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Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 2, Pages 141–182 (Mi izv2429)  

This article is cited in 42 scientific papers (total in 42 papers)

The fundamental solution of a diffusion-wave equation of fractional order

A. V. Pskhu

Scientific Research Institute of Applied Mathematics and Automation, Kabardino-Balkar Scientific Centre of the Russian Academy of Sciences

Abstract: We construct a fundamental solution of a diffusion-wave equation with Dzhrbashyan–Nersesyan fractional differentiation operator with respect to the time variable. We prove reduction formulae and solve the problem of sign-determinacy for the fundamental solution. A general representation for solutions is constructed. We give a solution of the Cauchy problem and prove the uniqueness theorem in the class of functions satisfying an analogue of Tychonoff's condition. It is shown that our fundamental solution yields the corresponding solutions for the diffusion and wave equations when the order of the fractional derivative is equal to 1 or tends to 2. The corresponding results for equations with Riemann–Liouville and Caputo derivatives are obtained as particular cases of our assertions.

Keywords: fundamental solution, diffusion equation of fractional order, wave equation of fractional order, diffusion-wave equation, Dzhrbashyan–Nersesyan fractional differentiation operator, Riemann–Liouville derivative, Caputo derivative, Tychonoff's condition, Wright's function, Cauchy problem.


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English version:
Izvestiya: Mathematics, 2009, 73:2, 351–392

Bibliographic databases:

UDC: 517.95
MSC: 26A33, 35A08, 35S10, 45K05
Received: 14.11.2006
Revised: 24.12.2007

Citation: A. V. Pskhu, “The fundamental solution of a diffusion-wave equation of fractional order”, Izv. RAN. Ser. Mat., 73:2 (2009), 141–182; Izv. Math., 73:2 (2009), 351–392

Citation in format AMSBIB
\by A.~V.~Pskhu
\paper The fundamental solution of a~diffusion-wave equation of fractional order
\jour Izv. RAN. Ser. Mat.
\yr 2009
\vol 73
\issue 2
\pages 141--182
\jour Izv. Math.
\yr 2009
\vol 73
\issue 2
\pages 351--392

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    This publication is cited in the following articles:
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  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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