New analytical method for gas dynamics equation arising in shock fronts

This work suggests a new analytical technique called the fractional homotopy analysis transform method (FHATM) for solving nonlinear homogeneous and nonhomogeneous time-fractional gas dynamics equations. The FHATM is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. The proposed technique solves the nonlinear problems without using Adomian polynomials and He's polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method. In this paper, it can be observed that the auxiliary parameter which controls the convergence of the HATM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more qualitative difference in analysis between HATM and other methods. The solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive. The proposed method is illustrated by solving some numerical examples.