AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS OF THE SECOND KIND

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1 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS,... 5 LE MATEMATICHE Vol. LXII (2007) - Fs. I, pp AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS OF THE SECOND KIND M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM In this rtile, A numeril method is used to solve the two dimensionl Fredholm integrl eqution of the seond kind with wek singulr kernel using the Toeplitz mtrix nd produt Nystrom method. The numeril results given in this pper re omputed using mple 8. The error, in eh se, is omputed.. Introdution. Singulr integrl equtions rise in mny problems of mthemtil physis. Its pplitions re in mny importnt fields like frture mehnis, erodynmis, the theory of porous filtering, ntenn problems in eletromgneti theory nd others. The solutions of their pplitions n be obtined nlytilly, using the theory developed by Muskhelishvili [], but in prtie pproximte methods re needed to solve the Fredholm integrl eqution in one dimensionl problem with different kernels. The diret numeril re preferred, whih ttk the eqution s it is written, without trnsforming it beforehnd into Fredholm eqution. Among Entrto in redzione il 6 Mrzo Key words: Toeplitz mtrix, Nystrom method, logrithmi kernel, Crlemn kernel, liner lgebri system A.M.S.: 45B05, 45E0, 65R.

2 6 M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM these, Glerkin method [2], blok by blok method, Nystrom method [3] nd Toeplitz mtrix method [4]. Afterwrds, mny numeril methods, whih solve problems in one dimensionl integrl equtions, n be found in the books [5], [6] nd [7]. Here, in this work, the existene nd uniqueness solution of the singulr Fredholm integrl eqution in two dimensionl re proved. Also, we use the Toeplitz mtrix nd produt Nystrom method, s fmous methods for solving singulr integrl equtions, to obtin numerilly the solution of the integrl eqution. The error, in eh se, is omputed when the kernel of Fredholm tkes logrithmi nd Crlemn form. 2. Existene nd uniqueness solution. We onsider the two dimensionl Fredholm integrl eqution (2.) µ (x, y) λ b d k(x, u; y,v) (u,v)dvdu = f (x, y), where µ is onstnt defined the kind of the integrl eqution, for µ = 0 nd µ = onstnt 0, we hve, respetively, the Fredholm first nd seond kind, while λ is onstnt, my be omplex, tht hs mny physil mening. The known funtions k(x, u; y, v) nd f (x, y) represent respetively, the disontinuous kernel of the integrl eqution nd its free term. While (x, y) represents the unknown funtion. Assume the following onditions: (i) The generl kernel k(x, u; y, v) stisfies the ondition (2.2) b b d d { k(x, u; y,v) 2 dxdudydv} 2 C, C is smll enough. (ii) The given funtion f (x, y) nd its prtil derivtives with respet to x, y re ontinuous nd its normlity in L 2 [, b] L 2 [, d] is given by (2.3) f (x, y) L2 [,b] L 2 [,d]= [ b d f (x, y) 2 dxdy] 2 = D (iii) The unknown funtion (x, y) stisfies Lipshitz ondition for the rguments x, y, where its norm is onsidered in L 2 [, b] L 2 [, d].

3 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS,... 7 Theorem 2. Using the previous onditions, where λ < µ C, the solution of (2.) is exist nd unique. Proof. To prove tht the solution of Eq.(2.) is exist, we use the Pird method, by piking up ny rel ontinuous funtion φ 0 (x, y) in L 2 [, b] L 2 [, d], then onstruting sequene φ n (x, y) to hve µφ n (x, y)= f (x, y)+λ b d (2.4) µφ 0 (x, y) = f (x, y). It is onvenient to introdue: (2.5) then µψ n (x, y) = µ[φ n (x, y) φ n (x, y)] = λ b d (2.6) φ n (x, y) = Using (2.5), we get (2.7) µψ n (x, y) = λ b d k(x, u; y,v)φ n (u,v)dvdu, n =, 2, 3,... k(x, u; y, v)(φ n (u,v) φ n 2 (u,v))dvdu, n ψ i (x, y). i=0 k(x, u; y,v)ψ n (u,v)dvdu, Tking the norm of Eq.(2.7), we obtin (2.8) µ ψ n (x, y) = λ b d µψ 0 = f (x, y). k(x, u; y,v)ψ n (u,v)dvdu. For n =, then using Cuhy-Shwrz inequlity, we hve (2.9) Hene, we get ψ (x, y) λ µ { b b d d (2.0) ψ (x, y) λ µ CD. k(x, u; y,v) 2 dxdudydv} 2 ψ0 (x, y).

