacob polynomal R n α,β x can be obtaned n the form of: n x n p n x, where p n are, p n 1 n n + α + β + n + α. n Proof p n can be obtaned as: p n 1! d dx Rα,β n x. x Now, by usng propertes 1 and 3 n above, the lemma can be proved. Lemma 2.2 For m >, one has: x m xw α,β x dx l p l Bm + l + β + 1, α + 1, where Bs, t s the Beta functon and s defned as: Bs, t v s 1 1 v t 1 dv Γs Γt Γs + t.

5 K. Sadr & Z. Ayat/ IJM 2 C, Proof Usng Lemma 2.1 and W α,β 1 x α x β one has: x m x W α, βx dx l l l p l p l x m x l 1 x α x β dx 1 x α x m+l+β dx p l Bm + l + β + 1, α Convergence Analyss In ths secton, some theorems on convergence analyss and error estmaton of the proposed method are provded. Let Ω, 1 and for r N N s the set of all non negatve ntegers, the weghted Sobolov space HW r Ω s defned n the usual way and s denoted α,β nner product, sem norm and norm by u, v r,w α,β, v r,w α,β and v r,w α,β, respectvely. In partcular, and L 2 W α,βω H W α,βω, u, v W α,β u, v,w α,β, v W α,β v,w α,β, HW r α,βω {f f s measurable and v r,w α,β r u 2 r,w k α,β x u 2 W α+k,β+k, u r,w α,β k r xu W α+r,β+r. < }, A functon ux HW r Ω can be expanded n P N,α,β α,β span{ x, 1 x,..., x} as the below formula: N ux where the coeffcents c are gven by: c 1 θ c x, 4 x ux W α,β x dx,, 1, 2, By notng n practce, only the frst n + 1 terms shfted polynomals are consd-

6 6 K. Sadr & Z. Ayat/ IJM 2 C, ered, then one has: where ux u N x N c x Φ T x C, 6 C [c, c 1,..., c N ] T, Φx [ x, 1 x,..., N x] T. 7 Snce P N,α,β s a fnte dmensonal vector space, ux has a unque best approxmaton from P N,α,β, say u N x P N,α,β, that s: y P N,α,β, ux u N x W α,β ux y W α,β. In [28] s shown that for any ux HW r Ω, r N and µ r, a postve α,β constant c ndependent of any functon, N, α, and β exst that: ux u N x µ,w α,β cnn + α + β µ r 2 u r,w α,β. 8 Let ux s N + 1 tmes contnuously dfferentable. The followng theorem can present an upper bound for estmatng the error. Theorem 3.1 Let ux : for x >, and [x, 1] R s N + 1 tmes contnuously dfferentable P N,α,β span{ x, 1 x,..., x}. If u N Φ T C s the best approxmaton to ux from P N,α,β then the error bound s presented as follows: N where ux u N x W α,β MSN+1 Bα + 1, β + 1, N + 1! M max x [x un+1 x, S max{1 x, x }.,1] Proof Let consder the Taylor expanson of functon ux namely ũ N x. Therefore, one has: ux ũ N x x x N+1 u N+1 ξ, ξ x, 1. N + 1! Snce Φ T C s the best approxmaton to ux from P N,α,β, and yx P N,α,β one has: ux u N x 2 W α,β ux ũ Nx 2 W α,β M 2 N + 1! 2 x x 2N+1 W α,β x dx.

7 K. Sadr & Z. Ayat/ IJM 2 C, Snce W α,β x s always postve n, 1, choosng S max{1 x, x } leads to: ux u N x 2 W α,β M 2 S 2N+1 Bα + 1, β + 1. N + 1! 2 Ths error bound shows approxmaton of polynomals converges to ux as N. The followng theorem shows the coeffcents, c, tend to zero as N ncreases. Theorem 3.2 Let ux be a functon such that W α,β xux s ntegrable over Ω, 1, and let: ux then lm c. c x, c x ux W α,β x, 9 Proof Let s N x be the partal sum of the seres 9 through terms of Nth degree s N x N x. Then, the defnton of the c s and the propertes of the orthogonalty of s leads to the followng relaton: s N x ux W α,β x dx N ux x W α,β x dx N θ c 2. If W α,β xu 2 x as well as W α,β xux s ntegrable, then one has: W α,β x[ux s N x] 2 dx 2 + Therefore, W α,β x[ux] 2 dx W α,β uxs N x dx W α,β x[s N x] 2 dx W α,β x[ux] 2 dx N θ c 2. N θ c 2 W α,β x[ux] 2 dx. Consequently, θ c 2 s convergent and lm c.

