On the solution of nonlinear fractional-order shock wave equation via analytical method

Abstract

<abstract><p>In this study, we propose a method to study fractional-order shock wave equations and wave equations arising from the motion of gases. The fractional derivative is taken in Caputo manner. The approaches we used are the combined form of the Yang transform (YT) together with the homotopy perturbation method (HPM) called homotopy perturbation Yang transform method (HPYTM) and also Yang transform (YT) with the Adomian decomposition method called Yang transform decomposition method (YTDM). The HPYTM is a combination of the Yang transform, the homotopy perturbation method and He's polynomials, whereas the YTDM is a combination of the Yang transform, the decomposition method and the Adomian polynomials. Adomian and He's polynomials are excellent tools for handling nonlinear terms. The manipulation of the recurrence relation, which generates the series solutions in a limited number of iterations, is the essential innovation we describe in this study. We give several graphical behaviors of the exact and analytical results, absolute error graphs, and tables that highly agree with one another to demonstrate the reliability of the suggested methodologies. The results we obtained by implementing the proposed approaches indicate that it is easy to implement and computationally very attractive.</p></abstract>