Using analytical methods for finding the approximate solutions to fractional differential equations

Abstract

This essay focuses on studying the nonlinear fractional integral equation. Various methods, including Akbari-Ganji's Method (AGM), Homotopy Perturbation Method (HPM), and Vibrational Iteration Method (VIM), are utilized to obtain its solution. We introduce an innovative approach to obtain rough approximations for fractional differential equations. These equations play a significant role in the field of fluid dynamics and find widespread application. In this article, we have used analytical methods to check the correctness of the answers. Maple mathematical software is used to solve all fractional equations. Ordinary equations and fractional differential equations have connections to entropy, wavelets, and other related concepts. To demonstrate the method, a few examples are employed, chosen for their accuracy and simplicity of implementation. The solutions are explained using convergent series. According to the calculations performed with analytical methods on the fractional integral equations of the nonlinear oscillator, Due to the swing movements of the oscillator, as the swing movement's increase, the velocity gradient becomes an upward trend.