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Atypical Case of the Dielectric Relaxation Responses and its Fractional Kinetic Equation

  • Aleksander Stanislavsky EMAIL logo and Karina Weron
Published/Copyright: March 9, 2016
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 1

Abstract

We present a probabilistic model of the microscopic scenario of dielectric relaxation relating to the atypical case of two-power-law responses.The surveyed approach extends the cluster model concept used for the description of the typical, Havriliak-Negami (HN) law. Within the proposed framework, all empirical two-power-law relaxation patterns may be derived. Their relaxation functions are expressed in terms of the three-parameter Mittag-Leffler function, and the kinetic equation takes the pseudodifferential form generalizing the Riemann-Louiville fractional calculus. This provides a clue to explain the universality observed in relaxation phenomena.

Key Words and Phrases:: fractional calculus; Mittag-Leffler type functions; fractional ordinary and pseudo differential equations; dielectric susceptibility; fractional two-power relaxation
MSC: Primary 26A33, 33E12, 34A08, 34K37; Secondary 60Fxx, 60G18, 82D30, 82D60

Acknowledgements

The author A.S. is grateful to the Faculty of Fundamental Problems of Technology and the Hugo Steinhaus Center for pleasant hospitality during his visit in Wroc_law University of Technology. He is also grateful for a partial support from the NCN Maestro Grant No. 2012/06/A/ST1/00258.

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  1. Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 19, No 1 (2016), pp. 212-228, DOI: 10.1515/fca-2016-0012

Received: 2015-3-7
Revised: 2015-9-22
Published Online: 2016-3-9
Published in Print: 2016-2-1

© 2016 Diogenes Co., Sofia

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https://doi.org/10.1515/fca-2016-0012
Keywords for this article
fractional calculus; Mittag-Leffler type functions; fractional ordinary and pseudo differential equations; dielectric susceptibility; fractional two-power relaxation

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA Related News, Events and Books (FCAA–Volume 19–1–2016)
  4. Research Paper
  5. Smallest Eigenvalues for a Right Focal Boundary Value Problem
  6. Research Paper
  7. High-Order Algorithms for Riesz Derivative and their Applications (III)
  8. Research Paper
  9. Existence and Uniqueness for a Class of Stochastic Time Fractional Space Pseudo-Differential Equations
  10. Research Paper
  11. Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data
  12. Research Paper
  13. Bogolyubov-Type Theorem with Constraints Generated by a Fractional Control System
  14. Research Paper
  15. Numerical Solution of Nonstationary Problems for a Space-Fractional Diffusion Equation
  16. Research Paper
  17. Solving 3D Time-Fractional Diffusion Equations by High-Performance Parallel Computing
  18. Discussion Survey
  19. Physical and Geometrical Interpretation of Grünwald-Letnikov Differintegrals: Measurement of Path and Acceleration
  20. Discussion Survey
  21. Some Applications of Fractional Velocities
  22. Research Paper
  23. Maximum Principles for Multi-Term Space-Time Variable-Order Fractional Diffusion Equations and their Applications
  24. Survey Paper
  25. Atypical Case of the Dielectric Relaxation Responses and its Fractional Kinetic Equation
  26. Research Paper
  27. Operator Method for Construction of Solutions of Linear Fractional Differential Equations with Constant Coefficients
  28. Research Article
  29. On the Existence and Multiplicity of Solutions for Dirichlet’s problem for Fractional Differential equations
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  31. Approximate controllability for semilinear composite fractional relaxation equations
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