A Generalized Operational Method for Solving Integro Partial. Differential Equations Based on Jacobi Polynomials

Transcription

1 A Generalzed Operatonal Method for Solvng Integro Partal Dfferental Equatons Based on Jacob Polynomals A Borhanfar,a, Kh Sadr a a Department of Mathematcs, Faculty of Mathematcal Scences, Unversty of Mohaghegh Ardabl, Ardabl,Iran E-mal (Corresp): borhan@umaacr E-mal: hsadr@umaacr Abstract In ths paper, a numercal method s developed for solvng lnear and nonlnear ntegro-partal dfferental equatons n terms of the two varables Jacob polynomals Frst, some propertes of these polynomals and several theorems are presented then a generalzed approach mplementng a collocaton method n combnaton wth two dmensonal operatonal matrces of Jacob polynomals s ntroduced to approxmate the soluton of some ntegro partal dfferental equatons wth ntal or boundary condtons Also, t s shown that the resulted approxmate soluton s the best approxmaton for the consdered problem The man advantage s to derve the Jacob operatonal matrces of ntegraton and product to acheve the best approxmaton of the two dmensonal ntegro dfferental equatons umercal results are gven to confrm the relablty of the proposed method for solvng these equatons Keywords: Collocaton method, Integro partal dfferental equatons, Operatonal matrx, Shfted Jacob polynomals, Best approxmaton MSC: 47B36, 14R15, 39B5, 65D15, 15B99, 65R2, 45K5, 47G1,34K28 1 Introducton Fndng the analytcal solutons of functonal equatons has been devoted attenton of mathematcans s nterest n recent years Several methods are proposed to acheve ths purpose, such as [7, 8, 9, 1, 11, 13] Many problems n theoretcal physcs and other scences lead to ntegro partal dfferental equatons In order to solve these

2 equatons, several numercal methods have been proposed such as [21, 22, 25, 26, 28, 29] The solutons of ths nd of equatons are often qute complcated For ths reason n many cases, t s requred to obtan the approxmate solutons For example the Jacob collocaton method has been appled to solve varous dfferental equatons, [3, 14, 15, 17, 18, 19, 2, 23] Also, Bhrawy and et al have presented a new Legendre spectral collocaton method for fractonal Burgers equatons, [6] In [16], authors have used Jacob Gauss Lobatto collocaton method for the numercal soluton of nonlnear Schrödnger equaton Also, Bhrawy n [1] has presented an pseudospectral approxmaton based on Jacob polynomals for generalzed Zaharov system Two spectral tau algorthms based on Jacob polynomals have been appled to solve mult term tme space fractonal partal dfferental equatons and tme fractonal dffuson Wave equatons, [4, 5] Bhrawy and et al have been presented two dfferent collocaton scheme for both temporal and spatal dscrmnatons of moble mmoble advecton dsperson model (TVFO MIAD model), [2] Borhanfar and Sadr have utlzed a Jacob operatonal collocaton method for systems of two dmensonal ntegral equatons, [12] In ths paper, the Jacob polynomals are used as a bass functon for solvng lnear and nonlnear ntegro partal dfferental equatons, the numercal soluton, u(x, y), s approxmated n terms of two varables Jacob polynomals as x, y [, 1] In order to realze ths am, the shfted Jacob polynomals together the collocaton technque are used The Jacob operatonal matrces of the ntegraton and product are constructed on the nterval [, 1] The man am s to mprove Jacob operatonal matrces to the spectral soluton of partal ntegro-dfferental equatons For solvng the resulted algebrac system, the ( + 1) roots of one varable Jacob polynomals (α,β) + 1 (x) and P + 1 (y) are consdered n the x, y drectons The doman of two dmensonal s represented by a tensor product ponts {x } = and {y j } j= whch are roots of + 1 (α,β) (x) and P (y) Each the equatons of the algebrac system s collocated n the resulted tensor ponts + 1 {(x, y j)},j= and s gven lnear or nonlnear systems of algebrac equatons whch can be solved usng the well nown ewtons teratve method Thus, the Jacob coeffcents are obtaned and the approxmate soluton s determned The remander of ths paper s organzed as follows: The Jacob polynomals, some of ther propertes and one dmensonal operatonal matrx of ntegraton are ntroduced n Secton 2 Afterwards, some propertes of two varables Jacob polynomals are stated and the operatonal matrces of ntegraton and product wll be extended to two dmensonal case n Secton 3 In Secton 4, the exstence and unqueness of the best approxmaton are studed and an error estmator s ntroduced Secton 5 s devoted to applyng two dmensonal Jacob operatonal matrces for solvng the partal ntegro dfferental equatons For ths purpose, four examples are presented A concluson s presented n Secton 6 2

3 2 Jacob polynomals and ther operatonal matrx of ntegraton The well-nown Jacob polynomals are defned on the nterval z [ 1, 1], consttute an orthogonal system wth respect to the weght functon w (α,β) (z) = (1 z) α (1 + z) β, and can be determned wth the followng recurrence formula: + 1 (α,β) (z) = A(α, β, ) P (z) + z B(α, β, ) (z) D(α, β, ) (z), = 1, 2,, (1) 1 where A(α, β, ) = B(α, β, ) = D(α, β, ) = and (2 + α + β + 1)(α 2 β 2 ) 2( + 1)( + α + β + 1)(2 + α + β), (2 + α + β + 2)(2 + α + β + 1), 2( + 1)( + α + β + 1) ( + α)( + β)(2 + α + β + 2) ( + 1)( + α + β + 1)(2 + α + β), (z) = 1, 1 (z) = α + β z + α β 2 The orthogonalty condton of Jacob polynomals s 1 where j (z) (z) w (α,β) (z) dz = h δ j, h = 2α+β+1 Γ( + α + 1)Γ( + β + 1) (2 + α + β + 1)! Γ( + α + β + 1) The analytc form of Jacob polynomals s gven by, [27], (z) = = ( 1) ( ) Γ( + β + 1)Γ( + + α + β + 1) Γ( + β + 1)Γ( + α + β + 1) ( )!! ( 1 + z 2 ), =, 1, For practcal use of Jacob polynomals on the nterval x [, 1], t s necessary to shft the defnng doman by means of the followng change varable: z = 2 x 1, x [, 1] The shfted Jacob polynomals are generated from the three-term recurrence relaton +1 (α,β) (x) = A(α, β, ) P (x) + (2 x 1) B(α, β, ) (x) D(α, β, ) (x), = 1, 2,, (2) (x) = 1, (α + β + 2)(2 x 1) 1 (x) = 2 + α β 2 The orthogonalty condton and weght functon are respectvely, 1 (x) (x) w (α,β) (x) dx = θ δ j, j where θ = Γ( + α + 1)Γ( + β + 1) (2 + α + β + 1)! Γ( + α + β + 1), 3

