Geodesic
Steady-state oscillations in continuously inhomogeneous medium described by a~generalized Darboux equation
Sibirskij žurnal industrialʹnoj matematiki, Tome 13 (2010) no. 1, pp. 140-149.

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

We refine the result of Ovsyannikov on the general form of second order linear differential equations with a nonzero generalized Laplace invariant admitting a Lie group of transformations of maximal order with $n>2$ independent variables for which the associated Riemannian spaces have nonzero curvature. We show that the set of these equations is exhausted by the generalized Darboux equation and the Ovsyannikov equation. We find the operators acting on the set of solutions inside every one-parameter family of generalized Darboux equations. For the elliptic generalized Darboux equation possessing the maximal symmetry and describing steady-state oscillations in continuously inhomogeneous medium with a degeneration hyperplane, by group analysis methods we obtain exact solutions to boundary value problems for certain domains (generalized Poisson formulas), which in particular can be test solutions in simulating steady-state oscillations in continuously inhomogeneous media.
Keywords: generalized Darboux equation, intertwining operators, steady-state oscillations in continuously inhomogeneous medium, generalized Poisson formulas
Mots-clés : Ovsyannikov equation, radiation conditions. 
@article{SJIM_2010_13_1_a12,
     author = {Yu. A. Chirkunov},
     title = {Steady-state oscillations in continuously inhomogeneous medium described by a~generalized {Darboux} equation},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {140--149},
     publisher = {mathdoc},
     volume = {13},
     number = {1},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2010_13_1_a12/}
}
TY  - JOUR
AU  - Yu. A. Chirkunov
TI  - Steady-state oscillations in continuously inhomogeneous medium described by a~generalized Darboux equation
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2010
SP  - 140
EP  - 149
VL  - 13
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2010_13_1_a12/
LA  - ru
ID  - SJIM_2010_13_1_a12
ER  - 
%0 Journal Article
%A Yu. A. Chirkunov
%T Steady-state oscillations in continuously inhomogeneous medium described by a~generalized Darboux equation
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2010
%P 140-149
%V 13
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2010_13_1_a12/
%G ru
%F SJIM_2010_13_1_a12
Yu. A. Chirkunov. Steady-state oscillations in continuously inhomogeneous medium described by a~generalized Darboux equation. Sibirskij žurnal industrialʹnoj matematiki, Tome 13 (2010) no. 1, pp. 140-149. http://geodesic.mathdoc.fr/item/SJIM_2010_13_1_a12/

[1] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 | MR | Zbl

[2] Kurant R., Uravneniya s chastnymi proizvodnymi, Mir, M., 1964 | MR

[3] Uitteker E. T., Vatson Dzh. N., Kurs sovremennogo analiza, Ch. 2, Nauka, M., 1963