Numerical solution for systems of two dimensional integral equations by using Jacobi operational collocation method

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1 Sohag J Math, o, 5-26 (24) 5 Sohag Journal of Matheatcs An Internatonal Journal uercal soluton for systes of two densonal ntegral equatons by usng Jacob operatonal collocaton ethod Abdollah Borhanfar and Khadeh Sadr Faculty of Matheatcal Scences, Departent of Matheatcs, Unversty of Mohaghegh Ardabl, Ardabl, Iran Receved: 9 Feb 24, Revsed: 28 Mar 24, Accepted: 29 Mar 24 Publshed onlne: Sep 24 Abstract: In ths paper, the nuercal soluton of two densonal Fredhol and Volterra ntegral equatons wll be nvestgated For ths order, two densonal collocaton ethod s appled to solve syste of two densonal lnear and nonlnear Fredhol and Volterra ntegral equatons Usng the Jacob polynoals, two densonal ntegral equatons reduce to a syste of algebrac equatons The an a s the developng the Jacob operatonal atrces of ntegraton and product for the solvng syste of two densonal Fredhol and Volterra ntegral equatons These atrces together wth the collocaton ethod are appled to reduce the soluton of these probles to the soluton of a syste of algebrac equatons The nuercal exaples llustrate the effcency and accuracy of ths ethod Keywords: collocaton ethod, shfted Jacob polynoals, two densonal Fredhol and Volterra ntegral equatons, operatonal atrces of ntegraton and product, lnear and nonlnear systes, convergence Introducton Two densonal ntegral equatons provde an portant tool for odelng a nuerous probles n engneerng and echancs [, 2] There are any dfferent nuercal ethods for solvng one densonal ntegral equatons, such as [3,4,5,6,7,8,9] Soe of these ethods can be used for solvng two densonal ntegral equatons Coputatonal coplexty of atheatcal operatons s the ost portant obstacle for solvng ntegral equatons n hgher densons Malenead and et al n [] have appled the Adoan decoposton ethod to solve the nonlnear xed Volterra-Fredhol ntegral equatons Guoqang, [], has used the yströ ethod for a nonlnear Volterra-Fredhol ntegral equatons Babolan and et al have used the Hootopy perturbaton ethod and dfferental transfor ethod for two densonal lnear and nonlnear Volterra ntegral equatons [2] Hatazadeh and et al, [3], appled the bloc-pulse functons to solve two densonal lnear ntegral equatons In ths study, frst two densonal Jacob operatonal atrces of ntegraton and product are obtaned ext, the collocaton ethod s developed for solvng the systes of two densonal ntegral equatons The reander of ths paper s organzed as follows: The Jacob polynoals and soe ther propertes and one densonal atrces of ntegraton and product are ntroduced n Secton 2 Afterwards, these atrces wll be extended to two densonal case In Secton 3, the convergence of the ethod s studed Secton 4 s devoted to applyng two densonal Jacob operatonal atrces for solvng systes of two densonal ntegral equatons In Secton 5, the proposed ethod s appled to solve several exaples A concluson s presented n Secton 6 2 Jacob polynoals and Jacob operatonal atrces The Jacob polynoals, assocated wth the real paraeters (α,β > ) are a sequence of polynoals (t)(,,2,), each of degree, are orthogonal Correspondng author e-al: borhan@uaacr c 24 SP atural Scences Publshng Cor