4 and w (α,β) (x) = (1 x) α x β, x [, 1] Also, the analytc form of shfted Jacob polynomals wll be as follows, [27], (x) = = ( 1) ( ) Γ( + β + 1)Γ( + + α + β + 1) x, =, 1, (3) Γ( + β + 1) Γ( + α + β + 1) ( )!! A contnuous functon u(x), square ntegrable n [, 1], can be expressed n terms of shfted Jacob polynomals as u(x) = j= C j j (x), (4) where the coeffcents C j are gven by C j = 1 θ j u(x) j (x) w (α,β) (x) dx, j =, 1, In practce, only the frst ( + 1) terms shfted Jacob polynomals are consdered Therefore, one has u (x) = j= C j j (x) = Φ T (x) C = C T Φ(x), (5) where the vectors C and Φ(x) are gven by C = [C, C 1,, C ] T, Φ(x) = [ (x), 1 (x),, (x)] T (6) Some other propertes of the shfted Jacob polynomals are presented as follows (1) (2) d dx + α () = ( 1), P n (α,β) Γ(n + α + β + + 1) (x) = Γ(n + α + β + 1) P (α+,β+) n (x) Lemma 1 The shfted Jacob polynomal (x) can be obtaned n the form of: (x) = = γ () x, where γ () γ () are = ( 1) + + α + β + α Proof γ () γ () d = 1! dx can be obtaned as, (x) x= ow, usng propertes (1) and (2), the lemma can be proved Lemma 2 If p > β 1, then x p β n (x) w (α,β) (x) dx = n l= ( 1) n l Γ(n + β + 1) Γ(n + l + α + β + 1) Γ(p + l + 1) Γ(α + 1) Γ(l + β + 1) Γ(n + α + β + 1) Γ(p + l + α + 2) (n l)! l! 4

5 Proof For p β < n one has x p β n (x) w (α,β) (x) dx = Hence, we suppose p β n From analytc form of shfted Jacob polynomals, Eq(3), one has xp β P n (α,β) (x) w (α,β) (x) dx = n ( 1) n l Γ(n+β+1) Γ(n+l+α+β+1) l= B(p + l + 1, α + 1) Γ(l+β+1) Γ(n+α+β+1) (n l)! l! = n ( 1) n l Γ(n+β+1) Γ(n+l+α+β+1) Γ(p+l+1) Γ(α+1) l=, Γ(l+β+1) Γ(n+α+β+1) Γ(p+l+α+2) (n l)! l! where B(s, t) s the Beta functon and s defned as B(s, t) = v s 1 (1 v) t 1 dv = Γ(s) Γ(t) Γ(s + t) Lemma 3 If j (x) and (x) are j th and th shfted Jacob polynomals, respectvely, then the product of j (x) and (x) can be wrtten as Q (α,β) j+ j+ (x) = r= λ (j,) r x r Proof Defnng the Q (α,β) (α,β) (x) = P (x) (x) as a polynomal of degree j + that can be wrtten as j+ j j Q (α,β) j+ (x) = ( γ m (j) x m ) ( m= n= j+ γ n () x n ) = λ (j,) r x r r= The relaton between coeffcents λ (j,) n wth coeffcents γ (j) m and γ () m wll be as follows If j : r =, 1,, j +, f r > j then λ (j,) r = l=r j γ(j) r l γ() l, else (7) r 1 = mn{r, }, λ (j,) r = r 1 l= γ(j) r l γ() l, end 5

6 If j < : r =, 1,, j +, f r j then r 1 = mn{r, j}, else λ (j,) r = r 1 l= γ(j) r l γ() l, (8) r 2 = mn{r, }, end λ (j,) r = r 2 l=r j γ(j) r l γ() l, Thus, the coeffcents λ (j,) r s determned Lemma 4 If then (x), (x) and (x) are, j and th shfted Jacob polynomals, respectvely, j (x) j (x) (x) w (α,β) (x) dx = j+ n= l= ( 1) l λ (j,) n Γ( + β + 1) Γ( + l + α + β + 1) Γ(n + l + β + 1) Γ(α + 1), Γ(l + β + 1) Γ( + α + β + 1) Γ(n + l + α + β + 2) ( l)! l! where λ (j,) n has been ntroduced n Lemma 3 Proof Assumng that j (x) (x) = Q (α,β) (x) Usng of Eq(3), Lemmas 2 and 3 leads to j+ (x) j (x) (x) w (α,β) (x) dx = = j+ n= λ(j,) n (x) Q (α,β) (x) w(α,β) (x) dx j+ xn (x) w (α,β) (x) dx = j+ ( 1) l Γ(+β+1) Γ(+l+α+β+1) n= l= B(n + l + β + 1, α + 1), Γ(l+β+1) Γ(+α+β+1) ( l)! l! Thus, desrable result s obtaned In performng arthmetc and other operatons on the Jacob bases, we frequently encounter the ntegraton of the vector Φ(x) defned n Eq(6) whch s called the operatonal matrx of the ntegraton Hence, the matrx relatons must be obtaned In ths secton, ths operatonal matrx wll be derved, then t wll be extended to two dmensonal case n next secton To ths end, some useful lemmas and theorems are stated Theorem 5 Let Φ(x) be shfted Jacob vector n Eq(6) Then Φ(t) dt P Φ(x), 6

7 where P s the ( + 1) ( + 1) operatonal matrx of the ntegraton and s defned by: Ω(, ) Ω(, 1) Ω(, ) Ω(1, ) Ω(1, 1) Ω(1, ) P =, Ω(, ) Ω(, 1) Ω(, ) where Ω(, j) = l= and ω jl are gven by ω jl,, j =, 1,,, (9) ω jl = ( 1) l Γ( + β + 1) Γ( + l + α + β + 1) Γ(l + β + 1) Γ( + α + β + 1) (l + 1)! ( l)! j = ( 1) j Γ(j + + α + β + 1) Γ(j + β + 1) Γ( + l + β + 2) Γ(α + 1),, j =, 1,,, l =, 1,, θ j Γ( + β + 1) Γ(j + α + β + 1) Γ( + l + α + β + 3)! (j )! Proof Integratng of the analytcal form of (x), e Eq(3), from to x leads to (t) dt = l= ( 1) l Γ( + β + 1) Γ( + l + α + β + 1) x l+1 Γ(l + β + 1) Γ( + α + β + 1) (l + 1)! ( l)! (1) ow, one can approxmate x l+1 n terms of shfted Jacob polynomals as x l+1 = a l,j j (x), where = a l,j = 1 θ j x l+1 j (x) w (α,β) (x) dx But accordng to Lemma 2 one has, x l+1 j (x) w (α,β) (x) dx = j = ( 1) (j ) Γ(j + + α + β + 1) Γ(j + β + 1) Γ( + l + β + 2) Γ(α + 1) Γ( + β + 1) Γ(j + α + β + 1) Γ( + l + α + β + 3)! (j )! Therefore, Eq(1) wll be as follows: (t) dt = = { j= = l= ( 1) l Γ( + β + 1) Γ( + l + α + β + 1) Γ(l + β + 1) Γ( + α + β + 1) (l + 1)! ( l)! j ( 1) j Γ(j + + α + β + 1) Γ(j + β + 1) Γ( + l + β + 2) θ j Γ( + β + 1) Γ(j + α + β + 1) Γ( + l + α + β + 3) (j )! ()! } j (x) j= Ω(, j) j (x) (11) where Ω(, j) are gven n Eq(9) Accordngly, rewrtng Eq(11) as a vector form gves (t) dt = [Ω(, ), Ω(, 1),, Ω(, )] Φ(x), =, 1,, Ths leads to the desred result 7