4 8 M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM By indution, one hs (2.) ψ n (x, y) α n D, α = λ µ C. This bound under the used ondition λ < µ C ψ n (x, y) uniformly onvergent. Hene, we get (2.2) (x, y) = ψ i (x, y). i=0 mkes the sequene Sine eh of ψ i (x, y) in (2.2) is ontinuous, therefore φ i (x, y) is lso ontinuous, onvergent nd represents the existene of the solution of Eq. (2.). To prove (x, y) is the unique solution of Eq. (2.), ssume (x, y) is nother solution, hene, we get (2.3) (x, y) (x, y) = λ b d k(x, u; y,v)[ (u,v) (u,v)]dvdu. µ Applying Cuhy-Shwrz inequlity nd onditions (i) nd (iii), we get (2.4) (x, y) (x, y) α (x, y) (x, y), α <, whih leds to =. 3. Integrl opertor. The normlity nd ontinuity of the integrl opertor re very importnt to prove the existene nd uniqueness solution of the integrl eqution of the first kind or for homogeneous integrl eqution, where the Pird method fils. For this, ssume the integrl opertor (2.5) W = f µ + K where b d (2.6) K = λ k(x, u; y,v) (u,v)dvdu. µ Hene, the formul (2.), yields (3.7) W =

5 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS,... 9 The normlity of K n be showed s follows (3.8) K = λ µ b d k(x, u; y,v) (u,v)dvdu, then, using Cuhy-Shwrz inequlity nd ondition (i), we hve (3.9) K λ µ C = α, α = λ µ C. So by the ondition α<,k is bounded, whih leds to sy tht W is lso bounded opertor. The ontinuity of the integrl opertor W, given by (3.5), n be proved by ssuming n (x, y), m (x, y) stisfy Eq.(3.5) then (3.20) W n W m = K n K m α n m. Therefore, if n m 0, then W n W m 0, whih yields W is ontinuous opertor. Hene, W is ontrtion mpping, then, by Bnh fixed point theorem, the Eq. (2.) hs unique solution. 4. Method of Solution. Here, we disuss the solution of integrl eqution, in two dimensionl problem, using two different methods Toeplitz mtrix method. Toeplitz mtrix method is used to obtin the numeril solution of two dimensionl integrl eqution of the seond kind with singulr kernel. The ide of this method is to obtin, in generl, system of (2N +) (2M+) liner lgebri eqution, where 2N + nd 2M + re the numbers of disretiztion points used in x nd y dimensions respetively. The oeffiients mtrix is expressed s the sum of two mtries, one of them is the Toeplitz mtrix nd the other is mtrix with zero elements exept the first nd lst rows nd olumns.

6 20 M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM (4.2) For this, we ssume = N n= N l= M b d k(x, u; y,v) (u,v)dvdu0 M (n+)h (l+)h =nh =lh k(x, u; y,v) (u,v)dvdu, where h = b nd h = d. The integrl of the right hnd side N M of Eq.(4.2) n be evluted by ssuming x = mh, inresing by rte h, nd y = ph, inresing by rte h. Therefore, (4.22) (n+)h (l+)h k(x, u; y,v) (u,v)dvdu nh lh = A(x, y) (nh, lh ) + B(x, y) (nh,lh + h ) +C(x, y) (nh + h, lh ) + D(x, y) (nh + h, lh + h ) + R. where the weights of the integrtion A, B, C nd D re funtions of x, y will be determined, nd R is the error term. For the prinipl of Toeplitz mtrix method, to solve Eq.(4.22), we ssume (x, y) =, x, y, xy, in this se R = 0, whih led to I = I 2 = nh+h lh +h nh nh+h lh +h nh lh k(x, u; y,v)dvdu = A + B + C + D, lh + C(nh + h) + D(nh + h), uk(x, u; y,v)dvdu = Anh + Bnh (4.23) nh+h lh +h I 3 = vk(x, u; y,v)dvdu = Alh + B(lh + h ) nh lh + Clh + D(lh + h ), nh+h lh +h I 4 = uvk(x, u; y,v)dvdu = Anhlh nh lh + Bnh(lh + h ) + Clh (nh + h) + D(nh + h)(lh + h ).