8 8 K. Sadr & Z. Ayat/ IJM 2 C, Theorem 3.2 shows that a good approxmate soluton can be obtaned by means of the fnte numbers of Jacob polynomals n the seres 9. Now, the theorem on convergence of the proposed method s provded. Theorem 3.3 The seres soluton Eq.6 usng Jacob collocaton method converges towards ux n Eq.4. Proof Let L 2 W Ω be the Hlbert space and let ux N α,β c x where, c 1 ux, R θ α, β x Defne the sequence of partal sums S n as follows: S n x n c x.,w α,β. Let S n are S m are arbtrary partal sums wth n > m. It s gong to prove that S n s a Cauchy sequence n Hlbert space L 2 W α,β Ω. So, one has: n m+1 That s, c ux, S n x,w α,β x 2,W α, β ux, n c x,w α,β n c ux, x n θ c c n θ c 2. n m+1 n m+1 m+1 n m+1 c x, n θ c 2. S n x S m x 2,W α,β n m+1 c c n m+1,w α,β c x,w α,β x, θ c 2, for n > m. x,w α,β From Bessel nequalty, one has θ c 2 s convergent and hence S n x S m x 2,W as m, n. So, S α,β n s a Cauchy sequence and t converges

9 to say s. It s asserted that ux s. So, s ux, x K. Sadr & Z. Ayat/ IJM 2 C, ,W α,β s, x ux, x,w α,β,w α,β lm n S n x, x θ c,w α,β lm n S n x, x θ c,w α,β θ c θ c s ux, x.,w α,β Hence, ux s and n c x converges to ux. 4. The Jacob Operatonal Matrces In performng arthmetc and other operatons on the Jacob bass, we frequently encounter the ntegraton of the vector Φx defned n Eq.7 and t s necessary to evaluate the product of Φx and Φ T x, whch called the product matrx for the Jacob polynomals bass. In ths secton, these operatonal matrces are derved. 4.1 The Jacob Operatonal Matrx of Integraton In ths subsecton, Jacob operatonal matrx of the ntegraton s derved. Let, x Φt dt P Φx, 1 where matrx P N+1 N+1 s called the Jacob operatonal matrx of the ntegraton. The elements of ths matrx are obtaned as follows: Theorem 4.1 Let P s N + 1 N + 1 operatonal matrx of ntegraton. Then the elements of ths matrx are obtaned as: P 1 θ m n p m p n m + 1 Bm + n + β + 2, α + 1,,, 1, 2,..., N. Proof Usng Eq. 1 and orthogonalty property of Jacob polynomals one has: x P Φt dt, Φ T x 1, W α,β where x Φt dt, ΦT x W α,β and 1 are two N +1 N +1 matrces defned

10 1 K. Sadr & Z. Ayat/ IJM 2 C, as follows: x { x Φt dt, Φ T x W α,β dag { 1 θ } N. t dt, } N x, W α,β, Now, set: x ρ { x t dt, x W α,β } t dt W α,β x dx. x Rα,βt dt and x by usng Lemma 2.1 can be obtaned as follows: x t dt t m m p m x m+1 m + 1, p m x n,,, 1,..., N. Therefore, ρ by usng Lemma 2.2 can be obtaned as: ρ m m n n p m p n m + 1 p m p n m + 1 x m+1 x n 1 x α x β dx Bm + n + β + 2, α + 1. So, the elements of matrx P s obtaned as: P 1 θ m n p m p n m + 1 Bm + n + β + 2, α + 1,,, 1, 2,..., N. Now the followng theorem can present an upper bound for estmatng the error of ntegral operator. Frst, the error vector E s defned as: where E x Φtdt P Φx [E, E 1,..., E N ], E k x k t dt N P k x, k, 1,..., N.

