A space-time collocation scheme for modified anomalous subdiffusion and nonlinear superdiffusion equations

Abstract

This paper reports a new spectral collocation technique for solving time-space modified anomalous subdiffusion equation with a nonlinear source term subject to Dirichlet and Neumann boundary conditions. This model equation governs the evolution for the probability density function that describes anomalously diffusing particles. Anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur. A space-time Jacobi collocation scheme is investigated for solving such problem. The main advantage of the proposed scheme is that, the shifted Jacobi Gauss-Lobatto collocation and shifted Jacobi Gauss-Radau collocation approximations are employed for spatial and temporal discretizations, respectively. Thereby, the problem is successfully reduced to a system of algebraic equations. The numerical results obtained by this algorithm have been compared with various numerical methods in order to demonstrate the high accuracy and efficiency of the proposed method. Indeed, for relatively limited number of Gauss-Lobatto and Gauss-Radau collocation nodes imposed, the absolute error in our numerical solutions is sufficiently small. The results have been compared with other techniques in order to demonstrate the high accuracy and efficiency of the proposed method.