Numerical Solution of System of Two Nonlinear Volterra Integral Equations

In this paper, using the implicit trapezoidal rule in conjunction with Newton's method to solve nonlinear system.We have used a Maple 17 program to solve the System of two nonlinear Volterra integral equations. Finally, several illustrative examples are presented to show the effectiveness and accuracy of this method.


INTRODUCTION
In this paper, we consider the Volterra integral equation of the second kind Where and are vector-valued functions with components. If and are continuous and ( ( )) satisfies a Lipcshitz condition with respect to , then a unique solution ( ) of (1) exists [1,4,7]. Volterra integral equations have been found to be effective to describe some application such as population dynamics, renewal equations, nuclear reactor dynamics, viscoelasticity, study of epidemics, super fluidity, damped vibrations, heat conduction and diffusion [1,7].
In this paper, we present the computation of numerical solution of system of two nonlinear Volterra integral equation of the second kind.

PRELIMINARIES
In this section, we recall the main theorems [7]. Theorem 1.Consider the equation

THE MATHEMATICSOF THE VOLTERRA PROCEDURE
In this section, we use the technique of the Volterraeqution [2,7] to find an approximates the solution ( ) of (1) at the equally spaced points for where and is the total number of steps of size denotes the approximation of ( ) at . Setting in (1), we have By the composite trapezoidal rule an approximation of the integral in (4) is Replacing ( ) in (4) and (5) by , we obtain the implicit trapezoidal rule Where We can rewrite (6) as ( ) Where denotes the zero vector. From (8), we see that is the solution of the vector equation Where is the vector-valued function We will obtain an approximation to the solution of (9) by way of the matrix-valued function defined in (11). If ( ) is an by matrix-valued function that is invertible in a neighborhood of , then is a fixed point of Assuming the components of ( ) have continuous first and second order partial derivatives and that the first order partial derivatives and that the first order partial derivatives at are equal to zero, it can be shown that if ( ) is set equal to the Jacobian matrix of the function , the iterates ( ) defined by (13) below will usually converge quadratically to provided the starting value is sufficiently close to . The Jacobian matrix of is the by matrix ( ) with the element In row and column, where is the Kronecker delta. Details of the statements made here follow from the discussion of Newton's method for nonlinear systems in [2]. Linz gives a brief outline of the trapezoidal rule and Newton's method for Volterra integral systems of the second kind in Section of [7].
We obtain from by setting ( ) and then generating the iterates For . (This is Newton's method for nonlinear systems.) Let denote the solution of the matrix equation Then the iteration formula (13) becomes We compute the solution ( ( ) ) ( ( ) ) using the command Linear Solve. The iterates ( ) are computed until the infinity norm of the vector is less than a prescribed tolerance Tol. Then is assigned the value of the last iterate [2,7].

NUMERICAL EXAMPLES
In this section, we solve some examples, and we can compare the numerical results with the exact solution.
Example1. Consider the system of Volterra integral equations With the exact solution ( ) and ( ) .
With the exact solution ( ) and ( )  Example3. Consider the system of Volterra integral equations With the exact solution ( ) and ( ) Table.3 Numerical results and exact solution of systems of two Nonlinear Volterra integral equations for example 3.  Example4. Consider the system of Volterra integral equations With the exact solution ( ) and ( ) .

Conclusion
In this paper, we compute the numerical solution of some examples and compare it with their exact solution.
The computed values and graphics, illustrated by the results, agree well with the exact solution.