%0 Journal Article %T WENO schemes with Z-type non-linear weighting procedure for fractional differential equations %J Journal of Mathematical Modeling %I University of Guilan %Z 2345-394X %A Abedian, Rooholah %D 2022 %\ 12/01/2022 %V 10 %N 4 %P 555-567 %! WENO schemes with Z-type non-linear weighting procedure for fractional differential equations %K finite difference scheme %K Fractional differential equations %K WENO-Z scheme %R 10.22124/jmm.2022.22535.1988 %X 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 $L_1$ norm, yielding improved WENO scheme with more accurate resolution. The Caputo fractional derivative of order $\alpha$ 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. %U https://jmm.guilan.ac.ir/article_5892_31bc383e7160229ae7d3e6280493e694.pdf