Superconvergence Results for the Iterated Discrete Legendre Galerkin Method for Hammerstein Integral Equations

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1 Journal of Computer Science & Computational athematics, Volume 5, Issue, December 05 DOI: /jcscm Superconvergence Results for the Iterated Discrete Legendre Galerkin ethod for Hammerstein Integral Equations Payel Das*, Gnaneshwar Nelakanti Department of athematics, Indian Institute of Technology, Kharagpur -7 30, India * Corresponding author daspayel@gmail.com Abstract: In this paper we analyse the iterated discrete Legendre Galerkin method for Fredholm-Hammerstein integral equation with a smooth kernel. Using a sufficiently accurate numerical quadrature rule, we obtain super-convergence rates for the iterated discrete Legendre Galerkin solutions in both infinity and L -norm. Numerical examples are given to illustrate the theoretical results. Keywords: Hammerstein integral equations, Spectral method, Iterated discrete Galerkin, Numerical quadrature, Superconvergence.. Introduction Let X be a Banach space and consider the following Hammerstein integral equation: x(t) Ω k(t, s)ψ(s, x(s))ds = f(t), t Ω, x X, where Ω is a closed bounded region in R. Hammerstein integral equations arise as a reformulation of various physical phenomena in different branches of study such as mathematical physics, vehicular traffic, biology, economics etc. There has been a notable interest in the numerical analysis of solutions of intergal equations (see [-]). The Galerkin, collocation, Petrov-Galerkin, degenerate kernel and Nystro m methods are the most frequently used projection methods for solving the equations of type (). We are mainly interested in iterated discrete Galerkin method in this paper. The projection methods for solving equation () lead to algebraic nonlinear system, in which the coefficients are integrals, it appeared due to inner products and integral operator. Replacement of these integrals by numerical quadrature rule gives rise to the discrete projection methods. The effect of quadrature error on the convergence rates of the approximate solution is considered in these discrete projection methods (see [], [-]). Discrete projection methods for Fredholm nonlinear integral equations with spline bases and their super-convergence results have been studied by many authors such as Atkinson and Potra [3], Atkinson and Flores [5] and many others. Atkinson and Bogomolny [6] have shown that sufficiently accurate numerical quadrature rules can preserve the rates of convergence of the spline based Galerkin method. However, to get better accuracy in spline based discrete projection () methods, the number of partition points should be increased. Hence in such cases, one has to solve a large system of nonlinear equations, which is computationally very much expensive. To overcome the computational complexities encountered in the existing piecewise polynomial based projection methods, we apply polynomially-based projection methods to nonlinear Fredholm integral equations ([, 5, 7, 8]). We choose the approximating subspaces X n to be global polynomial subspaces of degree n which has dimension n +. The advantage of using global polynomials is that the projection method will imply smaller nonlinear systems, something which is highly desirable in practical computations. In particular here, we choose to use Legendre polynomials, which can be generated recursively with ease and possess nice property of orthogonality. In a recent paper [8], we