Pseudospectral analysis and approximation of two‐dimensional fractional cable equation

Abstract

In recent years, major attention has been devoted to fractional‐order partial differential equations since they seem to be more efficient in modeling complex processes. Fractional‐order partial differential equations are considered as generalizations of classical integer‐order partial differential equations. In many areas of electrophysiology and in modeling neuronal dynamics, the cable equation plays a major role. In this article, the authors constructed to approximate the numerical solutions of the two‐dimensional fractional cable equation using the time‐space Jacobi pseudospectral method. The proposed method is established in both time and space to approximate the solutions. Moreover, we use fractional Lagrange interpolants polynomial as a test function, which satisfies the Kronecker delta property at Jacobi–Gauss–Lobatto (JGL) points. The fractional derivative is defined in the modified Riemann–Liouville fractional derivatives formula at JGL points. Using the proposed method, the approximate solution is obtained by solving a diagonally block system of nonlinear algebraic equations. The stability analysis of the proposed method, uniqueness of the solution, and error estimate are also provided. Finally, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. The pseudospectral solutions are more accurate as compared to the available results to date in the same vicinity.

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

“…In 2008, Henry and Langlands [33] introduced the FCE for the first time to study the anomalous diffusion phenomenon in the nerve cell. Since then, FCE has been employed to report on the behavior of different phenomena occurring in several fields, such as neuronal dynamics, control theory, the heterogeneous nature of neuronal tissue, and viscoelastic materials (see [34] and references cited therein). Consequently, the solution of FCE is of practical importance in different fields of application.…”

“…The Sinc-Bernoulli collocation method was considered by Moshtaghi and Saadatmandi [51] to study the one-dimensional CO FCE. Mittal et al [34] introduced a time-space Jacobi pseudospectral method for the numerical solution of the two-dimensional FCE. Stability, uniqueness, and error analysis were also given in their work.…”