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A boundary-value problem for the equation of mixed type with generalized operators of fractional differentiation in the boundary conditions. (English) Zbl 1343.35177

Summary: The paper is devoted to the study of a boundary-value problem for an equation of mixed type with generalized operators of fractional differentiation in boundary conditions. We prove uniqueness of solutions under some restrictions on the known functions and on the different orders of the operators of generalized fractional differentiation appearing in the boundary conditions. Existence of solutions is proved by reduction to a Fredholm equation of the second kind, for which solvability follows from the uniqueness of the solution of our original problem.

MSC:

35M12 Boundary value problems for PDEs of mixed type
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
35R11 Fractional partial differential equations
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[1] F. I. Frankl, Selected Works on Gas Dynamics (in Russian), Nauka, Moscow, 1973.; Frankl, F. I., Selected Works on Gas Dynamics (1973)
[2] S. K. Kumykova, A problem with nonlocal boundary conditions in characteristics for the equations of mixed type (in Russian), Differ. Equ. 10 (1974), 1, 78-88.; Kumykova, S. K., A problem with nonlocal boundary conditions in characteristics for the equations of mixed type, Differ. Equ., 10, 1, 78-88 (1974) · Zbl 0291.35060
[3] S. K. Kumykova, A boundary value problem for the equation \({\operatorname{sign}y |y|^mu_{xx}+u_{yy}=0} \), Differ. Equ. 12 (1976), 1, 79-88.; Kumykova, S. K., A boundary value problem for the equation \({\operatorname{sign}y |y|^mu_{xx}+u_{yy}=0} \), Differ. Equ., 12, 1, 79-88 (1976) · Zbl 0321.35059
[4] S. K. Kumykova, A boundary value problem for the equation with displacement (in Russian), Differ. Equ. 15 (1979), 1, 55-61.; Kumykova, S. K., A boundary value problem for the equation with displacement, Differ. Equ., 15, 1, 55-61 (1979) · Zbl 0343.35064
[5] S. K. Kumykova, A boundary value problem with shifted for the degenerative within the domain hyperbolic type equation, Differ. Equ. 16 (1980), 1, 93-104.; Kumykova, S. K., A boundary value problem with shifted for the degenerative within the domain hyperbolic type equation, Differ. Equ., 16, 1, 93-104 (1980) · Zbl 0426.35070
[6] N. I. Muskhelishvili, Singular Integral Equations, Nauka, Moscow, 1968.; Muskhelishvili, N. I., Singular Integral Equations (1968) · Zbl 0174.16202
[7] A. M. Nakhushev, Equations of Mathematical Biology (in Russian), Vyisshaya Shkola, Moscow, 1995.; Nakhushev, A. M., Equations of Mathematical Biology (1995)
[8] A. M. Nakhushev, Element of Fractional Calculus and Their Applications (in Russian), Kabardino-Balkarckiy Nauchnyiy Tsentr Rossiyskoy Akademii Nauk, Nalchik, 2000.; Nakhushev, A. M., Element of Fractional Calculus and Their Applications (2000)
[9] O. A. Repin, Boundary Value Problems with Shift for Equations of Hyperbolic and Mixed Type (in Russian), Saratov University, Saratov, 1992.; Repin, O. A., Boundary Value Problems with Shift for Equations of Hyperbolic and Mixed Type (1992) · Zbl 0867.35002
[10] M. Saigo, A remark on integral operators involving the Gauss hypergeometric function, Math. Rep. College General Educ. Kyushu Univ. 11 (1978), 2, 135-143.; Saigo, M., A remark on integral operators involving the Gauss hypergeometric function, Math. Rep. College General Educ. Kyushu Univ., 11, 2, 135-143 (1978) · Zbl 0399.45022
[11] S. G. Samko, A. A. Kilbbas and O. I. Marichev, Fractional Integrals and Derivatives and Some of Their Applications (in Russian), Nauka i Tekhnika, Minsk, 1987.; Samko, S. G.; Kilbbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives and Some of Their Applications (1987)
[12] M. M. Smirnov, Equations of Mixed Type, Nauka, Moscow, 1970.; Smirnov, M. M., Equations of Mixed Type (1970)
[13] F. Tricomi, Lezioni sulle equazioni a derivate pazziali, Editrice Gheroni, Turin, 1954.; Tricomi, F., Lezioni sulle equazioni a derivate pazziali (1954) · Zbl 0057.07502
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