A multi-step differential transform method and application to non-chaotic or chaotic systems

Abstract

The differential transform method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. In this paper, we propose a reliable new algorithm of DTM, namely multi-step DTM, which will increase the interval of convergence for the series solution. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. This new algorithm is applied to Lotka-Volterra, Chen and Lorenz systems. Then, a comparative study between the new algorithm, multistep DTM, classical DTM and the classical Runge-Kutta method is presented. The results demonstrate reliability and efficiency of the algorithm developed.

Application of the multi-step differential transform method to solve a fractional human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells

“…The approximate solutions obtained by using DTM are valid only for a short time. While the ones obtained by using the MSDTM [16] are more valid and accurate during a long time, and are in good agreement with the RK4-5 numerical solution when the order of the derivative (α = 1). The rest of the paper is organized as follows.…”

Numerical solution and stability analysis of a nonlinear vaccination model with historical ejects

“…Although the GDTM is used to provide approximate solutions for a wide class of nonlinear problems in terms of convergent series, it has some drawbacks. The series solution always converges in a very small region and it has been shown that the obtained approximate solution is valid only for a short time in some problems [34,29]. Here, we solve the nonlinear fractional system (2.9) with initial values (2.2) by applying the multi-step generalized differential transform method (MSGDTM) which is a modified version of GDTM.…”

“…Here, we solve the nonlinear fractional system (2.9) with initial values (2.2) by applying the multi-step generalized differential transform method (MSGDTM) which is a modified version of GDTM. It is shown that the approximate solutions obtained by MSGDTM algorithm are more accurate than GDTM during a longer time [29,30]. Also, it is shown that MSDTM has a significant performance compared with the Runge-Kutta method when the order of the derivative is one [35].…”

“…We also solve the nonlinear fractional system by using the multi-step generalized differential transform method (MSGDTM) which is a modified version of GDTM [28]. It is shown that the approximate solutions obtained by MSGDTM algorithm are more accurate than GDTM during a longer time [29,30].…”

A new piecewise-quasilinearization method for solving chaotic systems of initial value problems

“…Examples of multi-stage methods that have been developed recently to solve IVPs for chaotic and nonchaotic systems include the, multi-stage homotopy analysis method [2,4,5], piecewise homotopy perturbation methods [9,25], multi-stage Adomian decomposition method [1,20], multi-stage differential transformation method, [3,15,24], multi-stage variational iteration method [14,19]. Because these methods attempt to obtain analytical solutions at each interval they involve timeconsuming and tedious computational operations and if too many small intervals are considered, as may be the case when dealing with highly oscillatory systems, the analytical integration process will be too much to handle even with the use of symbolic scientific software.…”