Geodesic
A~Boundary-value Problem with Shift for a~Hyperbolic Equation Degenerate in the Interior of a~Region
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2014), pp. 37-47.

Voir la notice de l'article provenant de la source Math-Net.Ru

For a degenerate hyperbolic equation in characteristic region (lune) a boundary-value problem with operators of fractional integro-differentiation is studied. The solution of this equation on the characteristics is related point-to-point to the solution and its derivative on the degeneration line. The uniqueness theorem is proved by the modified Tricomi method with inequality-type constraints on the known functions. Question of the problem solution’s existence is reduced to the solvability of a singular integral equation with Cauchy kernel of the normal type.
Keywords: Cauchy problem, boundary-value problem with shift, fractional integro-differentiation operators, singular equation with Cauchy kernel, regularizer, Gauss hypergeometric function, Euler gamma function. 
@article{VSGTU_2014_1_a3,
     author = {O. A. Repin and S. K. Kumykova},
     title = {A~Boundary-value {Problem} with {Shift} for {a~Hyperbolic} {Equation} {Degenerate} in the {Interior} of {a~Region}},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {37--47},
     publisher = {mathdoc},
     number = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2014_1_a3/}
}
TY  - JOUR
AU  - O. A. Repin
AU  - S. K. Kumykova
TI  - A~Boundary-value Problem with Shift for a~Hyperbolic Equation Degenerate in the Interior of a~Region
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2014
SP  - 37
EP  - 47
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2014_1_a3/
LA  - ru
ID  - VSGTU_2014_1_a3
ER  - 
%0 Journal Article
%A O. A. Repin
%A S. K. Kumykova
%T A~Boundary-value Problem with Shift for a~Hyperbolic Equation Degenerate in the Interior of a~Region
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2014
%P 37-47
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2014_1_a3/
%G ru
%F VSGTU_2014_1_a3
O. A. Repin; S. K. Kumykova. A~Boundary-value Problem with Shift for a~Hyperbolic Equation Degenerate in the Interior of a~Region. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2014), pp. 37-47. http://geodesic.mathdoc.fr/item/VSGTU_2014_1_a3/

[1] S. G. Samko, A. A. Kilbas, O. I. Maritchev, Integraly i proizvodnyye drobnogo poryadka i nekotoryye ikh prilozheniya [Integrals and Derivatives of the Fractional Order and Some of their Applications], Nauka i Tekhnika, Minsk, 1987, 688 pp. (In Russian) | MR | Zbl

[2] A. M. Nakhushev, Zadachi so smeshcheniyem dlya uravnenii v chastnykh proizvodnykh [Problems with shifts for partial differential equations], Nauka, Moscow, 2006, 287 pp. (In Russian) | Zbl

[3] M. M. Smirnov, Vyrozhdayushchiyesya giperbolicheskiye uravneniya [Degenerate Hyperbolic Equations], Vysshaya Shkola, Minsk, 1977, 158 pp. | MR | Zbl

[4] S. K. Kumykova, “Boundary-value problem with translation for a hyperbolic equation degenerate in the interior of a region”, Differ. Equations, 16:1 (1980), 68–76 | MR | Zbl | Zbl

[5] O. A. Repin, S. K. Kumykova, “Nonlocal problem for a equation of mixed type of third order with generalized operators of fractional integro-differentiation of arbitrary order”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2011, no. 4(25), 25–36 (In Russian) | DOI

[6] S. K. Kumykova, F. B. Nakhusheva, “A boundary-value problem for a hyperbolic equation degenerate in the interior of a region”, Differ. Equations, 14:1 (1978), 35–46 | MR | Zbl | Zbl

[7] O. A. Repin, Krayevyye zadachi so smeshcheniyem dlya uravneniy giperbolicheskogo i smeshannogo tipov [Boundary value problems with shift for equations of hyperbolic and mixed type], Saratov State Univ., Samara Branch, Samara, 1992, 164 pp. (In Russian) | MR | Zbl

[8] N. I. Muskhelishvili, Singulyarnyye integral'nyye uravneniya [Singular Integral Equations], Nauka, Moscow, 1968, 512 pp. (In Russian) | MR