Solution for nonlinear Duffing oscillator using variable order variable stepsize block method

Abstract

Real life phenomena found in various fields such as engineering, physics, biology and communication theory can be modeled as nonlinear higher order ordinary differential equations, particularly the Duffing oscillator. Analytical solutions for these differential equations can be time consuming whereas, conventional numerical solutions may lack accuracy. This research propose a block multistep method integrated with a variable order step size (VOS) algorithm for solving these Duffing oscillators directly. The proposed VOS Block method provides an alternative numerical solution by reducing computational cost (time) but without loss of accuracy. Numerical simulations are compared with known exact solutions for proof of accuracy and against current numerical methods for proof of efficiency (steps taken).

Numerical analysis on chaos attractors using a backward difference formulation

“…In previous research, authors successfully adapted the variable order step size backward difference formulation (VOSBF) for solving orbital problems with periodic solutions (see [35][36][37][38][39]). After the success of tackling ODEs with periodic solutions, we initiated this study to attempt approximating chaos attractors which proven to be a more challenging type of ODE.…”

Chebyshev Pseudospectral Method for Duffing Nonlinear Differential Equations

“…Several numerical solutions have been studied so far dealing with the Duffing differential equation such as the modified differential transform method to obtain the approximate solutions of the nonlinear Duffing oscillator [11]; the collocation method is based on the radial basis functions to approximate the solution of the nonlinear controlled Duffing oscillator [12]; M. A. Al-Jawary proposed the Daftardar-Jafari method to solve the Duffing equations and to find the exact solution and numerical solutions [13]; M. Gorji-Bandpy applied Modified Homotopy Perturbation Method and the Max-Min approach to study the generalized Duffing equation [14]; in [15], the authors employed the new perturbation technique to solve strongly nonlinear Duffing oscillators; in [16], the authors used the Taylor Expansion to find approximate solution of Nonlinear Duffing Oscillator; to find numerical solution of the Duffing oscillator, the authors in [17][18] used the Legendre pseudospectral method, the authors in [19] used the spectral method, the authors in [20] used the Taylor matrix method; in [21], the authors proposed the post-verification method for solving the forced Duffing oscillator problems without prescribed periods; the analytical approximation technique basing on the energy balance method was used to determine approximate solutions for highly nonlinear Duffing oscillator [22][23]; the block multistep method is integrated with a variable order step size algorithm to find numerical solutions of the nonlinear Duffing oscillator [24]. This article uses the pseudospectral method based on Chebyshev differential matrix [25] to determine approximate solutions with the boundary conditions on the interval   1,1  take the form ( 1) , (1) .…”

Numerical Solutions of Duffing Van der Pol Equations on the Basis of Hybrid Functions

“…Hence, it needs of numerical techniques for approximate solutions. Some of these numerical methods applied by various researchers include the following: Haar wavelet [11], Adomian decomposition method [12], successive linearisation method [13], iterative method [14], restarted Adomian decomposition method [15], variational iteration method [16], parameter expansion method [17], the chaotic motions [18], and block multistep method [19]. Numerical integration is a particularly effective and multipurpose technique in solving the solution to arbitrary nonlinear differential equations.…”