8 3 Two varables Jacob polynomals and ther operatonal matrces ow n ths secton, two varables Jacob polynomals can be defned by means of one varable Jacob polynomals as follows: Defnton 6 Let {P n (α,β) (x)} n= be the sequence of one varable shfted Jacob polynomals on D = [, 1] Two varables Jacob polynomals, {P m,n (α,β) (x, y)} m,n= are defned on D 2 = [, 1] [, 1] as: m,n (x, y) = P m (α,β) (x) P n (α,β) (y), (x, y) D 2 (12) The famly { m,n (x, y)} m,n= s orthogonal wth weghted functon w (α,β) (x, y) = w (α,β) (x) w (α,β) (y) on D 2 and forms a bass for L 2 (D 2 ) Theorem 7 The bass { m,n (x, y)} m,n= s orthogonal on D 2 Proof One has P m,n (α,β) (x, y),l (x, y) w (α,β) (x, y) dxdy = m (x) (x)w (α,β) (x) dx n θ m θ n, (m, n) = (, l), =, (m, n) (, l) or m or n l (y) l Smlar to one varable case, a two varables contnuous functon u(x, y) defned over D 2 may be expanded by the (y) w (α,β) (y) dy two varables Jacob polynomals as: u(x, y) = C j,j (x, y), = j= where the Jacob coeffcents, C j, are obtaned as: C j = 1 θ θ j,j (x, y) u(x, y) w (α,β) (x, y) dxdy (13) If the nfnte seres n Eq(13) s truncated up to ther ( + 1) terms n terms of both two varables x and y then t can be wrtten as: u(x, y) u (x, y) = C j,j (x, y) = Φ T (x, y) C, (14) = j= where C and Φ(x, y) are Jacob coeffcents and Jacob polynomals vectors, respectvely: C = [C, C 1,, C, C 1,, C 1,, C 1,, C ] T, Φ(x, y) = [, (x, y),,1 (x, y),, (α,β), (x, y), P 1, (x, y),, (α,β) (α,β) 1, (x, y),, P, (x, y),, P, (x, y)]t A functon of four varables, (x, y, t, s), on D 4 may be approxmated based on Jacob operatonal matrx as: (15) (x, y, t, s) Φ T (x, y) K Φ(t, s), 8

9 where Φ(x, y) s two varables Jacob vector defned by Eq(15) and K s a ( + 1) 2 ( + 1) 2 nown matrx Before proceedng, let us represent the partal seres (14) as followng form: S (x, y) = d Q (α,β) (x, y), where d = C rs, = Q (x, y) = P r,s (α,β) (x, y), r = + 1, s = r( + 1), M = ( + 1)2 1 (16) Hence, one has Q (α,β) = { (x, y) Q (α,β) (x, y) w (α,β) (x, y) dxdy l +1 (α,β) (x) P l (x) w(α,β) (x) dx} { +1 (α,β) (+1)(y) P l l +1 = θ +1 θ +1 (+1) = θ r θ s, r =, s = r( + 1) (+1)(y) w(α,β) (y) dy} Remar 8 The relaton (16) can be rewrtten as: S (x, y) = ω R (α,β) (x, y), where = ω = d θrθ s, R (α,β) (x, y) = Q(α,β) (x, y), r =, s = r( + 1) θr θ s + 1 (17) Ths shows the sequence {R (α,β) (x, y)} M = s orthonormal That s R (α,β) (x, y) R (α,β) l (x, y) w (α,β) (x, y) dxdy = 1, = l,, otherwse ow, the two dmensonal operatonal matrces of ntegraton n x and y drecton are defned by followng theorem: Theorem 9 If P s the operatonal matrx n one dmensonal case then the operatonal matrces of ntegraton n x and y drecton are defned as follows a) b) y Φ(t, y) dt P x Φ(x, y) = (P I) Φ(x, y), Φ(x, s) ds P y Φ(x, y) = (I P ) Φ(x, y), where P x and P y are ( +1) 2 ( +1) 2 operatonal matrces of ntegraton n the drectons x and y, respectvely, I s ( + 1) ( + 1) dentty matrx and denotes the Kronecer product and s defned for two arbtrary matrces A and B as A B = (a jb) Proof a) Suppose r j be jth row of matrx P One has j (t) dt = rj T Φ(x) Also, notng the defnton of the vector Φ(x, y) one has Φ(x, y) = [ (x) (y),, (x) (y),, (x) (y),, (x) (y)] T (18) 9

10 Integratng of Eq(18) from to x yelds Φ(t, y)dt = [ (y) (t)dt,, (y) (t)dt,, (y) (t)dt] T = [r Φ(x) (y),, r Φ(x) (y),, r Φ(x) (y),, r Φ(x) (y)] T = [r [ (x) (y),, (x) (y)],, r [ (x) (y),, (x) (y)]] T, P P 1 P P P 1 P P P 1 P = P P 1 P P P 1 P P P 1 P (y) (x) 1 (y) (x) (y) (x) (y) (x) 1 (y) (x) (y) (x) 1