obtain that the discrete Legendre Galerkin solution of the equation () converges with the order O(n r+ ) in both infinity and L -norm, n being the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the kernel k the nonlinear function ψ, the right hand side function f and the solution, with n r. In this paper, we investigate the superconvergence property of the iterated discrete Legendre Galerkin method for Fredholm- Hammerstein integral equation (). Our purpose in this paper is to obtain similar superconvergence results in polynomially-based discrete Galerkin method for Fredholm- Hammerstein integral equation () with a smooth kernel as in the case of piecewise polynomial based discrete Galerkin method. By choosing a numerical quadrature rule of certain degree of precesion, we show that the iterated discrete Legendre Galerkin solution converges with the order O(n r ) in both infinity and L -norm. The organization of this paper is as follows. In Section and 3, we discuss the discrete and iterated discrete Legendre spectral Galerkin methods and obtain superconvergence results. In Section, numerical examples are given to illustrate the theoretical results. Throughout this paper, we assume that c is a generic constant.. Discrete Legendre Galerkin ethod In this section, we describe the discrete Galerkin method for solving Fredholm-Hammerstein integral equations using

2 76 Superconvergence Results for the Iterated Discrete Legendre Galerkin ethod for Hammerstein Integral Equations global polynomial basis functions. Let L [,] be the space of real square-integrable functions on [,] and X = C[,] L [,]. Consider the following Hammerstein integral equation x(t) k(t, s)ψ(s, x(s))ds = f(t), t, where k, f and ψ are known real functions and x is the unknown function to be determined. For a fixed t [,], we denote k t (s) = k(t, s). Throughout the paper, the following assumptions are made on f, k(.,. ) and ψ: I. f C[,], II. k(t, s) C([,] [,]) and = sup t,s [,] k(t, s) <, III. IV. Let lim k t t t(. ) k t (. ) = 0, t, t [,], the nonlinear function ψ(s, x) is continuous in s [,] and is Lipschitz continuous in x, i.e., for any x, x R, c > 0 such that ψ(s, x ) ψ(s, x ) c x x, V. the partial derivative ψ (0,) (s, x) of ψ w.r. to the second variable exists and is Lipschitz continuous in x, i.e., for any x, x R, c > 0 such that ψ (0,) (s, x ) ψ (0,) (s, x ) c x x. Ky(t) = k(t, s)y(s)ds, t [,], y X. Note that, using Holder's inequality, we have for any y X and This implies Ky = sup Ky(t) t [,] = sup k(t, s)y(s)ds t [,] sup t,s [,] k(t, s) y(s) ds () y L, (3) Ky L Ky y L. () K L. (5) We will use Kumar and Sloan s [8] technique for finding the approximate solution of the equation (). To do this, we let Then the solution x of Hammerstein integral equation () is obtained by x = f + Kz. (7) For our convenience, we define a nonlinear operator Ψ: X X by Then (6) takes the form Ψ(x)(t) ψ(t, x(t)). (8) z = Ψ(Kz + f). (9) Let T(u) Ψ(Ku + f), u X, then the equation (9) can be written as z = Tz. Theorem. Let X = C[,], f X and k(.,. ) C([,] [,]) with = sup t,s [,] k(t, s) <. Let ψ(s, y(s)) C([,] R) satisfy the Lipschitz condition in the second variable, i.e. ψ(s, y ) ψ(s, y ) c y y, y, y R, with c <. Then the operator equation z = Tz has a unique solution z 0 X, i.e., we have z 0 = Tz 0. Proof of the above theorem can be easily done using similar technique given in Theorem. of [9]. We denote x 0 be the solution of equation (7) corresponding to the solution z 0 of (9), i.e., x 0 = Kz 0 + f. Now, we describe the discrete