A Collocation Method for Solving System of Volterra-differential-difference Equations with Terms of Chebyshev Polynomials

In this study, we present a numerical algorithm for solving systems of Volterra-differential-difference equations with variable coefficients by collocation method. This algorithm based on polynomial approximation, using the first kind Chebyshev polynomial basis with collocation method. This method transforms the system of Volterra-differential-difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. In addition, convergence analysis of the method is presented. Some cases of the mentioned equations are solved as examples to illustrate the reliability of the method. The results reveal that the method is very effective and accuracy.


INTRODUCTION
Systems of linear integral equations and their solutions are great importance in science and engineering [1][2][3]. Most physical and biological problems, such as biological applications in population dynamics and genetics can be modeled by the differential equation, an integral equation or an integro-differential equation or a system of these equations [4,5]. Moreover, the competition between tumor cells and the immune system, electromagnetic theory lead to the problem of solving integro-differential equation systems [6].
Chebyshev polynomials are encountered in several areas of numerical analysis and they hold particular importance in various subjects such as orthogonal polynomials, polynomial approximation, numerical integration and spectral methods [23][24][25]. Moreover, It is interesting to note that they also play an important part in the representation theory of algebras and polynomial factorization [26][27][28].
The Chebyshev polynomials ) (t T r of the first kind are the polynomials in t of degree r , defined by relation [23][24][25] iii) It is well known that the relation between the powers n t and the Chebyshev polynomials

FUNDAMENTAL MATRIX RELATIONS
To solve Eq.(1), we construct the following matrix realation. Using the Eq.(3), we have the matrix relations of solutions ] ... = A By using the expression (5-6) and taking N r , , 1 , 0 K = we find the corresponding matrix relation as follows To obtain the matrix we can use the following relation: Consequently, by substituting the matrix forms (10) and (11) into (9) and its derivatives, we get the approximate solution and its first-derivative of the matrix relations Moreover, Using Eq.(7) and Eq.(9), we obtain the following matrix representation of and using Binomial expansion, we obtain the following matrix relation between the matrices ) (

Matrix Representation of Volterra Integral Part
Let assume that can be expanded to univariate Chebyshev series with respect to t as follows: Then the matrix representations of the kernel function

METHOD OF SOLUTION
We are now ready to construct the fundamental matrix equation corresponding to Eq.(1). For this purpose, we substitute the matrix relations Eqs. (11), (14), (17) into Eq.(1) and obtain the matrix equation: (18) then, it can be written as: When the points of Chebyshev-Gauss grid are substituting in Eq.(16), the fundamental matrix equation of the system of Fredholm-differential-difference equations Eq.(1) is Hence, the matrix equation (17) correponding to Eq.(1) can be written in the form Here, Eq.(21) corresponding to a system of We can obtain the corresponding matrix forms for conditions (2), by means of the relation (11) , as On the other hand, the matrix form for conditions can be written as G UA = (23) where for m a , , We can obtain the approximate solutions of Eq.(1) with the conditions Eq.(2) by terms of Chebyshev polynomials. By replacing the conditions matrices (21) by the last l rows of the matrix (23) we obtain the new augmented matrix

Convergence Analysis
In this section, we present convergence analysis of the mention method. We assume that Therefore, we have [23,25] ∞ + + We can easly check the accuracy of the method. Since the truncated Chebyshev series (3) [25]; that is, for and we can rewrite       (27) and conditions (28), the new augmented matrix ___ W and * F is the systems of linear equations with ten unknowns, then solving this system, Chebyshev coefficients matrix are obtained as: In Tables 1 and 2, we compare the exact solutions and approximate values for various N . Figs. 1 and 2 display the comparison of absolute errors for various N . Moreover, we compare the absolute errors and error estimation function in Fig.3. The numerical results show that the accuracy improves when N is increased. Tables and figures indicate that as N increases the errors decrease; hence for better results, using large number N is recommended.    Table 3 contains a numerical comparision of absolute errors between our solution Chebyshev collocation method and the solutions obtained by HPM [12] and ADM [44]. Present method is incisive, because for the same number basis functions it obtains beter results. Moreover, we give the comparison of absolute errors for present methods for various N in Figs. 3 and 4, and error estimation function display in Fig. 5.
The exact solution of this system is In Fig.   6, we display the absolute error obtained by present method for The exact solution of this system is

CONCLUSION
In the present paper we used Chebyshev collocation method to solve systems of Volterradifferential-difference equations. The main idea of the proposed method is to convert the problem including linear algebraic equation and find the Chebyshev coefficients in truncated Chebyshev sum. Numerical examples reveal that the present method is very accurate and convenient for solving systems of high order linear VFIDEs.
Tables and figures indicate that as N increases, the errors decrease more rapidly; hence for better results, using large number N is recommended. We compare the some computational errors for Examples 2 and 4 suchs as maximum errros, truncation errors and 2 Lnorm errors in Table 6.