POLYNOMIAL APPROACH FOR THE MOST GENERAL LINEAR FREDHOLM INTEGRODIFFERENTIAL-DIFFERENCE EQUATIONS USING TAYLOR MATRIX METHOD

A Taylor matrix method is developed to find an approximate solution of the most general linear Fredholm integrodifferential-difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. This method transforms the given general linear Fredholm integrodifferential-difference equations and the mixed conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equations, the Taylor coefficients can be easily computed. Hence, the finite Taylor series approach is obtained. Also, examples are presented and the results are discussed.


Introduction
An important problem in function theory is the problem of expanding a function in a series of polynomials.Several extensions of the classical theory of Taylor series to differential, differential-difference operators on the real line have shown up recently.Boundaryvalue problems involving integrodifferential-difference equations arise in studying variational problems of control theory where the problem is complicated by the effect of time delays [4,5], signal transmission [9], biological problems as the problem of determining the expected time for the generation of action potentials in nerve cells by random synaptic inputs in the dendrites [1][2][3].
Taylor methods to find the approximate solutions of differential equations have been presented in many papers [6,8,10,11].In this paper, the basic ideas of these methods are developed and applied to the high-order general linear differential-difference equation with variable coefficients, which is given in [7, page 229], m k=0 p j=0 P k j (x)y (k) x − τ k j = f (x) + b a q i=0 s l=0 K il (x,t)y (i) t − τ il dt, τ k j ≥ 0, τ il ≥ 0, and the solution is expressed as the Taylor polynomial Here P k j (x), K il (x,t), and f (x) are functions that have suitable derivatives on a ≤ x, t ≤ b; and c r lk , c r , c and τ k j , τ il are suitable coefficients; y (n) (c) are Taylor coefficients to be determined.

Fundamental matrix relations
Let us convert the expressions defined in (1.1), (1.2), and (1.3) to matrix forms.We first consider the solution y(x) defined by the truncated Taylor series (1.3) and then we can put it in the matrix form where, y (1) (c) Now we substitute quantities (x − τ k j ) instead of x in (1.3) and differentiate it N times with respect to x.Then we obtain M. Sezer and M. Gülsu 3 and the matrix form, for where On the other hand, we consider terms P k j (x)y (k) (x − τ k j ), k = 0,1,...,m, j = 0,1,..., p, in (1.1) and can write them as the truncated series expansions of degree N at x = c in the form By means of Leibnitz's rule we have and substitute in expression (2.6).Thus expression (2.6) becomes and its matrix form or from (2.4) where where (2.12) M. Sezer and M. Gülsu 5 Let the function f (x) be approximated by a truncated Taylor series Then we can put this series in the matrix form where the matrices X and M 0 are defined in (2.1); the matrix F is (2.15)

2.19)
On the other hand, we can obtain the matrix form of the function y (i) (t) as, and thereby the matrix form of y (i) (t − τ il ) as 6 Taylor matrix method where . (2.23) Substituting the matrix forms (2.18) and (2.21) into the integral part of (1.1), we have the matrix relation where (2.25) Substituting the matrix forms (2.10), (2.14), and (2.24) corresponding to the expressions in (1.1) and then simplifying the resulting equation, we have the fundamental matrix equation Next, let us form the matrix representation for the conditions (1.2) as follows [6].
The expression (1.3) and its derivatives are equivalent to the matrix equations where the matrix M k is defined in the expressions (2.21).By using these equations, the quantities y (k) (c r ), k = 0,1,...,m − 1, r = 1,2,...,R, a ≤ c r ≤ b, can be written as where (2.29) Substituting quantities (2.28) into (1.2) and then simplifying, we obtain the matrix forms corresponding to the m mixed conditions as where and the constants u ln , n = 0,1,...,N, l = 1,2,...,m, are related to the coefficients c r lk and c r .
Briefly we can write (2.26) in the form where The augmented matrix of (3.1) becomes Consequently, to find the unknown Taylor coefficients y (n) (c), n = 0(1)N, related with the approximate solution of the problem consisting of (1.1) and conditions (1.2), by replacing the m row matrices (2.30) by the last m rows of augmented matrix (3.3), we have a new augmented matrix or the corresponding matrix equation If det W * = 0, we can write (2.14) as and the matrix Y is uniquely determined.Thus our problem has a unique solution.This solution is given by the truncated Taylor series (1.3).Also we can easily check the accuracy of the obtained solutions as follows [6,8].Since the Taylor polynomial (1.3) is an approximate solution of (1.1), when the function y(x) and its derivatives are substituted in (1.1), the resulting equation must be satisfied approximately; that is, for If max |10 −ki | = 10 −i (i is any positive integer) is prescribed, then the truncation limit N is increased until the difference D(x i ) at each of the points becomes smaller than the prescribed 10 −i .

Illustrations
The method of this study is useful in finding the solutions of general linear Fredholm integrodifferential-difference equations in terms of Taylor polynomials.We illustrate it by the following examples.
Thereby the solution of the given problem under the condition (1.2) becomes x 2 − 0.036356832x 3 + 0.019107892x 4 + 0.00054980308x 5 − 0.00023939909x 6 . (4.9) We use the absolute error to measure the difference between the numerical and exact solutions.In Table 4.1 the errors obtained for N = 6,7,8 are given with the exact solution y(x) = cos(x).
Example 4.3.Our last example is the second-order linear differential-difference equation with the conditions and approximate the solution y(x) by the polynomial M. Sezer and M. Gülsu 13 where Then for N = 7, the fundamental matrix equation from (2.26) becomes To find a Taylor polynomial solution of the problem above, we first take c = 0, N = 7 and then proceed as before.Then we obtain the desired augmented matrix ;

Conclusions
The method presented in this study is useful in finding approximate and also exact solutions of general linear Fredholm integrodifferential-difference equations.Equation (1.1) can be reduced to differential equations, difference equations, integral equations, integrodifferential equations, and also the given method can be applied to all these equations.Differential-difference equations with variable coefficients are usually difficult to be solved analytically.In this case, the presented method is required for the approximate solutions.On the other hand, it is observed that the method has the best advantage when the known functions in equation can be expanded to Taylor series with rapid convergence.In addition, an interesting feature of this method is to find the analytical solutions if the equation has an exact solution that is a polynomial of degree N or less than N.
A considerable advantage of the method is that Taylor coefficients of the solution are found very easily by using the computer programs.We can use the symbolic algebra program, Maple, to find the Taylor coefficients of the solution.
Also, the method can be developed and applied to system of linear integrodifferentialdifference equations, but some modifications are required.

Table 4 .
1. Error analysis of Example 4.1 for the x value.

Table 4 .
2. Error analysis of Example 4.2 for the x value.