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.
Citation:
K. U. Khubiev, “Boundary value problem with shift for loaded hyperbolic-parabolic type equation involving fractional diffusion operator”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 28:1 (2018), 82–90
\Bibitem{Khu18}
\by K.~U.~Khubiev
\paper Boundary value problem with shift for loaded hyperbolic-parabolic type equation involving fractional diffusion operator
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2018
\vol 28
\issue 1
\pages 82--90
\mathnet{http://mi.mathnet.ru/vuu622}
\crossref{https://doi.org/10.20537/vm180108}
\elib{https://elibrary.ru/item.asp?id=32697218}
Linking options:
https://www.mathnet.ru/eng/vuu622
https://www.mathnet.ru/eng/vuu/v28/i1/p82
This publication is cited in the following 8 articles:
A. V. Dzarakhokhov, E. L. Shishkina, “Zadacha dlya smeshannogo uravneniya s drobnoi stepenyu operatora Besselya”, Vestnik KRAUNTs. Fiz.-mat. nauki, 42:1 (2023), 37–57
B. I. Islomov, F. M. Juraev, “Local boundary value problems for a loaded equation of
parabolic-hyperbolic type degenerating inside the domain”, Ufa Math. J., 14:1 (2022), 37–51
K. U. Khubiev, “Boundary-Value Problem for a Loaded Hyperbolic-Parabolic Equation with Degeneration of Order”, J Math Sci, 260:3 (2022), 387
K. U. Khubiev, “Kraevye zadachi dlya kharakteristicheski nagruzhennogo uravneniya giperbolo-parabolicheskogo tipa”, Materialy mezhdunarodnoi konferentsii po matematicheskomu modelirovaniyu v prikladnykh naukakh “International Conference on Mathematical Modelling in Applied Sciences — ICMMAS'19”. Belgorod, 20–24 avgusta 2019 g., Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 195, VINITI RAN, M., 2021, 127–138
K. U. Khubiev, “Zadacha Bitsadze—Samarskogo dlya nagruzhennogo giperbolo-parabolicheskogo uravneniya c vyrozhdeniem poryadka v oblasti ego giperbolichnosti”, Differentsialnye uravneniya i matematicheskaya fizika, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 198, VINITI RAN, M., 2021, 123–132
A. K. Urinov, A. O. Mamanazarov, “Odnoznachnaya razreshimost odnoi nelokalnoi zadachi so smescheniem dlya parabolo-giperbolicheskogo uravneniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 30:2 (2020), 270–289
K. U. Khubiev, “Kraevaya zadacha dlya nagruzhennogo giperbolo-parabolicheskogo uravneniya s vyrozhdeniem poryadka”, Materialy IV Mezhdunarodnoi nauchnoi konferentsii “Aktualnye problemy prikladnoi matematiki”. Kabardino-Balkarskaya respublika, Nalchik, Prielbruse, 22–26 maya 2018 g. Chast III, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 167, VINITI RAN, M., 2019, 112–116
A. Kh. Attaev, “On a characteristic problem for a loaded hyperbolic equation”, Bull. Karaganda Univ-Math., 96:4 (2019), 15–21
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