Spectral method for the two-dimensional time distributed-order diffusion-wave equation on a semi-infinite domain

Abstract

The time distributed-order diffusion-wave equation describes radial groundwater flow to or from a well. In the paper, an alternating direction implicit (ADI) Legendre–Laguerre spectral scheme is proposed for the two-dimensional time distributed-order diffusion-wave equation on a semi-infinite domain. The Gauss quadrature formula has a higher computational accuracy than the Composite Trapezoid formula and Composite Simpson formula, which is presented to approximate the distributed order time derivative so that the considered equation is transformed into a multi-term fractional equation. Moreover, the transformed equation is solved by discretizing in space by the ADI Legendre–Laguerre spectral scheme to avoid introducing the artificial boundary and in time using the weighted and shifted Grünwald–Letnikov difference (WSGD) method. A stability and convergence analysis is performed for the numerical approximation. Some numerical results are illustrated to justify the theoretical analysis.