A Strong Method for Solving Systems of Integro-Differential Equations

The introduced method in this paper consists of reducing a system of integro-differential equations into a system of algebraic equations, by expanding the unknown functions, as a series in terms of Chebyshev wavelets with unknown coefficients. Extension of Chebyshev wavelets method for solving these systems is the novelty of this paper. Some examples to illustrate the simplicity and the effectiveness of the proposed method have been presented.


Introduction
In recent years, many different orthogonal functions and polynomials have been used to approximate the solution of various functional equations.The main goal of using orthogonal basis is that the equation under study reduces to a system of linear or non-linear algebraic equations.This can be done by truncating series of functions with orthogonal basis for the solution of equations and using the operational matrices.In this paper, Chebyshev wavelets basis, on the interval [0, 1], have been considered for solving systems of integro-differential equations.There are some applications of Chebyshev wavelets method in the literature [1][2][3].
Systems of integro-differential equations arise in mathematical modeling of many phenomena.Some techniques have been used for solving these systems such as, Adomian decomposition method (ADM) [4], He's homotopy perturbation method (HPM) [5,6], variational iteration method (VIM) [7], The Tau method [8,9], differential transform method (DTM ) [10], power series method, rationalized Haar functions method and Galerkin method for linear systems [11][12][13].The general form of these systems are considered as follows 1) Systems of Volterra integro-differential equations 2) Systems of Fredholm integro-differential equations where , and are positive integers, , This paper is organized as follows: in Section 2, Chebyshev wavelets method is explained.In Section 3, the application of the method for introduced systems (1) and ( 2) is studied.Some numerical examples are presented in Section 4, Conclusions are presented in Section 5.

Wavelets and Chebyshev Wavelets
Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet [14,15].When the dilation parameter a and the translation parameter b vary continuously we have the following family of continuous wavelets Chebyshev wavelets , is an arbitrary positive integer and m is the order of Chebyshev polynomials of the first kind.They are defined on the interval [0,1], as fo , 2 2 0, otherwise where

For
and .
m are the famous Chebyshev polynomials of the first kind of degree m, which construct an orthogonal system with respect to the weight function , and satisfy the following recursive formula: The set of Chebyshev wavelets is an orthogonal set with respect to the weight function where . Series presentation (7), can be approximated by the following truncated series.
where C and   and (10) Also a function   , f x y defined on     0,1 0,1  can be approximated as the following Here the entries of matrix The integration of the vector   x  , defined in (10), can be achieved as where P is the  operational matrix of integration [1,2].This matrix is determined as follows. , where F, and

Application of Chebyshev Wavelets Method
In this section the introduced method will be applied to solve systems (1) and (2).

Application to Solve System (1)
Consider system of integro-differential Equations ( 1), with the following conditions The property of the product of two Chebyshev wavelets vector functions will be as follows where is a given vector and is a  matrix.This matrix is called the operational matrix of product.The integration of the product of two Chebyshev wavelets vector functions with respect to the weight function is derived as First we assume the unknown functions are approximated in the following forms Therefore we have Using ( 20) and ( 21) other terms will be considered as the following general expansions where i F and ij K are known matrices, ij X and are column vectors of elements of the vectors .
where are matrices.

Multiplying
, in to both sides of the  a linear or a nonlinear system, in terms of the entries of , will be obtained and the elements of vectors can be obtained by solving this system.,

Solving the System (2)
Consider the system of integro-differential Equations (2) subject to the conditions (20).To solve this system, let's considered the unknown functions as a linear combination of Chebyshev wavelets, by the following and we have Therefore the following general expansions are achieved.
Substitution of these approximations into the system (2), would be obtained results in where is a D  the following linear or non-linear system will be obtained Solving system (27) the entries of , will be obtained., the error values can be obtained.

Numerical Results
In this section some systems of integro-differential equations are considered and solved by the introduced method.Parameters and k M are considered to be 1 and 6 respectively.
Example 1: Consider the following linear system of Fredholm integro-differential equations Subject to initial conditions and .The exact solutions are and [8].
, 2 , 3 2 Substituting into (28), the following linear system will be obtained .  Solving this system; the solutions would be achieved as follows This is the exact solution.
Example 2: Consider the following non-linear system of Fredholm integro-differential equations with the exact solutions and In this example, let's take Applying the Chebyshev wavelets method, the following non-linear system will be obtained Solving this system the following results would be achieved.
The exact solutions are and [7].
C and 2 are computed by solving a system of nonlinear equations with six unknowns and six equations as follows

Conclusions
This paper proposes a powerful technique for solving systems of integro-differential equations using Chebyshev wavelets method.Comparison of the approximate solutions and the exact solutions shows that the proposed method is more efficient tool and more practical for solving linear and non-linear systems of integro-difierential equations, and plots confirm.Researches for finding more applications of this method and other orthogonal basis functions are one of the goals in our research group.The package Maple 13 has been used to carry the computations associated with these examples.

Acknowledgements
Authors are grateful to the anonymous reviewers for his (her) influential comments which have improved the quality of the paper.
Also one can check the accuracy of the method.Since the truncated Chebyshev wavelets series are approximate the solutions of the systems (1) and (2), so the error function

x
Plots of the exact and approximate solutions are pre-Copyright © 2011 SciRes.AM sented in Figure 1 and also plots of error functions are shown in Figure 2. Example 3: consider non-linear system of Volterra integro-differential equations with conditions

Approximate and exact solutions are plotted in Figure 3 . 4 . Example 4 :
Error functions are plotted in Figure Consider the following non-linear system of Volterra integro-differential equations with the exact solutions ,

Figure 1 .
Figure 1.(a) and (b) plots of exact and approximate solutions of Example 2.

Figure 2 .
Figure 2. Plots of error functions of Examples 2.

Figure 3 .
Figure 3. (a) and (b) plots of exact and approximate solutions of Example 3.

Figure 4 .
Figure 4. Plots of error functions of Examples 3.

Figure 5 .
Figure 5. (a), (b) and (c) plots of exact and approximate solutions of Example 4.