7 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS,... 2 In ft, one n esily evlute weights expliity, but we prefer to evlute suh integrl numerilly, then the vlues of A, B, C nd D re diretly obtined, where (4.24) A = (nl + n + l + )I l + h I 2 n + h I 3 + hh I 4, B = (nl + l)i + l h I 2 + n + h I 3 hh I 4, C = (nl + n)i + l + h D = nl I l h I 2 n h I 3 + hh I 4. Let x = mh, y = ph, so Eq.(4.2) beomes (4.25) b d k(x, u; y,v) (u,v)dvdu = I 2 + n h I 3 hh I 4, N M n= N l= M χ l,p n,m N,M(nh, lh ), where C n,l + D n,l, if n = N B n,l + D n,l, if l = M (4.26) χn,m l,p = A n,l +B n,l +C n,l +D n,l, if M + l M, if N + n N A n,l + B n,l, if n = N A n,l + C n,l, if l = M nd N,M is the numeril pproximte solution, whih stisfy the following formul (4.27) (x, y) N,M 0 s N, M. Hene, Eq.(2.) is pproximtely equivlent to the following (4.28) N M µ N,M (mh, ph ) λ χn,m l,k N,M(nh, lh ) = f (mh, ph ) n= N l= M whih represents system of liner lgebri equtions. The mtrix χ l,p n,m n be written s (4.29) χ l,p n,m = χ l p n m G l,p n,m,

8 22 M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM where (4.30)χn m= l p!a n,l +B n,l +C n,l +D n,l,, M l M,, N n N, whih is the Toeplitz mtrix of order (2N + ) (2M + ) nd A n,l + B n,l + C n,l, n = N, l = M A n,l + C n,l, l = M A n,l + B n,l, n = N (4.3) G l,p n,m = 0, N + n N, M + l M B n,l + D n,l, l = M C n,l + D n,l, C n,l + B n,l + D n,l, n = N l = M, n = N The formul (4.30) represents the elements of Toeplitz mtrix of order (2N + ) (2M + ),while (4.3) is mtrix of order (2N + ) (2M + ) whose elements re zeros exept the first nd lst rows nd olumns. However, the liner lgebri system of (4.28) n be redued to the following mtrix form (4.32) [µi λ(χ G)] = F, µi λ(χ G) = 0. Definition 4.. The estimte lol error R N,M n be determined by the following eqution (x, y) N,M (x, y) = (4.33) N M χn m[ (nh, l p lh ) N,M (nh, lh )] + R N,M, where (4.34) n= N l= M b R N,M = d k(x, u; y,v) (u,v)dvdu N M n= N l= M χ l p n m (nh, lh ). Definition 4.2. The Toeplitz mtrix method is sid to be onvergent of order r + r 2 in the domin [, b] [, d], if nd only if for lrge N, M,

9 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS, there exist onstnt D > 0 independent of N, M suh tht (4.35) (x, y) N,M (x, y) DN r M r 2 5. Convergene of lgebri system. Theorem 5.. The liner lgebri system of Eq. (4.28), when N, M is bounded nd hs unique solution under the following onditions (i) f (x, y) = (ii) n= l= l= p= n= l= f 2 (nh, lh ) 2 < H, His onstnt. (5.36) (χ l p n m) 2 2 < Q, Q is onstnt. (5.37) (iii) λ < µ Q. (5.38) Proof. The onvergene of the liner system (4.28) n be proved by onsidering the two bounded sets x, ȳ { s}, where { s} is the fmily set of ll bounded vetors. Here, the metri spe will defined by (5.39) ρ( x, ȳ) = sup x l ȳ l, l nd (5.40) x = (x, x 2 ), ȳ = (y, y 2 ), x i ={x (i) l } l=, y i ={y (i) l } l=, i =, 2. Set the opertor sum L { s} : R 2 R 2 suh tht (5.4) z = L x, z = (z, z 2 ) R 2. Let (5.42) z = n,l= L l,n x + C,

