Geodesic
The Dirichlet problem for a~mixed-type equation with characteristic degeneration in a~rectangular domain
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2009), pp. 43-52

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We study the first boundary problem for the following mixed-type equation of the second kind: $$ u_{xx}+yu_{yy}+au_y-b^2u=0 $$ in the domain $\{(x,y)\mid0$, where $a,b,\alpha$, and $\beta$ are given real numbers, and $0$, $b\geq0$, $\alpha>0$, $\beta>0$. Based on the completeness of the system of eigenfunctions of one-dimensional spectral problem we establish a uniqueness criterion. We construct a solution to the problem as the sum of the series in eigenfunctions.
Keywords: Dirichlet problem, mixed-type equation, spectral method, uniqueness
Mots-clés : existence. 
@article{IVM_2009_11_a4,
     author = {K. B. Sabitov and A. Kh. Suleimanova},
     title = {The {Dirichlet} problem for a~mixed-type equation with characteristic degeneration in a~rectangular domain},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {43--52},
     publisher = {mathdoc},
     number = {11},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2009_11_a4/}
}
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K. B. Sabitov; A. Kh. Suleimanova. The Dirichlet problem for a~mixed-type equation with characteristic degeneration in a~rectangular domain. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2009), pp. 43-52. http://geodesic.mathdoc.fr/item/IVM_2009_11_a4/