An Explicit Numerical Method for the Fractional Cable Equation

Abstract EN

An explicit numerical method to solve a fractional cable equation which involves two temporal Riemann-Liouville derivatives is studied.

The numerical difference scheme is obtained by approximating the first-order derivative by a forward difference formula, the Riemann-Liouville derivatives by the Grünwald-Letnikov formula, and the spatial derivative by a three-point centered formula.

The accuracy, stability, and convergence of the method are considered.

The stability analysis is carried out by means of a kind of von Neumann method adapted to fractional equations.

The convergence analysis is accomplished with a similar procedure.

The von-Neumann stability analysis predicted very accurately the conditions under which the present explicit method is stable.

This was thoroughly checked by means of extensive numerical integrations.