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Finite Difference Method for Two-Sided Space-Fractional Partial Differential Equations

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Finite Difference Methods,Theory and Applications (FDM 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9045))

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Abstract

Finite difference methods for solving two-sided space-fractional partial differential equations are studied. The space-fractional derivatives are the left-handed and right-handed Riemann-Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. The Hadamard finite-part integrals are approximated by using piecewise quadratic interpolation polynomials and a numerical approximation scheme of the space-fractional derivative with convergence order \(O(\varDelta x^{3- \alpha }), \, 1<\alpha <2\) is obtained. A shifted implicit finite difference method is introduced for solving two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is \(O (\varDelta t + \varDelta x^{\min (3- \alpha , \beta )}), 1< \alpha <2, \, \beta >0\), where \(\varDelta t, \varDelta x\) denote the time and space step sizes, respectively. Numerical examples are presented and compared with the exact analytical solution for its order of convergence.

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Acknowledgements

We wish to express our sincere gratitude to Professor Neville. J. Ford for his encouragement, discussions and valuable criticism during the research of this work.

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Authors and Affiliations

  1. Department of Mathematics, University of Chester, Thornton Science Park, Chester, CH2 4NU, UK

    Kamal Pal, Yubin Yan & Graham Roberts

  2. Department of Mathematics, Luliang University, Lishi, PR China

    Fang Liu

Authors
  1. Kamal Pal
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  2. Fang Liu
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  3. Yubin Yan
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  4. Graham Roberts
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Corresponding author

Correspondence to Yubin Yan .

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Editors and Affiliations

  1. Bulgarian Academy of Sciences, Institute of Information and Communication Technologies, Sofia, Bulgaria

    Ivan Dimov

  2. Institue of Mathematics and MTA-ELTE Numerical Analysis and Large Networks Research Group, Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. University of Rousse, Rousse, Bulgaria

    Lubin Vulkov

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© 2015 Springer International Publishing Switzerland

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Pal, K., Liu, F., Yan, Y., Roberts, G. (2015). Finite Difference Method for Two-Sided Space-Fractional Partial Differential Equations. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods,Theory and Applications. FDM 2014. Lecture Notes in Computer Science(), vol 9045. Springer, Cham. https://doi.org/10.1007/978-3-319-20239-6_33

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  • DOI: https://doi.org/10.1007/978-3-319-20239-6_33

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20238-9

  • Online ISBN: 978-3-319-20239-6

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