11 P I P 1I P I P 1 I P 11 I P 1 I = P I P 1I P I (x) Φ(y) 1 (x) Φ(y) = (P I)Φ(x, y) (x) Φ(y) Where P j denotes (, j) th entry of the matrx P The case (b) s proven smlarly 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) C C Φ(x, y), (19) where C and C are a ( + 1) 2 1 vector and a ( + 1) 2 ( + 1) 2 operatonal matrx of product, respectvely Theorem 1 The entres of the matrx C, n Eq(19), are computed as: C m(+1)+n,(+1)+l = 1 θ θ l r= s= Proof The left sde of equalty (19) s as follows: Φ(x, y) Φ T (x, y) C = r= s= C rs q mr q nsl, m, n,, l =, 1,, (α,β) r= s= Crs P, (x, y) r,s (x, y) r,s (x, y) (α,β) (x, y) P r,s (x, y) r= s= C rs,1 (x, y) (α,β) Crs P, Consder the (p + 1)th row of above vector One puts,j (x, y) P r,s (α,β) (x, y) = u l,l (x, y), (2) = l= Multplyng both sdes of Eq(2) by m,n (x, y), m, n =, 1,,, and ntegratng of the result from to 1 yelds:,j (x, y) P r,s (α,β) (x, y) P m,n (α,β) (x, y) w (α,β) (x, y) dxdy = u l = l= = u mn θ m θ n,l (x, y) (x, y) w (α,β) (x, y) dxdy m,n Therefore, u mn = 1 θ m θ n { ow suppose u mn = qrm qjsn θ m θ n (x) P r (α,β) (x) P m (α,β) (x) w (α,β) (x)dx} { j (x) (x) (x) w (α,β) (x) dx = q rm one gets r m 11 (y) (y) (y) w (α,β) (y) dy} s n

12 Substtutng u mn nto Eq(2) one has:,j (x, y) P r,s (α,β) q rm q jsn (x, y) = θ m θ n = l=,l (x, y), Hence each component n the left sde of relaton (19) wll be as follows: { = l= r= s= C rs q mr q nsl },l (x, y) = θ θ l Thus, the desrable result s obtaned = l= C m(+1)+n,(+1)+l,l (x, y), m, n =, 1,, The ext theorem presents the general formula approxmatng the nonlnear term v r (x, y) u s (x, y) whch may appear n nonlnear equatons Theorem 11 If c and υ are the ( + 1) 2 vectors, c and υ are the ( + 1) 2 ( + 1) operatonal matrces of the product such that u(x, y) Φ T (x, y) c = c T Φ(x, y), v(x, y) Φ T (x, y) υ = υ T Φ(x, y), Φ(x, y) Φ T (x, y) c c Φ(x, y) and Φ(x, y) Φ T (x, y) υ υ Φ(x, y), then the followng proposton s hold: v r (x, y) u s (x, y) υ T ( υ) r 1 Bs 1 Φ(x, y), B s 1 = ( c T ) s 1 c, r, s = 1, 2, Proof One has u 2 (x, y) (Φ T (x, y) c) 2 = c T Φ(x, y) Φ T (x, y) c c T c Φ(x, y), So, by use of nducton, u s (x, y) wll be approxmated as u s (x, y) c T ( c) s 1 Φ(x, y), s = 1, 2, To smlar way, v r (x, y) s approxmated as v r (x, y) υ T ( υ) r 1 Φ(x, y), r = 1, 2, By usng the expressed relatons and nducton s easly seen, v r (x, y) u s (x, y) υ T ( υ) r 1 Bs 1 Φ(x, y), B s 1 = ( c T ) s 1 c, r, s = 1, 2, 4 Best approxmaton and Convergence analyss In ths secton, the theorems on exstence and unqueness of best approxmaton, convergence analyss and error estmaton of the proposed method are provded For ths reason, frst the space P M s consdered as follows: Defnton 12 The set of all the lnear combnatons of R (α,β) (x, y), R (α,β) 1 (x, y),, R (α,β) (x, y), whch M = ( + 1) 2 1, s represented by P M On the other hand, M P M = span {R (α,β) (x, y), R (α,β) 1 (x, y),, R (α,β) (x, y)}, (21) M 12

13 where two varables orthonormal polynomals R (α,β) (x, y) are ntroduced by Eq(17) The followng lemma s useful to prove the convexty and completeness propertes of space P M Lemma 13 There s a number η > such that for every choce of scalars α, α 1,, α M one has α R (α,β) (x, y) + α 1 R (α,β) 1 (x, y) + + α M R (α,β) (x, y) η ( α + α1 ++ αm ) M Proof See [24] Theorem 14 The space P M, defned by (21), s convex and complete Proof Suppose v 1 (x, y) and v 2 (x, y) P M One has for < λ < 1 λ v 1 (x, y) + (1 λ) v 2 (x, y) = Ths shows that P M s convex = (λ ω 1 + (1 λ) ω 2 ) R (α,β) (x, y) P M For provng the completeness property, let us consder Cauchy sequence {w n (x, y)} n= P M Then each w n (x, y) s a unque representaton of the form w n(x, y) = λ (n) R (α,β) (x, y) = Snce {w n (x, y)} n= s a Cauchy sequence, for every ε > there s a such that w m (x, y) w n (x, y) < ε where m, n > From ths and Lemma 13, one has for η > ε > w m (x, y) w n (x, y) = = (λ (m) λ (n) ) R (α,β) (x, y) η = λ (m) λ (n) Dvson by η > gves λ (m) λ (n) λ (m) = λ (n) < ε η, Ths shows that each of the M + 1 sequences {λ (n) } n=, =, 1,, M, s Cauchy n R Hence t converges Let λ denotes the lmt Usng ths M + 1 lmts λ, λ 1,, λ M, one defnes w(x, y) = λ R (α,β) (x, y) = Clearly, w(x, y) P M Furthermore, w n (x, y) w(x, y) = = (λ (n) λ ) R (α,β) (x, y) = λ (n) λ R (α,β) (x, y) On the rght, λ (n) λ Hence w n (x, y) w(x, y), that s, w n (x, y) w(x, y) Ths shows that {w n (x, y)} n= s convergent n P M, and the completeness of P M s proven Theorem 15 For every gven contnuous functon u(x, y) there exsts a unque u M (x, y) P M such that δ = nf u(x, y) ũ(x, y) = u(x, y) um (x, y) ũ PM 13

