A Semianalytical Approach for the Solution of Nonlinear Modified Camassa–Holm Equation with Fractional Order

Abstract

This paper presents the approximate solution of the nonlinear acoustic wave propagation model is known as the modified Camassa–Holm (mCH) equation with the Caputo fractional derivative. We examine this study utilizing the Laplace transform ( $\mathrm{ℒ}$ T) coupled with the homotopy perturbation method (HPM) to construct the strategy of the Laplace transform homotopy perturbation method ( $\mathrm{ℒ}$ T-HPM). Since the Laplace transform is suitable only for a linear differential equation, therefore $\mathrm{ℒ}$ T-HPM is the suitable approach to decompose the nonlinear problems. This scheme produces an iterative formula for finding the approximate solution of illustrated problems that leads to a convergent series without any small perturbation and restriction. Graphical results demonstrate that $\mathrm{ℒ}$ T-HPM is simple, straightforward, and suitable for other nonlinear problems of fractional order in science and engineering.