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Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions

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Abstract

Fractional partial differential equations with more than one fractional derivative in time describe some important physical phenomena, such as the telegraph equation, the power law wave equation, or the Szabo wave equation. In this paper, we consider two- and three-dimensional multi-term time and space fractional partial differential equations. The multi-term time-fractional derivative is defined in the Caputo sense, whose order belongs to the interval (1,2],(2,3],(3,4] or (0,m], and the space-fractional derivative is referred to as the fractional Laplacian form. We derive series expansion solutions based on a spectral representation of the Laplacian operator on a bounded region. Some applications are given for the two- and three-dimensional telegraph equation, power law wave equation and Szabo wave equation.

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Authors and Affiliations

  1. Department of Applied Mathematics, Donghua University, Shanghai, 201620, PR China

    H. Ye

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

    F. Liu, I. Turner, V. Anh & K. Burrage

  3. Department of Computing Science and OCISB, Oxford University, Oxford, OXI 3QD, UK

    K. Burrage

Authors
  1. H. Ye
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  2. F. Liu
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  3. I. Turner
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  4. V. Anh
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  5. K. Burrage
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Correspondence to F. Liu.

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Ye, H., Liu, F., Turner, I. et al. Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions. Eur. Phys. J. Spec. Top. 222, 1901–1914 (2013). https://doi.org/10.1140/epjst/e2013-01972-2

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  • DOI: https://doi.org/10.1140/epjst/e2013-01972-2

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