NUMERICAL SOLUTION OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS BY AKBARI-GANJI’S METHOD

. In this study, Akbari-Ganji’s Method (AGM) was applied to solve Volterra Integro-Differential Difference Equations (VIDDE) using Legendre polynomials as basis functions. Here, a trial solution function of unknown constants that conform with the differential equations together with the initial conditions were assumed and substituted into the equations under consideration. The unknown coefficients are solved for using the new proposed approach, AGM which principally involves the application of the boundary conditions on successive derivatives and integrals of the problem to obtain a system of equations. The system of equation is solved using any appropriate computer software, Maple 18. Some examples were solved and the results compared to the exact solutions.


INTRODUCTION
Modeling of some physical phenomena often result to differential or integro-differential equations. The numerical solutions of these equations with either fractional or non-fractional order have been the hallmark of researchers in applied mathematics such as Numerical Analysis and other related fields. Approximate solutions of problems in the field of mathematics is centered on the use of approximate polynomials among which is Legendre polynomial. Legendre polynomials are well known member of orthogonal polynomials on the interval [-1,1]. They are mostly used because of their good properties in the approximation of functions [1]. In the task approximation, a lots methods abound for implementation which includes and limited to Homotopy Perturbation Method (HPM), Variational Iterative Method (VIM), Adomian Decomposition Method (ADM), Differential Transform Method (DTM) among others.
Homotopy Perturbation Method (HPM) is simply the merging of Perturbation and the Homotopy methods. It has the advantage of reducing the deficiency and difficulties of the traditional perturbation methods [2]. The method is suitable for solving linear and nonlinear differential equations. Many authors including [3] and [4] agreed that the method yield accurate results effectively while solving differential and integro-differential equations In a study to establish the existence of a unique solution of Partial Integro-Differential Equation (PIDE) as well as to show the convergence of the iterated sequence of approximate solutions, [5] use the Variation Iteration Method (VIM) to solve some PIDE and noted that the method is a powerful one for solving large amount of problems and concluded that it can handle different kind of linear and nonlinear problems easily, effectively and accurately. [6] described Variational Iteration Method as a modified form of the general Lagrange multiplier method which has been improved upon to ba able to solve effectively a large class of nonlinear problems. The method yields approximate solutions that rapidly converge to the exact solutions [7]. While solving differential equations, [8] applied the general Adomian Decomposition Method (ADM) to some fourth order linear differential equations and obtained good results compared to the analytical solution. [9] used ADM to solve some functional integral equations with similar conclusion on the efficiency of the method. According to [10], the Differential Transformation Method (DTM) is another semi analytical technique that depends on Taylor series and does not discretize the given problem before solving it. It gives successive and rapid convergent approximations to analytical solutions and can solve nonlinear system including algebraic equations, ordinary and partial differential, equations with initial data effectively.
[11] worked on the numerical solution of differential equations of Lane-Emden type using multilayer perceptron neural network method. The study produced a good approximate solutions to differential equations with less computations using multilayer perceptron artificial neural network technique. The results obtained showed that the technique has the capacity to yield good results with less computational time and memory space. Using Perturbed Collocation Method (PCM) [12] solved Singular Multi-order Fractional Differential Equations (SMFDE) of Lane-Emden Type with results which converged to the exact solutions. Akbari-Ganji Method (AGM) was used by [13], [14], [15] and [16] to find the numerical solution of differential equations arising from different physical problems. The studies concluded that AGM is a strong method to solve linear, nonlinear and partial differential equations, especially vibrational, wave and heat transfer problems. According to the studies, the main purpose of AGM is to obtain an approximate solution with less calculations and minimal errors compared with other existing methods. They all described the method as a semi-analytical method compatible for solving linear and nonlinear differential equations. A relatively new method in comparison with other schemes that requires boundary conditions that agrees with the order of the differential equation. However, additional new boundary conditions can be created with regard to the differential equations and their derivatives.

Definition of terms
Here, some terminologies that would aid better understanding of this article are defined according to [13]. Definition 1: Integro-differential Equations: the type of equations where both differential and integral operators are together in one equation which needs initial conditions to determine the solutions. The general form of integro-differential equation is given as ( ) ( ) = ( ) + ∫ ( , ) ( ) (1) subject to conditions (0) = , and being limits of integration.
Definition 2: An integro-differential equation is called Integro-Differential Difference Equation (IDDE) if the kernel ( , ) under the integral sign depends on the difference − so that (1) becomes and the kennel is called difference kernel Definition 3: An integro-differential equation is known as Volterra integro-differential difference equation when the limits of the integral part is from 0 to . The general form of fractional integro-differential equation is: subject to the conditions: It is a semi-analytic method for solving linear and nonlinear differential equations. It admits trial solutions in series form and that makes its implementation easy.
To solve Eq. (4) and (5) we assume a trial solution of the form: Where ( ) here, is Legendre Polynomial Substituting the initial conditions given in Eq. (6) into the assumed solution (5) gives and ′ (0) = ∑ =0 ′ (0) = 2 1 − 6 2 + 12 3 +. . . = Next, we put Eq. (6) into (4) and have Rewrite (9) as Numerical Solution Of Voltera Integro-Differential Equations By…..… We now compute the derivatives successively and evaluating each at the initial condition; = = 0 up to − 2 as follows Such that the number of equations equals the number of unknown constants in the assumed solution.
After applying initial conditions to the considered answer, we exit from the field of differential equations to solve the set of algebraic equations. Solving Eq. (7), (8), (11)- (14) with a simple procedure of the Gaussian elimination method or any mathematical software, the constant coefficients are easily obtained. The values of the constants are substituted back into the assumed solution (6), which yields the approximate solution.

Example 1
Consider the Volterra Integro-differential Difference equation

Example 2
Consider the Volterra Integro-differential Difference equation

Example 3
Consider the Volterra Integro-differential Difference equation

Example 4
Consider the Volterra Integro-differential Difference equation    The four examples were solved, and the results of the approximate solutions are presented in Tables 1-4 and Figures 1-4, respectively. It is seen from the tables and figures that the results converge closely to the analytical solutions. From the table of results also, it is observed that the method gives better results on second order than third order integro-differential equations. However, the result of example 2 is rather affected by the trigonometrical source function.

CONCLUSION
In this study, the proposed method was used to solve Volterra integro-differential equations successfully. We compared the approximate and exact solutions in all the examples considered. The system of equations is solved using any appropriate computer software, such as Maple 18. Some examples were