In this work two non-local problems for the parabolic-hyperbolic type equation with non-characteristic line of changing type are considered. Unique solvability of these problems is proven. The uniqueness of the solution is proven by the method of energy integrals and the existence is proven by the method of integral equations.
@article{bwmeta1.element.doi-10_2478_s11533-006-0007-8,
author = {A. Berdyshev and E. Karimov},
title = {Some non-local problems for the parabolic-hyperbolic type equation with non-characteristic line of changing type},
journal = {Open Mathematics},
volume = {4},
year = {2006},
pages = {183-193},
zbl = {1098.35116},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0007-8}
}
A. Berdyshev; E. Karimov. Some non-local problems for the parabolic-hyperbolic type equation with non-characteristic line of changing type. Open Mathematics, Tome 4 (2006) pp. 183-193. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0007-8/
[1] Gel’fand I.M.: “Some questions of analysis and differential equations”, UMN, Ser. 3(87), Vol. XIV, (1959), pp. 3–19.
[2] Nakhushev A.M.: The equations of mathematical biology, Vishsaya shkola, Moscow, 1995, p. 301.
[3] Nakhushev A.M.: “On one approximate method of solving boundary value problems for differential equations and its application to the dynamics of land and ground waters”, Differensialniye Uravneniya, Vol. 18(1), (1982), pp. 72–81.
[4] S.V. Nerpin and A.F. Chudnovskij: Energy and mass exchange in the system flora - soil - air, Gidromechizdat, Leningrad, 1975.
[5] P. Bassani and M. Calaverni: “Cantrazioni multi sistemi iperbolica, iproblema del lazer”, Atti.Semin.mat.e.fis., Univ.Modena, Vol. 35(2), (1982), pp. 32–50.
[6] L. Bers: Mathematical questions of subsonic and transonic gas dynamics, IL, Moscow, 1961, p. 208.
[7] M.S. Salakhitdinov and A.K. Urinov: “About one boundary value problem for the mixed type equation with non-smooth line of degeneracy”, Doklady AN SSSR, Vol. 262(3), (1982), pp. 539–541.
[8] V.A. Eleev: “Analogue of the Tricomi problem for the mixed parabolic-hyperbolic equations with non-characteristic line of changing type”, Differensialniye Uravneniya, Vol. 13(1), (1977), pp. 56–63.
[9] N.Y. Kapustin: “The Tricomi problem for the parabolic-hyperbolic equation with degenerating hyperbolic part”, Differensialniye Uravneniya, Vol. 24(8), (1988), pp. 1379–1386.
[10] G.D. Tojzhanova and M.A. Sadybekov: “About spectral properties of one analogue of the Tricomi problem for the mixed parabolic-hyperbolictype equation”, Izvestija AN KazSSR, ser.phys.-math.nauk, Vol. 3, (1989), pp. 48–52.
[11] A.S. Berdyshev: “Nonlocal boundary problems for the mixed type equation in the domain with deviation from the characteristic”, Differensilaniye Uravneniya, Vol. 29(12), (1993), pp. 2118–2125.
[12] A.S. Berdyshev: “On uniqueness of the solution of general Tricomi problem for the parabolic-hyperbolic equation”, Doklady ANRUz., Vol. 10, (1994), pp. 5–7.
[13] A.N. Tikhonov and A.A. Samarskij: Equations of mathematical physics, Nauka, Moscow, 1977, p. 736.
[14] M.S. Salakhitdinov and A.K. Urinov: Boundary value problems for the mixed type equations with spectral parameter, Fan, Tashkent, 1997, p. 166.
[15] G. Bateman and A. Erdelji: Higher transcendent functions, Nauka, Moscow, 1965, p. 296.
[16] A. Fridman: Partial differential equations of parabolic type, Izdatelstvo Mir, Moscow, 1968, p. 428.