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Mittag-Leffler function and fractional differential equations

  • Katarzyna Górska EMAIL logo , Ambra Lattanzi and Giuseppe Dattoli
Published/Copyright: March 13, 2018
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 21 Issue 1

Abstract

We adopt a procedure of operational-umbral type to solve the (1 + 1)-dimensional fractional Fokker-Planck equation in which time fractional derivative of order α (0 < α < 1) is in the Riemann-Liouville sense. The technique we propose merges well documented operational methods to solve ordinary FP equation and a redefinition of the time by means of an umbral operator. We show that the proposed method allows significant progress including the handling of operator ordering.

MSC 2010: Primary 35R11; Secondary 26A33; 05A40; 60G52
Key Words and Phrases: fractional Fokker-Planck equation; fractional calculus; moments; umbral (operational) method

Acknowledgments

The authors thanks Prof. Andrzej Horzela and Dr. Silvia Liccardi for helpful discussion and suggestions.

K.G and A.L. were supported by the NCN, OPUS-12, Program No. UMO-2016/23/B/ST3/01714. Moreover, K.G. thanks for support from MNiSW (Poland), Iuventus Plus 2015-2016, Program No. IP2014 013073.

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Appendix A. Proof of Eq. (5.10)

Let us recall

eλκ2=∫−∞∞e−u24λe−κudu2πλ,(7.1)

in which we take λ = τ and κ=xddx. According to the operational method Eq. (7.1) with such chosen λ and κ reads as, [10, 11],

F1(x,τ)=eτ(xddx)2f(x)=∫−∞∞e−u24τe−uxddxf(x)du2πλ.(7.2)

The use of example (2) enable us to obtain Eq. (7.2) as the left hand site of Eq. (5.10) times eτ.

Observe that the solution (7.2) satisfies the partial differential equation in the form

∂∂τF1(x,τ)=x2d2dx2+xddxF1(x,τ).
Received: 2017-10-30
Published Online: 2018-3-13
Published in Print: 2018-2-23

© 2018 Diogenes Co., Sofia

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  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 21–1–2018)
  4. Survey Paper
  5. From continuous time random walks to the generalized diffusion equation
  6. Survey Paper
  7. Properties of the Caputo-Fabrizio fractional derivative and its distributional settings
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  9. Exact and numerical solutions of the fractional Sturm–Liouville problem
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  11. Some stability properties related to initial time difference for Caputo fractional differential equations
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  13. On an eigenvalue problem involving the fractional (s, p)-Laplacian
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  15. Diffusion entropy method for ultraslow diffusion using inverse Mittag-Leffler function
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  24. Research Paper
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  29. Mittag-Leffler function and fractional differential equations
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  31. Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point
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https://doi.org/10.1515/fca-2018-0014
Keywords for this article
fractional Fokker-Planck equation; fractional calculus; moments; umbral (operational) method

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 21–1–2018)
  4. Survey Paper
  5. From continuous time random walks to the generalized diffusion equation
  6. Survey Paper
  7. Properties of the Caputo-Fabrizio fractional derivative and its distributional settings
  8. Research Paper
  9. Exact and numerical solutions of the fractional Sturm–Liouville problem
  10. Research Paper
  11. Some stability properties related to initial time difference for Caputo fractional differential equations
  12. Research Paper
  13. On an eigenvalue problem involving the fractional (s, p)-Laplacian
  14. Research Paper
  15. Diffusion entropy method for ultraslow diffusion using inverse Mittag-Leffler function
  16. Research Paper
  17. Time-fractional diffusion with mass absorption under harmonic impact
  18. Research Paper
  19. Optimal control of linear systems with fractional derivatives
  20. Research Paper
  21. Time-space fractional derivative models for CO2 transport in heterogeneous media
  22. Research Paper
  23. Improvements in a method for solving fractional integral equations with some links with fractional differential equations
  24. Research Paper
  25. On some fractional differential inclusions with random parameters
  26. Research Paper
  27. Initial boundary value problems for a fractional differential equation with hyper-Bessel operator
  28. Research Paper
  29. Mittag-Leffler function and fractional differential equations
  30. Research Paper
  31. Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point
  32. Research Paper
  33. Differential and integral relations in the class of multi-index Mittag-Leffler functions
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