On numerical solution of two-dimensional variable-order fractional diffusion equation arising in transport phenomena

Abstract

<abstract><p>In recent years, the application of variable-order (VO) fractional differential equations for describing complex physical phenomena ranging from biology, hydrology, mechanics and viscoelasticity to fluid dynamics has become one of the most hot topics in the context of scientific modeling. An interesting aspect of VO operators is their capability to address the behavior of scientific and engineering systems with time and spatially varying properties. The VO fractional diffusion equation is a fundamental model that allows transitions among sub-diffusive, diffusive and super-diffusive behaviors without altering the underlying governing equations. In this paper, we considered the two-dimensional fractional diffusion equation with the Caputo time VO derivative, which is essential for describing anomalous diffusion in real-world complex systems. A new Crank-Nicolson (C-N) difference scheme and an efficient explicit decoupled group (EDG) method were proposed to solve the problem under consideration. The proposed EDG method is based on a skewed difference scheme in conjunction with a grouping procedure of the solution grid points. Special attention was devoted to investigating the stability and convergence of the proposed methods. Three numerical examples with known exact analytical solutions were provided to illustrate our considerations. The proposed methods were shown to be stable and convergent theoretically as well as numerically. In addition, a comparative study was done between the EDG method and the C-N difference scheme. It was found that the proposed methods are accurate in simulating the considered problem, while the EDG method is superior to the C-N difference method in terms of Central Processing Unit (CPU) timing, verifying the efficiency of the former method in solving the VO problem.</p></abstract>

On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics

“…Consequently, VO FPDEs have attracted the attention of numerous researchers and scholars as accurate models for describing a large variety of phenomena in various branches of science and engineering, such as mechanics, viscoelasticity, anomalous diffusion, wave propagation, control theory, ecology, and many others. Similar to the CO FPDEs, it is difficult to solve VO FPDEs analytically, and numerical techniques are very often resorted to, for instance, see [26][27][28][29]. A profound discussion of the definitions, applications, and numerical simulations of the VO fractional operators can be seen in the insightful review papers [30,31].…”