14 Proof Exstence By the defnton of an nfmum, there s a sequence {w n (x, y)} n= n P M such that δ n δ where δ n = u(x, y) w n(x, y) We show that {w n(x, y)} n= s Cauchy Wrtng v n(x, y) = u(x, y) w n(x, y), one has v n(x, y) = δ n and v m(x, y) + v n(x, y) = 2 1 (wm(x, y) + wn(x, y)) u(x, y) 2 δ, 2 because P M s convex, so that 1 2 (w m(x, y)+w n (x, y)) P M Furthermore, one has v m (x, y) v n (x, y) = w n (x, y) w m (x, y) Hence by the parallelogram equalty w n(x, y) w m(x, y) 2 = v m(x, y) v n(x, y) 2 = v m (x, y) + v n (x, y) 2 +2 ( v m (x, y) 2 + v n (x, y) 2 ) (2 δ) 2 + 2(δ 2 m + δ 2 n) < ε 2 Ths mples that {w n (x, y)} n= s Cauchy Snce P M s complete, {w n (x, y)} n= converges, say, w n (x, y) w(x, y) P M Snce w(x, y) P M, one has u(x, y) w(x, y) δ Also, u(x, y) w(x, y) u(x, y) w n (x, y) + w n (x, y) w(x, y) = δ n + w n (x, y) w(x, y) δ Ths shows that u(x, y) w(x, y) = δ Unqueness Let us assume that w(x, y) P M and w (x, y) P M both satsfy u(x, y) w(x, y) = δ, u(x, y) w (x, y) = δ By the parallelogram equalty, w(x, y) w (x, y) 2 = ( w(x, y) u(x, y)) (w (x, y) u(x, y)) 2 = 2 w(x, y) u(x, y) 2 +2 w (x, y) u(x, y) 2 w(x, y) u(x, y))+(w (x, y) u(x, y)) 2 = 4 δ ( w(x, y) + w (x, y)) u(x, y) 2, because 1 2 ( w(x, y) + w(x, y)) PM So that w(x, y) = w (x, y) Orthogonalty We assume there be a w 1(x, y) P M such that (z(x, y), w 1(x, y)) = γ where z(x, y) = u(x, y) w(x, y) and (, ) denotes the nner product Furthermore, for any scalar η, z(x, y) η w 1(x, y) 2 = (z(x, y) η w 1(x, y), z(x, y) η w 1(x, y)) = z(x, y) 2 η γ η ( γ η w 1(x, y) 2 ), Choosng η = γ w 1 (x,y) 2 yelds z(x, y) η w 1 (x, y) 2 = z(x, y) 2 γ w 1 (x, y) = γ 2 δ2 δ 2, w 1 (x, y) 2 But ths s mpossble because one has z(x, y) η w 1(x, y) δ by the defnton of δ Hence the assumpton can not be hold So (z(x, y), ũ(x, y)) =, ũ(x, y) P M 14

15 ow t s shown that w(x, y) = u M (x, y) It was proven that w(x, y) s the best approxmaton for u(x, y) So, j, j =, 1,, M, (u(x, y) w(x, y), R j (x, y)) =, where two varables polynomals R (α,β) (x, y) are ntroduced by Eq(17) One has from ths pont, ( w(x, y) u M (x, y), R j(x, y)) = (u(x, y) (u(x, y) w(x, y)) u M (x, y), R j(x, y)) = (u(x, y), R j(x, y)) (u(x, y) w(x, y), R j(x, y)) (u M (x, y), R j(x, y)) = ω j ω j =, j =, 1,, M, Therefore w(x, y) u M (x, y) = Ths shows that u M (x, y) = w(x, y) and proof s completed Two followng theorems state the decayng of the Jacob coeffcents and the convergence of the best approxmaton Theorem 16 The Jacob coeffcents ω, ntroduced by (17), decay when the number of the terms of the partal sum of the seres soluton,, ncreases Proof Employng the ω and the propertes of the orthogonalty of R (x, y), we have S M (x, y) u(x, y) w (α,β) (x, y) dxdy = ω = = If u 2 (x, y) w (α,β) (x, y) as well as u(x, y) w (α,β) (x, y) s ntegrable, then R (x, y) u(x, y) w (α,β) (x, y) dxdy = ω 2 [u(x, y) S M (x, y)] 2 w (α,β) (x, y) dxdy = Therefore, = ω 2 + = u 2 (x, y) w (α,β) (x, y) dxdy,, Consequently = ω2 s convergent and lm ω = u 2 (x, y) w (α,β) (x, y) dxdy 2 S 2 M (x, y) w (α,β) (x, y) dxdy u 2 (x, y) w (α,β) (x, y) dxdy ω 2, = u(x, y) S M (x, y) w (α,β) (x, y) dxdy Theorem 16 states the gven functon u(x, y) may be approxmated usng only the fnte numbers of two varables Jacob polynomals Theorem 17 The seres soluton (14) converges towards u(x, y) n (13) Proof Consder the relaton (17) and defne the sequence partal sums {S M (x, y)} M= as follows, S M (x, y) = ω R (x, y) = Let suppose that the S M (x, y) and S L (x, y) are partal sums wth M > L It s gong to prove that {S M (x, y)} M= s a Cauchy sequence n P M For ths purpose, t s wored out as follows: =L+1 ω R (x, y) 2 = ( =L+1 ω R (x, y), j=l+1 ω j R j (x, y)) = =L+1 j=l+1 ω ω j (R (x, y), R j (x, y)) = =L+1 ω 2 15

16 That s S M (x, y) S L (x, y) 2 = M =L+1 ω 2 From Bessel nequalty = ω 2 s convergent and hence S M (x, y) S L(x, y) 2 ε 2 Ths shows {S M (x, y)} M= s a Cauchy sequence Snce P M s complete one has S M (x, y) S(x, y) P M We show S(x, y) = u(x, y): (S(x, y) u(x, y), R j (x, y)) = (S(x, y), R j (x, y)) (u(x, y), R j (x, y)) Hence S(x, y) = = ω R (x, y) = = = ( lm SM (x, y), Rj(x, y)) (u(x, y), Rj(x, y)) M = lm M (S M (x, y), R j (x, y)) (u(x, y), R j (x, y)) = ω j ω j =, S(x, y) u(x, y) =, j =, 1,, M j= C j (x) j (y) = u(x, y) Whenever the soluton of a problem s not nown, specally n nonlnear phenomena, an error estmator s needed as an essental component of the computatonal algorthm To ths end, an error estmator for the proposed method s presented n ths secton For smplcty, dscussed equatons are wrtten n the operator form Du(x, y) = f(x, y) (22) where D s a ntegro-partal dfferental operator Defne the error functon as e (x, y) = u(x, y) u (x, y) Substtutng u (x, y) nto gven equatons yelds Du (x, y) = f(x, y) + H (x, y), (23) where H (x, y) s a perturbaton term Subtractng Eq(23) from Eq(22) gves De (x, y) = H (x, y) ow, t can be proceeded by the same way as the basc problem s solved to get the estmaton e,m (x, y) to the error functon e (x, y) ote the stated approxmatons are also substtuted n the condtons of the gven problem Subsequently, the condtons of new problem wll be homogeneous 5 umercal results In ths secton, four examples are gven to certfy the effcency and accuracy of the proposed method where the maxmum absolute and estmate errors are reported for dfferent values of parameters α and β Also, the absolute and estmate errors are computed at some arbtrary selected ponts Example 1 Consder the followng lnear Volterra-Fredholm ntegro-partal dfferental equaton u xx (x, y) + sn(xy) u(x, y) = xy sn(xy) 1 6 y3 + y 16 ts u t (t, s) dtds, (24)

