Fractional cable problem in the frame of meshless singular boundary method

Abstract

In this study, singular boundary method (SBM) is employed for solving fractional cable problem in two dimensional space with initial and Dirichlet-type boundary conditions. The process is modeled as a two dimensional time-fractional equation in sense of Riemann–Liouville fractional derivatives. A splitting scheme is applied to split the solution of the inhomogeneous governing equation into homogeneous solution and particular solution. We present the numerical operation for calculating the particular solution and homogeneous solution. For achieving approximation particular solution and homogeneous solution, we employ Method of Particular solution (MPS) and SBM, respectively. We use θ-weighted and finite difference method as time discretization for time derivatives. A comparison between the present method and other methods is given to show the accuracy of SBM applying on this equation. Consequently, some numerical examples with different domains are tested and compared with the exact analytical solutions to display the validity and accuracy of the numerical method in comparison with other methods.