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A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation

  • Gianni Pagnini EMAIL logo and Paolo Paradisi EMAIL logo
Published/Copyright: May 4, 2016
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 2

Abstract

The stochastic solution with Gaussian stationary increments is established for the symmetric space-time fractional diffusion equation when 0 < β < α ≤ 2, where 0 < β ≤ 1 and 0 < α ≤ 2 are the fractional derivation orders in time and space, respectively. This solution is provided by imposing the identity between two probability density functions resulting (i) from a new integral representation formula of the fundamental solution of the symmetric space-time fractional diffusion equation and (ii) from the product of two independent random variables. This is an alternative method with respect to previous approaches such as the scaling limit of the continuous time random walk, the parametric subordination and the subordinated Langevin equation. A new integral representation formula for the fundamental solution of the space-time fractional diffusion equation is firstly derived. It is then shown that, in the symmetric case, a stochastic solution can be obtained by a Gaussian process with stationary increments and with a random wideness scale variable distributed according to an arrangement of two extremal Lévy stable densities. This stochastic solution is self-similar with stationary increments and uniquely defined in a statistical sense by the mean and the covariance structure.

Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental solution of the symmetric space-time fractional diffusion equation.

MSC 2010: Primary 26A33; Secondary 60G20, 60G22, 82C31
Key Words and Phrases: anomalous diffusion; space-time fractional diffusion equation; stochastic solution; Gaussian processes; fractional Brownian motion; self-similar stochastic process; stationary increments

Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 19, No 2 (2016), pp. 408–440, DOI: 10.1515/fca-2016-0022

Acknowledgements

This research is supported by MINECO under Grant MTM2013-40824-P, by Bizkaia Talent and European Commission through COFUND programme under Grant AYD-000-252, and also by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323.

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Received: 2015-5-5
Revised: 2015-11-30
Published Online: 2016-5-4
Published in Print: 2016-4-1

© 2016 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA-Volume 19-2-2016)
  4. Survey Paper
  5. A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations
  6. Survey Paper
  7. A free fractional viscous oscillator as a forced standard damped vibration
  8. Research Paper
  9. On a Legendre Tau method for fractional boundary value problems with a Caputo derivative
  10. Research Paper
  11. Nonlinear Dirichlet problem with non local regional diffusion
  12. Research Paper
  13. Generalized fraction evolution equations with fractional Gross Laplacian
  14. Research Paper
  15. A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
  16. Research Paper
  17. Convolutional approach to fractional calculus for distributions of several variables
  18. Research Paper
  19. Existence theorems for semi-linear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions
  20. Research Paper
  21. Nonlinear Riemann-Liouville fractional differential equations with nonlocal Erdelyi-Kober fractional integral conditions
  22. Research Paper
  23. Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity
  24. Research Paper
  25. Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain
  26. Short Paper
  27. Measurement of para-xylene diffusivity in zeolites and analyzing desorption curves using the Mittag-Leffler function
  28. Short Paper
  29. Monotonicity of functions and sign changes of their Caputo derivatives
  30. Short Note
  31. On the Volterra μ-functions and the M-Wright functions as kernels and eigenfunctions of Volterra type integral operators
https://doi.org/10.1515/fca-2016-0022
Keywords for this article
anomalous diffusion; space-time fractional diffusion equation; stochastic solution; Gaussian processes; fractional Brownian motion; self-similar stochastic process; stationary increments

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA-Volume 19-2-2016)
  4. Survey Paper
  5. A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations
  6. Survey Paper
  7. A free fractional viscous oscillator as a forced standard damped vibration
  8. Research Paper
  9. On a Legendre Tau method for fractional boundary value problems with a Caputo derivative
  10. Research Paper
  11. Nonlinear Dirichlet problem with non local regional diffusion
  12. Research Paper
  13. Generalized fraction evolution equations with fractional Gross Laplacian
  14. Research Paper
  15. A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
  16. Research Paper
  17. Convolutional approach to fractional calculus for distributions of several variables
  18. Research Paper
  19. Existence theorems for semi-linear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions
  20. Research Paper
  21. Nonlinear Riemann-Liouville fractional differential equations with nonlocal Erdelyi-Kober fractional integral conditions
  22. Research Paper
  23. Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity
  24. Research Paper
  25. Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain
  26. Short Paper
  27. Measurement of para-xylene diffusivity in zeolites and analyzing desorption curves using the Mittag-Leffler function
  28. Short Paper
  29. Monotonicity of functions and sign changes of their Caputo derivatives
  30. Short Note
  31. On the Volterra μ-functions and the M-Wright functions as kernels and eigenfunctions of Volterra type integral operators
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