A local differential transform approach for the cubic nonlinear Duffing oscillator with damping term

Abstract

Nonlinear behaviour of various problems is described by the Duffing model interpreted as a forced oscillator with a spring which has restoring force. In this paper, a new numerical approximation technique based on differential transform method has been introduced for the nonlinear cubic Duffing equation with and without damping effect. Since exact solutions of the corresponding equation for all initial guesses do not exist in the literature, we have first produced an exact solution for specific parameters by using the Kudryashov method to measure the accuracy of the currently suggested method. The innovative approach has been compared with the semi-analytic differential transform method and the fourth order RungeKutta method. Although the semi-analytic differential transform method is valid only for small time intervals, it has been proved that the innovative approach has ability to capture nonlinear behaviour of the process even in the long-time interval. In a comparison way, it has been shown that the present technique produces more accurate and computationally more economic results than the rival methods.

Novel mechanism for primordial perturbations in minimal extensions of the Standard Model

“…indicated by the dashed lines in figure 1. The approximation, which works exceptionally well across a large range of scales, was obtained by realising that the logistic function is often used to approximate the solution of similar differential equations [63]. The coefficients are determined through a fit of the numerical result.…”

A new implicit-explicit local differential method for boundary value problems

“…Generally, those semi-analytic approaches give good agreement with exact solution only about one of the boundaries due to the structure of the Taylor series expansion. The idea of discretization of finite domain and then applying DTM procedure numerically is known as local DTM (LDTM) or multi-step DTM [34,35,44]. Although the LDTM is better than the DTM in terms of convergence, as will be shown later, the LDTM does not lead to stable results as a shooting method [44] for especially stiff problems.…”

“…. θ = 0 yields the explicit forward scheme and known as local differential transform method which is used in the literature [34,35].…”

“…In line with our theoretical analysis, the accuracy of the present IELDTM is increasing with either dx -refinement or K -refinement for all problems. Both the semi-analytical DTM and the numerical LDTM[34,35,44] suffer from instability, especially in solving stiff BVPs. Just at this time, the IELDTM is here constructed for elimination of this serious drawback.…”

An implicit-explicit local method for parabolic partial differential equations

“…The automatic differentiation method (ADM) and the differential transform method (DTM) are the Taylor series-based multi-derivative approximation methods in either semi-analytic form (Abdel-Halim Hassan, 2004; Rashidi, 2009) or numerical form (Abdel-Halim Hassan, 2008; Tunc and Sari, 2019; Abbasbandy and Bervillier, 2011). Even if the debates are increasing about the relation between the ADM and the DTM, they are indeed mathematically equivalent (Bervillier, 2012).…”