Solving Systems of Volterra Integral and Integrodifferential Equations with Proportional Delays by Differential Transformation Method

In this paper, the differential transformation method is applied to the system of Volterra integral and integrodifferential equations with proportional delays. The method is useful for both linear and nonlinear equations. By using this method, the solutions are obtained in series forms. If the solutions of the problem can be expanded to Taylor series, then the method gives opportunity to determine the coefficients of Taylor series. Hence, the exact solution can be obtained in Taylor series form. In illustrative examples, the method is applied to a few types of systems.


Introduction
Integral and integrodifferential equations have found applications in engineering, physics, chemistry, and insurance mathematics [1][2][3].In particular, functional-differential equations with proportional delays have described some models such as motion of particle in liquid and polymer crystallization which can be found in [4].
There are a lot of methods of approach for solutions of systems of integral and integrodifferential equations.For example, the linear and nonlinear systems of integrodifferential equations have been solved by Haar functions [5]; Maleknejad and Tavassoli Kajani [6] used the hybrid Legendre functions, the Chebyshev polynomial method [7], the Bessel collocation method [8,9], the Taylor collocation method [10], the homotopy perturbation method [11,12], the variational iteration method [13], the differential transformation method [14], and the Taylor series method [15].Biazar et al. [16] have obtained the solutions of systems of Volterra integral equations of the first kind by the Adomian method.In addition, the homotopy perturbation method has been used for systems of Abel's integral equations [17].On the other hand, the special systems of integral equations have been solved by the differential transformation method [18].Katani and Shahmorad [19] have presented Romberg quadrature for the systems of Urysohn type Volterra integral equations.The nonlinear systems of Volterra integrodifferential equations with delay arguments have been studied by Yalc ¸ınbas ¸and Erdem [20].

Differential Transformation Method
In 1987, the differential transformation method is introduced by Zhou [21] in the study of electric circuits.The method based on Taylor series and yields of differential transformation are difference equations which solutions give the exact values of derivatives of origin function at the given point.The method has been used for a wide class of problems [22][23][24][25].The main advantage of differential transformation from Laplace and Fourier transformations is that it can be applied easily to linear equations with constant and variable coefficients and some nonlinear equations.The differential transformation of the th derivative of function () is defined by and the inverse transformation is defined as follows: The following theorems can be obtained from definitions (2) and (3).
The proofs of Theorems 1 and 2 are given in [22,25].

Illustrate Examples
Example 1.Let us consider the following linear system of Volterra integrodifferential equations with proportional delays and separable kernels: with the initial conditions  1 (0) = 1 and  2 (0) = 0.
The differential transformation of the last system is Substituting  = 0 in Example 1, we obtain values of first derivatives of unknown functions; that is,  1 (1) = 0 and  2 (1) = 1.
Continue this process and use inverse transformation; we get  1 () = cos  and  2 () = sin  which are the exact solutions of Example 1.
Example 4. In last we consider the linear system with variable coefficients of Volterra integral equations with proportional delays: with exact solutions  1 () =  and  2 () = cos .