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This article is cited in 4 scientific papers (total in 4 papers)
Differential Equations and Mathematical Physics
The nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation of the first kind
S. Z. Djamalov V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, 100041, Uzbekistan
(published under the terms of the Creative Commons Attribution 4.0 International License)
Abstract:
In this paper the unique solvability and smoothness of generalized solution of a nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation of the first kind in Sobolev spaces $W_{2}^{l }(Q)$, ($2\le l $ is integer number), have been proved. First, the unique solvability of the generalized solution from space $W_{2}^{2 }(Q)$ has been studied. Further, the uniqueness of the generalized solution of nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation was proved by a priory estimates. For the proof of the existence of the generalized solution, we used method of "$\varepsilon$-regularization" together with Galerkin method. Precisely, first, we study regular solvability of the nonlocal boundary value problem for the multidimensional mixed type equation by functional analysis methods, i.e. we obtained necessary a priory estimates for the considered problems. Using these estimates we solve composite type equation, then by the theorem on weak compactness, we pass to the limit and deduce to the multidimensional mixed type equation of the first kind. At the end, smoothness of the generalized solution of the considered problems has been discussed.
Keywords:
multidimensional mixed type equations, nonlocal boundary value problem with constants coefficients, unique solvability, smoothness of the generalized solution, $\varepsilon$-regularization method, Galerkin method.
Received: March 22, 2017 Revised: November 28, 2017 Accepted: December 18, 2017 First online: December 22, 2017
Citation:
S. Z. Djamalov, “The nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation of the first kind”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 21:4 (2017), 597–610
Linking options:
https://www.mathnet.ru/eng/vsgtu1536 https://www.mathnet.ru/eng/vsgtu/v221/i4/p597
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