11 K. Sadr & Z. Ayat/ IJM 2 C, Theorem 4.2 If E k x Rα,β k t dt N P k x HW r Ω, then α,β an error bound of ntegral operator of vector Φ can be expressed by: E k µ,w α,β c 2 NN+α+β µ r k k ρ k ρ k B++β r+3, α+r+1. Proof By usng nequalty 8, Lemma 2.1, and settng ux x has: Rα,β k t dt one x u r,w α,β Dr { k Dr k k k t dt 2 W α,β } pk x +1 W α+r,β+r k! 2 Γ r + 2 pk x r+1 W α+r,β+r k k W α+r,β+r x ρ k x r+1 ρ k x r+1 dx k k ρ k ρ k ρ k 1 x α+r x ++β r+2 dx ρ k B + + β r + 3, α + r + 1, where ρ k! Γ r+2 pk and the theorem can be proved. 4.2 The Jacob Operatonal Matrx of Produc The followng property of the product of two Jacob functon vector wll also be appled to solve the Volterra and Volterra Fredholm ntegro dfferental equatons. Φx Φ T x Y Ỹ Φx, 11 where Ỹ s a N + 1 N + 1 product operatonal matrx and t s entres are determned n terms of the vector Y s components. By usng Eq.11 and by the orthogonalty property of the Jacob polynomals the entres Ỹ can be calculated

12 12 K. Sadr & Z. Ayat/ IJM 2 C, as follows: Ỹ 1 θ N k 1 θ N k 1 θ N k Y k Φx Φx Φx k W α,β x dx Y k Y k h k, x x x W α,β x dx k where h k x x x W α,β x dx. k 5. Applcatons of Operatonal Matrces of Integraton and Product In ths secton, the presented operatonal matrces are appled to solve the system of lnear and nonlnear Fredholm, Volterra and Volterra Fredholm ntegro dfferental equatons. 5.1 The System of Fredholm Volterra Integor Dfferental Equatons In ths paper, a system of Fredholm Volterra ntegro dfferental equatons s consdered as follows: m 1 x + F k x, u 1 x, u 1x,..., u m 1 x,..., u n x,..., u m n x u m k1 m 2 x + 1 m 3 + l1 k x, t G u 1 t, u 1t,..., u m 1 t,..., u n t,..., u m n tdt h l x, t L l u 1 t, u 1t,..., u m 1 t,..., u n t,..., u m n tdt f x, x 1, 1, 2,..., n. 12 Where k x, t and h l x, t L 2 [, 1] [, 1], f are known functons, and G and L l are lnear or nonlnear functons n terms of unknown functons u 1 x, u 2 x,..., u n x and ther dervatves. Consder system 12 wth the followng condtons: u s x a, 1,..., n, s, 1,..., m 1. To solve system 12, the functons u x, u r x, G t, H l t, k x, t, and h l x, t can be approxmated as follows: Frst t s assumed the unknown functons u m x, 1,..., n, are approxmated

13 n the followng forms: The teratve ntegratng leads to: K. Sadr & Z. Ayat/ IJM 2 C, u m x Φ T x C. 13 u s m 1 s x Φ T x P m s T x C + a s, 1,..., n, s,..., m 1. 14! Usng Eqs.13 and 14 other terms wll be consdered as the followng general expansons: F k x, u 1 x, u 1x,..., u m 1 x,..., u n x,..., u m n x Φ T x X k, G u 1 t, u 1t,..., u m 1 t,..., u n t,..., u m n t Φ T x Y, L l u 1 t, u 1t,..., u m 1 t,..., u n t,..., u m n t Φ T x Z, k x, t Φ T x K Φt, h l x, t Φ T x H l Φt, 1,..., n, k 1,..., m 1, 1,..., m 2, l 1,..., m 3, where X k, Y, and Z l are the column vectors of the components of unkown vectors C,,..., n, and K and H l are known matrces. Also, C [c, c 1,..., c N] T, Φx [ x, 1 x,..., N x]t. Substtutng above approxmatons nto system 12, leads to the followng algebrac system: m 1 { m2 Φ T x C + Φ T x X k + Φ T x K Ỹ }P Φx k1 1 m 3 + Φ T xd H l Z l f x, 1, 2,..., n, l1 15 where Ỹ are operatonal matrces of product and ther elements are n terms of the elements of vectors Y, P s operatonal matrx of ntegraton, and D N+1 N+1 s the followng known matrx, D Φt Φ T t dt. The system 15 has nn + 1 unknown coeffcents c. So, nn + 1 collocatng ponts are needed to collocate. For ths purpose, the frst nn + 1 roots of Jacob polynomals nn+1+1 are appled and the equatons are collocated at them. Unknown coeffcents are determned wth solvng the resultant system of lnear or nonlnear algebrac equatons. Fnally, the approxmate solutons are obtaned as