Legendre Galerkin method for the solution of Hammerstein integral equation (). To do this, we let X n = span{φ 0, φ, φ,, φ n } be the sequence of Legendre polynomial subspaces of X of degree n, where {φ 0, φ, φ,, φ n } forms an orthonormal basis for X n. Here φ i s are given by φ i (s) = i+ L i (s), i = 0,,, n, (0) where L i s are the Legendre polynomials of degree i. These Legendre polynomials can be generated by the following three-term recurrence relation L 0 (s) =, L (s) = s, and for i =,,, n and s [,] s [,], (i + )L i+ (s) = (i + )sl i (s) il i (s). In practice, the integrals in Galerkin method for solving (7) and (9), appearing due to inner products and the integral operator K cannot be evaluated exactly. The replacement of these integrals by numerical quadrature gives rise to the discrete Galerkin method, which we describe below. First we choose a numerical integration scheme z(t) = ψ(t, x(t)), t [,]. (6)

3 Payel Das, Gnaneshwar Nelakanti 77 where (i) (ii) (n) f(t)dt = w p f(t p ), () p= the weights w p are such that w p > 0, p =,,, (n), () the degree of precision d of the quadrature rule is at least n, that is (n) f(t)dt = w p f(t p ), (3) p= for all polynomials of degree n d. Proof: Let us consider the error function E d (f) = f(s)ds w j f(t j ). Using the quadrature rule defined by ()-(3), we see that E d (p) = 0, p P d, where P d is the space of all polynomials of degree d. For all p P d, consider j= (K n K)z(t) = k(t, s)z(s)ds w j k(t, t j )z(t j ) j= = E d (k(t,. )z(. )) E d (p) For the notational convenience, from now on we set (n) =. Using the above quadrature rule ()-(3) (see Golberg [], Sloan [0]), we define the discrete inner product < f, g > = w p f(t p )g(t p ), f, g C[,]. () p= For the approximation of the integral operator K, using the quadrature rule ()-(3), we consider the Nystro m operator K n defined by (K n z)(t) = w p k(t, t p )z(t p ). (5) p= For the rest of the paper, we assume that z 0 C d [,], k(.,.) C d ([,] [,]), ψ C d ([,] R) and f C d [,], where C d [,] denotes the space of d-times continuously differentiable functions on [-,] and d is the degree of precision of the numerical quadrature rule. For z C d [,], let z d, = max{ z (j) : 0 j d}. We denote where k d, = max{ D (l,p) k : 0 l, p d}, l+p (D (l,p) k)(t, s) = t l sp k(t, s), t, s [,]. In the following theorem, we give the error bounds for the integral operator K and the Nystro m operator K n defined by (5). Theorem. Let k(.,. ) C d ([,] [,]), then for any z C d ([,]), there hold (K n K)z cn d k d, z d,, where c is a constant independent of n and d is the degree of precision of the quadrature rule. = E d [k t (. )z(. ) p]. (6) Since weight functions w j > 0, choosing p(x) =, we j= have ds = w j =. Using this we get E d [k t (. )z(. ) p] = [k(t, s)z(s) p(s)]ds w j [k(t, t j )z(t j ) p(t j )] k t (. )z(. ) p [ k t (. )z(. ) p. j= ds + w j ] As p P d is arbitrary, from the above estimate and using Jackson's Theorem [], we obtain j= E d [k t (. )z(. ) p] inf p P d k t (. )z(. ) p c (k t (. )) d z d, d d. (7) Since from (3), we have d n, it follows that E d [k t (. )z(. ) p] c k d, z d, (n) d, (8) where c is a constant independent of n. Now combining estimates (6) and (8), we get (K n K)z cn d k d, z d,, where c is a constant independent of n. This completes the proof. Note that from Theorem, we see that K n converges pointwise in infinity norm. Hence, K n is pointwise bounded and since X = C[,] is a Banach space with. norm, by Uniform Boundedness Principle we have K n is uniformly bounded, i.e.

