An efficient hybrid numerical approach for solving two-dimensional fractional cable model involving time-fractional operator of distributed order with error analysis

In this article, we study and examine an efficient numerical approach to obtain approximate solutions of the two-dimensional fractional cable model involving the time-fractional operator of distributed order. A hybrid numerical approach is used to approximate the proposed fractional model. For approximating the integral part of the distributed order including Caputo fractional derivative, the combination of Gauss quadrature rule and finite difference are used. As well as, for the integral part of the distributed order including Riemann Liouville fractional derivatives, from the mid-point quadrature rule and shifted Grünwald estimation are applied. Also, to approximate the proposed model in the space direction, the Legendre spectral numerical approach is used in order to calculate the full-discrete numerical approach. In this work, error analysis and convergence are studied. In the end, to show the effectiveness of the proposed approach, two numerical examples are stated and checked.