Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

Abstract

We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems.

Using differentiation matrices for pseudospectral method solve Duffing Oscillator

“…It is important to determine the iterative equation (33). The iterative procedure is simple, we assumed u (0) = const, then found u (1) , u (2) , ..., stop it until the error ε = |u (k) − u (k−1) | < ε 0 .…”

“…Several approaches have been studied so far dealing with the nonlinear Duffing Oscillators such as The differential transform method [12]; The Jacobi elliptic function cn [16]; The analysis method [1,6,8]; The Taylor Expansion [5]; The Legendre pseudospectral method [14,15]; A Chebyshev collocation algorithm [13]; The Enhanced Cubication Method [4]; The Improved Taylor Matrix Method [2]; The Postverification Method [10], the energy balance method [9].…”

A reduced computational matrix approach with convergence estimation forsolving model differential equations involving specific nonlinearities ofquartic type

“…So far, with this aim, Kürkçü et al [18] employed the Dickson matrix-collocation method to solve some model problems arising in science. Bülbül and Sezer [4,5] established the Taylor polynomial method to find the approximate solutions of nonlinear differential equations of Abel and Duffing types. Rajaraman and Hariharan [26] proposed the shifted second-kind Chebyshev wavelet method to solve singular boundary value problems.…”

A Numerical Approach Based on Taylor Polynomials for Solving a Class of Nonlinear Differential Equations

“…They implement a relatively new analytic iterative technique to get approximate solutions of differential algebraic equations system based on generalized Taylor series formula and the analysis proposes an analytical-numerical approach for providing solutions of a class of nonlinear fractional Klein-Gordan equation subjected to appropriate initial conditions in Caputo sense by using the Fractional Reduced Differential Transform Method (FRDTM) in [13,14], respectively. In this study, by means of the matrix methods based on collocation points given by M. Sezer and coworkers [15][16][17], we developed a new numerical method to find the approximate solutions of (1) in the truncated Taylor series form…”