4 78 Superconvergence Results for the Iterated Discrete Legendre Galerkin ethod for Hammerstein Integral Equations K n p <, (9) where p is a constant independent of n. Discrete orthogonal projection operator: Discrete orthogonal projection namely hyperinterpolation operator Q n : X X n (see Sloan [0]) is defined by and Q n satisfy n Q n u = < u, φ j > φ j, (0) j=0 < Q n u, φ > = < u, φ >, φ X n. () We quote some crucial properties of Q n from Sloan [0] which are needed for the convergence analysis of the approximate solutions. Lemma. Let Q n : X X n be the hyperinterpolation operator defined by (0). Then for any u X, the following result holds < u Q n u, u Q n u > = min χ X n < u χ, u χ >. Lemma. Let Q n : X X n be the hyperinterpolation operator defined by (0). Then the following results hold i. Q n u L p u, where p is a constant ii. iii. independent of n. There exists a constant c > 0 such that for any n N and u X, Q n u u L c inf u φ L 0, as n. φ X n () In particular, if u C r [,], then there holds, Q n u u L cn r u r,, n r, (3) where c is a constant independent u and n. Now the discrete Legendre Galerkin method for (9) is to find z n X n, such that z n = Q n ѱ(k n z n + f). () Letting T n (u) Q n ѱ(k n u + f), u X, the equation () can be written as z n = T n z n. The corresponding discrete approximate solution x n of x is defined by x n = K n z n + f. (5) We quote the following results from [8], which gives the convergence rates of the discrete Legendre Galerkin solutions in both infinity and L -norm. Theorem 3. Let z 0 C r [,], n r, be an isolated solution of the equation (9). Let Q n be the discrete orthogonal projection operator defined by (0). Assume that is not an eigenvalue of the linear operator T (z 0 ), then for sufficiently n, the approximate solution z n defined by the equation () is the unique solution in the sphere B(z 0, δ) = {z: z z 0 δ} for some δ > 0. oreover, there exists a constant 0 < q <, independent of n such that α n + q z n z 0 α n q, where α n = (I T n (z 0 )) (T n (z 0 ) T(z 0 )), and z n z 0 = O(n r+ ). Theorem. Let z 0 C r [,] be an isolated solution of the equation (9) and x 0 be a isolated solution of the equation (7) such that x 0 = Kz 0 + f. x n be the discrete Legendre Galerkin approximation of x 0. Then the following holds x 0 x n, x 0 x n L = O(n r+ ). 3. Iterated Discrete Legendre Galerkin ethod In order to obtain more accurate approximation solution, we further consider the iterated discrete Legendre Galerkin approximation of z 0 in this section. To this end, we define the iterated discrete solution as z n = ѱ(k n z n + f). (6) Applying Q n on both sides of the equation (6), we obtain Q n z n = Q n ѱ(k n z n + f). (7) From equations () and (7), it follows that Q n z n = z n. Using this, we see that the iterated solution z n satisfies the following equation z n = ѱ(k n Q n z n + f). (8) Letting T n(u) ѱ(k n Q n u + f), u X, the equation (8) can be written as z n = T nz n. Corresponding approximate solution x n of x is given by x n = K n z n + f. (9) Next we discuss the existence of the iterated approximate solutions and their error bounds. To do this, we first recall the following definition of υ-convergence and a lemma from []. Definition. Let X be Banach space and BL(X) be space of bounded linear operators from X into X. Let F n, F BL(X). We say F n is υ-convergent to F if F n c <, (F n F)F 0, (F n F)F n 0, as n. Lemma 3. (Ahues et al. []) Let X be a Banach space and F n, F be bounded linear operators on X. If F n is υ-convergent to F and (I F) exists, then (I F n ) exists and uniformly bounded on X, for sufficiently large n. Lemma. Let k(.,. ) C d ([,] [,]), d n > n r, then for any u X, the following hold

