A linearized finite difference/spectral-Galerkin scheme for three-dimensional distributed-order time–space fractional nonlinear reaction–diffusion-wave equation: Numerical simulations of Gordon-type solitons

Abstract

In this paper, we construct a novel linearized finite difference/spectral-Galerkin scheme for three-dimensional distributed-order time–space fractional nonlinear reaction–diffusion-wave equation. By using Gauss–Legendre quadrature rule to discretize the distributed integral terms in both the spatial and temporal directions, we first approximate the original distributed-order fractional problem by the multi-term time–space fractional differential equation. Then, we employ the finite difference method for the discretization of the multi-term Caputo fractional derivatives and apply the Legendre–Galerkin spectral method for the spatial approximation. The main advantage of the proposed scheme is that the implementation of the iterative method is avoided for the nonlinear term in the fractional problem. Additionally, numerical experiments are conducted to validate the accuracy and stability of the scheme. Our approach is show-cased by solving several three-dimensional Gordon-type models of practical interest, including the fractional versions of sine-, sinh-, and Klein–Gordon equations, together with the numerical simulations of the collisions of the Gordon-type solitons. The simulation results can provide a deeper understanding of the complicated nonlinear behaviors of the 3D Gordon-type solitons.