Fourier spectral exponential time-differencing method for space-fractional generalized wave equations

Abstract

This manuscript deals with a space-fractional generalized wave problem involving the fractional Laplacian operator of order α for 1 < α ≤ 2. We propose an accurate numerical method to solve the mentioned fractional wave problem. The problem is discretized in spatial direction by the Fourier spectral method, and in temporal direction by utilizing the fourth-order exponential time-differencing Runge-Kutta (ETDRK4) method. One of the main features of this method is to reduce the mentioned fractional wave model to an ODE by using the Fourier transform, and then the ETDRK4 method is used to solve this ODE. We define the discrete energy function and check the energy-conserving properties. The convergence of this method is proved. Various numerical experiments are conducted to confirm the accuracy and dependability of the suggested approach.Keywords Generalized wave equation • Fractional Laplacian operator • Fourier spectral method • Fourth-order exponential Runge-Kutta method • convergence.