Numerical Solution of the System of Linear

In this paper, a method for solving linear system of Volterra integro – differential equation of the first order numerical presented based on Monte – Carlo techniques. Numerical examples illustrate the pertinent features of the method with the proposed system. The computer program are written in (TURBO MATLAB) language (Version 6).


Introduction
It is in general, very difficult to find a useful solution of a linear integrodifferential equations of the first order if the solution depends on several variables or if the equation is coupled with other integrodifferential equations.Well known methods of solution are mostly in effective because the amount of computation involved is too great, even for the latest machines.In many cases, however, especially in particle transport problems, one can use a statistical procedurethe Monte -Carlo method to find a solution which is sufficient accurate for practical purposes.[2] The application of a well known Monte -Carlo method to the solution of an equation with non-negative kernel is already illustrated in the paper [2, 3,2].
In this study, the basic ideas of the previous works are developed and applied to the system of the linear Volterra Integro -Differential Equation of the first order.Consider the following system of linear Volterra Integro-Differential Equation of order n is an equation of the form: Here, K(x, t) , f(x), p i (x) (h=1,1,……., n-1) are known functions, u(x) is the known function, and λ is a scalar parameter [4].
With the initial condition u(a) = u o , where the functions f and p are assumed to be continuous on I and K denotes given continuous functions. Are

Numerical Approximation
The method we propose is based on a similar concept of the So-Called Monte -Carlo method to approximate integrodifferential when the integrand is known the method is developed as iteration technical where the approximation is over a finite interval for the unknown function, At each step we have a new partition for the interval; such partition comes from the generation of random numbers on the interval [1].
We can summarize the method as follows: Step 9: We start with a value in the interval where we want to find the approximation of the function (the unknown function).
Step 2: We assume that the function is a constant then it can be put outside the integrodifferential and we solve a system linear equation.
Step 3: We generate a random number on the interval we want to find the approximation we assume that the functions are stepwise function.We take the solution of the system of linear equation of step 2 as the value of the function a round some subinterval.We assume a new unknown value for the function around some other interval.Again the unknown value for the function can be put outside the integrodifferential, we sole a new system linear equation to find a new value, then we come back step 2. This iteration come from the fact that we generate random numbers on the interval where we want to find the approximation.

Approximation for system of linear Volterra Integro -Differential Equation of the First Order
To find an approximation for the solution u i =(x), i=1,…, m of (2) on the

Algorithm (AQ)
In this algorithm we expose the step for solve VIDE s of 1 st order using Monte-Carlo method.
Step 8 In this section, two examples are presented for demonstrating the methods and a depending on the ,least square errors.Example8: [7] Consider the following VIDE The exact solution is u 1 (x)= 1+ x and u 1 (x)= 1-0.0x-0.61667x 1  Take n = 10 and x i = a+ih, i=0, 1, ….., n The numerical results obtained for example 1 are shown in Table 1 and Table 1.

Example 2
Consider the following Volterra Integro -Differential Equation: Table (1) presents results from a computer program that solves this problem over the interval x=0 to x=1 with u 1 (x) = x 1 , and u 1 (x) = x 1 -1.6667x 0 for which the analytical solution h=0.1 [7] Numerical Solution of the System of Linear Volterra Integro -Differential Equations of the First Order using Monte-Carlo Method………………………………………………… Attifa J. Al-Tmeme Rawaa E. Al-Mashhadany The numerical results obtained for example 1 are shown in Table 3