2 6 A Borhanfar, Kh Sadr: uercal Soluton by usng Jacob Operatonal Collocaton Method wth Jacob weghted functon, w(t) ( t) α (+ t) β over I [,], and P n (α,β) (t) P (α,β) (t)w(t)dt h n δ n, where δ n s Kroneer functon and h n 2α+β+ Γ(n+α+ )Γ(n+β + ) (2n+α+ β + )n!γ(n+α+ β + ) These polynoals can be generated wth the followng recurrence forula: (α+ β + 2 ) (t) 2(α+ β + )(α+ β + 2 2) α 2 β 2 +t(α+ β + 2)(α+ β + 2 2)} (t) (α+ )(β + )(α+ β + 2) (α+ β + )(α+ β + 2 2) 2 (t), 2,3,, where (t) and (t)(α+ β + 2)t/2+ (α β)/2 In order to use these polynoals on the nterval[,], shfted Jacob polynoals are defned by ntroducng the change of varable t 2x In what followng, the shfted Jacob polynoals (2x ) are denoted by (x), for convenence Then the shfted Jacob polynoals (x) can be generated fro followng forula: (α+ β + 2 ) (x) 2(α+ β + )(α+ β + 2 2) α 2 β 2 +(2x )(α+ β + 2)(α+ β + 2 2)} (x) (α+ )(β + )(α+ β + 2) (α+ β + )(α+ β + 2 2) where (x), and 2 (x), x D[,], 2,3,, (x)(α+ β + 2)(2 x )/2+(α β)/2 Rear Of ths polynoals, the ost coonly used are the shfted Gegenbauer polynoals, CS, α (x), the shfted Chebyshev polynoals of the frst nd, T S, (x), the shfted Legendre polynoals, P S, (x), the shfted Chebyshev polynoals of the second nd, U S, (x) These orthogonal polynoals are related to the shfted Jacob polynoals by the followng relatons CS, α (x)!γ(α+ 2 ) 2,β 2 ) Γ(+α+ 2 )P(α (x), T S, (x)!γ( 2 ) 2, 2 ) Γ(+ 2 )P( (x), P S, (x)p (,) (x), U S, (x) (+)!Γ( 2 ) Γ(+ 3 2 ) P ( 2, 2 ) (x) The analytc for of the shfted Jacob polynoals, (x), s gven by (x) ( ) ( ) Γ(+β + )Γ(++ α+ β + )x, Γ(+ β + )Γ(+α+ β + )( )!! Soe propertes of the shfted Jacob polynoals are as follows: ( ) () ()( ) +α, ( (2) ()( ) +β (3) d ), dx P n (α,β) (x) Γ(n+α+β++) Γ(n+α+β+) P(α+,β+) n (x) The orthogonalty condton of shfted Jacob polynoals s: (x) (x)w (α,β) (x)dxθ δ, where w (α,β) (x), shfted weghted functon, s as follows: and, θ h /2 α+β+ w (α,β) (x)x β ( x) α, Lea 2 The shfted Jacob polynoal P n (α,β) (x) can be obtaned n the for of: where p (n) p (n) Proof The p (n) are n (x) n p (n) x, ( ) n ( n+α+ β + can be obtaned as, p (n) d )( ) n+α n! dx P(α,β) n (x) x ow, usng propertes () and (3) n above relaton, the lea can be proved Lea 22 For >, one has x (x)w (α,β) (x)dx l p ( ) l B(+l+β+,α+), c 24 SP atural Scences Publshng Cor

3 Sohag J Math, o, 5-26 (24) / wwwnaturalspublshngco/journalsasp 7 where B(s,t) s the Beta functon and s defned as B(s,t) v s ( v) t dv Γ(s)Γ(t) Γ(s+t) Proof Usng Lea 2 and w (α,β) ( x) α x β one has x (x)w (α,β) (x)dx l l l p ( ) l x x l ( x) α x β dx p ( ) l ( x) α x (+l+β) dx p ( ) l B(+l+β+,α+) A functon u(x) L 2 (D) can be expanded as the below forula: u(x) c (x), where the coeffcents c are gven by c θ (x)u(x)w (α,β) (x)dx,,,2, By notng n practce only the frst ( + ) ters shfted polynoals are consdered, then one has u(x) u (x) where C[c,c,,c ] T, and c (x)φ T (x)c, Φ(x)[ (x), (x),, (x)] T ow, two varables Jacob polynoals can be defned by eans of one varable Jacob polynoals as follows: Defnton 23 Let P n (α,β) (x)} n be the sequence of one varable shfted Jacob polynoals on D [,] Two varables Jacob polynoals, R,n (α,β) (x,y)},n, are defned on D 2 [,] [,] as:,n (x,y) (x) n, (x,y) D 2 The faly R,n (α,β) (x,y)},n s orthogonal wth weghted functon W (α,β) (x,y) w (α,β) (x) w (α,β) on D 2 and fors a bass for L 2 (D 2 ) Theore 24 The bass,n (x, y)} s orthogonal on D 2 Proof One has R,n (α,β) (x,y) (x,y)w (α,β) (x,y)dxdy n,l (x) l (x)w (α,β) (x)dx w (α,β) dy θ θ n, (,n)(,l),, (,n) (,l) or or n l A functon u(x,y) defned over D 2 ay be expanded by the two varables Jacob polynoals as: u(x,y) n c n,n (x,y), (x,y) D 2 () where the Jacob coeffcents, c n, are obtaned as: c n θ θ n R,n (α,β) (x,y)u(x,y)w (α,β) (x,y)dxdy If the