Modified Laplace Based Variational Iteration Method for the Mechanical Vibrations and its Applications

Abstract

In this paper, we are putting forward the periodic solution of non-linear oscillators by means of variational iterative method (VIM) using Laplace transform. Here, we present a comparative study of the new technique based on Laplace transform and the previous techniques of maximum minimum approach (MMA) and amplitude frequency formulation (AFF) for the analytical results. For the non-linear oscillators, MMA, AFF and VIM by Laplace transform give the same analytical results. Comparison of analytical results of VIM by Laplace transform with numerical results by fourth-order Runge–Kutta (RK) method conforms the soundness of the method for solving non-linear oscillators as well as for the time and boundary conditions of the non-linear oscillators.

A modern analytic method to solve singular and non-singular linear and non-linear differential equations

“…Many analytical, numerical, and numero-analytical techniques have been proposed previously and recently to provide solutions for differential equations with initial or boundary conditions, such as the Laplace and Fourier transforms method [2], the Adomian decomposition method [3][4][5], the variational iteration method [6][7][8], the homotopy perturbation method [9][10][11], the homotopy analysis method [12,13], the differential transformation method [14][15][16], the finite difference method [17], the predictor-corrector method [18,19], the first integral method [20,21], the Adams-Bashforth Molten method [22], the new iterative method [23,24], the Crank-Nicolson method [25,26], the reproducing kernel method [27,28], the Laplace Adomian decomposition method [11,29,30], the He-Laplace method [31][32][33], and others [34][35][36].…”

“…Indeed, the LRPS method is similar to the He-Laplace method's [31][32][33] idea of searching for the solution of differential equations in Laplace space. The He-Laplace technique uses the variational iteration method or homotopy perturbation method to solve the just-transformed Laplace space.…”

Combination of modified Lindstedt-Poincare and homotopy perturbation methods

“…¼ a, _ xð0Þ ¼ 0 oscillators (Qie et al, 17 Koochi A and Goharimanesh, 18 Anjum et al, 19,20 Rehman et al, 21 Suleman et al, 22 Anjum et al, 23 Li and Nadeem, 24 and Askari et al 25 ). The goal of this article is to present a combined technique of the LP and homotopy perturbation methods to solve some strongly nonlinear oscillators.…”

“…In a recent article, 7 Alam et al provided a generalized form of the modified LP 6 method and used it to solve various nonlinear oscillators. On the other hand, the homotopy perturbation method has been modified in some recent articles presented by Anjum et al, 12 He et al, 13 Anjum et al, 14 Ji et al, 15 and He et al 16 Moreover, some approximate techniques are available to solve strongly nonlinear oscillators (Qie et al, 17 Koochi A and Goharimanesh, 18 Anjum et al, 19,20 Rehman et al, 21 Suleman et al, 22 Anjum et al, 23 Li and Nadeem, 24 and Askari et al 25 ).…”

Investigation of Nonlinear Vibrational Analysis of Circular Sector Oscillator by Using Cascade Learning

“…A series of analytical and semiexact techniques that do not require a small parameter was proposed in the middle of the twentieth century to overcome the problem. Some well-known techniques used for the numerical and analytical solution of nonlinear problems are Adomian decomposition method (ADM) [16,17], energy balance method (EBM) [18], variational iteration method (VIM) [19][20][21], extended direct algebraic method (EDAM) [22], Adams-Bashforth method [23], differential transform method [24], Hamiltonian approach [25,26], Backlund method [27] rational harmonic balance method [28], min-max approach (MMA) [29,30], power series technique [31], one-step hybrid block method [32], amplitude-frequency formulation (AFF) [33,34], and modified homotopy perturbation method (MHPM) [35][36][37]. Ebrahimi Khah [38] studied the motion of a rigid rod rocking back over a circular surface using He's energy balance method.…”