Abstract
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.
1. Introduction
Integral and integrodifferential equations have found applications in engineering, physics, chemistry, and insurance mathematics [1–3]. 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 Yalçınbaş and Erdem [20].
In this paper, we consider the system of Volterra integral and integrodifferential equations with proportional delays: where , are given functions, , , and .
2. 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 [22–25]. 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).
Theorem 1. Assume that , , are the differential transformations, at the , of the functions , , , respectively; then one has the following. If , then . If , then . If , then is the Kronecker delta symbol. If , then . If , then . If , then . If , then .
Theorem 2. Assume that , and , , are the differential transformations, at the , of the functions , and , respectively, and , . Then, one has the following. If , then . If , then . If , then . If , then . If , then . If , then where .
The proofs of Theorems 1 and 2 are given in [22, 25].
3. 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 and .
The differential transformation of the last system is
Substituting in Example 1, we obtain values of first derivatives of unknown functions; that is, and .
For in (6), we have the following system:
Solving the last system, we get and .
Substituting in (6), we obtain the following system of equations:
Solving the last system with two unknown, we have and .
Continue this process and use inverse transformation; we get and which are the exact solutions of Example 1.
Example 2. Consider the following system of nonlinear Volterra integrodifferential equations with proportional delays: with the initial conditions and .
Analogously, for in (9), we get and .
Applying the differential transformation to (9), we have the following system of difference equations:
For in (10), we get the following system:
Solving the last system, we obtain and .
Substituting in (10) and solving corresponding system, we have and , and for , and . Then using (3), we gain and which are the exact solutions of system (9).
Example 3. Consider the following system of nonlinear Volterra integral equations: with the initial condition .
Now, applying the differential transformation to (12), we get the following system of difference equations:
From initial conditions, we have .
Using (13), we get the following values: and for , .
Using (3), we have and which are exact solutions of system (12).
Example 4. In last we consider the linear system with variable coefficients of Volterra integral equations with proportional delays: with exact solutions and .
Applying DTM, we have
Solving the last system we have and which are Taylor series of exact solutions of Example 4.
4. Conclusions
In this study, the differential transformation method has been presented for solving system of integral and integrodifferential equations with proportional delays. The major benefits of method from integral transformations are that the method can be applied for linear equations with variable coefficients and nonlinear equations and the method gives the exact solutions in series forms.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to thank the reviewers for their constructive comments and suggestions to improve the paper.