ABSTRACT

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.

1. Introduction

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

(1.1)

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.

2. Sinc Function Properties

The sinc function properties are discussed thoroughly in [3-10]. The sinc function is defined on the real line by

(2.1)

For, and The translated sinc functions with evenly spaced nodes are given by

(2.2)

The sinc function form for the interpolating point is given by

(2.3)

Let

(2.4)

If a function is defined on the real axis, then for h > 0 the series

(2.5)

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 wplane, 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,

(2.6)

Thus we may define the inverse images of the real line and of evenly spaced nodes as

and

(2.7)

We consider the following definitions and theorems in [8-10].

Definition 2.1:

Let be the set of all analytic functions, for which there exists a constant, C, such that

(2.8)

Theorem 2.1:

Let, let N be appositive integer, and let h be

(2.9)

Then there exists positive constant C₁, independent of N, such that

(2.10)

Proof: See [8,9].

Theorem 2.2:

Let Let N be a positive integer and let h be selected by the relation (2.9) then there exist positive constant C₂, independent of N, such that

(2.11)

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].

3. 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:

(3.1)

where

(3.2)

where the basis functions defined by

(3.3)

(3.4)

(3.5)

We denote then basis function must satisfy the following conditions:

(3.6)

(3.7)

Obviously by using Equations (2.3) and (3.1) we have

Lemma 3.1:

, let N be a positive integer and

, Then (see Equation (3.8)), where is defined in (2.14) and C₄ is a positive constant, independent of N.

Proof: See [9].

Lemma 3.2:

For defined in (3.1), let

and h be selected from (2.9) then (see Equation(3.9))

(3.8)

(3.9)

Now let be the exact solution (1.1) that is approximated by following expansion.

(3.10)

Upon replacing in the Fredholm-Volterra integral equation (1.1), applying lemma 3.1 and Lemma 3.2, setting sinc collocation points and Then, considering we obtain the following system

(3.11)

We write the above system of equations in the matrix forms:

(3.12)

where

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

By solving the above system we obtain, , then, by using such solution we can obtain the approximate solution un as

(3.18)

4. Convergence Analysis

Now we discuss the convergence each of sinc collocation method. Suppose that is the exact solution of the Fredholme-Volterra integral equation (1.1). For each N, we can find which is our solution of the liner system (3.12), also by using we obtain the approximate solution, In order to derive a bound for |u(x) - u_n(x)| we need to estimate the norm of the vector, where is a vector defined by

where is the value of the exact solution of integral equation at the sinc points. There for we need the following lemma.

Lemma 4.1:

Let u(x) be the exact solution of the integral (1.1) and let

for, then there exists a constant C₅ independent of N, such that

(4.1)

Proof: See [10].

Theorem 4.1:

Let us consider all assumptions of lemma 3.1 and let be the approximate solution of Fredholme-Volterra integral equation given by (3.3) then we have

(4.2)

where C₆ a constant independent of N, and

Proof:

Suppose defined this following form:

(4.3)

So we have

(4.4)

By using lemma 3.1, we obtain

Obviously by using equations (4.3) and (3.3) we have.

(4.5)

And we have from definition of the We obtain

(4.6)

That C₇ a constant independent of N.

Now, by using equations (4.5) and (4.6) we get

(4.7)

In this case by using the system (3.12) and lemma 4.1 we obtain

(4.8)

Now by using Equations (4.7) and (4.8) we get

(4.9)

Obviously by using Equations (4.9) and (4.2) we obtain

5. Numerical Examples

In this section, we apply the sinc collocation method for solving Fredholm-Volterra integral equation example.

Example 5.1: consider the following Fredholm-Volterra integral equation of the second kind with exact solution u(x) = x.

We solved example 5.1 for different of

And we consider the sinc grid points as:

where

The errors on the given points are denoted by

(5.1)

Computational results are given in Tables 1-5.

Example 5.2: we consider the following FredholmVolterra integral equation of the second kind with exact solution.

Computational results are given in Tables 6-9.

REFERENCES