A new multi-step technique with differential transform method for analytical solution of some nonlinear variable delay differential equations.

Brahim Benhammouda,Hector Vazquez-Leal

doi:10.1186/s40064-016-3386-8

Brahim Benhammouda, Hector Vazquez-Leal

Open Access

https://doi.org/10.1186/s40064-016-3386-8

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Journal: SpringerPlus	Publication Date: Oct 6, 2016
Citations: 10	License type: CC BY 4.0

Affiliation: Higher Colleges of Technology

Abstract

This work presents an analytical solution of some nonlinear delay differential equations (DDEs) with variable delays. Such DDEs are difficult to treat numerically and cannot be solved by existing general purpose codes. A new method of steps combined with the differential transform method (DTM) is proposed as a powerful tool to solve these DDEs. This method reduces the DDEs to ordinary differential equations that are then solved by the DTM. Furthermore, we show that the solutions can be improved by Laplace–Padé resummation method. Two examples are presented to show the efficiency of the proposed technique. The main advantage of this technique is that it possesses a simple procedure based on a few straight forward steps and can be combined with any analytical method, other than the DTM, like the homotopy perturbation method.

Highlights

Differential equations are relevant tools to model a wide variety of physical phenomena across all areas of applied sciences and engineering
Variable delays differential equations (DDEs) with arbitrary types of nonlinear functions for the time delay is an open area of research that require new numerical and analytical methods in order to deal with their solution
The straight forward procedure was able to deal with different algebraic time delays as quadratic and cubic term highlighting the malleability of the technique presented to solve nonlinear delay differential equations (DDEs) with variable delays

Summary

Background

Differential equations are relevant tools to model a wide variety of physical phenomena across all areas of applied sciences and engineering. Case studies we will demonstrate the effectiveness and accuracy of the multi-step technique proposed in “Multi-step technique and DTM for nonlinear variable DDEs” section with the differential transform method (DTM) (Zhou 1986; Keskin et al 2007; Benhammouda et al 2014b; Benhammouda and Vazquez-Leal 2015; Biazar and Eslami 2010; Chen and Liu 1998; Ayaz 2004; Kangalgil and Ayaz 2009; Kanth and Aruna 2009; Arikoglu and Ozkol 2007; Chang and Chang 2008; Kanth and Aruna 2008; Lal and Ahlawat 2015; Odibat et al 2010; El-Zahar 2013; Gökdoğan et al 2012) through two nonlinear variable delay differential equations (DDEs).

Discussion

Conclusions