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

Differential equations are relevant tools to model a wide variety of physical phenomena across all areas of applied sciences and engineering. Analytical techniques are applied to find the exact solutions of some cases of differential equations; nevertheless, when the differential equations are nonlinear there are no general techniques of solutions. Therefore, numerical methods are important tools to study and understand the quantitative behavior of the nonlinear differential equations with unknown exact solutions. However, such methods can exhibit numerical instabilities, oscillations or false equilibrium states, among others (Gumel 2002, 2003; de Markus and Mickens 1999). This means that the numerical solution may not correspond to the real solution of the original problem. This situation is further aggravated for some types of differential equations like differential-algebraic equations, fractional differential equations and time delay differential equations (DDEs), among others (Ford and Wulf 2000; Engelborghs et al. 2000; Ascher and Petzold 1998; Campbell et al. 2008; Wanner and Hairer 1998).

Time DDEs appear in propagation and transport phenomena, population dynamics, bioscience problems, neural network model, control problems, electrical networks containing lossless transmission lines and economical systems where decisions and effect are separated by some time intervals, among many other applications (Martín and García 2002a, b; Aiello and Freedman 1990; Gourley and Kuang 2004a, b; Kuang 1993; Li et al. 2006; Li and Jiang 2013; Zhang and Zhang 2013). Given the importance of this kind of equations, some numerical methods have been developed to solve them; among them one can mention: variable multi-step methods (Martín and García 2002a, b), Chebyshev polynomials for pantograph differential equation (Sedaghat et al. 2012), a spectral Galerkin method for nonlinear delay convection-diffusion reaction equations (Liu and Zhang 2015), power series method (Benhammouda et al. 2014a) and the differential transform method (DTM) (Karako and Bereketoglu 2009).

In this work, we propose some case studies of nonlinear DDEs with variable time delays. For such case studies, there are no known numerical methods available. Therefore, we propose a multi-step method with a modified version of the 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; Gökdo?an et al. 2012) in order to find the solutions of variable time DDEs.

In the literature, there are some works reporting variable time delays. For instance, Taylor series method is applied to solve time-dependent stochastic DDEs (Miloševi? and Jovanovi? 2011). In addition, a study of asymptotic neutral type differential equations with variable time delay is given in Skvortsova (2015). Besides, a study of asymptotic behavior of first order differential equations with variable delay is presented in Dix (2005). In Ding et al. (2010), some new conditions for the boundness and stability by means of the contraction mapping principle are given for nonlinear scalar DDEs with variable delays. The issue of uniqueness of variable DDEs is investigated in Winston (1970), Eloe et al. (2005), Liu and Clements (2002) and Luo et al. (2013). Finally, a research for the existence of attractors for differential equations with a variable delay is presented in Graef and Qian (2000) and Caraballo et al. (2001).

Nonetheless, in the present study, we propose different types of variable delays in terms of algebraic expressions of the time. What is more, given that the approximate solutions of the DTM are power series solutions, we propose to extend the domain of convergence by using a combined scheme of a multi-step technique and the Laplace–Padé resummation method (Vazquez-Leal and Guerrero 2014; Filobello-Nino et al. 2013; Jiao et al. 2002; Sweilam and Khader 2009; Momani et al. 2009; Khan and Faraz 2011; Momani and Ertürk 2008; Tsai and Chen 2010; Ebaid 2011).

This paper is organized as follows: in “Differential transform method” section, we introduce the basic concept of the DTM. Then, in “Multi-step technique and DTM for nonlinear variable DDEs” section, the proposed multi-step technique with the use of the DTM to deal with variable time DDEs is presented. “Padé approximant” and “Laplace–Padé resummation method” sections are devoted to present the basic concepts of Padé and Laplace–Padé resummation methods. In “Case studies” section, the type of delay differential equations with variable delays under study is presented and solved using the proposed technique. Finally, discussion and conclusions are given in “Discussion” and “Conclusions” sections, respectively.