Geodesic
Unique solvability of a nonlocal problem with shift for a parabolic-hyperbolic equation
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 30 (2020) no. 2, pp. 270-289

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

In the paper, a parabolic-hyperbolic equation with a singular coefficient and a spectral parameter in the domain which consists of a characteristic triangle and a half strip has been considered. A nonlocal problem connecting the values of the desired function at the two points of boundary characteristics and the line of equation type changing by means of two operators, the first of which depends on the coefficient of the singularity and the second one — on the spectral parameters, is formulated. The considered problem is investigated by reducing it to the system of equations in the trace of the desired function and its derivative with respect to $x$ on the line of equation type changing. The uniqueness of the solution is proved by the method of energy integrals, for this we use integral representations of Euler gamma-function and Bessel function of the first kind. The existence of the solution is proved by the method of integral equations, for this we equivalently reduce the considered problem to the Fredholm integral equation of the second kind which solvability follows from the uniqueness of the problem solution. Sufficient conditions for unique solvability of the considered problem are found.
Keywords: parabolic-hyperbolic equation, spectral parameter, noncharacteristic line of type changing, nonlocal problem, unique solvability.
Mots-clés : singular coefficient 
@article{VUU_2020_30_2_a9,
     author = {A. K. Urinov and Mamanazarov A.O.},
     title = {Unique solvability of a nonlocal problem with shift for a parabolic-hyperbolic equation},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {270--289},
     publisher = {mathdoc},
     volume = {30},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2020_30_2_a9/}
}
TY  - JOUR
AU  - A. K. Urinov
AU  - Mamanazarov A.O.
TI  - Unique solvability of a nonlocal problem with shift for a parabolic-hyperbolic equation
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2020
SP  - 270
EP  - 289
VL  - 30
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VUU_2020_30_2_a9/
LA  - ru
ID  - VUU_2020_30_2_a9
ER  - 
%0 Journal Article
%A A. K. Urinov
%A Mamanazarov A.O.
%T Unique solvability of a nonlocal problem with shift for a parabolic-hyperbolic equation
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2020
%P 270-289
%V 30
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VUU_2020_30_2_a9/
%G ru
%F VUU_2020_30_2_a9
A. K. Urinov; Mamanazarov A.O. Unique solvability of a nonlocal problem with shift for a parabolic-hyperbolic equation. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 30 (2020) no. 2, pp. 270-289. http://geodesic.mathdoc.fr/item/VUU_2020_30_2_a9/