Geodesic
Regularization of the solution of the Cauchy problem: the quasi-reversibility method
Sibirskij žurnal industrialʹnoj matematiki, Tome 21 (2018) no. 4, pp. 96-109.

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

Some regularization algorithm is proposed related to the problem of continuation of the wave field from the planar boundary into the half-plane. We consider a hyperbolic equation whose main part coincideswith the wave operator, whereas the lowest term contains a coefficient depending on the two spatial variables. The regularization algorithm is based on the quasi-reversibility method proposed by Lattes and Lions. We consider the solution of an auxiliary regularizing equation with a small parameter; the existence, the uniqueness, and the stability of the solution in the Cauchy data are proved. The convergence is substantiated of this solution to the exact solution as the small parameter vanishes. A solution of an auxiliary problem is constructed with the Cauchy data having some error. It is proved that, for a suitable choice of a small parameter, the approximate solution converges to the exact solution.
Keywords: Cauchy problem, continuation of the wave field, regularization. 
@article{SJIM_2018_21_4_a7,
     author = {V. G. Romanov and T. V. Bugueva and V. A. Dedok},
     title = {Regularization of the solution of the {Cauchy} problem: the quasi-reversibility method},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {96--109},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2018_21_4_a7/}
}
TY  - JOUR
AU  - V. G. Romanov
AU  - T. V. Bugueva
AU  - V. A. Dedok
TI  - Regularization of the solution of the Cauchy problem: the quasi-reversibility method
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2018
SP  - 96
EP  - 109
VL  - 21
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2018_21_4_a7/
LA  - ru
ID  - SJIM_2018_21_4_a7
ER  - 
%0 Journal Article
%A V. G. Romanov
%A T. V. Bugueva
%A V. A. Dedok
%T Regularization of the solution of the Cauchy problem: the quasi-reversibility method
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2018
%P 96-109
%V 21
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2018_21_4_a7/
%G ru
%F SJIM_2018_21_4_a7
V. G. Romanov; T. V. Bugueva; V. A. Dedok. Regularization of the solution of the Cauchy problem: the quasi-reversibility method. Sibirskij žurnal industrialʹnoj matematiki, Tome 21 (2018) no. 4, pp. 96-109. http://geodesic.mathdoc.fr/item/SJIM_2018_21_4_a7/

[1] John F., “Continuous dependence on data for solutions of partial differential equations with a prescribed bound”, Comm. Pure Appl. Math., 13:4 (1960), 551–585 | DOI | MR | Zbl

[2] John F., Differential Equations with Approximate and Improper Data, Lectures, New York Univ., N.Y., 1995

[3] Kurant R., Uravneniya s chastnymi proizvodnymi, Mir, M., 1964

[4] Romanov V. G., Nekotorye obratnye zadachi dlya uravnenii giperbolicheskogo tipa, Nauka, Novosibirsk, 1972

[5] Lavrentev M. M., Romanov V. G., Shishatskii S. P., Nekorrektnye zadachi matematicheskoi fiziki, Nauka, M., 1980

[6] Finch D., Patch S. K., Rakesh, “Determining a function from its mean values over a family of spheres”, SIAM J. Math. Anal., 35:5 (2004), 1213–1240 | DOI | MR | Zbl

[7] Natterer F., “Photo-acoustic inversion in convex domains”, Inverse Probl. Imaging, 6:2 (2012), 1–6 | DOI | MR

[8] Palamodov V. P., “Reconstruction from limited data of arc means”, J. Fourier Anal. Appl., 6:1 (2000), 25–42 | DOI | MR | Zbl

[9] Symes W. W., “A trace theorem for solutions of the wave equation, and the remote qdetermination of acoustic sources”, Math. Meth. Appl. Sci., 5 (1983), 131–152 | DOI | MR | Zbl

[10] Lattes R., Lions Zh.-L., Metod kvaziobrascheniya i ego prilozheniya, Mir, M., 1970

[11] Nirenberg L., Topics in Nonlinear Functional Analysis, Courant Inst. Math. Sci., N.Y., 1974 | MR | Zbl

[12] Ovsyannikov L. V., “Nelineinaya zadacha Koshi v shkalakh banakhovykh prostranstv”, Dokl. AN SSSR, 200:4 (1971), 789–792 | MR | Zbl

[13] Romanov V. G., “O lokalnoi razreshimosti nekotorykh mnogomernykh obratnykh zadach dlya uravnenii giperbolicheskogo tipa”, Differents. uravneniya, 25:2 (1989), 275–283 | MR | Zbl

[14] Romanov V. G., “O chislennom metode resheniya odnoi obratnoi zadachi dlya giperbolicheskogo uravneniya”, Sib. mat. zhurn., 37:3 (1996), 633–655 | MR | Zbl

[15] Romanov V. G., “Lokalnyi variant chislennogo metoda resheniya obratnoi zadachi”, Sib. mat. zhurn., 37:4 (1996), 904–918 | MR | Zbl

[16] Smirnov V. I., Kurs vysshei matematiki, v. II, Fizmatgiz, M., 1961

[17] Tikhonov A. N., Samarskii A. A., Uravneniya matematicheskoi fiziki, Nauka, M., 1966 | MR