Extension of the differential transformation method to nonlinear differential and integro-differential equations with proportional delays
© Šmarda et al.; licensee Springer. 2013
Received: 11 January 2013
Accepted: 4 March 2013
Published: 21 March 2013
In this paper, the differential transformation method is applied by providing new theorems to develop exact and approximate solutions of nonlinear differential and integro-differential equations with proportional delays represented by nonlinear multi-pantograph equations. Some examples are given to demonstrate the validity and applicability of the present method and a comparison is made with existing results.
Functional differential and integro-differential equations with proportional delays are usually referred to as pantograph equations or generalized equations, and, as well, they are often used to model some problems with aftereffect in mechanics and the related scientific fields [1–13]. Many typical examples such as stress-strain states of materials, motion of rigid bodies and models of polymer crystallization can be found in Kolmanovskii and Myshkis’s monograph  and the references therein.
2 Differential transformation method
The method that is developed in this work depends on the differential transformation method (DTM) introduced by Zhou  in a study of electric circuits. This method constructs a semi-analytical numerical technique that uses Taylor series for the solution of differential equations in the form of polynomials. It is different from the high-order Taylor series method which requires symbolic computation of the necessary derivatives of the data functions.
There is no need for linearization or perturbations, large computational work and round-off errors are avoided. It has been used to solve effectively, easily and accurately a large class of linear and nonlinear problems with approximations [16–25].
In fact, inverse transformation (2) implies that the concept of differential transformation is derived from the Taylor series expansion. Although DTM is not able to evaluate the derivatives symbolically, relative derivatives can be calculated in an iterative way which is described by the transformed equations of the original function.
From definitions (1), (2) we can derive the following.
The proof of Theorem 1 is given in .
3 Main results
In this section, we state the fundamental theorems of this paper.
- (ii)Using the Leibnitz formula, we obtain
- (iii)From the proof of formula (i), we get
- (iv)Analogously, from to previous part of the proof, we get
The proof is complete. □
- (II)Similarly as in previous part (I), we have
- (III)Put , then from the previous part we get(3)
The proof is complete. □
4 Numerical examples
The closed form of the above series solution is , which is the exact solution of equation (5).
which is the exact solution of equation (8). Evans and Raslan  solved equation (8) using the Adomian decomposition method and obtained a sequence of approximate solutions in the form of triple integrals with Adomian’s polynomials that required many symbolic calculations to obtain approximate solutions of (8). Saeed and Rahman  also solved equation (8) using the differential transformation method, but they transformed equation (8) into a system of three differential equations, which is a uselessly complicated approach to solving equation (8).
Thus , . Using the inverse transformation rule (2), we obtain the exact solution .
The closed form of the above series solution is , which is the exact solution of equation (14).
Similarly, we obtain , . Using the inverse transformation rule (2), we get the exact solution . A homogeneous form of equation (16) subject to initial conditions was solved by Abazari and Kilicman . They obtained the closed form of a series solution in the form .
In the present paper, we have shown that the differential transformation method can be successfully used for solving nonlinear differential and integro-differential equations with proportional delays. New theorems are introduced with their proofs, and as application some examples are carried out. The main advantage of this method is that it can be applied directly to functional differential and integro-differential equations without requiring linearization, discretization or perturbation. Another important advantage is that this method is capable of greatly reducing the size of computational work and, moreover, the proposed method reduces the solution of a problem to the solution of a system of recurrence algebraic equations.
The first author is supported by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. The second author is supported by the Grant P201/11/0768 of the Czech Grant Agency (Prague).
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