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
In this paper we consider the following neutral functional-differential equations with proportional delays.
Consider
Consider
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
The differential transform of the
The fundamental mathematical operations performed by one-dimensional differential transform can be obtained from (
Fundamental operations of differential transformation.
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In the following theorems, we find the differential transformations of given functions. These results are very useful in our approach for solving NFDEs.
Suppose that if if if if if
(I), (II) The proof follows immediately by substituting
(III) By using the definition of DTM (
(IV) By using the definition of DTM (
(V) Let the differential transform of
In this part, we will apply the DTM to solve NFDE with proportional delays.
The numerical solutions of Examples
Consider the following first-order NFDE with proportional delay:
Comparison of the absolute errors for Example
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Comparison of the absolute errors for Example
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DTM | |
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Consider the first-order NFDE with proportional delay
Consider the following second-order NFDE with proportional delay:
Consider the second-order NFDE with proportional delay:
Form the initial condition we can get
Consider the following third-order NFDE with proportional delays:
Form the initial condition we can get
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.