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Maximum norm error analysis of difference schemes for fractional diffusion equations

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  • Ren, Jincheng
  • Sun, Zhi-zhong

Abstract

In this study, we present several numerical approximation methods for solving the fractional diffusion equations with the Riesz space fractional derivative. Based on the fractional centered difference approximation to the Riesz fractional derivative, an explicit and an implicit finite difference schemes for solving the space fractional diffusion equations are obtained. The stability and global convergence of these two schemes in the maximum norm are analyzed rigorously. The convergence order is O(τ+h2), where τ is the temporal grid size and h is spatial grid size, respectively. Furthermore, a numerical scheme for the one-dimensional multi-term time–space fractional diffusion equations is obtained. The unconditionally stability and convergence of the scheme in maximum norm are established. The convergence orders are 2-α in the temporal direction and two in the spatial direction, where α is the maximum time fractional derivative orders in the equation. In addition, the extension to the two-dimensional case is also discussed. Finally, several numerical examples are provided to show the effectiveness and accuracy of our methods.

Suggested Citation

  • Ren, Jincheng & Sun, Zhi-zhong, 2015. "Maximum norm error analysis of difference schemes for fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 299-314.
    • Handle: RePEc:eee:apmaco:v:256:y:2015:i:c:p:299-314
    DOI: 10.1016/j.amc.2014.12.151
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    1. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    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.
    3. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
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