10 24 M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM where (5.43) C = (, 2 ), i ={ (i) l } l= R, i =, 2. The system of Eq.(4.28), when N, M, n be written in the form (5.44) x m,p = µ f m,p l + λ µ l,n= χ l p n mx l,n. From the two liner system of Eq.(5.42) nd (5.44), we tke (5.45) S m,p = λ χ l p n m = λ K l,n. µ µ l,n= l,n= Apply Cuhy-Minkowski inequlity, with the id of ondition (ii) of theorem (5.), we get (5.46) S m,p < λ Q <. µ Therefore, under the ondition λ < µ we hve unique solution of Q (5.44), nd the vlue of x m,p stisfies H (5.47) x m,p <, µ λ. µ λ To prove x m,p is the unique solution of Eq. (5.44), ssume xm,p is nother solution. Hene, we get (5.48) x m,p xm,p = λ χ l p µ n m[x l,n xm,p ]. l,n= Applying Cuhy-Shwrz inequlity nd onditions (i) nd (ii) of theorem (5.), we get (5.49) x m,p x m,p λ µ Q x m,p x m,p. Therefore, we hve x m,p = x m,p. Corollry 5.. then Assume tht, onditions of Theorem (5.) re verified, (5.50) lim N,M R N,M = 0.

11 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS, Proof. The formul (4.33), yields (5.5) R N,M = [ (x, y) N,M (x, y)] N M χn m[ (nh, l p lh ) N,M (nh, lh )]. n= N m= M Hene, we hve (5.52) R N,M (x, y) N,M (x, y) therefore + N M n= N m= M χ l p n m[ (nh, lh ) N,M (nh, lh )], (5.54) R N,M < (x, y) N,M (x, y) ( + Q). Sine (x, y) N,M (x, y) 0sN, M, then R n,m The produt Nystrom method. In this setion, we disuss the produt Nystrom method [4] by onsidering the integrl eqution (6.54) b d µ (x, y) λ p(x, u; y,v) k(x, u; y,v) (u,v)dvdu = f (x, y) where p nd k re respetively well behved nd bdly behved funtions of their rguments, nd f (x, y) is given funtion, while (x, y) is the unknown funtion. The formul (6.54) n be written in the form N M µ N,M (x i, y k ) λ w ijkl p(x i, u j ; y k,v l ) N,M (u j,v l ) (6.55) j=0 l=0 = f (x i, y k ), where x i = u i = +ih, nd y k = v k = +kh, i = 0,, 2,..., N with h = b N, Neven, k = 0,, 2,..., M with h = d M, Meven,

12 26 M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM nd the pproximte numeril solution N,M stisfy (6.56) (x, y) N,M (x, y) 0, s N, M. Here, the estimte error R N,M n be defined from the following N M (x, y) N,M (x, y) = w ijkl p(x i, u j ; y k,v l ) (6.57) (6.58) R N,M = b d N j=0 j=0 l=0 [ (u j,v l ) N,M (u j,v l )] + R N,M, k(x, u; y,v) (u,v)dvdu M w ijkl p(x i, u j ; y k,v l ) (u j,v l ). l=0 Therefore, in the light of orollry (5.), we n prove R N,M 0 s N, M. The produt Nystrom method is sid to be onvergent of order r +r 2 in the domin [, b] [, d], if nd only if for lrge N, M, there exist onstnt D > 0 independent of N, M suh tht (6.59) (x, y) N,M (x, y) DN r M r 2 The weights funtions w ijkl re onstruted by insisting tht the rule in (6.55) be ext when p(x i, u; y k,v) (u,v) is polynomil of degree r r 2, sy. Aording to the produt Nystrom method, we pproximte the integrl term in (6.54), when x = x i, y = y k, by produt integrtion form suh s Simpson rule, therefore we write (6.60) = N 2 2 j=0 b d M 2 2 l=0 p(x i, u; y k,v) k(x i, u; y k,v) (u,v)dvdu u2 j+2 v2l+2 u 2 j v 2l p(x i, u; y k,v) k(x i, u; y k,v) (u,v)dvdu.

