On the consistency of the finite difference approximation with the Riemann-Liouville fractional derivative for $0 < \alpha < 1$

Abstract

Fractional differential equations have become an important modeling technique in describing various natural phenomena. A variety of numerical methods for solving fractional differential equations has been developed over the last decades. Among them, finite difference methods are most popular owing to relative easiness for implementation.

In this paper, we show that the finite difference method with the Riemann-Liouville (RL) fractional derivative yields inconsistent and oscillatory numerical solutions to fractional differential equations of discontinuous problems for the fractional order alpha, 0 < alpha < 1. Such an inconsistency affects even smooth problems since the numerical solution can be regarded as a discontinuous function over grids. We show that the inconsistency inherited in discontinuous problems causes the numerical solution for smooth problems to be oscillatory under certain conditions although the magnitude of the oscillations decreases as the number of grids, N, increases. That is, although the truncation error is decaying as N -> infinity and the method is consistent for smooth problems, the numerical solution can be oscillatory for any value of N. To illustrate the inconsistency and the oscillation phenomenon with the RL method, we also consider the finite difference methods with the Caputo and Grunwald-Letnikov fractional derivatives and compare the results with by with the RL method. We also show that the integral approach for the RL method can resolve the issues of the inconsistency and the oscillation phenomenon. Numerical results are presented to support the statements.