17 wth the followng condtons, u x (, y) = y, u(, y) = (25) The exact soluton of ths problem s u(x, y) = xy Frst, let us consder the followng approxmaton, u xx(x, y) Φ T (x, y) C (26) Integratng Eq(26) wth respect to x from to x, one gets the followng approxmaton for u x (x, y) u x (x, y) Φ T (x, y) P T x C + u x (, y) Φ T (x, y) P T x C + Φ T (x, y) V, (27) where u x(, y) s approxmated by Φ T (x, y) V whch V s a ( + 1) 1 nown vector Agan, ntegratng Eq(27) wth respect to x from to x, one gets the followng approxmaton for u(x, y), u(x, y) Φ T (x, y) (P T x ) 2 C + Φ T (x, y) P T x V (28) In order to approxmate the ntegral part n the Eq(24), the ernel ts s approxmated as follows: ts Φ T (x, y) K Φ(t, s), (29) where K s a ( + 1) 2 ( + 1) 2 nown matrx and s determned by nner product ow, the ntegral part n (24) s approxmated as: y ts u t (t, s) dtds y Φ T (x, y) K Φ(t, s) {Φ T (t, s) P T x C +Φ T (t, s) V } dtds Φ T (x, y) K{Ṽ + B}P y A, where Ṽ s operatonal matrx of product and ts entres are determned n terms of the components of the vector V, B s operatonal matrx of product correspondng to vector B = P T x C, and A = Φ(t, y) dt Substtutng the approxmatons (26)-(3) nto Eq(24), leads to the followng lnear algebrac equaton (3) Φ T (x, y) C + sn(xy) Φ T (x, y) {(P T x ) 2 C + P T x V } Φ T (x, y) K{Ṽ + B}P y A xy sn(xy) y3 =, (31) Settng = 3 and usng the roots of 4 (x) and 4 (y) n the x and y drectons, Eq(31) s collocated n 16 nner tensor ponts for dfferent values of parameters α and β Hereby, the Eq(31) reduces the problem to solve a system of lnear algebrac equatons and unnown coeffcents are obtaned for some values of parameters α and β Table 1 shows the maxmum absolute and estmate errors of the approxmate solutons for dfferent values of α and β Table 2 dsplays dfferent values of the exact and approxmate solutons n ponts (x, y) = (1, 1), ( = 1, 2,, 1) for α = β = 1/4 As can be seen from Tables the results of the solutons obtaned by Jacob polynomals method are almost the same as the results of the exact solutons 17

18 Example 2 Consder the followng lnear Volterra ntegro-partal dfferental equaton u x (x, y) + u y (x, y) = 1 + e x + e y + e x+y + y u(t, s) ds dt, (32) wth the condtons u(x, ) = e x and u(, y) = e y The exact soluton of ths problem s u(x, y) = e x+y Let us consder the followng approxmaton, u xy (x, y) Φ T (x, y) C (33) Integratng Eq(33) wth respect to y from to y, one gets the followng approxmaton for u x (x, y) u x (x, y) Φ T (x, y) P T y C + u x (x, ) Φ T (x, y) P T y C + Φ T (x, y) V 1 (34) ow, ntegratng Eq(33) wth respect to x from to x, one gets the followng approxmaton for u y(x, y) as follows: u y (x, y) Φ T (x, y) P T x C + u y (, y) Φ T (x, y) P T x C + Φ T (x, y) V 2 (35) Also, by ntegratng of relaton (34) wth respect to x from to x an approxmaton yelds for u(x, y) as follows: u(x, y) Φ T (x, y) P T x P T y C +Φ T (x, y) P T x V 1+u(, y) Φ T (x, y) P T x P T y C +Φ T (x, y) P T x V 1+Φ T (x, y) V 1 (36) The ernel s approxmated as follows: 1 Φ T (x, y) K Φ(t, s) (37) ow, the ntegral part n (32) s approxmated as: y u(t, s) dtds y Φ T (x, y) K Φ(t, s) Φ T (x, y) {P T x P T y C+P T x V 1 +V 1 } dsdt Φ T (x, y) K à P y P x Φ(x, y), (38) where à s operatonal matrx of product correspondng to vector A = P x T Py T C + Px T V 1 + V 1 Substtutng the approxmatons (34)-(38) nto Eq(32), leads to the followng lnear algebrac equaton Φ T (x, y) P T y C +Φ T (x, y) V 1 +Φ T (x, y) P T x C +Φ T (x, y) V 2 Φ T (x, y) K à Py Px Φ(x, y)+1 ex e y e x+y =, Settng = 7 and usng the roots of 8 (x) and 8 (y) n the x and y drectons, Eq(39) s collocated n 64 nner tensor ponts for dfferent values of parameters α and β Hereby, the Eq(39) reduces the problem to solve a system of lnear algebrac equatons and unnown coeffcents are obtaned for some values of parameters α and β Table 3 dsplays the maxmum absolute and estmate errors of the approxmate solutons for dfferent values of α and β Table 4 shows dfferent values of the exact and approxmate solutons n ponts (x, y) = (2, 2), ( = 1, 2,, 5) for α = β = 1 and = 4, 7, 8 It can be observed from Table 4 that the errors decrease as ncreases Also, (39) 18

19 Example 3 Consder the followng nonlnear system of Fredholm ntegro-partal dfferental equaton u(x, y) v(x, y) + u(t, y) v t(t, y) dt = f 1 (x, y), v(x, y) + 3u(x, y) u t(t, y) v(t, y) dt = f 2 (x, y), (4) where f 1 (x, y) = x 2 cos(y) y sn(x) y cos(y) (sn(1) 2 cos(1)) and f 2 (x, y) = y sn(x) + 3 x 2 cos(y) 2 y cos(y) (sn(1) cos(1)) wth boundary condtons u(, y) = and v(, y) = The exact solutons of ths problem are u(x, y) = x 2 cos(y) and v(x, y) = y sn(x) The followng approxmatons are used for = 5, u x(x, y) Φ T (x, y) C 1, u(x, y) Φ T (x, y) P T x C 1 v x(x, y) Φ T (x, y) C 2, v(x, y) Φ T (x, y) P T x C 2 1 Φ T (x, y) K Φ(t, y), u(t, y) v t (t, y) A T Φ(t, y) Φ T (t, y) C 2 A T C2 Φ(t, y) = Φ T (t, y) C 2 T A, u t(t, y) v(t, y) C T 1 Φ(t, y) Φ T (t, y) B C T 1 B Φ(t, y) = Φ T (t, y) B T C 1, u(t, y) v t(t, y) dt Φ(x, y) K E C T 2 A, u t(t, y) v(t, y) dt Φ(x, y) K E B T C 1, where A = P T x C 1, B = P T x C 2, C2 and B are the operatonal matrces of product correspondng to the vectors C 2 and B, and E s the followng matrx: E = Φ(t, y) Φ T (t, y) dt ote for approxmatng the nonlnear terms u(t, y) v t (t, y) and u t (t, y) v(t, y) has been used the Theorem 11 Substtutng above approxmatons nto system (4), leads to the followng nonlnear system of algebrac equatons Φ T (x, y) Px T C 1 Φ T (x, y) Px T C 2 + Φ(x, y) K E C T 2 A = f1 (x, y), (41) Φ T (x, y) Px T C Φ T (x, y) Px T C 1 Φ(x, y) K E B T C 1 = f 2 (x, y), Settng = 5 and usng the roots of 6 (x) and 6 (y) n the x and y drectons, each equaton of the system (41) s collocated n 36 nner tensor ponts for dfferent values of parameters α and β Hereby, the system (41) reduces the problem to solve a system of nonlnear algebrac equatons and 72 unnown coeffcents are obtaned for some values of parameters α and β by usng ewton teratve method Table 5 shows dfferent values of the exact and approxmate solutons n ponts (x, y) = (1, 1), ( = 1, 2,, 1) for α = β = 1 Also, n 2 Fgure 1 the exact and approxmate solutons are compared for the case α = β = Also, the absolute errors functons obtaned by the proposed method are dsplayed n Fgure 1 for α = β = Example 4 Consder the followng nonlnear system Volterra ntegro-partal dfferental equaton u y(x, y) + v(x, y) y t sn(s) (u2 (t, s) v 2 (t, s)) dtds = f 1(x, y), u y (x, y) + v y (x, y) + u(x, y) y t cos(s) (u(t, s) v s(t, s)) dtds = f 2 (x, y), (42) 19