nfnte seres n equaton () s truncated up to ther (+ ) ters then t can be wrtten as: u(x,y) u (x,y) n c n R,n (α,β) (x,y)φ T (x,y)c, where C and Φ(x,y) are Jacob coeffcents and Jacob polynoals vectors, respectvely: C[c,c,,c,,c,,c ] T, Φ(x,y)[Φ (x,y),,φ (x,y),φ (x,y),,φ (x,y)] T [, (x,y),,, (x,y),r(α,β), (x,y),,r(α,β), (x,y)]t (2) Slarly, a functon of four varables, (x,y,t,s), on D 4 ay be approxated wth respect to Jacob polynoals such as: (x,y,t,s) Φ T (x,y) K Φ(t,s), where Φ(x,y) s two varables Jacob vector and K s a (+ ) 2 (+ ) 2 nown atrx 2 One densonal Jacob operatonal atrces In perforng arthetc and other operatons on the Jacob bass, we frequently encounter the ntegraton of the vector Φ(x) and t s necessary to evaluate the product of Φ(x) and Φ T (x), whch s called the product atrx for the Jacob polynoals bass In ths subsecton, these operatonal atrces are derved c 24 SP atural Scences Publshng Cor

4 8 A Borhanfar, Kh Sadr: uercal Soluton by usng Jacob Operatonal Collocaton Method 2 One densonal Jacob operatonal atrx of ntegraton In ths subsecton, Jacob operatonal atrx of the ntegraton s derved Let Φ(t)dt PΦ(x), (3) where atrx P s called the Jacob operatonal atrx of ntegraton The entres of ths atrx are obtaned as follows: Theore 25 Let P be ( + ) ( + ) operatonal atrx of ntegraton Then the eleents of ths atrx are obtaned as: P θ n + p() p ( ) n B(+n+β + 2,α+ ),,,,2,, Proof Usng equaton (3) and orthogonalty property of Jacob polynoals one has: P( Φ(t)dt,Φ T (x)), where( Φ(t)dt,ΦT (x)) and are two(+) (+ ) atrces defned as follows: ( Φ(t)dt,Φ T (x))( (t)dt, (x))},, Set ρ ( P(α,β)(t) dag θ } (t)dt, (x)) (t)dt} w (α,β) (x)dx dt and (x) by usng Lea 2 can be obtaned as: (t) (t)dt n p () x + +, p ( ) n x n,,,,, Therefore, ρ by usng Lea 22 can be obtaned as follows, ρ n n + + ) p( p n ( ) x + x n ( x) α x β dx p( ) p ( ) n B(+n+β + 2,α+ ) So, the entres of atrx P s obtaned as: P θ n + p() p ( ) n B(+n+β + 2,α+ ),,,,2,, 22 One densonal Jacob operatonal atrx of product The followng property of the product of two Jacob functon vector wll be also be appled to solve the Volterra and Volterra - Fredhol ntegral equatons Φ(x)Φ T (x)y Ỹ Φ(x), (4) where Ỹ s a(+ ) (+ ) product operatonal atrx and ts entres are deterned n ters of the coponents of the vector Y Usng equaton (4) and by the orthogonalty property of Jacob polynoals the entres Ỹ, can be calculated as follows: Ỹ θ Y Φ (x)φ (x)φ (x)w (α,β) (x)dx where θ Y θ Y h, h (x) (x) (x) (x) (x)w (α,β) (x)dx (x)w (α,β) (x)dx 22 Two densonal Jacob operatonal atrces In ths subsecton, two densonal operatonal atrces are presented 22 Two densonal Jacob operatonal atrx of ntegraton for x varable The operatonal atrx of ntegraton n x drecton s defned as follows: Theore 26 The operatonal atrx of ntegraton n x drecton s defned as follows Φ(t,y)dt P x Φ(x,y)(P I)Φ(x,y), where P x s a (+ ) 2 (+ ) 2 operatonal atrx of ntegraton, P s operatonal atrx of ntegraton c 24 SP atural Scences Publshng Cor

5 Sohag J Math, o, 5-26 (24) / wwwnaturalspublshngco/journalsasp 9 ntroduced n subsecton 2 and I s (+ ) (+ ) dentty atrx Proof Suppose R be th row of atrx P One has (t) dt R T Φ(x) Also, notng the defnton of the vector Φ(x,y) one has Φ(x,y)[ (x),,, (x),, (x) (x) Integratng of equaton (5) fro to x yelds, Φ(t,y)dt [ (t)dt,,, ] T (5) (t)dt, (t)dt] T [R Φ(x),,R Φ(x),,R Φ(x),,R Φ(x) ] T [R [ (x),, (x) ],,R [ (x),, (x) ]] T, P P P P P P P P P P P P P P P P P P P P P P P P P P P (x) (x) (x) (x) (x) (x) P I P I P I (x) Φ P I P I P I (x) Φ P I P I P I (x) Φ (P I)Φ(x,y) Where denotes the Kronecer product and s defned for two arbtrary atrces A and B as A B(a B) and P denotes (, )th entry of the atrx P 222 Two densonal Jacob operatonal atrx of ntegraton for y varable Theore 27 The operatonal atrx of ntegraton n y drecton s defned as: y Φ(x,s)ds P y Φ(x,y)(I P)Φ(x,y), where P y s a (+ ) 2 (+ ) 2 operatonal atrx of ntegraton Proof Agan, ntegratng of equatonq (5) (3) fro to y one has y Φ(x,s)ds[P(α,β) (x) y P(α,β) (s)ds,, (x) y P(α,β) (s)ds] T [ (x) R Φ,, (x) R Φ,, (x) R Φ,, (x) R Φ] T P P P P P P P P P P P P P P P P P P P P P P P P P P P (x) (x) (x) (x) (x) (x) c 24 SP atural Scences Publshng Cor

6 2 A Borhanfar, Kh Sadr: uercal Soluton by usng Jacob Operatonal Collocaton Method P O O (x) Φ O P O (x) Φ (I P) Φ(x,y) O O P (x) Φ Where O s a(+ ) (+ ) zero atrx 23 Two densonal Jacob operatonal atrx of product The followng property of the product of two vectors Φ(x,y) and Φ T (x,y) wll also be used Φ(x,y) Φ T (x,y)u ŨΦ(x,y), (6) where U and Ũ are a(+ ) 2 vector and a(+ ) 2 (+) 2 product operatonal atrx, respectvely One has u, (x,y) ]Φ(x,y)Φ T (x,y)u u u One puts, (x,y),l (x,y) r s, (x,y), (x,y),, (x, y), (x, y) (x,y) a rs r,s (x,y) (7) The coeffcents a rs are obtaned by the followng anner Multplyng both equaton (7) by,n (x,y),,n,,2,,, and ntegratng of the result fro to yelds:, (x,y),l r s a n θ θ n Therefore a rs a n θ θ n θ θ n r,s (x,y) (x,y)w (α,β) (x,y)dxdy,n (x,y) (x,y)w (α,β) (x,y)dxdy,n, (x,y),l ow suppose ω (x) (x)p (α,β) l P n (α,β) (x) (x)p (α,β) (x,y) (x, y),n W (α,β) (x,y)dxdy (x)w (α,β) (x)dx w (α,β) dy (x)w (α,β) (x)dx, one gets a n ω ω ln θ θ n Substtutng a n nto equaton (7) one has:, (x,y),l (x,y) n,,,,, So, ω ω ln θ θ n,n (x,y) If only the coponents of Φ(x,y) are retaned, then the atrx Ũ n the equaton (6) s obtaned as Ũ [ũ ],,,,, (8) In the equaton (8), ũ,,,,,, are(+) (+ ) atrces gven by ũ θ nb n ω n,,,,,, and B n are(+ ) (+ ) atrces as [B n ] l θ l u n ω l,,l,,, 3 Convergence analyss In ths secton, the theores on convergence analyss and error estaton of the proposed ethod are provded Theore 3 Suppose u(x,y) C [,] C [,] and c (x) be an approxaton for u(x,y) Then for the coeffcents c one has: c A(α,β) 22(+ ), ax x y (x,y) D + u(x,y), D[,] [,], where A (α,β), are ndependent of the functon u(x, y) Proof Accordng to the assupton, u(x,y) c (x), where the coeffcents c,,,,, are obtaned as follows: c θ θ u(x,y) (x) W (α,β) (x,y)dxdy, (9) Consder Taylor expanson about ponts x and y For each,,,, one has: + u(x,y) n x n y n ( n)!n! u(,) n y n c 24 SP atural Scences Publshng Cor

7 Sohag J Math, o, 5-26 (24) / wwwnaturalspublshngco/journalsasp n x + n y n (+ n)n! + u(ξ x,ξ y ) x + n y n, (ξ x,ξ y ) [,x] [,y] Substtutng equaton () n equaton (9) leads to: c + θ θ + + n n (+ n)!n! + θ θ + + n u(,) x n y n ( n)!n! W (α,β) (x,y)x n y n n w (α,β) (x)x n (x)dx (+ n)!n! + θ θ then n c + θ θ (x) () dxdy W (α,β) (x,y)x + n y n (x) + u(ξ x,ξ y ) x + n y n dxdy} u(,) x n y n ( n)!n! (+ n)!n! n w (α,β) y n dy W (α,β) (x,y)x + n y n (x) + u(ξ x,ξ y ) x + n y n dxdy} W (α,β) (x,y)x + n y n (x) + u(ξ x,ξ y ) x + n y n dxdy, (+ n)!n! ax + u(x,y) (x,y) D x + n y n w (α,β) (x)x + n (x)dx w (α,β) y n dy The last suaton wll be nonzero only for n Therefore c θ θ!! x y ax + u(x,y) (x,y) D w (α,β) y dy Thereupon c 2 2(+ ) 2 2(α+β+) w (α,β) (x)x (x)dx Γ(α+ β + 2+2)Γ(α+ β ) Γ(α+ +)Γ(α+ + )Γ(β + +)Γ(β + + )!! where u(x,y) ax (x,y) D + x y w (α,β) (x)x (x) dx w (α,β) y dy 2 u(x,y) A(α,β) 2(+ ), ax (x,y) D + x y, A (α,β) Γ(α+ β + 2+2), 