Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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


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Kurdyumov V. P. Classic and generalized solutions of the mixed problem for wave equation with a summable potential. Part I. Classic solution of the mixed problem. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2023, vol. 23, iss. 3, pp. 311-319. DOI: 10.18500/1816-9791-2023-23-3-311-319, EDN: GUFKKJ

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.08.2023
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Russian
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Article
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519.663
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GUFKKJ

Classic and generalized solutions of the mixed problem for wave equation with a summable potential. Part I. Classic solution of the mixed problem

Autors: 
Kurdyumov Vitalii Pavlovich, Saratov State University
Abstract: 

The resolvent approach and the using of the idea of A. N. Krylov on the acceleration of convergence of Fourier series, the properties of a formal solution of a mixed problem for a homogeneous wave equation with a summable potential and a zero initial function are studied. This method makes it possible to obtain deep results on the convergence of a formal series with arbitrary boundary conditions and without overestimating the requirements for the smoothness of the initial data. The different-order boundary conditions considered in the article are such that the operator corresponding to the spectral problem may have an infinite set of multiple eigenvalues and their associated functions. A classical solution is obtained without overstating the requirements for the initial velocity $u'_t(x,0) = \psi(x)$. It is shown that for $\psi(x) \in L[0,1]$ the formal solution, being the uniform limit of the classical ones, is a generalized solution, and when $\psi(x) \in L_p[0,1], ~ 1 <p\leqslant 2$, the formal solution has much smoother properties than the case $\psi(x) \in L[0,1]$.

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Received: 
22.04.2022
Accepted: 
01.09.2022
Published: 
31.08.2023