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Semi-fractional diffusion equations

  • Peter Kern EMAIL logo , Svenja Lage and Mark M. Meerschaert
Published/Copyright: May 11, 2019
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 22 Issue 2

Abstract

It is well known that certain fractional diffusion equations can be solved by the densities of stable Lévy motions. In this paper we use the classical semigroup approach for Lévy processes to define semi-fractional derivatives, which allows us to generalize this statement to semistable Lévy processes. A Fourier series approach for the periodic part of the corresponding Lévy exponents enables us to represent semi-fractional derivatives by a Grünwald-Letnikov type formula. We use this formula to calculate semi-fractional derivatives and solutions to semi-fractional diffusion equations numerically. In particular, by means of the Grünwald-Letnikov type formula we provide a numerical algorithm to compute semistable densities.

MSC 2010: Primary 35R11; 60E10; Secondary 26A33; 60E07; 60G22; 60G51; 60H30; 82C31
Key Words and Phrases: semistable Lévy process; log-characteristic function; log-periodic perturbation; semi-fractional derivative; Zolotarev fractional derivative; Grünwald-Letnikov type formula

Acknowledgements

M. Meerschaert was partially supported by ARO MURI Grant W911NF-15-1-0562 and USA National Science Foundation Grants DMS-1462156 and EAR-1344280.

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Received: 2018-06-02
Published Online: 2019-05-11
Published in Print: 2019-04-24

© 2019 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 22–2–2019)
  4. Discussion Paper
  5. The flaw in the conformable calculus: It is conformable because it is not fractional
  6. The failure of certain fractional calculus operators in two physical models
  7. Research Paper
  8. Inverse problems for a class of degenerate evolution equations with Riemann – Liouville derivative
  9. On Riesz derivative
  10. A note on fractional powers of strongly positive operators and their applications
  11. Semi-fractional diffusion equations
  12. Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations
  13. Well-posedness of fractional degenerate differential equations in Banach spaces
  14. Structure factors for generalized grey Browinian motion
  15. Linear stationary fractional differential equations
  16. Perfect control for right-invertible Grünwald-Letnikov plants – an innovative approach to practical implementation
  17. On fractional differential inclusions with Nonlocal boundary conditions
  18. On solutions of linear fractional differential equations and systems thereof
  19. Existence of mild solution of a class of nonlocal fractional order differential equation with not instantaneous impulses
  20. A CAD-based algorithm for solving stable parameter region of fractional-order systems with structured perturbations
  21. On representation and interpretation of Fractional calculus and fractional order systems
https://doi.org/10.1515/fca-2019-0021
Keywords for this article
semistable Lévy process; log-characteristic function; log-periodic perturbation; semi-fractional derivative; Zolotarev fractional derivative; Grünwald-Letnikov type formula

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 22–2–2019)
  4. Discussion Paper
  5. The flaw in the conformable calculus: It is conformable because it is not fractional
  6. The failure of certain fractional calculus operators in two physical models
  7. Research Paper
  8. Inverse problems for a class of degenerate evolution equations with Riemann – Liouville derivative
  9. On Riesz derivative
  10. A note on fractional powers of strongly positive operators and their applications
  11. Semi-fractional diffusion equations
  12. Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations
  13. Well-posedness of fractional degenerate differential equations in Banach spaces
  14. Structure factors for generalized grey Browinian motion
  15. Linear stationary fractional differential equations
  16. Perfect control for right-invertible Grünwald-Letnikov plants – an innovative approach to practical implementation
  17. On fractional differential inclusions with Nonlocal boundary conditions
  18. On solutions of linear fractional differential equations and systems thereof
  19. Existence of mild solution of a class of nonlocal fractional order differential equation with not instantaneous impulses
  20. A CAD-based algorithm for solving stable parameter region of fractional-order systems with structured perturbations
  21. On representation and interpretation of Fractional calculus and fractional order systems
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