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Computational solutions of the tempered fractional wave-diffusion equation

  • André Liemert EMAIL logo and Alwin Kienle
Published/Copyright: February 18, 2017
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 20 Issue 1

Abstract

In this paper, we consider the space-time fractional wave-diffusion equation

−∞Dxα,λu(x,t)=Dtβu(x,t),x∈R,t>0,

for the tempered Riemann-Liouville derivative of order 1 <α≤ 2 in space and the Caputo derivative of order 0 <β≤ α in time. The first fundamental solution of the Cauchy problem is derived by means of the Fourier transform and given in computable series form as well as under the restriction 0 <β≤ 1 in form of a fast converging integral. We additionally point out two integral representations for accurate evaluation of the two-sided M-Wright function. The derived equations are verified by comparisons with the Fourier spectral method.

MSC 2010: Primary 26A33; Secondary 33E12; 34A08; 34K37; 35R11; 60G22
Key Words and Phrases: tempered fractional wave-diffusion equation; fundamental solution; probability density; Fourier transform; Mittag-Leffler function; M-Wright function

Acknowledgements

One of the authors (A.L.) gratefully acknowledges the financial support by the Carl Zeiss Foundation, Germany. The authors thank Diego del–Castillo–Negrete for his useful comments.

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Received: 2015-10-30
Revised: 2016-9-3
Published Online: 2017-2-18
Published in Print: 2017-2-1

© 2017 Diogenes Co., Sofia

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https://doi.org/10.1515/fca-2017-0007
Keywords for this article
tempered fractional wave-diffusion equation; fundamental solution; probability density; Fourier transform; Mittag-Leffler function; M-Wright function

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 20–1–2017)
  4. Survey paper
  5. Ten equivalent definitions of the fractional laplace operator
  6. Research paper
  7. Consensus of fractional-order multi-agent systems with input time delay
  8. Research paper
  9. Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-Type Hadamard derivatives
  10. Research paper
  11. A preconditioned fast finite difference method for space-time fractional partial differential equations
  12. Research paper
  13. On existence and uniqueness of solutions for semilinear fractional wave equations
  14. Research paper
  15. Computational solutions of the tempered fractional wave-diffusion equation
  16. Research paper
  17. Completeness on the stability criterion of fractional order LTI systems
  18. Research paper
  19. Wavelet convolution product involving fractional fourier transform
  20. Research paper
  21. Solutions of the main boundary value problems for the time-fractional telegraph equation by the green function method
  22. Research paper
  23. A foundational approach to the Lie theory for fractional order partial differential equations
  24. Research paper
  25. Null-controllability of a fractional order diffusion equation
  26. Research paper
  27. New results in stability analysis for LTI SISO systems modeled by GL-discretized fractional-order transfer functions
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