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This paper describes an approximating solution, based on Lagrange interpolation and spline functions, to treat functional integral equations of Fredholm type and Volterra type. This method extended to functional integral and integro-differential equations. For showing efficiency of the method we give some numerical examples.

In recent years there has been a growing interest in the numerical treatment of the functional differential equations,

which is said to be of retarded type if. It is said to be natural type. If it is said to be advanced type. For more general functional equations see Arndt [

Rashed introduced new interpolation method for functional integral equations and functional integro-differential equations [

where

For this approximation we first interpolate with following interpolation formula:

where

and

where

where is a well known function.

Then for computing we use B-Spline approximation that we present its details in the next section [6-9]. In the third section we give our method for functional integral equations. Also the fourth section is devoted to numerical solution of integro-differential equations. Of course for computing integrals both in the third and the fourth section we used Clenshaw-Curtis rule [10,11]. Finally in the latest section we give some applications of both functional integral equations and integro differential equations with numerical solutions. In addition, we compared our results with Rashed method [5,12]. We present some additional conclusions in Section 6.

In this paper we use spline function with Lagrange interpolation to compute the numerical solution of functional linear integral equations of the second kind:

In fact, We seek to find an approximation to which satisfies some interpolation property or variational principle.

In this functional integral Equations (2)-(5), we may use Lagrange interpolation of by

where

and

and also

where

and

where is well known function. Also

The integral part of each functional Equations (2)-(5) is given as follows: Integrating (1) w.r.t from to

Integrating (10) w.r.t from to

Integrating (10) w.r.t from to

where

The integral in the relation (21) is approximated as:

where

Finally, the functional integral equation is approximated by system of linear equations. Also, the method is extended to treat the functional equations of advanced type (in or)

The last equation may not has analytically solution.

Consider the functional integro-differential equation of the second equation

where

With using Lagrange interpolation, the second derivative is given by

where

Integrating (35) w.r.t from to

Integrating (37) w.r.t from to

The integral is computed as (31). Also, integrating (36) w.r.t from to

Substitution from (37) to (40) into (34) lead to system of linear equations

where

and

The integrals in (40) and (41) may be computed by Clenshaw-Curtis rule. It is obviously that the method may be extended to functional linear differential equations of the second order if in (34).

We compared our results with Rashed results [

Example 1. Volterra integral equation of the second kind

Example 2. Fredholm integral equation of the second kind

Example 3. Volterra integral equation of the second kind

Example 4. Volterra integral equation of the second kind

Example 5. Volterra integro-differential equation of the second kind

1) The method give the approximate solution at the points

2) The method can be extended to the functional differential equation

3) The method may be used to treat boundary functional differential or integro-differential equations.

4) The small errors obtained shows that the method indeed successfully approximate the solution of problem.

Remark 1 The spline piecewise functions are very essential tools in approximation Theory. Then for this we applied B-Splines for approximation of unknown answer in integro-differential equations. From Tables, We see that for mesh points with, we have small errors. Therefore the mentioned results for have high quality and application of B-Splines is acceptable (of course for getting convenient answer we must had).