Finite difference/spectral methods for the two-dimensional distributed-order time-fractional cable equation

摘要

The cable equation plays a significant role in many areas of electrophysiology and in modeling neuronal dynamics. In recent years, considerable attention has been devoted to distributed-order differential equations because they appear to be more effective for modeling complex processes. In this work, a finite difference/Legendre spectral method is presented for the numerical simulation of the two-dimensional (2D) distributed order time-fractional cable equation, where the finite difference method is employed in the temporal discretization and Legendre spectral method is adopted in the spatial discretization. The midpoint quadrature rule is used to approximate the distributed-order, such that the considered equation could be transformed into a multi-term fractional equation. The stability and convergence analysis of the proposed scheme is established, which illustrates that the numerical solution converges to the exact solution with order O(tau(2) + sigma(2) + N-s), where tau, sigma, N are the time step size, the step length in the approximation of the distributed-order and the polynomial degree, respectively. Furthermore, to demonstrate the versatility and applicability of our method, we provide numerical results that show good agreement with the theoretical analysis. (C) 2020 Elsevier Ltd. All rights reserved.