14 14 K. Sadr & Z. Ayat/ IJM 2 C, follows: m 1 u x Φ T x P m T x C + a, 1,..., n.! 5.2 System of Volterra Integro Dfferental Equatons A system of Volterra ntegro dfferental equatons can be presented as follows: m 1 x + F k x, u 1 x, u 1x,..., u m 1 x,..., u n x,..., u m n x u m k1 m 2 x + 1 k x, t G u 1 t, u 1t,..., u m 1 t,..., u n t,..., u m n tdt f x, x 1, 1, 2,..., n. By usng the approxmate relatons n subsecton 5.1, one has: k m 1 { m2 Φ T x C + Φ T x X k + Φ T x K Ỹ }P Φx f x. 17 By usng the frst nn +1 roots of Jacob polynomals nn+1+1 x and collocated system 17, unknown coeffcents c are determned. 5.3 System of Fredholm Integro Dfferental Equatons A system of Fredholm ntegro dfferental equatons can be presented as follows, m 1 x + F k x, u 1 x, u 1x,..., u m 1 x,..., u n x,..., u m n x u m k1 m k x, t G u 1 t, u 1t,..., u m 1 t,..., u n t,..., u m n tdt f x, x 1, 1, 2,..., n. 18 By usng the approxmate relatons n subsecton 5.1, one has: m 1 m 2 Φ T x C + Φ T x X k + Φ T x K DY f x. 19 k1 By usng the frst nn +1 roots of Jacob polynomals nn+1+1 x and collocated system 19, unknown coeffcents c are determned Illustratve Examples In ths secton, some systems of ntegro dfferental equatons are consdered and solved by the proposed method. Comparson between the results of present method

15 K. Sadr & Z. Ayat/ IJM 2 C, wth the correspondng analytc solutons are gven. For ths purpose, the maxmum of absolute error s computed. Example 6.1 Consder the followng nonlnear Fredholm Volterra ntegro dfferental equaton. x 4 u 6 x + u 3 x + u x fx 2 where x fx x 4 cosx +.5 sn2x + 3x +.4.1e 1 + u 2 tdt + e t u 3 tdt, 2 cos1 + sn1cos e. Subect to the follong condtons, u u 4 1, u u u 5, u 1. The exact soluton s ux cosx. By the applyng the technque descrbed n pervous secton wth N 7, unknown functons and kernels are approxmated as: u 6 x Φ T xc, u 5 x Φ T xp T C, u 4 x Φ T xp 2 T C + 1 Φ T xp 2 T C + Φ T xu 1, u x Φ T xp 3 T C + x Φ T xp 3 T C + Φ T xu 2, u x Φ T xp 4 T C + x2 2 1 ΦT xp 4 T C + Φ T xu 3, u x Φ T xp 5 T C + x3 6 x ΦT xp 5 T C + Φ T xu 4, ux Φ T xp 6 T C + x4 24 x ΦT xp 6 T C + Φ T xu 5, 1 Φ T x K 1 Φt, e t Φ T x K 2 Φt, 1 + u 2 t Φ T xx, u 3 t Φ T xy. By usng above approxmatons, the equaton 2 s rewrtten as: x 4 Φ T x C + Φ T xp 3 T C + Φ T xu 2 + Φ T xp 5 T C + Φ T xu 4 + 2Φ T xk 1 XP Φx Φ T xk 2 DY fx. 21 Now, usng the roots of 8 x and collocatng the system 21, reduces the problem to solve a system of algebrac equatons. Unknown coeffcents and thereupon the approxmate solutons are obtaned for some values of parameters α and β as follows:

16 16 K. Sadr & Z. Ayat/ IJM 2 C, α, β : c , c , c , c , c , c c , c , ux Φ T xp 6 T C + x4 24 x x x x x x x x 4. α 1 2, β 1 2 : c , c , c , c , c , c , c e 4, c , ux x x x x x x x 2. α 1 2, β 1 2 : c , c , c , c , c , c , c , c , ux x x x x e 2x e 3x x 4.

