Abstract—In this paper we propose a new approach for solving linear and nonlinear Volterra integral equations (LVIE, NVIE) of the first and the second kinds. First, we define a new problem in calculus of variations, which is equivalent to this kind of problem. By taking this approach, one can solve a large number of problems in calculus of variations. We use a discretisation method to obtain a nonlinear programming problem and in some cases a linear programming problem. Then by using the optimal solution of the latest (LP or NLP) problem, we obtain an approximate solution with a controllable error for the original solution. Keywords—Volterra integral equations, Discretisation, Nonlinear programming. I. Introduction Finding the exact solution of the integral equations by classical methods is sometimes too difficult, and it is usually very useful to find a numerical estimation of the exact solution. Consider the following Volterra integral equation of the second kind: 0 ( ) ( ) ( ,)(()) , x ux f x kxt ut dt      (1) where k(.,.):R 2 →R is the kernel which is a known function, and λ is a given real parameter and f(.):R→R is a given function. We are trying to find an approximate solution of equation (1) where u(.):R→R. )) ( ( t u  is a linear or nonlinear function of u(t). Many different techniques have been presented so far for solving (1) such as Adomian's decomposition method, series solution method and successive substitution method [6],[9]. In recent years many numerical methods are also presented for solving VIEs [2],[5],[7],[8]. Homotopy perturbation method is applied in [1], [4] to solve such problems. _______________________ A.V. Kamyad is with the department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran. (e-mail: Kamyad@math.um.ac.ir ). M. Mehrabinezhad is a Ph.D. student in Applied Math. at Ferdowsi University of Mashhad, Mashhad, Iran. (corresponding author. Phone and fax: +98 (0) 511 8828606; e-mail: mmehrabinezhad@gmail.com , mehrabinezhad@stu- mail.um.ac.ir). J. Saberi-Nadjafi is with the department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran. (e-mail: najafi141@gmail.com ). In this paper we propose a new numerical approach for solving the above equation, both linear and nonlinear by discretisation and using an interpolation method to find a formula for solution of such integral equation. The organization of this paper is as follows: in Section II some theorems are presented that will be used in later sections. Our algorithm is illustrated In Section III. In Section IV some examples are provided and the results are compared with the exact solutions. Section V is the conclusion. II. Preliminaries. Consider the following theorem Theorem 2.1. If f(x,t) be a given function and a and b are constants, and let   1 2 , , , n t t t  be a set of suppo- rt points in [a,b], where 1 2 , n a t t t b       then 1 , 1 ( ,) lim ( , ) i n b i i a n i f x t dt f x t          (2) Proof. As we know from calculus 1 1 1 ( ,) lim ( ,) , i i i n b t a t n i f x t dt f x t dt          (3) where 1 i i i t t t     and i  are arbitrary points in the interval 1 [ , ], i i t t  ( 1, 2, , 1) i n    , notice that the subintervals are of equal length. The right-hand components of (3) contain integrals with small intervals (when n→∞ t i 's get close to each other), so we can substitute each integral by 1 ( ,) ( , ) , i i t i i t f x t dt f x t      (4) where i  ( 1, 2, , 1) i n    are arbitrary points in the interval 1 [ , ]. i i t t  Now we use (4) in the right-hand side of (3) to obtain 1 1 1 1 1 lim (,) lim (, ) , i i i n i n t i i t n n i i f x t dt f x t              (5) By considering (3) and (5) together, the proof is completed, and we have 1 1 1 1 1 (,) lim (,) lim (, ) . i i i n i n b t i i a t n n i i fxtdt fxtdt fx t                ▀ A Numerical Approach for Solving Linear and Nonlinear Volterra Integral Equations with Controlled Error A. Vahidian Kamyad, M. Mehrabinezhad, J. Saberi-Nadjafi IAENG International Journal of Applied Mathematics, 40:2, IJAM_40_2_04 (Advance online publication: 13 May 2010) ______________________________________________________________________________________