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A robust numerical method for a fractional differential equation

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  • Cen, Zhongdi
  • Le, Anbo
  • Xu, Aimin

Abstract

This paper is devoted to giving a rigorous numerical analysis for a fractional differential equation with order α ∈ (0, 1). First the fractional differential equation is transformed into an equivalent Volterra integral equation of the second kind with a weakly singular kernel. Based on the apriori information about the exact solution, an integral discretization scheme on an apriori chosen adapted mesh is proposed. By applying the truncation error estimate techniques and a discrete analogue of Gronwall’s inequality, it is proved that the numerical method is first-order convergent in the discrete maximum norm. Numerical results indicate that this method is more accurate and robust than finite difference methods when α is close to 0.

Suggested Citation

  • Cen, Zhongdi & Le, Anbo & Xu, Aimin, 2017. "A robust numerical method for a fractional differential equation," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 445-452.
    • Handle: RePEc:eee:apmaco:v:315:y:2017:i:c:p:445-452
    DOI: 10.1016/j.amc.2017.08.011
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    References listed on IDEAS

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    1. Meerschaert, Mark M. & Scalas, Enrico, 2006. "Coupled continuous time random walks in finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(1), pages 114-118.
    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. Atanacković, Teodor M. & Janev, Marko & Pilipović, Stevan, 2018. "Non-linear boundary value problems involving Caputo derivatives of complex fractional order," Applied Mathematics and Computation, Elsevier, vol. 334(C), pages 326-342.
    2. Meng, Zhijun & Yi, Mingxu & Huang, Jun & Song, Lei, 2018. "Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 454-464.

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