Geodesic
A resolvent approach in the Fourier method for the wave equation: The non-selfadjoint case
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 7, pp. 1156-1167 
V. V. Kornev; A. P. Khromov. A resolvent approach in the Fourier method for the wave equation: The non-selfadjoint case. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 7, pp. 1156-1167. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_7_a5/
@article{ZVMMF_2015_55_7_a5,
     author = {V. V. Kornev and A. P. Khromov},
     title = {A resolvent approach in the {Fourier} method for the wave equation: {The~non-selfadjoint} case},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1156--1167},
     year = {2015},
     volume = {55},
     number = {7},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_7_a5/}
}
TY  - JOUR
AU  - V. V. Kornev
AU  - A. P. Khromov
TI  - A resolvent approach in the Fourier method for the wave equation: The non-selfadjoint case
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2015
SP  - 1156
EP  - 1167
VL  - 55
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_7_a5/
LA  - ru
ID  - ZVMMF_2015_55_7_a5
ER  - 
%0 Journal Article
%A V. V. Kornev
%A A. P. Khromov
%T A resolvent approach in the Fourier method for the wave equation: The non-selfadjoint case
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2015
%P 1156-1167
%V 55
%N 7
%U http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_7_a5/
%G ru
%F ZVMMF_2015_55_7_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

Under minimum smoothness requirements for the initial data, the Fourier method in the mixed problem for the wave equation with a complex potential is justified by using the Cauchy–Poincare technique for the contour integration of the resolvent of the eigenvalue problem. Generic boundary conditions are used; one of them contains first-order derivatives, while the other does not. In this case, even for the benchmark situation, the operator in the eigenvalue problem can have any number of generalized eigenfunctions. A substantial use is made of the technique for accelerating Fourier series due to A. N. Krylov.

[1] Steklov V. A., Osnovnye zadachi matematicheskoi fiziki, Nauka, M., 1983

[2] Petrovskii I. G., Lektsii ob uravneniyakh s chastnymi proizvodnymi, GITTL, M.–L., 1953

[3] Smirnov V. I., Kurs vysshei matematiki, v 4 t., v. 4, Gostekhizdat, M., 1953

[4] Ladyzhenskaya O. A., Smeshannaya zadacha dlya giperbolicheskogo uravneniya, Gostekhizdat, M., 1953

[5] Ilin V. A., Izbrannye trudy, v. v 2 t., Izd-vo OOO “Maks-press”, M., 2008

[6] Ilin V. A., “O razreshimosti smeshannykh zadach dlya giperbolicheskikh i parabolicheskikh uravnenii”, Uspekhi matem. nauk, 15:2 (1960), 97–154 | MR | Zbl

[7] Chernyatin V. A., Obosnovanie metoda Fure v smeshannoi zadache dlya uravnenii v chastnykh proizvodnykh, Izd-vo MGU, M., 1991

[8] Krylov A. N., O nekotorykh differentsialnykh uravneniyakh matematicheskoi fiziki, imeyuschikh prilozheniya v tekhnicheskikh voprosakh, GITTL, M.–L., 1950

[9] Burlutskaya M. Sh., Khromov A. P., “Rezolventnyi podkhod v metode Fure”, Dokl. AN, 458:2 (2014), 138–140 | Zbl

[10] Burlutskaya M. Sh., Khromov A. P., “Smeshannye zadachi dlya giperbolicheskikh uravnenii pervogo poryadka s involyutsiei”, Dokl. AN, 441:2 (2011), 151–154

[11] M. A. Naimark, Lineinye differentsialnye operatory, Nauka, M., 1969 | MR

[12] Rasulov M. L., Metod konturnogo integrala, Nauka, M., 1964 | MR

[13] Vagabov A. I., Vvedenie v spektralnuyu teoriyu differentsialnykh operatorov, Izd-vo Rost. un-ta, Rostov n/D., 1994

[14] Marchenko V. A., Operatory Shturma–Liuvilya i ikh prilozheniya, Naukova dumka, Kiev, 1977 | MR