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Galerkin spectral method for nonlinear time fractional Cable equation with smooth and nonsmooth solutions

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  • Liu, Haiyu
  • Lü, Shujuan

Abstract

In this work, we study the numerical solutions of the time fractional Cable equations with nonlinear term, where the fractional derivatives are described in Riemann–Liouville sense. An explicit scheme is constructed based upon finite difference method in time and Legendre spectral method in space. Stability and convergence of scheme are proved rigorously. Moreover, an improved algorithm for the problem with nonsmooth solutions is proposed by adding correction terms to the approximations of first-order derivative, fractional derivatives and nonlinear term. Numerical examples are given to support theoretical analysis.

Suggested Citation

  • Liu, Haiyu & Lü, Shujuan, 2019. "Galerkin spectral method for nonlinear time fractional Cable equation with smooth and nonsmooth solutions," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 32-47.
    • Handle: RePEc:eee:apmaco:v:350:y:2019:i:c:p:32-47
    DOI: 10.1016/j.amc.2018.12.072
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    1. Lokenath Debnath, 2003. "Recent applications of fractional calculus to science and engineering," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-30, January.
    2. Raberto, Marco & Scalas, Enrico & Mainardi, Francesco, 2002. "Waiting-times and returns in high-frequency financial data: an empirical study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 749-755.
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    Cited by:

    1. Mahmoud A. Zaky & Ahmed S. Hendy & Rob H. De Staelen, 2021. "Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System," Mathematics, MDPI, vol. 9(2), pages 1-22, January.

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