5 Payel Das, Gnaneshwar Nelakanti 79 (K n Q n K)u 0, as n. In particular if u C r [,], then (K n Q n K)u = O(n r ). Proof: For any u n X n, it follows that p= ) ( w p [u n (t p )] We denote = ( [u n (s)] e(k t (. )) = (K K n )u n (t) ds) = u n L. (30) = k(t, s)u n (s)ds w p k(t, t p )u n (t p ). p= Using Cauchy-Schwartz inequality and estimate (30), we get e(k t (. )) = k(t, s)u n (s)ds w p k(t, t p )u n (t p ) = ( k(t, s) ds p= p= ) ( un (s) ds) + ( w p k(t, t p ) ) ( w p u n (t p ) ) ( + ( w p ) p= k u n L. ) p= Since for any y X n, e(y) = 0, we have k u n L e(k t (. )) = e(k t (. ) y) inf y X n k y u n L. Using this and Jackson's Theorem [], we obtain (K n K)u n cn r k r, u n L. (3) Since Q n u X n, u X, from (3) we obtain (K K n )Q n u cn r k r, Q n u L cn r k r, p u. (3) Since d r, using estimates (3), (3) and (3), we have for any u C r [,], (K n Q n K)u (K K n )Q n u + K(Q n I)u = O(n r ). (3) Note that form estimates () and (33), it follows that for any u X, (K n Q n K)u 0, as n. (35) This completes the proof. Theorem 5. Let z 0 C d [,], d n > n r, be an isolated solution of the equation (9). Assume that is not an eigenvalue of T (z 0 ). Then for sufficiently large n, the operator I T n (z 0 ) is invertible on X and there exist a constant L > 0 independent of n such that (I T n (z 0 )) L. Proof: Consider We have T n (z 0 ) = ѱ (K n Q n z 0 + f)k n Q n. (36) ѱ (K n Q n z 0 + f) ѱ (K n Q n z 0 + f) ѱ (K n z 0 + f) + ѱ (K n z 0 + f). (37) Form Lemma and Jackson's Theorem [], we have for any z C r [,], < (Q n I)z, (Q n I)z > = min < z χ, z χ > χ X n = min { w p (z χ) (t p )} χ X n p= ( w p ) p= inf χ X n z χ cn r z r,. (38) Using Lipschitz continuity of ψ (0,) (.,. ) and (38), we obtain ѱ (K n Q n z 0 + f) ѱ (K n z 0 + f) c K n (Q n I)z 0 = c sup K n (Q n I)z 0 (t) t [,] = c sup w p k(t, t p )(Q n I)z 0 (t p ) t [,] c sup t [,] p= p= ( w p k(t, t p ) ) ( w p [(Q n I)z 0 (t p )] ) p= cn r k r, p u + (Q n I)u L (33)

6 80 Superconvergence Results for the Iterated Discrete Legendre Galerkin ethod for Hammerstein Integral Equations c ( w p ) p= k < (Q n I)z 0, (Q n I)z 0 > c cn r k z 0 r, 0, as n. (39) Using Theorem, Lipschitz continuity of ψ (0,) (.,. ) and boundedness of ψ (Kz 0 + f), we have ѱ (K n z 0 + f) ѱ (K n z 0 + f) ѱ (Kz 0 + f) + ѱ (Kz 0 + f) c (K n K)z 0 + ѱ (Kz 0 + f) c cn d k d, z 0 d, + ѱ (Kz 0 + f) <. (0) From estimates (37), (39) and (0), it follows that ѱ (K n Q n z 0 + f) B <, () where B is a constant independent of n. Since Q n z X n, from estimate (30), we have p= ) ( w p [Q n z(t p )] = Q n z L. () Hence using Cauchy-Schwartz inequality and the estimate (), we have This implies K n Q n z = sup K n Q n z (t) t [,] = sup w p k(t, t p )Q n z(t p ) t [,] p= p= ( w p ) ( w p [Q n z(t p )] ) p= = Q n z L p z <. (3) K n Q n p. () Combining estimates (36), () and (), we have T n (z 0 ) B <, (5) where B is independent of n. Hence it follows that T n (z 0 ) is uniformly bounded. Using estimates (3), () and (5), we have [T n (z 0 ) T (z 0 )]T n (z 0 ) = [ѱ (K n Q n