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.

The Variational Iteration Method for a Pendulum with a Combined Translational and Rotational System

“…The idealised description of a mechanical device is a homogenous, uniform wheel rolling smoothly over a horizontal surface. A combined translational and rotational system is shown geometrically [36,46] in Figure 2.…”

“…However, analytical methods are preferable to numerical methods because they allow for easier understanding of the basic physics of the problem. The variational iteration technique was introduced in 1998 [35] and has been successfully used to solve a wide range of non-linear problems [36]. The main objective of this approach is to construct a correction function by utilising a general Lagrange multiplier that is carefully selected because its correctional solution is significant than the initial trail function.…”

Oscillator Simulation with Deep Neural Networks

“…Although many methods are used to model, simulate, and solve dynamical systems [3] and to discuss their stability [4], neural network-based techniques [5,6] to approximate the solution of DEs occurring in various systems [7] have, however, garnered a reputation in recent years. Other analytical methods for ordinary DEs [8] and partial DEs [9], semi-analytical methods [10], including the Variational Iteration Method (VIM) by using the Laplace Transform [11], and numerical methods have their own shortcomings in terms of convergence, precision, processing time, and computational complexity. However, a DNN [12] resolves these problems and, rather than through direct calculation, features are successfully acquired by the neural network through a training process in the layers of connected neurons [13].…”

The Homotopy Perturbation Method for Electrically Actuated Microbeams in Mems Systems Subjected to Van Der Waals Force and Multiwalled Carbon Nanotubes

“…Through the examination of different design configurations and optimization methodologies made possible by this methodology, the field of MEMS is ultimately advanced, and the functionality and efficiency of MEMSs in a variety of applications are improved. Highly nonlinear problems have been solved using the homotopy perturbation method (HPM) [35][36][37][38][39][40]. This method provides the answer in the form of a series that quickly approaches the approximate answer.…”