13 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS, Now, if we pproximte the nonsingulr prt of the integrnd over eh intervl [y 2 j, y 2 j+2 ] by the seond degree Lgrnge interpoltion polynomil whih interpoltes it t the points y 2 j, y 2 j+, y 2 j+2, we obtin (6.6) where, b d = p(u i, u; v k,v) k(u i, u; v k,v) (u,v)dvdu N j=0 M w ijkl p(u i, u j ; v k,v l ) (u j,v l ), l=0 w i,k,0,0 = β 2 (i, k,, ), w i,k,2 j+,0 = 2β 3 (i, k, j +, ), w i,k,2 j,0 = β (i, k, j, ) + β 2 (i, k, j +, ), w i,k,n,0 = β (i, k, N/2, ), w i,k,0,2l+ = 2γ 2 (i, k,, l + ), w i,k,2 j+,2l+ = 4γ 3 (i, k, j +, l + ), w i,k,2 j,2l+ = γ (i, k, j, l + ) + γ 2 (i, k, j +, l + ), (6.62) w i,k,n,2l+ = γ (i, k, N/2, l + ), w i,k,0,2l = α 2 (i, k,, l) + β 2 (i, k,, l + ), w i,k,2 j+,2l = 2[α 3 (i, k, j +, l) + β 3 (i, k, j +, l + )], w i,k,2 j,2l = α (i, k, j, l) + β 2 (i, k, j +, l + ), w i,k,n,2l = α (i, k, N/2, l) + β (i, k, N/2, l + ), w i,k,0,m = α 2 (i, k,, M/2), w i,k,2 j+,m = 2α 3 (i, k, j +, M/2), w i,k,2 j,m = α (i, k, j, M/2) + α 2 (i, k, j +, M/2), w i,k,n,m = α (i, k, N/2, M/2),

14 28 M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM suh tht α (i, k, j, l) = v2 l u2 j v 2l 2 4h 2 i, u; v k,v)(u u 2 j 2 ) h 2 u 2 j 2 k(u (u u 2 j )(v v 2l 2 )(v v 2l )dudv, β (i, k, j, l) = v2 l u2 j v 2l 2 i, u; v k,v)(u u 2 j 2 ) 4h 2 h 2 u 2 j 2 k(u (u u 2 j )(v v 2l )(v v 2l )dudv, γ (i, k, j, l) = v2 l u2 j v 2l 2 i, u; v k,v)(u u 2 j 2 ) 4h 2 h 2 u 2 j 2 k(u (u u 2 j )(v v 2l 2 )(v v 2l )dudv, α 2 (i, k, j, l) = v2 l u2 j v 2l 2 4h 2 i, u; v k,v)(u u 2 j ) h 2 u 2 j 2 k(u (u u 2 j )(v v 2l 2 )(v v 2l )dudv, (6.63) β 2 (i, k, j, l) = v2 l u2 j v 2l 2 4h 2 i, u; v k,v)(u u 2 j ) h 2 u 2 j 2 k(u (u u 2 j )(v v 2l )(v v 2l )dudv, γ 2 (i, k, j, l) = v2 l u2 j v 2l 2 4h 2 i, u; v k,v)(u u 2 j ) h 2 u 2 j 2 k(u (u u 2 j )(v v 2l 2 )(v v 2l )dudv, α 3 (i, k, j, l) = v2 l u2 j v 2l 2 i, u; v k,v)(u u 2 j 2 ) 4h 2 h 2 u 2 j 2 k(u (u u 2 j )(v v 2l 2 )(v v 2l )dudv, β 3 (i, k, j, l) = v2 l u2 j v 2l 2 i, u; v k,v)(u u 2 j 2 ) 4h 2 h 2 u 2 j 2 k(u (u u 2 j )(v v 2l )(v v 2l )dudv, γ 3 (i, k, j, l) = v2 l u2 j v 2l 2 4h 2 i, u; v k,v)(u u 2 j 2 ) h 2 u 2 j 2 k(u (u u 2 j )(v v 2l 2 )(v v 2l )dudv.

