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Russia Nalchik
Section Mathematics
Title Boundary value problem with shift for loaded hyperbolic-parabolic type equation involving fractional diffusion operator
Author(-s) Khubiev K.U.a
Affiliations Institute of Applied Mathematics and Automation, Kabardino-Balkarian Scientific Center of the Russian Academy of Sciencesa
Abstract The paper deals with non-local boundary-value problems with shift and discontinuous conjugation conditions in the line of type changing for a model loaded hyperbolic-parabolic type equation. The parabolic domain presents a fractional diffusion equation while the hyperbolic one presents a characteristically loaded wave equation. The uniqueness of the solution to the considered problems under certain conditions on the coefficients is proved by the Tricomi method. The existence of the solution involves solving the Fredholm integral equation of the second kind with respect to the trace of the sought solution in the line of type changing. The unique solvability of the integral equation implies the uniqueness of the solution to the problems. Once the integral equation is solved, the solution to the problems is reduced to solving the first boundary value problem for the fractional diffusion equation in the parabolic domain and the Cauchy problem for the inhomogeneous wave equation in the hyperbolic one. In addition, representation formulas are written out for solving the problems under study in the parabolic and hyperbolic domains.
Keywords nonlocal problem, problem with shift, loaded equation, equation of mixed type, hyperbolic-parabolic type equation, fractional diffusion operator
UDC 517.95
MSC 35M10, 35M12
DOI 10.20537/vm180108
Received 2 February 2018
Language Russian
Citation Khubiev K.U. Boundary value problem with shift for loaded hyperbolic-parabolic type equation involving fractional diffusion operator, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 1, pp. 82-90.
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