WENO schemes with Z-type non-linear weighting procedure for fractional differential equations

In this paper, a new fourth-order finite difference weighted essentially non-oscillatory (weno) scheme is developed for the fractional differential equations which may contain non-smooth solutions at a later time, even if the initial solution is smooth enough. a set of z-type non-linear weights is constructed based on the norm, yielding improved weno scheme with more accurate resolution. the caputo fractional derivative of order is split into a weakly singular integral and a classical second derivative. the classical gauss-jacobi quadrature is employed for solving the weakly singular integral. also, a new weno-type reconstruction methodology for approximating the second derivative is developed. some benchmark examples are prepared to illustrate the efficiency, robustness, and good performance of this new finite difference weno-z scheme.