15 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS, Therefore, the integrl eqution (6.54) is redued to the following system of liner lgebri equtions N M µ (6.64) N,M (x i, y k ) λ w ijkl p(x i, u j ; y k,v l ) N,M (u j,v l ) j=0 l=0 = f (x i, y k ), whih n be written in mtrix form. By following the sme wy of Toeplitz mtrix, the existene nd uniqueness solution of the liner lgebri system (6.64) n be proved under the following ondition (6.65) λ < µ P, P > (w ijkl p(x i, u j ; y k,v l )) 2 2. i=0 k=0 j=0 l=0 7. Numeril results. In this setion, we pply the Toeplitz method nd Nystrom method, for different ses. The error, in eh se, is omputed. Exmple : (Logrithmi kernels). Consider the two dimensionl Fredholm integrl eqution (7.66) µ (x, y) λ ln x u ln y v (u,v)dvdu = f (x, y), where the ext solution is (x, y) = 2x 2 y 2 + xy, µ =, λ = 0.0, nd f (x, y) is given by f (x, y) = µ(2x 2 y 2 + xy) 2λ 9 [( x 3 )ln x + ( + x 3 )ln + x 2x ] (7.67) [( y 3 )ln y +( + y 3 )ln + y 2y ] λ 4 [( x 2 )(ln x ln + x ) 2x] [( y 2 )(ln y ln + y ) 2y].

16 30 M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM Disussion (i) For N = M : In Tbles - 3, the numeril solution T nd its error R T for Toeplitz mtrix method, in Eq.(4.28), re obtined for N = M =, 2 nd 3. Also, we lulte N nd evlute R N for the sme orresponding points when N = M = 2, 4 nd 6, in Eq. (6.64). We find tht the error is deresing in the two methods for inresing N = M, see Figures -3. Tble x y φ φ T R T φ N R N R N R T Tble 2 x y φ φ T R T φ N R N R N R T

17 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS,... 3 Tble 3 x y φ φ T R T φ N R N R N R T E E E E E E E E E

18 32 M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM (ii) For N M : If N M, for exmple N = 2, M = with respet to Toeplitz mtrix method nd its orresponding vlues of Nystrom method, the error is bigger thn the previous results when N = M (shown by Tble 4 nd Figure 4). Tble 4 x y φ φ T R T φ N R N R N R T E E E Tble 5 x y φ φ T R T φ N R N R N R T E E E E E E E E E E

19 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS, Exmple 2: (Crlemn kernels) Consider the two dimensionl Fredholm integrl eqution (7.68) µ (x, y) λ x u ν y v ν 2 (u,v)dvdu = f (x, y), where the ext solution is (x, y) = 2x 2 y 2 + xy nd f (x, y) is given by [ f (x, y) = µ(2x 2 y 2 + xy) 2λ[ ( x ν + x ν ) ν 2 ( ν )(2 ν ) ( x 2 ν + + x 2 ν ) 2 + ( ν )(2 ν )(3 ν ) ( x 3 ν + x 3 ν )] ν 2 ( y ν 2 +y ν 2 ) 2 ( ν 2 )(2 ν 2 ) ( y 2 ν 2 + +y 2 ν 2 ) 2 + ( ν 2 )(2 ν 2 )(3 ν 2 ) ( y 3 ν 2 + y 3 ν 2 )] λ[ ( x ν + + x ν ) ν ( ν )(2 ν ) ( x 2 ν +x 2 ν )][ ν 2 ( y ν 2 + +y ν 2 ) 2 ( ν 2 )(2 ν 2 ) ( y 2 ν y 2 ν 2 )]. Disussion (i) For N = M : Let µ =, λ = 0.0,ν = 0.05,ν 2 = 0.04, then the pproximte solution T nd the error E T of the integrl eqution (7.68), using Toeplitz mtrix method by tking N = M =, 2 nd 3 in Eq.(4.28), is displyed by Tble 4, 5 nd 6 respetively. Also, it displys the vlues of the pproximte solution N nd the error E N t the sme points for the sme integrl eqution but by using the produt Nystrom method with N = M = 2, 4, nd 6 in Eq.(6.64). The errors of Toeplitz mtrix method nd Nystrom method re deresing for inresing M nd N, see Figures 5-7.

