1. Introduction

System of Volterra Integro-differential equation (IDEs) arise in scientific fields such as biology [1], Medicine [2], Ecology [4], Population growth [3], physics such as electromagnetism theory [5], one dimensional visco elasticity and reactor dynamics [6]. This class of equations plays an important role in modelling of various problems of engineering and natural science and hence, attracted much attention in numerical computation and analysis.

This paper is concerned with the dynamic of two interacting species which was first modelled by [7]. It is considered two separated two species with number and at time where first species increases and second decreases. If they put together that the second species will feed on the first, there will be increase in the rate of the second species which depends not only on the present population but also on all previous values of the first species. When a steady-state or equilibrium is reached between the two species, it is described by the following system of nonlinear delay Volterra Integro-differential equations:

(1)

(2)

where with initial conditions

(3)

where and are coefficients of increase and decrease of the first and the second species respectively. The parameters and are given functions while and are unknown functions and is assumed to be the finite heredity duration of both species.

Several numerical methods of approximating the solution of this model are known; Adomian decomposition method (ADM) [9], Variational Iteration Method (VIM) [10], Legendre Multiwavelent Method (LMM) [11], Differential Transform Method (DTM) [12] and Taylor Collocation Method (TCM) [13]. Recently, [14] have introduced a new Iterative Method (NIM) to solve general functional equation: where is specified function and a given nonlinear function of [15] obtained the solution of order linear and nonlinear Integro-differential equation using NIM. [16] applied NIM to system of Volterra Integro-differential equations

NIM is simple in its principles and easy to implement on computer packages such as Mathematica and Maple. This method is better than numerical methods as it is free from rounding off errors and does not require large computer power. NIM has proven successful over other methods in many cases [17], [18] [19], [20] [21], [22], [16].

Bhaleker, S. and Datterder-Gejj, V. [18] presented a modification of the NIM called the Revised NIM (RNIM), to solve the following system of functional equations with improved convergence:

(4)

The main purpose of this paper is to solve equations (1) – (3) using the revised new Iterative algorithm [18]. The RNIM has been applied to solve various examples, some of which have already been solved by other methods. A comparison with other solutions reveals the usefulness and rapid convergence of this method.

2. Description of the Methods

2.1. New Iterative Method

Consider the system of nonlinear functional equations in equation (4) above, where are known functions and are nonlinear operators. Let be a solution of system (4), where having the series form:

(5)

The nonlinear operator can be decompose as

(6)

By virtue of equations (5) and (6), system (4) is equivalent to

for we define the recurrence relation:

then

The order approximation of is given by

2.2. Revised New Iterative Method

In this section we present the algorithm of the RNIM suggested by [18] to illustrate the technique. We consider the system of equation (4).

Initial step

First Iteration

kth Iteration

Thus

Hence,

3. Revised NIM for the System (1) - (3)

In view of the RNIM, the system (1) - (3) is equivalent to the system of Integral equation:

where is an integral operator with respect to

We set

Therefore,

and

Now for n=1, we get

Second iteration: n=2 gives;

Third Iteration: n=3 results to;

and so on. The rest iterations can be obtained.

The solution is of the form

4. Illustrative Examples

To give a clear over view of this method, we present the following examples. We apply the revised NIM and compare the results with the other numerical methods. Results are shown with tables and figures below. All of them were performed on the computer using a program written in Maple 15.

Example 4.1. [13]: Consider the system of Integro-differential equation (1) and (2) with and with the exact solution as

By solving the system (1) – (2) using the RNIM algorithm with this data, we obtain approximate solution as

This is indeed the exact solution of the problem.

In figure (1) and (2), approximate and exact solutions are plotted.

Figure 1.

Figure 2.

Example 4.2. [10], [12], [13] and [11]: Now we consider the system (1) – (2) with

The exact solutions of this system are in the form

We obtain the approximate solutions by RNIM and get

These results show the high accuracy of the technique only by two Iterations.

In Table 1 – 2, the absolute errors obtained by the RNIM are compared with the results obtained by variational Iteration method [10], Taylor collocation method [13], Adomian decomposition method [9] and Pseudospectral Legendre method [11]. In figure 3 – 4, the exact and approximate value for and is plotted. It is seen from this figure and tables that the present method is closer to exact solution than other method.

Figure 3.

Figure 4.

Table 1. Comparison of the absolute errors obtained by the ADM, PLM, TCM, VIM and the RNIM for in Example 2

Table 2. Comparison of the absolute errors obtained by the ADM, PLM, TCM, VIM and the RNIM for in Example 2

Example 4.3: [10] and [13]

In this example, we solve the system (1) – (2) with and

The exact solution of this system is in the form

Using the RNIM algorithm, we calculate the approximate solutions and and get

The exact solutions, absolute errors obtained by other methods and the present method are given in table 3 – 4.

Table 3. Comparison of the absolute errors obtained by the ADM, PLM, TCM, VIM and the RNIM for in Example 3

Table 4. Comparison of the absolute errors obtained by the ADM, PLM, TCM, VIM and the RNIM for in Example 3

5. Conclusions

In this present paper, we employed the modification of NIM; termed as “revised NIM” for solving a system of nonlinear delay Integro-differential equations which arises in a model describing biological species living together. The method yields a series solution which converges faster than numerical methods. It is then observed from figures and tables that the method is simple and powerful tool to obtain the approximate solution of this system.

Maple 15 was used to carried out the computations.

6. Recommendations

This method can also be extended to other models in future.