@article { author = {Abedian, Rooholah}, title = {WENO schemes with Z-type non-linear weighting procedure for fractional differential equations}, journal = {Journal of Mathematical Modeling}, volume = {10}, number = {4}, pages = {555-567}, year = {2022}, publisher = {University of Guilan}, issn = {2345-394X}, eissn = {2382-9869}, doi = {10.22124/jmm.2022.22535.1988}, abstract = {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.}, keywords = {finite difference scheme,Fractional differential equations,WENO-Z scheme}, url = {https://jmm.guilan.ac.ir/article_5892.html}, eprint = {https://jmm.guilan.ac.ir/article_5892_31bc383e7160229ae7d3e6280493e694.pdf} }