Application of a hybrid pseudospectral method to a new two-dimensional multi-term mixed sub-diffusion and wave-diffusion equation of fractional order

Abstract

<abstract><p>In the current study, a novel multi-term mixed sub-diffusion and wave-diffusion model was considered. The new model has a unique time-space coupled derivative in addition to having the diffusion-wave and sub-diffusion terms concurrently. Typically, an elliptic equation in the space variable is obtained by applying a finite difference time-stepping procedure. The severe stability restrictions are the main disadvantage of the finite difference method in time. It has been demonstrated that the Laplace transform is an excellent choice for solving diffusion problems and offers a substitute to the finite difference approach. In this paper, a method based on Laplace transform coupled with the pseudospectral method was developed for the novel model. The proposed method has three main steps: First, the model was reduced to a time-independent model via Laplace transform; second, the pseudospectral method was employed for spatial discretization; and finally, the inverse Laplace transform was applied to transform the obtained solution in Laplace transform domain back into a real domain. We also presented the numerical scheme's stability and convergence analysis. To demonstrate our method's efficacy, four problems were examined.</p></abstract>