Abstract
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.
Highlights
The neutral functional-differential equation NFDE is m−1 utatu σm t m βu t bk t uk σk t f t, t ≥ 0, k0 under the conditions m−1 ciku k 0 λi, i 0, 1, . . . , m − 1, k0 where a, bk, σk are analytical functions; β, cik and λi ∈ C
We find the differential transformations of given functions
These results are very useful in our approach for solving NFDEs
Summary
The neutral functional-differential equation NFDE is m−1 utatu σm t m βu t bk t uk σk t f t , t ≥ 0, k0 under the conditions m−1 ciku k 0 λi, i 0, 1, . . . , m − 1, k0 where a, bk, σk are analytical functions; β, cik and λi ∈ C. Another interesting case 2 is ISRN Applied Mathematics σk t qkt, k 0, 1, . In this paper we consider the following neutral functional-differential equations with proportional delays. 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 9 and homotopy perturbation method 6. Original function u t f t ±g t u t cf t ut ftgt u t f1 t f2 t · · · fm−1 t fm t u t dn/dtn f t u t tm Transformed function
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