17 K. Sadr & Z. Ayat/ IJM 2 C, α 1 2, β 1 2 : c , c , c , c , c , c c , c , ux x x x x x x x 4. α 1 2, β 1 2 : c , c , c , c , c , c , c , c , ux x x x x x x x 7. α 1 1, β 1 1 : c , c , c , c , c , c , c , c , ux x x x x x x x

18 18 K. Sadr & Z. Ayat/ IJM 2 C, Table 1. Maxmum absolute error for N 7 and dfferent values of α and β for Example 6.1. α β Errorux α β Errorux α 1, β 1 : c , c , c , c , c , c , c , c , ux x x x x x x x 7. α 3 4, β 3 4 : c , c , c , c , c , c , c e, c , ux x x x x x x x 7. Maxmum absolute error for N 7 and dfferent values of α and β are lsted n Table 1. Example 6.2 Consder the followng system of lnear Fredholm ntegro dfferental equatons. { u x + v x + 2xtut 3vt dt 3x x + 8, v x + u x + 32x + t2 ut 2vt dt 21x + 4 5, 22 subect to ntal condtons u 1, u, v 1, v 2, and the exact solutons are ux 3x and vx x 3 + 2x 1. Wth N 5, the solutons

19 K. Sadr & Z. Ayat/ IJM 2 C, Table 2. Maxmum absolute error for N 5 and dfferent values of α and β for Example 6.2. α β Errorux Errorvx a b Fgure 1. a Comparson of exact and approxmate solutons, b absolute error functon for ux, α 1 2, β 1, and N 1 for Example and kernels are approxmated as: u x Φ T xc 1, u x Φ T xp T C 1, ux Φ T xp 2 T C Φ T xp 2 T C 1 + Φ T xu 1, v x Φ T xc 2, v x Φ T xp T C Φ T xp T C 2 + Φ T xu 2, vx Φ T xp 2 T C 2 + 2x 1 Φ T xp 2 T C 2 + Φ T xu 3, 2xt Φ T xk 1 Φt, 32x + t 2 Φ T xk 2 Φt. By usng above relatons the system 22 s rewrtten as follows: Φ T x{c 1 + P T C 2 + U 2 + K 1 DP 2 T C 1 + K 1 DU 1 3K 1 DP 2 T C 2 + U 3 } 3x x + 8, Φ T x{c 2 + P T C 1 + K 2 DP 2 T C 1 + U 1 2P 2 T C 2 + U 3 } 21x + 4 5, 23 Now, usng the roots of 13 x and collocatng the system 23, reduces the problem to solve a system of lnear algebrac equatons and unknown coeffcents are obtaned for some values of parameters α and β. Table 2 dsplays the maxmum absolute errors for varous values α and β parameters wth N 5. Comparson of the exact and approxmate solutons and the plot of absolue error functons are presented n Fgures 1 and 2 for α β 1 2. The fgures show the good agreement between the exact and approxmate solutons.

20 2 K. Sadr & Z. Ayat/ IJM 2 C, a b Fgure 2. a Comparson of exact and approxmate solutons, b absolute error functon for vx, α 1 2, β 1, and N 1 for Example Example 6.3 Thrd example covers the system of nonlnear Volterra ntegro dfferental equaton. { u x x u x x u 2 t + v 2 tdt, v x snx sn2 x + x u t vtdt, 24 subect to ntal condtons u, u 1, u, v, v, v 1 and the exact solutons are ux snx and vx cosx. Wth N 7, the unknown functons and kernels are approxmated as: u x Φ T xc 1, u x Φ T xp T C 1, u x Φ T xp 2 T C Φ T xp 2 T C 1 + Φ T xu 1, ux Φ T xp 3 T C 1 + x Φ T xp 3 T C 1 + Φ T xu 2, v x Φ T xc 2, v x Φ T xp T C 2 1 Φ T xp T C 2 + Φ T xu 3, v x Φ T xp 2 T C 2 x Φ T xp 2 T C 2 + Φ T xu 4, vx Φ T xp 3 T C 2 x ΦT xp 3 T C 2 + Φ T xu 5, 1 Φ T xkφt, u 2 x + v 2 x Φ T xx 1, u t vt Φ T xx 2 Usng above approxmatons leads to the followng nonlnear systems: { Φ T x{c 1 + P 2 T C 1 + U 1 + K X 1 P Φx} x, Φ T x{c 2 K X 2 P Φx} snx sn2 x, 25 where P s Jacob operatonal matrx of ntegraton and X 1 and X 2 are operatonal matrces of product whch ther entres are obtaned n terms of components of vectors X 1 and X 2. Usng the roots of 17 x and collocatng the system 25, the problem reduces to solve a system of nonlnear algebrac equatons whch wll be solved by means of Newton teratve method and unknown coeffcents are determned for some values of α and β parameters. By solvng algebrac system obtaned, unknown coeffcents are determned. Table 3 dsplays the maxmum ab-

