Using Predictor-Corrector Methods for Numerical Solution of System of Non-linear Volterra Integral Equations of Second Kind

The aim of this paper is solving system of non-linear Volterra integral equations of the second kind (NSVIEK2) numerically using Predictor-Corrector methods (P-CM). Two multistep methods (AdamsBashforth, Adams-Moulton). Convergence and stability of the methods are proved and some examples are presented to illustrate the methods. Programs are written in matlab program version 7.0.

(Ahmed, [1]) Solved system of non-linear Volterra integral equations of the second kind using computational methods, (Babolian and Biazar, [2]) used Adomian decomposition method to find the solution of a system of non-linear Volterra integral equations of the second kind, (Jumaa, [6]) find approximate solutions for a system of non-linear Volterra integral equations using B-Spline function, (Linz,[8]) Solve Volterra integral equations of the second kind using two (block-by-block) method, (Maleknejad and Shahrezaee, [9]) solve a system of Volterra integral equation numerically using Runge-Kutta method, (Waswas, [10]), used modified decomposition method to treatment non-linear integral equations and system of non-linear integral equations analytically, (Laurene, [7]) in this book derive the formula of Runge-Kutta method of order (three, four, five), Adams method (Moulton, Bashforth), and Adams Predictor-Corrector method and use this method to find numerical solution of ordinary and partial differential equation .
In this paper, the Adams Predictor-Corrector method is applied for the first time to find the numerical solution for a (NSVIEK2), which is defined by Jumaa, [6]: In this paper, the method is based on the explicit fourth-order Adams Bashforth method as Predictor and the implicit fourth-order Adams-Moulton method as Corrector, with the starting values from the fourth-order Runge-Kutta method (Laurene,[7]).[4]), (Hall.and Watt, [5])

Adams Method: (Delves and Walsh,
The general multistep method for approximating the solution to the initial-value problem: can be written in the form: for each , the method is called implicit.
The explicit Adams methods, known as Adams-Bashforth methods.The implicit Adams methods, known as Adams-Moultons method. .

Adams Predictor-Corrector Method (P-CM): (Laurene, [7])
The (P-CM) combines the fourth-order Adams Bashforth method as Predictor and the implicit fourth-order Adams-Moulton method as Corrector: 0 w is given;

Solution of NSVIEK2 using (P-CM):
Consider the ith equation of (1): The form of the explicit fourth-order Adams Bashforth equation (10) The form of the implicit fourth-order Adams Moulton equation ( 11) can be written as:   and all roots with absolute value are simple roots, then the difference method is said to satisfy the root condition.
Theorem 1: (William and Richard, [11]) A multistep method of the form is stable if and only if it satisfies root condition; Moreover, if the difference method is consistent with the differential equation, then the method is stable if and only if it is convergent.

Stability of (P-CM)
We have seen that in the equation (13), Then the characteristic equation for (13) is, consequently.

Illustrative Examples
In this section, two examples are presented for demonstrating the method and a comparison among the solutions obtained by this method against the exact solution which has been made depending on the least square errors (L.S.E).

Example 1: (Waswas, [10])
Solve a system of non-linear VIEK2's: The exact solution of this system is: After solving this system by Predictor-Corrector method with h= 0.1 in equations ( 13)-( 16) for A-BM and A-MM and equations ( 18)-( 21) for P-CM, we obtain the following numerical solution.
The exact solution of this system is: After solving this system by Predictor-Corrector method with h= 0.1 in equations ( 13)-( 16) for A-BM and A-MM and equations ( 18)-( 21) for P-CM, we obtain the following numerical solution.

Conclusions
According to the numerical results which obtaining from the illustrative examples we concludes the following: 1.The explicit fourth-order Adams-Bashforth method gave better results than the implicit fourth-order Adams-Moulton method.

If we use the explicit fourth-order Adams Bashforth method as
Predictor and the implicit fourth-order Adams-Moulton method as Corrector, then the method gave better results than explicit fourthorder Adams-Bashforth method and than the implicit fourth-order Adams-Moulton method.3. The A-BM, A-MM, and P-CM methods are stable by section (4-1).4. In P-CM, the error will be decreasing if we chose small values for h (step size) and it is the faster.

1
definition 1 equation (13) satisfies the root condition Then, by theorem 1 it is stable.Also, we have seen that in the equation (15), definition 1 equation (15) satisfies the root condition Then, by theorem 1 it is stable.

Table ( 4
) comparison between the exact solution x and the numerical solution f 1 (x) of Example 2 taking h=0.1.