2 2(α+β+) Γ(α+ +)Γ(α+ + ) Γ(α+ β ) Γ(β + +)Γ(β + + )!! w (α,β) (x)x (x) dx w (α,β) y dy The last nequalty shows that the coeffcents decrease when, (n fact ) ncrease Therefore, functon u(x,y) can be approxated usng the fnte nubers of the Jacob polynoals Theore 32 Suppose u(x,y) C [,] C [,] Then the bound of the error for the approxate soluton resulted s as follows: u(x,y) Φ T (x,y)c 22 (2!) M, where M axm,m,,m 2 } and M ax 2 u(x,y) (x,y) D x 2,,,,2 y Proof otng the least square property, consder polynoal S (x,y), of degree at ost wth respect to both varables x and y, whch nterpolates u(x,y) n the doan D Therefore W (α,β) (x,y)(u(x,y) Φ T (x,y)c) 2 dxdy W (α,β) (x,y)(u(x,y) S (x,y)) 2 dxdy ow, consder the Taylor expanson of functon u(x, y) about pont (,) n D The bound of the error s obtaned as follows: u(x,y) S (x,y) 2 x 2 y (2 )!! where (ξ x,ξ y ) [,x] [,y] Therefore u(x,y) S (x,y) 2 (2 )!! ax (ξ x,ξ y ) [,x] [,y] 2 u(ξ x,ξ y ) x 2 y, 2 u(ξ x,ξ y ) x 2 y c 24 SP atural Scences Publshng Cor

8 22 A Borhanfar, Kh Sadr: uercal Soluton by usng Jacob Operatonal Collocaton Method But ax (ξ x,ξ y ) [,x] [,y] Therefore 2 u(ξ x,ξ y ) u(x,y) x 2 y ax (x,y) D 2 x 2 y M u(x,y) S (x,y) 2 M (2 )!! Let set M axm,m,,m 2 } Hence u(x,y) S (x,y) M Therefore 2 M 22 (2)!, (2 )!! M (2)! u(x,y) Φ T (x,y)c 22 (2)! M 2 ( 2 4 Soluton of the systes of two densonal ntegral equatons In ths secton, the presented operatonal atrces are appled to solve the systes of lnear and nonlnear Fredhol, Volterra and Volterra-Fredhol ntegral equatons 4 Syste of two densonal Fredhol ntegral equatons In ths paper, a syste of Fredhol ntegral equatons s consdered as follows: u (x,y)+ (x,y,t,s)g (u (t,s),,u n (t,s)) dsdt f (x,y), (x,y) D,,,n () where (x,y,t,s) L 2 (D 2 ), f (x,y) are nown functons, and g (x,y,t,s) are lnear or nonlnear functons n ters of unnown functons u (x,y),,u n (x,y) To solve the syste (), the functons u (x,y), (x,y,t,s) and g (x,y,t,s) can be approxated as follows: u (x,y) Φ T (x,y)c, (x,y,t,s) Φ T (x,y)k Φ(t,s), g (x,y) Φ T (x,y)g,,,n,,, where G and K are nown vectors and atrces, respectvely Also C [c,c,,c,,c,,c ] T,,,n ) Substtutng above approxatons nto syste () leads to the followng algebrac syste: Φ T (x,y)c + Φ T (x,y) K AG } f (x,y), (2) where A s a(+ ) 2 (+ ) 2 atrx as follows: A Φ(t,s)Φ T (t,s)dsdt The algebrac syste (2) has n( + ) 2 unnown coeffcents c So, n( + )2 collocatng ponts are needed for collocatng the algebrac syste resulted For ths purpose, the ( + ) roots of Jacob polynoals + (x) and P(α,β) + are consdered n the x and y drectons The doan of two densonal s represented by a tensor product ponts x } and y } whch are roots of P(α,β) + (x) and P(α,β) + Each the equatons of the syste (2) s collocated n the resulted tensor ponts (x,y )}, Fnally, collocatng the equatons (2) gves n( + ) 2 lnear or nonlnear equatons whch nonlnear equatons can be solved usng the well-nown ewton s teratve ethod 42 Syste of two densonal Volterra ntegral equatons A syste of two densonal Volterra ntegral equatons can be presented as follows: + u (x,y)+ 2 y (x,y,t,s)g (u (t,y),,u n (t,y))dt h (x,y,t,s)l (u (x,s),,u n (x,s))ds f (x,y), (x,y) D,,,n, (3) where (x,y,t,s) and h (x,y,t,s) L 2 (D 2 ), f (x,y) are nown functons, and g, (x,y) and l, (x,y) are lnear or nonlnear functons n ters of unnown functons u (x,y),,u n (x,y) The functons u (x,y), g (x,y), l (x,y), (x,y,t,s) and h (x,y,t,s) can be approxated as follows: u (x,y) Φ T (x,y)c, g (x,y) Φ T (x,y)g, l (x,y) Φ T (x,y)l (x,y,t,s) Φ T (x,y)k Φ(t,y), h (x,y,t,s) Φ T (x,y)h Φ(x,s),,,n,,,,,, 2, where C, G and L are (+ ) 2 vectors and K and H are (+ ) 2 (+ ) 2 nown atrces Substtutng above approxatons nto syste (3) leads to: c 24 SP atural Scences Publshng Cor