21 K. Sadr & Z. Ayat/ IJM 2 C, Table 3. Maxmum absolute error for N 7 and dfferent values of α and β for Example 6.3. α β Errorux Errorvx a b Fgure 3. a Comparson of exact and approxmate solutons, b absolute error functon for ux of Example 6.3 for for α β and N 7. a b Fgure 4. a Comparson of exact and approxmate solutons, b absolute error functon for vx of Example 6.3 for for α β and N 7. solute errors for some values of α and β parameters for N 7. Also, the parts a of Fgures 3 and 4 show the comparson between the exact and approxmate solutons and the parts b of these fgures dsplay the absolute error functons for α β. Example 6.4 In ths example, a nonlnear Volterra ntegro dfferental equaton s consdered.

22 22 K. Sadr & Z. Ayat/ IJM 2 C, Table 4. Maxmum absolute error for N 7 and dfferent values of α and β for Example 6.4. α β Errorux Errorvx Errorwx u x x + 2x v 2 x x v 2 t + ut w tdt, v x 3x 2 xux + x xt v tu t + w tdt, w x x3 + u 2 x 2 u 2 x + x x2 vt + u 2 t + t 3 w tdt, 26 subect to ntal condtons u u v w w, v 1 and the exact solutons are ux x 2, vx x and wx 3 x 2, x 1. Wth N 7, the unknown functons and kernels are approxmated as: u x Φ T xc 1, u x Φ T xp T C 1, ux Φ T xp 2 T C 1 v x Φ T xc 2, v x Φ T xp T C Φ T xp T C 2 + Φ T xu 1, vx Φ T xp 2 T C 2 + x Φ T xp 2 T C 2 + Φ T xu 2, w x Φ T xc 3, w x Φ T xp T C 3, wx Φ T xp 2 T C 3 v 2 x Φ T xx 1, uxw x Φ T xx 2, v xu x Φ T xx 3 u 2 x Φ T xx 4, u 2 x Φ T xx 5, u 2 x Φ T xx 6 1 Φ T xk 1 Φt, xt Φ T xk 2 Φt, x 2 Φ T xk 3 Φt, t 3 Φ T xk 4 Φt. By usng above relatons, the system 26 s rewrtten as: Φ T x{c 1 + K 1 X1 P Φx + K 1 X2 P Φx 2X 1 } x + 2x 3, Φ T x{c 2 + xp 2 T C 1 K 2 X3 P Φx K 1 Ỹ 1 P Φx} 3x 2, Φ T x{c 3 X 5 + 2X 6 K 3 Ỹ 2 P Φx K 3 Ũ 2 P Φx K 1 X4 P Φx K 4 C3 Φx} x3, 27 where Ỹ1 and Ỹ2 are operatonal matrces of product whch ther entres are obtaned n terms of components of vectors Y 1 P T C 3 and Y 2 P 2 T C 2. Usng the roots of 25 x and collocatng the system 27, the problem reduces to solve a system of nonlnear algebrac equatons, whch wll be solved by means of Newton teratve method and unknown coeffcents are determned for some values of α and β parameters. Table 4 dsplays the maxmum absolute errors for some values of parameters α and β wth N 7. Also, the comparson between the exact and approxmate solutons are dsplayed n the parts a of Fgures 5 7 and the absolute error functons are ndecated n parts of b of these fgures for α β 1.

23 K. Sadr & Z. Ayat/ IJM 2 C, a b Fgure 5. a Comparson of exact and approxmate solutons, b absolute error functon for ux of Example 6.4 for for α β 1 and N 7. a b Fgure 6. a Comparson of exact and approxmate solutons, b absolute error functon for vx of Example 6.4 for for α β 1 and N Concluson In ths paper, the shfted Jacob collocaton method was employed to solve a class of systems of Fredholm and Volterra ntegro dfferental equatons. Frst, a general formulaton for the Jacob operatonal matrx of ntegraton has been derved. Ths matrx s used to approxmate numercal soluton of system of lnear and nonlnear Volterra ntegro dfferental equatons. Proposed approach was based on the shfted Jacob collocaton method. The solutons obtaned usng the proposed method shows that ths method s a powerful mathematcal tool for solvng the ntegro dffenrental equatons. Provng the convergence of the method, consstency and stablty are ensured automatcally. Moreover, only a small number of shfted Jacob polynomals s needed to obtan a satsfactory result. 7.1 Acknowledgements The authors would lke to thank the referees for useful suggestons that have contrbuted to mprove ths work.

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