High order finite difference WENO schemes for fractional differential equations

This letter develops high order finite difference weighted essentially non-oscillatory (WENO) schemes for fractional differential equations. First, the alpha th, 1 < alpha <= 2, Caputo fractional derivative is split into a classical second derivative and a weakly singular integral. Then the sixth-order finite difference WENO scheme is used to discretize the classical second derivative and the Gauss-Jacobi quadrature is applied to solve the weakly singular integral. The constructed scheme of approximation for the fractional derivative has high order accuracy in smooth regions and maintains a sharp discontinuity transition. Finally, numerical experiments are performed to demonstrate the effectiveness of the proposed schemes. (C) 2012 Elsevier Ltd. All rights reserved.