Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Kurdyumov V. P., Khromov A. P., Khalova V. A. A Mixed Problem for a Wave Equation with a Nonzero Initial Velocity. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2018, vol. 18, iss. 2, pp. 157-171. DOI: 10.18500/1816-9791-2018-18-2-157-171, EDN: XQFNQT

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
28.05.2018
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Russian
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519.633
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XQFNQT

A Mixed Problem for a Wave Equation with a Nonzero Initial Velocity

Autors: 
Kurdyumov Vitalii Pavlovich, Saratov State University
Khromov August Petrovich, Saratov State University
Khalova Victoria Anatol'evna, Saratov State University
Abstract: 

We study a mixed problem for the wave equation with a continuous complex potential in the case of a nonzero initial velocity ut (x, 0) = ψ(x) and two types of two-point boundary conditions: the ends are fixed and when each of the boundary boundary conditions contains a derivative with respect to x. A classical solution in the case ψ(x) ∈ W12[0, 1] is obtained by the Fourier method with respect to the acceleration of the convergence of Fourier series by the resolvent approach with the help of A. N. Krylov’s recommendations (the equation is satisfied almost everyw here). It is also shown that in the case when ψ(x) ∈ L[0, 1] the series of a formal solution for a problem with fixed ends converges uniformly in any bounded domain, and for the second problem it converges only everywhere and for both problems is a generalized solution in the uniform metric.

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Received: 
12.01.2018
Accepted: 
08.05.2018
Published: 
04.06.2018
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