A boundary-value problem for the equation of mixed type with generalized operators of fractional differentiation in the boundary conditions

• Autorzy: Repin O. A. , Kumykova S. K.

Identyfikatory

DOI

10.1515/jaa-2016-0003

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

boundary value problem; operator of fractional integration; operator of fractional differentiation; Fredholm equation; singular integral equation; regularization

PL

zagadnienie brzegowe; operator całkowania ułamkowego; operator różniczkowania ułamkowego; równanie Fredholma; osobliwe równanie całkowe; regularyzacja

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2016
• Tom: Vol. 22, nr 1
• Strony: 27--36
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Repin O. A.

• Samara State Economic University, Soviet Army Str. 141, 443090 Samara, Russia

autor

Kumykova S. K.

• Kabardino-Balkarian State University, Chernyshevsky Str. 173, 360004, Nalchik, Russia

Bibliografia

• [1] F. I. Frankl, Selected Works on Gas Dynamics (in Russian), Nauka, Moscow, 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), no. 1, 78-88.
• [3] S. K. Kumykova, A boundary value problem for the equation sign y\y\muxx + uyy = 0, Differ. Equ. 12 (1976), no. 1, 79-88.
• [4] S. K. Kumykova, A boundary value problem for the equation with displacement (in Russian), Differ. Equ. 15 (1979), no. 1, 55-61.
• [5] S. K. Kumykova, A boundary value problem with shifted for the degenerative within the domain hyperbolic type equation, Differ. Equ. 16 (1980), no. 1, 93-104.
• [6] N.I. Muskhelishvili, Singular integral Equations, Nauka, Moscow, 1968.
• [7] A. M. Nakhushev, Equations of Mathematical Biology (in Russian), Vyisshaya Shkola, Moscow, 1995.
• [8] A. M. Nakhushev, Element of Fractional Calculus and Their Applications (in Russian), Kabardino-Balkarckiy Nauchnyiy Tsentr Rossîyskoy Akademii Nauk, Nalchik, 2000.
• [9] O. A. Repin, Boundary Value Problems with Shift for Equations of Hyperbolic and Mixed Type (in Russian), Saratov University, Saratov, 1992.
• [10] M. Sa i go, A remark on integral operators involving the Gauss hypergeometric function, Math. Rep. College General Educ. Kyushu Univ. 11 (1978), no. 2,135-143.
• [11] S. G. Samko, A. A. Kilbbas and O.I. Marichev, Fractional Integrals and Derivatives and Some of Their Applications (in Russian), Nauka î Tekhnika, Minsk, 1987.
• [12] M. M. Smirnov, Equations of Mixed Type, Nauka, Moscow, 1970.
• [13] F. Tricomi, Lezioni sulle equazhni a derivate pazziali, Editrice Gheoni, Turin, 1954.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.