Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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


For citation:

Khromov A. P. Divergent series and generalized mixed problem for a wave equation of the simplest type. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 3, pp. 322-331. DOI: 10.18500/1816-9791-2022-22-3-322-331, EDN: PTNPTE

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.08.2022
Full text:
(downloads: 1292)
Language: 
Russian
Heading: 
Article type: 
Article
UDC: 
517.96;517.984
EDN: 
PTNPTE

Divergent series and generalized mixed problem for a wave equation of the simplest type

Autors: 
Khromov August Petrovich, Saratov State University
Abstract: 

With the use of the operation of integrating the divergent series of a formal solution in the separating variables method, there are obtained the results concerning a generalized mixed problem (homogeneous and non-homogeneous) for the wave equation. The key moment consists in finding the sum of the divergent series  which corresponds to the simplest mixed problem with a summable initial function. This result helps to get  the solution of the generalized mixed problem for a non-homogeneous equation under the assumption that non-homogeneity is characterized by a locally summable function. As an application, the mixed problem with a non-zero potential is considered, in which the differential equation is treated quite formally but the  mixed problem itself is no longer a generalized one: instead of the formal solution of the separating  variables method we get an integral equation which can be solved by the successive substitutions method. Thus we essentially simplify the arguments.

References: 
  1. Khromov A. P., Kornev V. V. Divergent series in the Fourier method for the wave equation. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 4, pp. 215–238 (in Russian). https://doi.org/10.21538/0134-4889-2021-27-4-215-238
  2. Eiler L. Differentsialnoe ischislenie [Differential Calculus]. Moscow, Leningrad, GITTL, 1949. 580 p. (in Russian).
  3. Khardi G. Raskhodyashchiesya ryady [Divergent Series]. Moscow, Foreign Languages Publishing House, 1951. 504 p. (in Russian).
  4. Khromov A. P. Divergent series and generalized mixed problem for wave equation. Contemporary Problems of Function Theory and Their Applications. Proceedings of the 21st International Saratov Winter School. Sarartov, Saratov State University Publ., 2022. Iss. 21, pp. 319–324 (in Russian). Available at: https://sgu.ru/node/184778 (accessed 15 February 2022). EDN: JPPSUX
  5. Khromov A. P. Necessary and sufficient conditions for the existence of a classical solution of the mixed problem for the homogeneous wave equation with an integrable potential. Differential Equations, 2019, vol. 55, no. 5, pp. 703–717. https://doi.org/10.1134/S0012266119050112
  6. Natanson I. P. Teoriia funktsiy veshchestvennoy peremennoy [The Theory of Functions of a Real Variable]. Moscow, Leningrad, GITTL, 1957. 552 p. (in Russian).
  7. Kornev V. V., Khromov A. P. Convergence of a formal solution by the Fourier method in a mixed problem for the simplest inhomogeneous wave equation. Mathematics. Mechanics. Saratov, Saratov State University Publ., 2017. Iss. 19, pp. 41–44 (in Russian). EDN: YWRJOO
Received: 
15.03.2022
Accepted: 
01.04.2022
Published: 
31.08.2022