Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays

Abstract

An algorithm using the differential transformation which is convenient for finding numerical solutions to initial value problems for functional differential equations is proposed in this paper. We focus on retarded equations with delays which in general are functions of the independent variable. The delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. Approximate solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. Practical implementation of the presented algorithm is demonstrated in an example of the initial value problem for a differential equation with nonlinear nonconstant delay. A two-dimensional neutral system of higher complexity with constant, nonconstant, and proportional delays has been chosen to show numerical performance of the algorithm. Results are compared against Matlab function DDENSD.

On the differential transform method of solving boundary eigenvalue problems: An illustration

“…The main advantage of the DTM is that it can be applied directly to nonlinear differential equations without requiring linearization, discretization, or perturbation. As mentioned earlier the DTM has been extensively used to solve differential equations [14,15] (both ordinary and partial), delay differential equations [16], integral equations [19], integro-differential equations [20], and so on. In this section, we introduce some essential concepts of differential transform and apply them to solve the BEVP (36)-(39).…”

“…Numerical methods such as finite difference methods, shooting method, spectral methods, variational methods (Galerkin and Rayleigh-Ritz methods) and similar methods that handle boundary value problems can be used, with suitable modifications, to solve BEVPs (see Rees [1] and Barletta and Rees [2]). Likewise approximate analytical methods such as homotopy methods (homotopy analysis method -HAM [3][4][5], homotopy perturbation method -HPM [6][7][8]), variational iteration method (VIM) [9][10][11], Adomian decomposition method (ADM) [12,13], DTM [14][15][16] and many other methods may also be suitably altered to deal with BEVPs. So far the DTM has been successfully applied in solving initial and boundary value problems involving ordinary and partial differential equations.…”

Applications of the differential transform to second-order half-linear Euler equations

“…It has been shown that DT is convenient for solving a variety of initial value problems (IVPs), covering the range from ordinary to functional, partial and fractional differential equations [23][24][25][26]. In particular, results on the differential equations with proportional, constant and non-constant delays can be found in [27][28][29].…”

Applications of the differential transformation to three-point singular boundary value problems for ordinary differential equations

“…It has applications in solving different types of problems for all classes of differential equations (ordinary, partial, delayed, fractional, fuzzy etc). The recent developments and applications of DTM are discussed in [15,[17][18][19] and references therein.…”