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.
\Bibitem{Psk09}
\by A.~V.~Pskhu
\paper The fundamental solution of a~diffusion-wave equation of fractional order
\jour Izv. Math.
\yr 2009
\vol 73
\issue 2
\pages 351--392
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\crossref{https://doi.org/10.1070/IM2009v073n02ABEH002450}
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Linking options:
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https://doi.org/10.1070/IM2009v073n02ABEH002450
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