9 Sohag J Math, o, 5-26 (24) / wwwnaturalspublshngco/journalsasp 23 Φ T (x,y)c + Φ T (x,y) K G }P x Φ(x,y) +Φ T 2 (x,y) H L }P y Φ(x,y) f (x,y), (4),,n,,,,,, 2, where G and L are operatonal atrces of product and ther entres are n ters of the coponents of vectors G and L P x and P y are operatonal atrces of ntegraton n x and y drectons, respectvely Each the equatons of the syste (4) s collocated n tensor ponts (x,y )}, (stated n subsecton 4) Fnally, equatons (4) gve n( + ) 2 lnear or nonlnear equatons whch nonlnear equatons can be solved usng the ewton s teratve ethod 43 Syste of two densonal Volterra-Fredhol ntegral equatons In ths paper, a syste of Volterra-Fredhol ntegral equatons s consdered as follows: u (x,y)+ y (x,y,t,s)g (u (t,s),,u n (t,s))dsdt f (x,y), (x,y) D,,,n (5) To solve syste (5) the functons u (x,y), g (x,y) and (x,y,t,s) can be approxated as follows: u (x,y) Φ T (x,y)c, g (x,y) Φ T (x,y)g, (x,y,t,s) Φ T (x,y)k Φ(t,s),,,n,,,, where C and G are ( + ) 2 vectors and K are (+ ) 2 (+ ) 2 nown atrces Substtutng above approxatons nto syste (5) leads to, Φ T (x,y)c + Φ T (x,y),,n,,, where B s(+ ) 2 atrx as follows: K G }P y B f (x,y), B Φ(t,y) dt, (6) and K, G and P y are ( + ) 2 ( + ) 2 nown atrces, operatonal atrces of product and operatonal atrx of ntegraton,respectvely Collocatng each the equatons of the syste (6) n tensor ponts (x,y )}, leads to n( + )2 lnear or nonlnear Unnown coeffcents c are deterned by solvng the syste resulted Table : Dfferent values of exact and approxate solutons n ponts (,) for 4 and α β of Exaple (x,y ) u exact u approxate (, ) (2, 2) (3, 3) (4, 4) (5, 5) (6, 6) (7, 7) (8, 8) (9, 9) (, ) Illustratve exaples In ths secton, the proposed ethod s appled to solve soe systes of two densonal ntegral equatons Exaple Consder the followng two densonal lnear Fredhol ntegral equaton u(x,y) where (ssn(x)+ty)u(t,s)dtds f(x,y), (7) f(x,y)x cos 3 4 y+ 5 6 sn(x) y 3 sn() sn(x) cos() 2 sn(x) sn(), 2 and exact soluton s u(x,y)xcos y Functon u(x,y) and ernel are approxated as: u(x,y) Φ T (x,y)c, ssn(x)+ty Φ T (x,y)kφ(t,s) Equaton (7) s wrtten by usng above relatons as: where Φ T (x,y)c Φ T (x,y)kbc f(x,y), (8) B Φ(t,s)Φ T (t,s)dtds Settng 4 and usng the roots of 5 (x) and 5 n the x and y drectons, equaton (8) s collocated n 25 nner tensor ponts for dfferent values of paraeters α and β Hereby, the equaton (7) reduces the proble to solve a syste of lnear algebrac equatons and unnown coeffcents are obtaned for soe values of paraeters α and β Table dsplays dfferent values of the exact and approxate solutons n ponts (x,y)(,), (,2,,) for α β Table 2 shows errors of the approxate solutons n L 2 (D) for dfferent values of α and β Fgure ndcates the axu absolute errors for soe values α and β and y5 c 24 SP atural Scences Publshng Cor

