Numerical Solution of the Fredholme-Volterra Integral Equation by the Sinc Function

In this paper, we use the Sinc Function to solve the Fredholme-Volterra Integral Equations. By using collocation method we estimate a solution for Fredholme-Volterra Integral Equations. Finally convergence of this method will be discussed and efficiency of this method is shown by some examples. Numerical examples show that the approximate solutions have a good degree of accuracy.


Introduction
The sinc function form for the interpolating point In recent years, many different methods have been used to approximate the solution of the Fredholme-Volterra Integral Equations, such as [1,2].In this paper, we first present the Sinc Function and their properties.Then we consider the Fredholme-Volterra Integral Equation types in the forms where , and f(x) are known functions, but u(x) is an unknown function.Then we use the Sinc Function and convert the problem to a system of linear equations.

Sinc Function Properties
The sinc function properties are discussed thoroughly in [3][4][5][6][7][8][9][10]. The sinc function is defined on the real line by For , and The translated sinc functions with evenly spaced nodes are given by

1, , x jh S j h x c h
x jh h x jh x jh h x jh .
If a function   u x is defined on the real axis, then for h > 0 the series called whittaker cardinal expansion of , whenever this series converges.The properties of the whittaker cardinal expansion have been extensively studied in [8].
These properties are derived in the infinite stripe D of the complex w-plane, where for , Approximations can be constructed for infinite, semiinfinite and finite intervals.To construct approximations on the interval [a,b], which is used in this paper, the eyeshaped domain in the z-plane.
Is mapped conformably onto the infinite strip D via The basis functions on [a,b] are taken to be composite translated sinc functions, Thus we may define the inverse images of the real line and of evenly spaced nodes as We consider the following definitions and theorems in [8][9][10].
Definition 2.1: be the set of all analytic functions, for which there exists a constant, C, such that , let N be appositive integer, and let h be Then there exists positive constant C 1 , independent of N, such that

10)
Proof: See [8,9].Theorem 2.2: Let N be a positive integer and let h be selected by the relation (2.9) then there exist positive constant C 2 , independent of N, such that and also for be defined as in (2.4) then there exists a constant, which is independent of N, such that (2.12) Proof: See [8].

The Sinc Collocation Method
The solution of linear Fredholm-Volterra integral Equation (1.1) is approximated by the following linear combination of the sinc functions and auxiliary functions: where , where the basis functions defined by We denote and then basis function must satisfy the following conditions: Obviously by using Equations (2.3) and (3.1) we have , let N be a positive integer and and h be selected from (2.9) then (see Equation(3.9)) Now let   u x be the exact solution (1.1) that is approximated by following expansion.
, , We write the above system of equations in the matrix forms: ( , where 2 1 where , , , , , By solving the above system we obtain, (3.17) , then, by using such solution we can obtain the appro ate solution un as

Convergence Analysis
Now we discuss the convergence each of sinc collocation the exact solution of the ation (1.1).For each N,  method.Suppose that   u x is Fredholme-Volterra integral Equ we can find j u which is our solution of the liner system (3.12), also by using j u obtain the approximate solution   n u x , In order to derive a bound for |u(x)u n (x)| we need to es te the norm of the vector Tu p   , where u  is a vector defined here ( ) w j u x is the value of the exact solution of integral e at the sinc points quation

 
n x be the appr u ter oximate solution of Fredholme-Voltion given by (3.3) then we have ra integral equa where C 6 a constant independent of N, and  defined this following form: By using Lemma 3.1, we obtain (4 Obviously by using Equations (4.3) and (3.3) we have.
And we have from definition of the Now, by using Equations (4.5) and (4.6) we get That C 7 a constant independent of N.
In this case by using the system (3.12) a e obtain   Now by using Equations (4.7) and (4.8) we get e obta Obviously by using Equations (4.9) and (4.2) w in

Numerical Examples
In this section, we apply the sinc collocation method for solving Fredholm-Volterra integral equation example.llowing Fredholm-Volnd with exact so- The errors on the given points are denote by d .

i Exact solution Approxima
Computational results are given in Tables 1-5 Computational results are given in Tables 6-9.

10 )
Upon replacing   u x in the Fredholm-Volterra integral Equation (1.1)   n u x , applying Lemma 3.1 and Lemma 3.2, setting sinc collocation points kx and Then, considering(0) Let u(x) be the exact solution o the integral (1.1) and

Example 5 . 1 :
Consider the fo terra integral equation of the second ki lution u