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.
Abedian, R. (2022). WENO schemes with Z-type non-linear weighting procedure for fractional differential equations. Journal of Mathematical Modeling, 10(4), 555-567. doi: 10.22124/jmm.2022.22535.1988
MLA
Rooholah Abedian. "WENO schemes with Z-type non-linear weighting procedure for fractional differential equations". Journal of Mathematical Modeling, 10, 4, 2022, 555-567. doi: 10.22124/jmm.2022.22535.1988
HARVARD
Abedian, R. (2022). 'WENO schemes with Z-type non-linear weighting procedure for fractional differential equations', Journal of Mathematical Modeling, 10(4), pp. 555-567. doi: 10.22124/jmm.2022.22535.1988
VANCOUVER
Abedian, R. WENO schemes with Z-type non-linear weighting procedure for fractional differential equations. Journal of Mathematical Modeling, 2022; 10(4): 555-567. doi: 10.22124/jmm.2022.22535.1988
)