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 ααth, 1<α≤21<α≤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.