Geodesic
A nonlocal problem for a third order parabolic-hyperbolic equation with a singular coefficient
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 15 (2022) no. 4, pp. 467-481.

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

Non-classical problem with an integral condition for parabolic-hyperbolic equation of the third order is formulated and studied in this paper. The unique solvability of the problem was proved using the method integral equations. To do this the problem is equivalently reduced to a problem for a parabolic-hyperbolic equation of the second order with an unknown right-hand side. To study the obtained problem the formula of the Cauchy problem for hyperbolic equation with a singular coefficient and a spectral parameter was used. The solution of the first boundary value problem for the Fourier equation was also used.
Keywords: parabolic-hyperbolic equation, integral condition, uniqueness of the solution, existence of the solution
Mots-clés : singular coefficient. 
@article{JSFU_2022_15_4_a5,
     author = {Akhmadjon K. Urinov and Kobiljon S. Khalilov},
     title = {A nonlocal problem for a third order parabolic-hyperbolic equation with a singular coefficient},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {467--481},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2022_15_4_a5/}
}
TY  - JOUR
AU  - Akhmadjon K. Urinov
AU  - Kobiljon S. Khalilov
TI  - A nonlocal problem for a third order parabolic-hyperbolic equation with a singular coefficient
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2022
SP  - 467
EP  - 481
VL  - 15
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2022_15_4_a5/
LA  - en
ID  - JSFU_2022_15_4_a5
ER  - 
%0 Journal Article
%A Akhmadjon K. Urinov
%A Kobiljon S. Khalilov
%T A nonlocal problem for a third order parabolic-hyperbolic equation with a singular coefficient
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2022
%P 467-481
%V 15
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2022_15_4_a5/
%G en
%F JSFU_2022_15_4_a5
Akhmadjon K. Urinov; Kobiljon S. Khalilov. A nonlocal problem for a third order parabolic-hyperbolic equation with a singular coefficient. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 15 (2022) no. 4, pp. 467-481. http://geodesic.mathdoc.fr/item/JSFU_2022_15_4_a5/

[1] J.R. Cannon, “The solution of heat equation subject to the specification of energy”, Quarterly of Applied Math., 21:2 (1963), 155–160

[2] L.I. Kamynin, “A boundary-value problem in the theory of heat conduction with a nonclassical boundary condition”, Zh. Vychisl. Mat. Mat. Fiz., 4:6 (1964), 1006–1024 (in Russian)

[3] N.I. Ionkin, “The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition”, Differential Equations, 13:2 (1977), 294–304 (in Russian)

[4] L.A. Muravei, A.V. Filinovskii, “On the non-local boundary-value problem for a parabolic equation”, Mathematical Notes, 54:4 (1993), 98–116 (in Russian)

[5] A.M. Nakhushev, Equations of Mathematical Biology, Vysshaya Shkola, M., 1995 (in Russian)

[6] A.B. Golovanchikov, I.E. Simonova, B.V. Simonov, “The solution of diffusion problem with integral boundary condition”, Fundam. Prikl. Mat., 7:2 (2001), 339–349 (in Russian)

[7] T.K. Yuldashev, “On an optimal control of inverse thermal processes with an integral condition of redefinition”, Vestnik TvGU. Seriya: Prikladnaya matematika, 2019, no. 4, 65–87 (in Russian)

[8] T.K. Yuldashev, “Nonlinear Optimal Control of Thermal Processes in Nonlinear Inverse Problem”, Lobachevskii Journal of Mathematics, 41:1 (2020), 124–136

[9] L.S. Pulkina, “A Mixed Problem with Integral Condition for the Hyperbolic Equation”, Mathematical Notes, 74:3 (2003), 435–445 (in Russian)

[10] A.I. Kozhanov, L.S. Pulkina, “On the solvability of some boundary value problems with displacement for linear hyperbolic equations”, Matematicheskiy zhurnal, 9:2 (2009), 78–92 (in Russian)

[11] L.S. Pulkina, A.E. Savenkova, “A problem with nonlocal integral condition of the second kind for one-dimensional hyperbolic equation”, Vestn. Samar. Gos. Techn. Un-ta. Ser. Fiz.-mat. nauki (J. Samara State Tech. Univ., Ser. Phys. and Math. Sci., 20:2 (2016), 276–289 (in Russian) | DOI

[12] Y.T. Mehraliyev, “On the identification of a linear source for the second order elliptic equation with integral condition”, Trudy Instituta Matematiki, 21:2 (2013), 128–141 (in Russian)

[13] A.K. Urinov, Sh.T. Nishonova, “Problems with integral condition for equations of elliptic type”, Uzbek Mathematical Journal, 2016, no. 2, 124–132 (in Russian)

[14] Z.A. Nakhusheva, Nonlocal boundary value problems for basic and mixed types of differential equations, Nalchik, 2011 (in Russian)

