Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

By Qianqian Yang, Fawang Liu, Ian L Turner. 2010

621 citations recorded by the source corpus.

Subfields: Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9affc68a7683436ed72d84e0b35c33438618bceb

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Citations in the corpus

A series of high‐order quasi‐compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives (2013, neutral): “, [5,12,16,30,31].”
What is the fractional Laplacian? A comparative review with new results (2018, background): “where RL −∞Dx and x D∞ are the well-known one-dimensional left- and right-sided Riemann-Liouville fractional derivatives, respectively [53].”
Fourier spectral methods for fractional-in-space reaction-diffusion equations (2014, methodology): “Spectral decomposition plays a central role in the interpretation of the fractional Laplacian – see [15] and [20].”
A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations (2015, neutral): “[62].”
High-Order Algorithms for Riesz Derivative and Their Applications $(I)$ (2014, methodology): “[22] obtained some numerical solutions for the space Riesz fractional diffusion and advection-dispersion equations on a finite domain by using three numerical methods.”
Numerical studies for the variable-order nonlinear fractional wave equation (2012, methodology): “This method is used to solve many of similar problems to the proposed problem, for example, the variable-order nonlinear cable equation [1], the variable-order nonlinear Stokes’ first problem for a heated generalized second grade fluid [2], the variable-order anomalous sub-diffusion equation [4], th”
Adaptive finite element method for fractional differential equations using hierarchical matrices (2016, background): “However, as the size of the problem increases, the time required to solve the final system of equations increases considerably due to the nonlocality of the fractional differential operators [43, 23, 10, 11, 30, 9, 4, 36, 37, 40, 26].”
Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation (2013, methodology): “For numerical methods and simulations, see [3, 4, 28, 29, 33, 34, 52] and references therein.”
On Riesz derivative (2019, methodology): “[29] tried to prove the pointwise equivalence between these two operators through performing integration by parts on the Fourier definition of fractional Laplacian, and utilizing the Laplace transform.”