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Green Functions of the First Boundary-Value Problem for a Fractional Diffusion—Wave Equation in Multidimensional Domains

Institute of Applied Mathematics and Automation, Kabardin-Balkar Scientific Center of Russian Academy of Sciences, 89-A Shortanov Street, 360000 Nalchik, Russia
Mathematics 2020, 8(4), 464; https://doi.org/10.3390/math8040464 (registering DOI)
Received: 12 February 2020 / Revised: 13 March 2020 / Accepted: 22 March 2020 / Published: 26 March 2020
(This article belongs to the Special Issue Direct and Inverse Problems for Fractional Differential Equations)
We construct the Green function of the first boundary-value problem for a diffusion-wave equation with fractional derivative with respect to the time variable. The Green function is sought in terms of a double-layer potential of the equation under consideration. We prove a jump relation and solve an integral equation for an unknown density. Using the Green function, we give a solution of the first boundary-value problem in a multidimensional cylindrical domain. The fractional differentiation is given by the Dzhrbashyan–Nersesyan fractional differentiation operator. In particular, this covers the cases of equations with the Riemann–Liouville and Caputo derivatives.
Keywords: diffusion-wave equation; boundary-value problem; Green function; double-layer potential; fractional derivative; Dzhrbashyan–Nersesyan operator diffusion-wave equation; boundary-value problem; Green function; double-layer potential; fractional derivative; Dzhrbashyan–Nersesyan operator
MDPI and ACS Style

Pskhu, A. Green Functions of the First Boundary-Value Problem for a Fractional Diffusion—Wave Equation in Multidimensional Domains. Mathematics 2020, 8, 464.

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