20 34 M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM Figure Figure 2 Figure 3 Figure 4

21 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS, Figure 5 Figure 6 Figure 7 Figure 8

22 36 M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM Tble 6 x y φ φ T R T φ N R N R N R T Tble 7 x y φ φ T R T φ N R N R N R T E E E E E E E E E E E

23 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS, segue: Tble 7 x y φ φ T R T φ N R N R N R T E E E E E E E E E E E E

24 38 M. M. EL-BORAI - M. A. ABDOU - M. BASSEEM (ii) For N M : When N M, for exmple N = 2, M = with respet to Toeplitz mtrix method nd its orresponding vlues, N = 4, M = 2, of Nystrom method, the error is bigger thn the previous results in whih N = M (see Tble 8 nd Figure 8). Tble 8 x y φ φ T R T φ N R N R N R T Conlusion: - The error funtion of Toeplitz mtrix method is smller thn the orresponding error of Nystrom method in the most ses of logrithmi kernel form exept for the first vlues of N, M, see Tbles In Crlemn kernels form (ν = 0.05, ν 2 = 0.04), Nystrom method is better thn Toeplitz mtrix method. 3 - In the following Tble, we tke different vlues of λ = {0., 0.0, 0.00}, where N = M = 2 (with respet to Toeplitz mtrix method). The error, in both methods, is deresing whenever λ dereses. 4 - In Crlemn kernel form, when ν or ν 2 pprohes, for exmple ν 2 = 0.04 nd ν ={0.05, 0.5, 0.8}, the error is inresing in two methods (see Tble 0).

25 AN ANALYSIS OF TWO DIMENSIONAL INTEGRAL EQUATIONS, Tble 9 Tble 0 λ mx R T mx R N mx R T mx R N λ mx R T mx R N E Aknowledgment We thnks the referee for his effetive omments. REFERENCES [] N. I. Muskhelishvili, Singulr Integrl Equtions, Noordhoff, Groningen, The Netherlnds, 953. [2] E. Venturino, The Glerkin method for singulr integrl equtions revisited, J. Comp. Appl. Mth. 40 (992) [3] Youngmok Jeon, A qudrture method for the Cuhy singulr integrl equtions, J. Integrl equtions nd pplitions, V. 7 N0o. 2 (995) [4] M. A. Abdou, K. I. Mhmed, A. S. Isml, Toeplitz mtrix nd produt Nystrom methods for solving the singulr integrl eqution, Le Mtemtihe, Vol. LVII (2002) - Fs. I, pp [5] L. M. Delves, Mohmed J. L., Computtionl Methods for Integrl Equtions, New York. London, 985. [6] E. K. Atkinson, A Survey of Numeril Method for the Solution of Fredholm Integrl Eqution of the Seond Kind, Phildelphi, 976. [7] M. A. Golberg, ed., Numeril Solution of Integrl Equtions, Nelsson, New York, 990. M. M. El-Bori M. A. Abdou Alexndri University Alexndri University Fulty of Siene Fulty of Edution Deprtment of Mthemtis Deprtment of Mthemtis Egypt Egypt E-mil: m m elbori@ yhoo.om E-mil: bdell 77@ yhoo.om M. Bsseem Alexndri Thrwt 9 Mohmmed Eqbl St. Egypt E-mil: bsseem777@ yhoo.om