z 0 + f)k n Q n ѱ (Kz 0 + f)k]t n (z 0 ) ѱ (K n Q n z 0 + f)(k n Q n K)T n (z 0 ) ѱ (K n Q n z 0 + f) (K n Q n K)T n (z 0 ) + ѱ (K n Q n z 0 + f) ѱ (Kz 0 + f) KT n (z 0 ) B (K n Q n K)T n (z 0 ) + B ѱ (K n Q n z 0 + f) ѱ (Kz 0 + f). (6) Using Lipschitz continuity of ψ (0,) (.,. ) and Lemma, we obtain ѱ (K n Q n z 0 + f) ѱ (Kz 0 + f) c (K n Q n K)z 0 ) 0, as n. (7) Next, Let B {x X: x } be the closed unit ball in X. We have T n (z 0 ) = ѱ (K n Q n z 0 + f)k n Q n. Since {K n Q n } is a sequence of compact operators and ѱ (K n Q n z 0 + f) is uniformly bounded, T n (z 0 ) are compact operators. Thus S = {T n (z 0 )x x B, n N} is relatively compact set. Using estimate (35), we can conclude (K n Q n K)T n (z 0 ) = sup{ (K n Q n K)T n (z 0 )x : x B } = sup{ (K n Q n K)y : y S}, 0, as n. (8) Combining the estimates (6), (7) and (8), we have [T n (z 0 ) T (z 0 )]T n (z 0 ) 0, as n. Following the similar steps and using the fact that T (z 0 ) is compact, it can be proved that [T n (z 0 ) T (z 0 )]T (z 0 ) 0, as n. Hence T n (z 0 ) is υ-convergent to T (z 0 ) in. norm. By direct application of Lemma 3, it follows that for sufficiently large n, I T n (z 0 ) is invertible, i.e., there exist a constant L > 0, independent of n such that (I T n (z 0 )) L. This completes the proof. Theorem 6. Let z 0 C d [,], d n > n r, be an isolated solution of the equation (9). Let Q n : X X n be the discrete orthogonal projection operator defined by (0). Assume that is not an eigenvalue of T (z 0 ), then for sufficiently large n, the iterated solution z n defined by (8) is the unique solution in the sphere B(z 0, δ) = {z: z z 0 δ}. oreover, there exists a constant 0 < q <, independent of n such that β n + q z n z 0 β n q, where β n = (I T n (z 0 )) (T n(z 0 ) T(z 0 )). Proof: From Theorem 5, we have (I T n (z 0 )) exists and it is uniformly bounded in infinity norm, i.e., L > 0 such that (I T n (z 0 )) L. + [ѱ (K n Q n z 0 + f) ѱ (Kz 0 + f)]kt n (z 0 )

7 Payel Das, Gnaneshwar Nelakanti 8 Using Lipschitz continuity of ψ (0,) (.,. ) and estimate (3), we have for any z B(z 0, δ), T n(z 0 ) T(z 0 ) = ѱ(k n Q n z 0 + f) ѱ(kz 0 + f) c (K n Q n K)z 0 [T n (z) T n (z 0 )]v = [ѱ (K n Q n z + f) ѱ (K n Q n z 0 + f)]k n Q n v ѱ (K n Q n z + f) ѱ (K n Q n z 0 + f) K n Q n v We have c [ (K n Q n K n )z 0 + (K n K)z 0 ]. (50) This implies c K n Q n (z 0 z) K n Q n v c p z z 0 v c p δ v. sup (I T n (z 0 )) (T n (z) T n (z 0 )) Lc p δ z z 0 δ q, (say), where we choose δ in such a way that q (0,). This proves the estimate (.) of Theorem in [3]. Now using the Lipschtiz continuity of ψ(., x(. )) and Lemma, we have T n(z 0 ) T(z 0 ) = ѱ(k n Q n z 0 + f) ѱ(kz 0 + f) Hence c (K n Q n K)z 0 0, as n. β n = (I T n (z 0 )) (T n(z 0 ) T(z 0 )) Lc (K n Q n K)z 0 0, as n. Choose n large enough such that β n δ( q). We choose δ in such a way that q (0,). This proves the estimate (.5) of Theorem in [3]. Hence applying Theorem of [3], we obtain β n + q z n z 0 β n q, where β n = (I T n (z 0 )) (T n(z 0 ) T(z 0 )). This completes the proof. Theorem 7. Let z 0 C d [,], d n > n r, be an isolated solution of the equation (9) and x 0 be a isolated solution of the equation (7) such that x 0 = Kz 0 + f. x n be the iterated discrete Legendre Galerkin approximation of x 0. Then the following hold x 0 x n L, x 0 x n = O(n r ). Proof: From Theorem 6, we have β n z n z 0 q c (I T n (z 