10 24 A Borhanfar, Kh Sadr: uercal Soluton by usng Jacob Operatonal Collocaton Method Table 2: Coparson of the errors n L 2 (D) for dfferent values α and β of Exaple (α,β) Error(L 2 ) (,) ( 5, 5) (,) (5,5) ( 5, 5) (, ) where C and C 2 are operatonal atrces of product, P x and P y are operatonal atrces of ntegraton n the x and y drectons, respectvely ow usng the roots of 4 (x) and 4 n the x and y drectons, each equatons of syste (2) s collocated n 6 nner tensor ponts for dfferent values of paraeters α and β, the coeffcents are obtaned as follows Thereupon, the exact solutons are acqured α β : C [ 2, 2,,, 3 4, 3 4,,, 4, 4,,,,,,], C 2 [ 7 8, 9 4, 8, 4, 8, 9 4, 8, 4,,,,,,,,], α β 5 : C [ 5 32, 5 6,,, 2, 3,,, 3 2,,,,,,,], C 2 [ 57 64, 7 48, 3 4, 7, 7 96, 7 72, 2, 5,,,,,,,,], α β 5 : C [ 3 32, 3 6,,, 9 32, 9 6,,, 8, 4,,,,,,], Fg : Plot of the axu of the absolute errors n Exaple Exaple 2 Consder the followng lnear syste of Volterra ntegral equatons u (x,y) f (x,y) (2u (t,y)+u 2 (t,y))dt, u 2 (x,y) f 2 (x,y)+ y (u 2(x,s) u (x,s))ds, (x,y) D, where f (x,y)3x 2 y+2x 3 y+ 2 x2 y 3 x, f 2 (x,y)xy 3 4 xy x2 y 2 + y (9) and exact solutons are u (x,y)3x 2 y and u 2 (x,y)xy 3 Wth 3, functons and ernels are approxated as: u (x,y) Φ T (x,y)c, u 2 (x,y) Φ T (x,y)c 2, Φ T (x,y)kφ(t,y) Substtutng above approxatons n syste (9) leads to the followng algebrac syste: Φ T (x,y)c f (x,y)+φ T (x,y)k2 C + C 2 }P x Φ(x,y), Φ T (x,y)c 2 f 2 (x,y) Φ T (x,y)k C 2 C }P y Φ(x,y), (2) C 2 [ , 9 28, 5 96, 8, 5 28, 9 64, 5 48, 4,,,,,,,,], α β 5 : C [ 45 32, 5 45,,, 6 32, 5 6,,, 3 8, 4,,,,,,], C 2 [ 5 256, 63 28, 7 32, 3 8, 35 28, 2 64, 7 48, 4,,,,,,,,], α β 5 : C [ 9 6, 9 8,,, 3 2,3,,, 2,,,,,,,], C 2 [ 27 32, 5 32, 4, 2, 5 6, 5 6, 2,,,,,,,,,], α β : C [ 9 2, 9 4,,, 3 8, 3 6,,,, 2,,,,,,], C 2 [ 9, 3 28, 2, 2, 2, 3 56, 4, 224,,,,,,,,], c 24 SP atural Scences Publshng Cor

11 Sohag J Math, o, 5-26 (24) / wwwnaturalspublshngco/journalsasp 25 Exaple 3 Thrd exaple covers the syste of nonlnear Volterra - Fredhol ntegral equaton u (x,y) f (x,y)+ y u2 (t,s)dtds, u 2 (x,y) f 2 (x,y)+(x y 2 ) y u2 2 (t,s)dtds, (x,y) D, (2) where f (x,y) y 2 + 2xy 9 y3 (445y+8y 2 ), f 2 (x,y)+y 2 sn(x) 667y( y y 4 )(y 2 x) and exact solutons are u (x,y) y xy and u 2 (x,y)+y 2 sn(x) Wth 4, solutons and ernels are approxated as: u (x,y) Φ T (x,y)c, u 2 (x,y) Φ T (x,y)c 2, Φ T (x,y)kφ(t,s), u 2 (x,y) Φ T (x,y)u, u 2 2(x,y) Φ T (x,y)u 2 Substtutng above approxatons n syste (2) leads to the followng algebrac syste Φ(x,y)C f (x,y) Φ T (x,y)kũ P y B, Φ(x,y)C 2 f 2 (x,y) (x y 2 )Φ T (x,y)kũ 2 P y B, (22) where Ũ and Ũ 2 are operatonal atrces of product, P y s operatonal atrx of ntegraton and B s a(+) 2 vector as: B Φ(t,y) dt Usng roots of 5 (x) and 5 n the x and y drectons, each equatons of syste (22) s collocated n 25 nner tensor ponts The proble reduces to solve a syste of nonlnear algebrac equatons whch wll be solved by eans of ewton teratve ethod and 5 unnown coeffcents are deterned for soe values of paraeters α and β For u (x,y), the exact soluton s obtaned The nonzero coponents of vector C for the varous values of paraeters α and β are as follows: α β : c 6, c 2 6, c 5 2, c 6 2, α β 2 : c 3 6, c 2, c 5 3, c 6 2 9, α β 2 : c 8, c 2 3, c 5, c 6 2, α β 2 : c 8, c 2 6, c 5 4, c 6 2, α β 2 : c 2, c 2 8, c 5 3 4, c 6 2, Table 3: Dfferent values of exact and approxate solutons n ponts (,) for 4 and α β 5 of Exaple 3 (x,y ) u 2exact u 2approxate (, ) (2, 2) (3, 3) (4, 4) (5, 5) (6, 6) (7, 7) (8, 8) (9, 9) (, ) Table 4: coparson of the errors n L 2 (D) for dfferent values α and β of Exaple 3 (α,β) Error(L 2 ) (,) ( 5, 5) ( 5,5) (5,5) (5, 5) (,) α β : c 5, c 2 5, c 5 4, c 6 8, α β 2 : c 3 4, c 2 28, c 5 6, c 6 8, Table 3 shows dfferent values of the exact and approxate solutons for u 2 (x,y) n ponts (x,y )(,), (,2,,) for α β 5 Table 4 dsplays the axu absolute errors for values of α and β 6 Concluson Analytc soluton of the two densonal ntegral equatons are usually dffcult In any cases, t s requred to approxate solutons In ths