20 where f 1 (x, y) = 1 12 (1+2 cos3 (y) 3 cos(y)) x 4 and f 2 (x, y) = x (2 cos(y) sn(y)) wth the condtons u(x, ) = x and v(x, ) = The exact solutons of ths problem are u(x, y) = x cos(y) and v(x, y) = x sn(y) The followng approxmatons are used for = 5, u y(x, y) Φ T (x, y) C 1, u(x, y) Φ T (x, y) P T y C 1 + u(x, ) Φ T (x, y) P T y C 1 + Φ T (x, y) V, v y(x, y) Φ T (x, y) C 2, v(x, y) Φ T (x, y) P T y C 2, t sn(s) Φ T (x, y) K 1 Φ T (t, s), t cos(s) Φ T (x, y) K 2 Φ T (t, s), u 2 (x, y) A T 1 Φ(x, y) Φ T (x, y) A 1 A T 1 Ã 1 Φ(x, y) = Φ T (x, y) B 1, v 2 (x, y) A T 2 Φ(x, y) Φ T (x, y) A 2 A T 2 Ã 2 Φ(x, y) = Φ T (x, y) B 2, y y t sn(s) (u 2 (t, s) v 2 (t, s)) dtds Φ T (x, y) K 1 ( B 1 B 2 ) P x P y Φ(x, y), t cos(s) (u(t, s) v s (t, s)) dtds Φ T (x, y) K 2 (Ã1 C 2 ) P x P y Φ(x, y), where K 1 and K 2 are nown matrces, A 1 = P T y C 1 + V, A 2 = P T y C 2, B 1 = A T 1 Ã1 and B 2 = A T 2 Ã2 Substtutng above approxmatons nto system (42), leads to the followng nonlnear system of the algebrac equatons Φ T (x, y) C 1 + Φ T (x, y) Py T C 2 Φ T (x, y) K 1 ( B 1 B 2) P x P y Φ(x, y) = f 1(x, y), Φ T (x, y) C 1 + Φ T (x, y) C 2 + Φ T (x, y) (Py T C 1 + V ) Φ T (x, y) Py T C 2 Φ T (x, y) K 2 (Ã1 C 2 ) P x P y Φ(x, y) = f 2 (x, y), Settng = 5 and usng the roots of 6 (x) and 6 (y) n the x and y drectons, each equaton of the system (43) s collocated n 36 nner tensor ponts for dfferent values of parameters α and β Hereby, the system (43) reduces the problem to solve a system of nonlnear algebrac equatons and 72 unnown coeffcents are obtaned for α = β = by usng ewton teratve method In Fgure 2 the exact and approxmate solutons are compared for the case α = β = and the absolute and estmate errors functons obtaned by the proposed method are also dsplayed n Fgure 2 for α = β = The exact and approxmate solutons and ther error functons are seen n Fgure 3 for varous values of y = 2, 4, 5, 7, 9 (43) 6 Concluson In ths paper, a computatonal method based on the generalzed collocaton method was presented for solvng some of lnear and nonlnear ntegro-partal dfferental equatons n terms of two varable Jacob polynomals, by convertng them to a lnear or nonlnear system of algebrac equatons The llustratve examples wth the satsfactory results were acheved to demonstrate the applcaton of ths method The results ndcate the proposed approach can be regarded as the smple approach and those are applcable to the numercal soluton of these type of equatons It s predcted that the Jacob collocaton method wll be a powerful tools for nvestgatng approxmate 2

21 solutons and even analytc to lnear and nonlnear functonal equatons For numercal purposes the computer programmes have been wrtten n Maple 13 Acnowledgment The authors are deeply grateful to the referee for hs careful readng and helpful suggestons on the paper whch have mproved the paper References [1] Bhrawy, A H, An effcent Jacob pseudospectral approxmaton for nonlnear complex generalzed Zaharov system, Appl Math Comput, 214; 247: 3 46 [2] Bhrawy, A H, Abdelawy, M A, Zay, M A, Baleanu, D, umercal smulaton of tme varable fractonal order moblemmoble advecton-dsperson model, Rom Rep Phys, 215; 67 (3) (In Press) [3] Bhrawy, A H, Alof, A S, Ezz Elden, S S, A quadrature tau method for fractonal dfferental equatons wth varable coeffcents, Appl Math Lett, 211; 24: [4] Bhrawy, A H,Doha, E H, Baleanu, D, Ezz Elden, S S, A spectral tau algorthm based on Jacob operatonal matrx for numercal soluton of tme fractonal dffuson wave equatons, J Comput Phys, 214; do:1116/jjcp [5] Bhrawy, A H, Zay, M A, A method based on the Jacob tau approxmaton for solvng mult term tme space fractonal partal dfferental equatons, J Comput Phys, 215; 281, [6] Bhrawy, A H, Zay, M A, Baleanu, D, ew umercal Approxmatons for Space-Tme Fractonal Burgers Equatons va a Legendre Spectral Collocaton Method, Rom Rep Phys, 215; 67(2) (In Press) [7] Borhanfar, A, Abazar, R, Exact solutons for nonlnear Schrödnger equatons by dfferental transformaton method, App Math Computng, 211; 35 (1), [8] Borhanfar, A, Jafar, H, Karm, S A, ew soltary wave solutons for the bad boussnesq and good boussnesq equatons, umer Meth part Dff Equ, 29; 25: [9] Borhanfar, A, Kabr, M M, ew perodc and solton solutons by applcaton of applcaton of exp functon method for nonlnear evoluton equaton, Comput Appl Math, 29; 229: [1] Borhanfar, A, Kabr, M M, Vahdat, M, ew perodc and solton wave solutons for the generalzed zaharov system and (2 + 1) dmensonal nzhn-novov-veselov system, Chaos Solt Fract, 29; 42: [11] Borhanfar, A, Sadr, Kh, A new operatonal approach for numercal soluton of generalzed functonal ntegro dfferental equatons, J Comput Appl Math, 215; 279: 8 96 [12] Borhanfar, A, Sadr, Kh, umercal soluton for systems of two dmensonal ntegral equatons by usng Jacob operatonal collocaton method, Sohag J Math, 214; 1: [13] Borhanfar, A, zamr, A, Applcaton of ( G ) expanson method for the Zhber Shabat equaton and other related G equatons, Math Comput Model, 211; 54: [14] Doha, E H, Abd-Elhameed, W M, Youssr, Y H, Effcent spectral Petrov Galern methods for the ntegrated forms of thrd and ffth order ellptc dfferental equatons usng general parameters generalzed Jacob polynomals, Appl Math 21