0 )) (T n(z 0 ) T(z 0 )). (9) (K n Q n K n )z 0 = sup (K n Q n K n )z 0 (t) t [,] = sup w p k(t, t p )(Q n I) z 0 (t p ) t [,] p= = sup < k t (. ), (Q n I)z 0 >. (5) t [,] Using the estimate (38), orthogonality property of Q n and Cauchy-Schwartz inequality, we obtain < k t (. ), (Q n I)z 0 > = < (Q n I)k t (. ), (Q n I)z 0 > = w p (Q n I)k(t, t p )(Q n I)z 0 (t p ) p= ( w p [(Q n I)k(t, t p )] p= ) ( w p [(Q n I)z 0 (t p )] ) p= = < (Q n I)k t (. ), (Q n I)k t (. ) > < (Q n I)z 0, (Q n I)z 0 > cn r k r, z 0 r,. (5) Since d r, using Theorem and estimates (9), (50), (5) and (5), we have z n z 0 = O(n min{r,d} )= O(n r ). (53) From estimates (7), (9) and (9), we get x n x 0 = K n z n Kz 0 K n (z n z 0 ) + (K n K)z 0 p (z n z 0 ) + (K n K)z 0. Hence using Theorem and estimate (53) and the fact that d r, we obtain x 0 x n = O(n min{r,d} ) = O(n r ). Also as x 0 x n L x 0 x n, we get x 0 x n L = O(n r ). Now using Lipschtiz continuity of ψ(.,. ), we have This completes the proof.

8 8 Superconvergence Results for the Iterated Discrete Legendre Galerkin ethod for Hammerstein Integral Equations. Conclusions From Theorem, we see that the discrete Legendre Galerkin solution of Hammerstein integral equation converges with the optimal order, O(n r+ ) in both L and infinity norm. Result of Theorem 7 imply that, if the numerical quadrature is of certain degree of precision, the iterated discrete Legendre Galerkin solution converges with the order O(n r ) in both L and infinity norm. This shows that the iterated discrete Legendre Galerkin approximation exhibits superconvergence. 5. Numerical Example In this section, we present the numerical results. For that we take the Legendre polynomials as the basis functions for the subspace X n. We present the errors of the approximate solutions under the discrete Legendre Galerkin and iterated discrete Legendre Galerkin methods in both infinity and L - norm. For computations we use Gauss quadrature rule and the Newton-Raphson method to solve the nonlinear systems. The numerical algorithms are compiled by using atlab. In the following tables, n represents the highest degree of the Legendre polynomials employed in the computation. Example. We consider the following integral equation x(t) k(t, s)ψ (s, x(s))ds = f(t), t, with the kernel function k(t, s) = 3π 6 ψ(s, x(s)) = x(s) and the function f(t) = exact solution is given by (t) = cos ( πt ). Table cos (π s t ), ). The cos (πt n x 0 x n L x 0 x n x 0 x n L x 0 x n e-0 0.0e e e e e-0 0.7e-0 0.0e e e e-0 0.7e e e e- 0.e e e e e- Example We consider the following integral equation x(t) k(t, s)ψ (s, x(s))ds = f(t), t, with the kernel function k(t, s) = cos(πt) sin(πs), ψ(s, x(s)) = x(s) 3 and the function f(t) = sin (πt). The exact solution is given by x(t) = sin(πt) + (0 39)cos (πt). 3 From Tables and, we see that the iterated discrete Legendre Galerkin approximation converges much faster than the discrete Legendre Galerkin solution. 