The Fractional Analysis of a Nonlinear mKdV Equation with Caputo Operator

Abstract

In this study, we aim to provide reliable methods for the initial value problem of the fractional modified Korteweg–de Vries (mKdV) equations. Fractional differential equations are essential for more precise simulation of numerous processes. The hybrid Yang transformation decomposition method (YTDM) and Yang homotopy perturbation method (YHPM) are employed in a very simple and straightforward manner to handle the current problems. The derivative of fractional order is displayed in a Caputo form operator. To illustrate the conclusion given from the findings, a few numerical cases are taken into account for their approximate analytical solutions. We looked at two cases and contrasted them with the actual result to validate the methodologies. These techniques create recurrence relations representing the proposed problem’s solution. It is possible to find the series solutions to the given problems, and these solutions have components that converge to precise solutions more quickly. Tables and graphs are used to describe the new results, which demonstrate the present methods’ adequate accuracy. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. The innovation of the current work resides in the application of effective methods that require less calculation and achieve a greater level of accuracy. Additionally, the suggested approaches can be applied in the future to resolve other nonlinear fractional problems, which will be a scientific contribution to the research community.