He’s Variational Iteration Method for Nonlinear Jerk Equations: Simple but Effective

Abstract

Abstract. The variational iteration method (VIM) for determining periodic solutions of non-linear jerk equations involving third-order time-derivative is presented. This method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. We propose first-order approximate VIM solution for this equation and we compare this solution with the solution obtained by harmonic balance method.

The simplest approach to solving the cubic nonlinear jerk oscillator with the non‐perturbative method

“…The approximate numerical solutions, the analytical perturbation methods, the harmonic balance technique are utilized in, 5 whereas the high‐order ordinary differential equation in terms of its differential invariants is addressed in 6 . Bhrawy et al 7 has applied the algorithm to the numerical solutions of nonlinear jerk equations through used the Jacobi–Gauss paving method, and the variational iteration approach was applied in Raftari 8 for nonlinear jerk equations. The modification of the harmonic balance process was utilized for nonlinear third‐order equations in Rahman and Hasan 9 .…”

Two-Step Hybrid Block Method for Solving Nonlinear Jerk Equations

“…Many authors have studied the numerical solutions of the Jerk equation, harmonic balance approach to periodic solutions is used in [3], in [4] they have written the high-order ordinary differential equation in terms of its differential invariants. New algorithm for the numerical solutions of nonlinear third-order differential equations was used jacobi-gauss collocation method in [5], He's variational iteration method was used in [6] for nonlinear Jerk equations. Modified harmonic balance method was used for nonlinear Jerk equations in [7].…”