22 Comput, 212; 218: [15] Doha, E H, Bhrawy, A H, An effcent drect solver for mult dmensonal ellptc Robn boundary value problems usng a Legendre spectral Galern method, Comput Math Appl, 211; do: 1116/jcamwa [16] Doha, E H, Bhrawy, A H, Abdelawy, M A, Van Gorder, R A, Jacob Gauss Lobatto collocaton method for the numercal soluton of nonlnear Schrödnger equatons, J Comput Phys, 214; 261, [17] Doha, E H, Bhrawy, A H, Ezz Elden, S S, A new Jacob operatonal matrx: An applcaton for solvng fractonal dfferental equatons, Appl Math Model, 212; 36 (1): [18] Doha, E H, Bhrawy, A H, Ezz-Elden, S S, Effcent Chebyshev spectral methods for solvng mult term fractonal orders dfferental equatons, Appl Math Model, 211; 35: [19] Doha, E H, Bhrawy, A H, Hafez, R M, A Jacob Jacob dual Petrov Galern method for thrd and ffth order dfferental equatons, Math Com Model, 211; 53: [2] Doha, E H, Bhrawy, A H, Hafez, R M, On Shfted Jacob Spectral Method For Hgh Order Mult Pont Boundary Value Problems, Commun onln Sc umer Smulat, 212; 17 (1): [21] Guezane-Laoud, A, Bendjaza,, Khald, R, Galern method appled to telegraph ntegro-dfferental equaton wth a weghted ntegral condton, Bound Val Prob, 213; 12: 1 12 [22] Hossen, S M, Shahmorad, S, umercal soluton of a class of ntegro dfferental equatons by the Tau method wth an error estmaton, Appl Math Comput, 23; 136: [23] Karm Vanan, S, Amnatae, A, Tau approxmate soluton of fractonal partal dfferental equatons, Comput Math Appl, 211; 62: [24] Kreyszg, E, Introducton Functonal Analyss wth Applcatons, USA: Wley, 1978 [25] Luo, M, Xu, D, L, L, Yang, X, Quas wavelet based on numercal method for Volterra ntegro dfferental equatons on unbounded spatal domans, J Appl Math Comput, 214; 227: [26] Ma, J, Fnte element methods for partal Volterra ntegro dfferental equatons on two dmensonal unbounded spatal domans, Appl Math Comput, 27; 186: [27] Szegö, G, Orthogonal Polynomals, Amercan Mathematcal Socety, Provdence, Rhode Island, 1939 [28] Tar, A, Rahm, M Y, Shahmorad, S, Talat, F, Development of the Tau Method for the umercal Soluton of Two dmensonal Lnear Volterra Integro-dfferental Equatons, Compu Meth Appl Math, 29; 9: [29] Tar, A, Shahmorad, S, Dfferental transform method for the system of two dmensonal nonlnear Volterra ntegro dfferental equatons, J Comput Math Appl, 211; 61:

23 (α, β) Error Abs Error Est (α, β) Error Abs Error Est (, ) ( 1 4, 1 4 ) (1, 1) ( 1 4, 1 4 ) (2, 2) ( 3 4, 3 4 ) ( 1 2, 1 2 ) ( 1 1, 1 1 ) Table 1 Maxmum absolute and estmate errors of Example 1 for dfferent values α and β (x, y ) Exact value Approxmate value Error Abs Error Est (1, 1) (2, (3, 3) (4, 4) (5, 5) (6, 6) (7, 7) (8, 8) (9, 9) (1, 1) Table 2 Maxmum absolute and estmate errors of Example 1 for varous values of α = β = 1 4 (α, β) (, ) ( 1 2, 1 2 ) (1, 1) ( 1 4, 1 4 ) (2, 2) ( 1 4, 1 4 ) Error Abs Error Est Table 3 Maxmum absolute and estmate errors of Example 2 for = 7 and varous values of α and β 23

24 (x, y ) u Exact Error(u 4 ) u 7 (x, y) Error(u 7 ) u 8 (x, y) Error(u 8 ) (2, 2) (4, 4) (6, 6) (8, 8) (1, 1) Table 4 Comparson of the exact and approxmate solutons of Example 2 for = 4, 7, 8 and α = β = 1 (x, y ) u Exact u 5 (x, y) Error Abs v Exact v 5 (x, y) Error Abs (1, 1) e (2, 2) e (3, 3) e (4, 4) (5, 5) (6, 6) (7, 7) (8, 8) (9, 9) (1, 1) Table 5 Comparson of the exact and approxmate solutons of Example 3 for = 5 and α = β =

25 Fgure 1: Comparson of the exact and approxmate solutons and ther error functons for α = β = n Example 3: Plots of (a) u 5(x, y), (b) v 5(x, y), (c) error functon of u(x, y), (d) error functon of v(x, y) 25

26 Fgure 2: Comparson of the exact and approxmate solutons and ther error functons for α = β = n Example 4: Plots of (a) u 5 (x, y), (b) v 5 (x, y), (c) error functon of u(x, y), (d) error functon of v(x, y) 26

27 Fgure 3: (a) Comparson of the u(x, y) and u 6 (x, y), (b) Comparson of the v(x, y) and v 6 (x, y), (c) Plots of absolute error functon u(x, y), (d) Plots of absolute error functon v(x, y) for y = 2, 4, 5, 7, 9 of Example 4 27