Entropy, Vol. 20, Pages 321: Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative Entropy doi: 10.3390/e20050321 Authors: Sadia Arshad Dumitru Baleanu Jianfei Huang Maysaa Mohamed Al Qurashi Yifa Tang Yue Zhao In this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection&ndash;diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated by the second-order fractional weighted and shifted Gr&uuml;nwald&ndash;Letnikov formula. Based on the equivalence between the fractional differential equation and the integral equation, we have transformed the fractional differential equation into an equivalent integral equation. Then, the integral is approximated by the trapezoidal formula. Further, the stability and convergence analysis are discussed rigorously. The resulting scheme is formally proved with the second order accuracy both in space and time. Numerical experiments are also presented to verify the theoretical analysis.
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