Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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


For citation:

Gurevich A. P., Kurdyumov V. P., Khromov A. P. Justification of Fourier Method in a Mixed Problem for Wave Equation with Non-zero Velocity. Izv. Sarat. Univ. Math. Mech. Inform., 2016, vol. 16, iss. 1, pp. 13-28. DOI: 10.18500/1816-9791-2016-16-1-13-29

Published online: 
14.03.2016
Full text:
(downloads: 46)
Language: 
Russian
Heading: 
UDC: 
517.95;517.984
DOI: 
10.18500/1816-9791-2016-16-1-13-29

Justification of Fourier Method in a Mixed Problem for Wave Equation with Non-zero Velocity

Autors: 
Gurevich Alexandr Petrovich, Saratov State University
Kurdyumov Vitalii Pavlovich, Saratov State University
Khromov Avgust Petrovich, Saratov State University
Abstract: 

In the paper, using contour integration of the resolvent of the corresponding spectral problem operator, justification of Fourier method in two mixed problems for wave equation with trivial initial function and non-zero velocity is given. The boundary conditions of these problems, together with fixed endpoint conditions, embrace all cases of mixed problems with the same initial conditions for which the corresponding spectral operators in Fourier method have regular boundary conditions. The problems are considered under minimal requirements on initial data. A. N. Krylov’s idea of accelerating Fourier series convergence is essentially employed.

References: 
  1. Steklov V. A. Osnovnye zadachi matematicheskoi fiziki [The main tasks of mathematical physics]. Moscow, Nauka, 1983, 432 p. (in Russian).
  2. Krylov A. N. O nekotorykh differentsial’nykh uravneniiakh matematicheskoi fiziki, imeiushchikh prilozheniia v tekhnicheskikh voprosakh [On some differential equations of mathematical physics with applications in technical matters]. Leningrad, GITTL, 1950, 368 p. (in Russian).
  3. Chernyatin V. A. Obosnovanie metoda Fur’e v smeshannoi zadache dlya uravnenii v chastnykh proizvodnykh [Justification of the Fourier method in a mixed problem for partial differential equations]. Moscow, Moscow Univ. Press, 1991, 112 p. (in Russian).
  4. Burlutskaya M. Sh., Khromov A. P. The resolvent approach for the wave equation. Comput. Math. Math. Phys., 2015, vol. 55, iss. 2, pp. 227–239. DOI: https://doi.org/10.1134/S0965542515020050.
  5. Kornev V. V., Khromov A. P. Resolvent approach to the Fourier method in a mixed problem for the wave equation. Comput. Math. Math. Phys., 2015, vol. 55, iss. 4, pp. 618–627. DOI: https://doi.org/10.1134/S0965542515040077.
  6. Kornev V. V., Khromov A. P. A resolvent approach in the Fourier method for the wave equation: The non-selfadjoint case. Comput. Math. Math. Phys., 2015, vol. 55, iss. 7, pp. 1138–1149. DOI: https://doi.org/10.1134/S0965542515070088.
  7. Burlutskaya M. Sh., Khromov A. P. Initialboundary value problems for first-order hyperbolic equations with involution. Doklady Math., 2011, vol. 84, no. 3, pp. 783–786.
  8. Kamke E. Spravochnik po obyknovennym differentsial’nym uravneniiam [Handbook of Ordinary Differential Equations]. Moscow, Nauka, 1971, 538 p. (in Russian).
  9. Naimark M. A. Linear Differential Operators. New York, Ungar, 1967; Moscow, Nauka, 1969, 828 p.
  10. Rasulov M. L. Metod konturnogo integrala [The method of the contour integral]. Moscow, Nauka, 1964, 462 p. (in Russian).
  11. Vagabov A. I. Vvedenie v spektral’nuiu teoriiu differentsial’nykh operatorov [Introduction to the spectral theory of differential operators]. Rostovon-Don, Rostov Univ. Press, 1994, 106 p. (in Russian).
  12. Marchenko V. A. Sturm – Liouville Operators and Applications. Kiev, Naukova Dumka, 1977, 332 p. (in Russian).