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Convolutional approach to fractional calculus for distributions of several variables

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Published/Copyright: May 5, 2016
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 2

Abstract

We extend the semigroup {Φλ}λ∈ℂ of Gel’fand and Shilov to the semigroup {Φλ}λ∈ℂn and the one-dimensional fractional calculus of distributions to the case of distributions on Rn with supports in the nonnegative cone

ℝ¯+n. We use distributional mixed derivatives of an arbitrary complex order to solve Abel’s multidimensional integral equation and a distributional fractional differential equation of wave type.

Key Words and Phrases: convolution of distributions; partial fractional derivatives of distributions; fractional partial differential equations
MSC 2010: Primary 26A33; Secondary 46F10; 46F05

Dedicated to Professor Ivan H. Dimovski

Acknowledgements

This work was partly supported by the Center for Innovation and Transfer of Natural Sciences and Engineering Knowledge, University of Rzeszów.

  1. Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 19, No 2 (2016), pp. 441–462, DOI: 10.1515/fca-2016-0023

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Received: 2015-7-21
Revised: 2015-12-15
Published Online: 2016-5-5
Published in Print: 2016-4-1

© 2016 Diogenes Co., Sofia

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  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
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  7. A free fractional viscous oscillator as a forced standard damped vibration
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  17. Convolutional approach to fractional calculus for distributions of several variables
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  19. Existence theorems for semi-linear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions
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  21. Nonlinear Riemann-Liouville fractional differential equations with nonlocal Erdelyi-Kober fractional integral conditions
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  23. Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity
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  25. Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain
  26. Short Paper
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https://doi.org/10.1515/fca-2016-0023
Keywords for this article
convolution of distributions; partial fractional derivatives of distributions; fractional partial differential equations

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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