Research Article | Open Access
Sabir Widatalla, "Extension of Zhou's Method to Neutral Functional-Differential Equation with Proportional Delays", International Scholarly Research Notices, vol. 2012, Article ID 518361, 10 pages, 2012. https://doi.org/10.5402/2012/518361
Extension of Zhou's Method to Neutral Functional-Differential Equation with Proportional Delays
The differential transform method (DTM) is a reliable method applied by providing new theorems to develop exact and approximate solutions of neutral functional-differential equation (NFDE) with proportional delays. The results obtained with the proposed methods are in good agreement with one obtained by other methods. The advantages of this technique are illustrated. It is easy to see that the DTM is very accurate and easy to implement in finding analytical solutions of wide classes of linear and nonlinear NFDEs.
The neutral functional-differential equation (NFDE) is under the conditions where are analytical functions; and . A classical case  where is positive. Another interesting case  is where . Both cases are playing an interesting role in many applications (see [1, 2] and references therein). In recent years, there has been a growing interest in the numerical treatment of NFDE, some of which are the Adams method , continuous Runge-Kutta methods , segmented Tau approximation , Homotopy perturbation method , one-leg -methods [7, 8], and variational iteration method .
In this paper we consider the following neutral functional-differential equations with proportional delays.
Problem 1. Consider
Problem 2. Consider where are analytical functions; and are real or complex constants; .
The basic motivation of this work is to extend the differential transform method (DTM) by presenting and proving new theorems to create the exact or approximate solutions to a high degree of accuracy to the Problems 1 and 2. The DTM is a numerical-analytical technique that was first proposed by Zhou (1986) , who solved problems in electric circuit analysis. Since then, DTM was successfully applied for a large variety problems. For example, differential-difference equations , Volterra integral equation with separable kernels , MHD boundary-layer equations , linear and nonlinear systems of partial differential equations , and nonlinear oscillators with fractional nonlinearities . To the best of our knowledge differential transform method has not be used by any researcher before to solve NFDE. By this method it is possible to obtain highly accurate results when compared with existing results from variational iteration method  and homotopy perturbation method .
2. Basic Idea of Differential Transform Method
The differential transform of the th derivative of a function is defined as follows: The original and transformed functions are denoted throughout this paper by small and capital letters, respectively. The differential inverse transform of is defined as And from (2.1) and (2.2): which is actually the Taylor series expansion of about the point .
3. Main Results
In the following theorems, we find the differential transformations of given functions. These results are very useful in our approach for solving NFDEs.
Theorem 3.1. Suppose that , and are the differential transformations of the functions , respectively, and and , for : (I)if , then , (II)if , then , (III)if , then (IV)if , then (V)if , then
Proof. (I), (II) The proof follows immediately by substituting into (2.1).
(III) By using the definition of DTM (2.1), we have and from (II), we have By utilizing this value, we get
(IV) By using the definition of DTM (2.1), we have then from (II), we have By utilizing these values
(V) Let the differential transform of at for be , then by using operations of differential transformation given in Table 1, we have and from (II), we have Substituting those values into (3.10), we obtain (3.3).
4. Illustrative Examples
In this part, we will apply the DTM to solve NFDE with proportional delays.
The numerical solutions of Examples 4.3, 4.4, and 4.5 have been calculated by variational iteration method  and homotopy perturbation method , which did not yield the exact solutions. However, applying DTM gives the exact solutions of those examples, as we will show later.
Example 4.1 (see [6, 9]). Consider the following first-order NFDE with proportional delay: Taking the differential transform of (4.1) as given in (2.1), we get which can be rewritten as follows: The differential transform of the initial condition of at , is , form (4.3) for , we can get substituting these values into (2.2), to get The closed form of the above solution, when is , which is the exact solution. In Table 2 the absolute errors of DTM for , VIM  with eight terms and HPM  with eight terms (Table 3) are compared.
Example 4.2 (see [6, 9]). Consider the first-order NFDE with proportional delay Let The differential transforms of , , and at be , and , respectively: We can obtain the differential transform of (4.6) as At , the initial condition transformed to , so from (4.8), we have Substituting these values into (2.2), we get The closed form of the above solution, when , is , which is the exact solution. In Table 2 we compare the absolute errors of DTM for , VIM  with eight terms, and HPM  with eight terms.
Example 4.3 (see[6, 9]). Consider the following second-order NFDE with proportional delay: The differential transform for (4.11) is found as form the initial condition we can get and . Form (4.12), we get Then, by using (2.2), , which is the exact solution.
Example 4.4 (see [6, 9]). Consider the second-order NFDE with proportional delay:
The differential transform for (4.14) at is given by
and can be rewritten as
Form the initial condition we can get , and form (4.16), we get Then, by using (2.2), , which is the exact solution.
Example 4.5 (see [6, 9]). Consider the following third-order NFDE with proportional delays:
The differential transform of (4.18) can be written as
Form the initial condition we can get , so from the (4.19), we get Substituting (4.20) in (2.2) gives , which is the exact solution.
In this study, we extended DTM to the solution of NFDE with proportional delays. New theorems are presented with their proofs. All examples results show that the DTM is more effective than VIM and HPM for solving NFDE with proportional delays. We believe that the ease of implementation and efficiency of the DTM gives it much wider applicability.
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Copyright © 2012 Sabir Widatalla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.