0 Table n x 0 x n L x 0 x n x 0 x n L x 0 x n e-0 0.e e e-07 0.e-09 References 0.556e e-0 0.9e e e e e e e e- 0.9e e e e- 0.78e-3 [] K. E. Atkinson, F. A. Potra, F. A, The discrete Galerkin method for nonlinear integral equations, Journal of Integral Equations and Applications, vol., no, pp. 7-5, 988. [] K. E. Atkinson, A survey of numerical methods for solving nonlinear integral equations, Journal of Integral Equations and Applications, vol., no., pp. 5-6, 99. [3] Z. Chen, G. Long, G. Nelakanti, Y. Zhang, Iterated Fast Collocation ethods for Integral Equations of the Second Kind, Journal of Scientific Computing, vol. 57, no. 3, pp , 03. [] P. Das, G. Nelakanti, Convergence Analysis of Discrete Legendre Spectral Projection ethods for Hammerstein Integral Equations of ixed Type, Applied athematics and Computation, vol. 65, pp , 05. [5] P. Das,.. Sahani, G. Nelakanti. Convergence analysis of Legendre spectral projection methods for Hammerstein integral equations of mixed type, Journal of Applied athematics and Computing, vol. 9, no. -, pp , 0. [6] G. Ben-yu, Spectral methods and their applications World Scientific, 998. [7] H. Kaneko, Y. Xu, Superconvergence of the iterated Galerkin methods for Hammerstein equations, SIA Journal on Numerical Analysis, vol. 33, no. 3, pp , 996. [8] S. Kumar, I. H. Sloan, A new collocation-type method for Hammerstein equations, athematics of Computation, vol. 8, no. 78, pp , 987. [9] S. Kumar, Superconvergence of a collocation-type method for Hammerstein equations, IA Journal of Numerical Analysis, vol. 7, no. 3, pp , 987. [0] S. Kumar, A discrete collocation-type method for Hammerstein equations, SIA Journal on Numerical Analysis, vol. 5, no., pp. 38-3, 988. [] G. Long, G. Nelakanti, X. Zhang, Iterated fast multiscale Galerkin methods for Fredholm integral equations of second kind with weakly singular kernels, Applied Numerical athematics, vol. 6, no.3, pp. 0-, 0. [] G. ichael, Improved convergence rates for some discrete Galerkin methods, Journal of Integral Equations and Applications, vol. 8, no. 3, pp , 996.

9 Payel Das, Gnaneshwar Nelakanti 83 [3]. A. Golberg, C. S. Chen, Discrete projection methods for integral equations, Computational echanics Publications, Southampton, 997. []. Golberg, H. Bowman., Optimal convergence rates for some discrete projection methods, Applied mathematics and computation, vol. 96, no., pp. 37-7, 998. [5] K. Atkinson, J. Flores, The discrete collocation method for nonlinear integral equations, IA Journal of Numerical Analysis, vol. 3, no.., pp. 95-3, 993. [6] K. E. Atkinson, A. Bogomolny. The discrete Galerkin method for integral equations, athematics of computation, vol. 8, no. 78, pp , 987. [7] P. Das,.. Sahani, G. Nelakanti, Legendre spectral projection methods for Urysohn integral equations, Journal of Computational and Applied athematics, vol. 63, pp. 88-0, 0. [8] P. Das, G. Nelakanti, G. Long., Discrete Legendre spectral projection methods for Fredholm-Hammerstein integral equations, Journal of Computational and Applied athematics, vol. 78, pp , 05. [9] H. Kaneko, R. D. Noren, Y. Xu Regularity of the solution of Hammerstein equations with weakly singular kernels, Integral Equations and Operator Theory, vol. 3, no. 5, pp , 990. [0] I. H. Sloan, Polynomial interpolation and hyperinterpolation over general regions, Journal of Approximation Theory, vol. 83, no., pp. 38-5, 995. [] L. L. Schumaker, Spline functions: basis theory, John Wiley and Sons, New York, 98. []. Ahues, A. Largillier, B. Limaye, Spectral computations for bounded operators, CRC Press, 00. [3] G.. Vainikko, Galerkin's perturbation method and the general theory of approximate methods for nonlinear equations, USSR Computational athematics and athematical Physics, vol. 7, no., pp. -, 967.