Discrete-time orthogonal spline collocation method with application to two-dimensional fractional cable equation

Abstract

Discrete-time orthogonal spline collocation (OSC) methods are presented for the two-dimensional fractional cable equation, which governs the dynamics of membrane potential in thin and long cylinders such as axons or dendrites in neurons. The proposed scheme is based on the OSC method for space discretization and finite difference method for time, which is proved to be unconditionally stable and convergent with the order O(τmin(2−γ1,2−γ2)+hr+1) in L2-norm, where τ,h and r are the time step size, space step size and polynomial degree, respectively, and γ1 and γ2 are two different exponents of fractional derivatives with 0<γ1,γ2<1. Numerical experiments are presented to demonstrate the results of theoretical analysis and show the accuracy and effectiveness of the method described herein, and super-convergence phenomena at the partition nodes is also exhibited, which is a characteristic of the OSC methods, namely, the rates of convergence in the maximum norm at the partition nodes in ux and uy are approximately hr+1 in our numerical experiment.