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Differential quadrature method for space-fractional diffusion equations on 2D irregular domains

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Abstract

In mathematical physics, the space-fractional diffusion equations are of particular interest in the studies of physical phenomena modelled by Lévy processes, which are sometimes called super-diffusion equations. In this article, we develop the differential quadrature (DQ) methods for solving the 2D space-fractional diffusion equations on irregular domains. The methods in presence reduce the original equation into a set of ordinary differential equations (ODEs) by introducing valid DQ formulations to fractional directional derivatives based on the functional values at scattered nodal points on problem domain. The required weighted coefficients are calculated by using radial basis functions (RBFs) as trial functions, and the resultant ODEs are discretized by the Crank-Nicolson scheme. The main advantages of our methods lie in their flexibility and applicability to arbitrary domains. A series of illustrated examples are finally provided to support these points.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions. This research was supported by National Natural Science Foundations of China (Nos. 11471262 and 11501450).

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Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    X. G. Zhu, Z. B. Yuan & Y. F. Nie

  2. School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, QLD, 4001, Australia

    F. Liu

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  1. X. G. Zhu
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Zhu, X.G., Yuan, Z.B., Liu, F. et al. Differential quadrature method for space-fractional diffusion equations on 2D irregular domains. Numer Algor 79, 853–877 (2018). https://doi.org/10.1007/s11075-017-0464-0

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