paper, the syste of two densonal lnear and nonlnear ntegral equatons was solved by usng collocaton ethod For ths purpose, the shfted Jacob collocaton ethod was eployed to solve a class of systes of Fredhol and Volterra ntegral equatons Frst, a general forulaton for two densonal Jacob operatonal atrx of ntegraton has been derved Ths atrx s used to approxate nuercal soluton of syste of lnear and nonlnear Volterra ntegral equatons Proposed approach s based on the shfted Jacob collocaton ethod The solutons obtaned usng the proposed ethod shows that ths ethod s a powerful atheatcal tool for solvng the ntegral equatons Provng the convergence of the ethod, consstency and stablty are ensured autoatcally Moreover, only a sall nuber of shfted c 24 SP atural Scences Publshng Cor

12 26 A Borhanfar, Kh Sadr: uercal Soluton by usng Jacob Operatonal Collocaton Method Jacob polynoals s needed to obtan a satsfactory result References [] K E Atnson, The uercal Soluton of Integral Equatons of the Second Knd, Cabrdge Unversty Press, (997) [2] A J Jerr, Introducton to Integral Equatons wth Applcatons, John Wley and Sons, IC, (999) [3] Sahn, S Yuzbas, M Gulsu, A collocaton approach for solvng systes of lnear Volterra ntegral equatons wth varable coeffcents, Coputers and Matheatcs wth Applcatons, 62, (2) [4] G Capobanco, A Cardone, A parallel algorth for large systes of Volterra ntegral equatons of Abel type, Coputatonal and Appled Matheatcs, 22, (28) [5] J Rashdna, M Zarebna, ew approach for nuercal soluton of Haersten ntegral equatons, Appled Matheatcs and Coputaton, 85, (27) [6] H Adb, P Assar, Chebyshev wavelet ethod for nuercal soluton of Fredhol ntegral equatons of the frst nd, Matheatcal probles n Engneerng, (2), do: 55/2/3848 [7] Z M Odbat, Dfferental transfor ethod for solvng Volterra ntegral equaton wth separable ernels, Matheatcal and Coputer Modellng, 48, (28) [8] J Bazar, M Esla, Modfed HPM for solvng systes of Volterra ntegral equatons of the second nd, Journal of Kng Saud Unversty (Scence), 23, (2) [9] F Mrzaee, uercal coputatonal soluton of the lnear Volterra ntegral equatons syste va ratonalzed Haar functons, Journal of Kg Saud Unversty (Scence), 22, (2) [] K Malenead, M Hadzadeh, A new coputatonal ethod for Volterra-Fredhol ntegral equatons, Coputers and Matheatcs wth Applcatons, 37, -8 (999) [] H Guoqang, Asyptotc error expanson for the yströ ethod for a nonlnear Volterra-Fredhol ntegral equaton, Journal of Coputatonal and Appled Matheatcs, 59, (995) [2] E Babolan, Dastan, uercal solutons of twodensonal lnear and nonlnear Volterra ntegral equatons: Hootopy perturbaton ethod and dfferental transfor ethod, Int J Industral Matheatcs, 3, (2) [3] S Hatazadeh, Z Masour, uercal ethod for analyss of one- and two-densonal electroagnetc scatterng based on usng lnear Fredhol ntegral equaton odels, Matheatcal and Coputer Modellng, 54, (2) Abdollah Borhanfar receved the PhD degree fro Moscow Unversty n 24 on nonlnear PDE Hs research nterest s n the areas of appled atheatcs and atheatcal physcs and uercal ethods He has publshed research artcles n reputed nternatonal ournals of atheatcal and physcs scence He s referee and edtor of atheatcal ournal At now he s assocate degree Khadeh Sadr s a PhD student of Appled Matheatcs at Mohaghegh Ardabl of Ardabl, Iran Her research nterest s n the area of appled atheatcs ncludng the nuercal analyss, nuercal lnear algebrac and coputatonal ethods for nonlnear partal dfferental equatons and ntegral equatons c 24 SP atural Scences Publshng Cor