Insight into the study of some nonlinear evolution problems: Applications based on Variation Iteration Method with Laplace

Abstract

In this study, we look at the solutions of nonlinear partial differential equations and ordinary differential equations. Scientists and engineers have had a hard time coming up with a way to solve nonlinear differential equations. Almost all of the nature’s puzzles have equations that aren’t linear. There aren’t any well-known ways to solve nonlinear equations, and people have tried to improve methods for a certain type of problems. This doesn’t mean, however, that all nonlinear equations can be solved. With this in mind, we’ll look at how well the variation approach works for solving nonlinear DEs. Different problems can be solved well by using different methods. We agree that a nonlinear problem might have more than one answer. Factorization, homotropy analysis, homotropy perturbation, tangent hyperbolic function and trial function are all examples of ways to do this. On the other hand, some of these strategies don’t cover all of the nonlinear problem-solving methods. In this paper, a new method called the variation iterative method with Laplace transformation is used to find a solution to the highly nonlinear evolution of a simple pendulum whose rotation revolves around its fixed position. When the Laplace operator is used to change the Maximum Minimum Approach, Amplitude Frequency Formulation and Variation Iteration Method (VIM) nonlinear oscillators, the results of the analysis are all the same. The method for solving nonlinear oscillators, as well as their time and boundary conditions, can be shown to be correct by comparing analytical results of VIM obtained through the Laplace transformation.