[15] A. Friedman, “Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions”, Quart. Appl. Math., 44:3 (1986), 401–407

[16] T. Sh.Kalmenov, D. Suragan, “To spectral problems for the volume potential”, Dokl. Ross. Akad. Nauk, 428:1 (2009), 16–19 (in Russian)

[17] A.I. Kozhanov, “On the solvability of a boundary-value problem with a non-local boundary condition for linear parabolic equations”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki (J. Samara State Tech. Univ., Ser. Phys. Math. Sci., 30 (2004), 63–69 (in Russian)

[18] A.I. Kozhanov, L.S. Pulkina, “On the solvability of boundary value problems with a nonlocal boundary condition of integral form for multidimensional hyperbolic equations”, Differential equations, 42:9 (2006), 1166–1179 (in Russian)

[19] L.S. Pulkina, A.E. Savenkova, “A problem with a nonlocal with respect to time condition for multidimensional hyperbolic equations”, Russian Math. (Iz. VUZ), 2016, no. 10, 41–52 (in Russian)

[20] A.M. Abdrakhmanov, A.I. Kozhanov, “A problem with a nonlocal boundary condition for one class of odd-order equations”, Russian Math. (Iz. VUZ), 2007, no. 5, 3–12 (in Russian)

[21] O.S. Zikirov, “On a problem with integral conditions for a third-order equation”, Uzbek Mathematical Journal, 2006, no. 4, 26–31 (in Russian)

[22] O.S. Zikirov, D.K. Kholikov, “Mixed problem with an integral condition for third-order equations”, Matematicheskiye zametki SVFU, 21:2 (2014), 22–30 (in Russian)

[23] O.S. Zikirov, M.M. Sagdullayeva, “Solvability of a non-local problem for a third-order equation with the heat operator in the main part”, Vestnik KRAUNTS. Fiziko-matematicheskiye nauki, 30:1 (2020), 20–30 (in Russian)

[24] Ya.T. Mehraliev, U.S. Alizade, “On the problem of identifying a linear source for the third-order hyperbolic equation with integral condition”, Problemy fiziki, matematiki i tekhniki, 40:3 (2019), 80–87 (in Russian)

[25] T.K. Yuldashev, “Nonlinear integro-differential equation of pseudoparabolic type with nonlocal integral condition”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1: Fizika. Matematika, 32:1 (2016), 11–23 (in Russian)

[26] A.K. Urinov, Sh.T. Nishonova, “A problem with integral conditions for an elliptic-parabolic equation”, Mathematical Notes, 102:1 (2017), 68–80

[27] Yu.K. Sabitova, “Boundary-value problem with nonlocal integral condition for mixed-type equations with degeneracy on the transition line”, Mathematical Notes, 98:3 (2015), 393–406 (in Russian)

[28] K.B. Sabitov, “Boundary Value Problem for a Parabolic-Hyperbolic Equation with a Nonlocal Integral Condition”, Differential equations, 46:10 (2010), 1468–1478

[29] A.K. Urinov, A.O. Mamanazarov, “Problems with an integral condition for a parabolic-hyperbolic equation with a non-characteristic line of change of type”, Vestnik Natsional'nogo Universiteta Uzbekistana, 2017, no. 2/2, 227–238 (in Russian)

[30] A.K. Urinov, A.O. Mamanazarov, “A problem with integral condition for a parabolic-hyperbolic equation with non-characteristic line of type changing”, Contemp. Anal. Appl. Math., 3:2 (2015), 170–183

[31] A.K. Urinov, K.S. Khalilov, “Some nonclassial problems for a class parabolic-hyperbolic equations”, Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi Akademii Nauk, 16:4 (2014), 42–49 (in Russian)

[32] A.K. Urinov, K.S. Khalilov, “Problems with non-local conditions for parabolic-hyperbolic equations”, Doklady Akademii Nauk Respubliki Uzbekistan, 2014, no. 5, 8–10 (in Russian)

[33] K.S. Khalilov, “Problems with integral conditions for a third order of parabolic-hyperbolic equation”, Bulletin of the Institute of Mathematics, 4:1 (2021), 71–81 (in Russian)

[34] Q.S. Khalilov, Lobachevskii Journal of Mathematics, 42:6 (2021), 1274–1285 | DOI

[35] M.M. Smirnov, Mixed type equation, Vysshaya Shkola, M., 1985 (in Russian)

[36] M.S. Salakhitdinov, A.K. Urinov, “On properties of some operators of Volterra type”, Dokl. Akad. Nauk UzSSR, 1988, no. 4, 3–5 (in Russian)

[37] G.N. Watson, Theory of Bessel functions, Cambridge, London, 1966

[38] G. Bateman, A. Erdelyi, Higher transcendental functions. Hypergeometric function. Legendre functions. Math reference library, Nauka, M., 1965 (in Russian)

[39] M.S. Salakhitdinov, A.K. Urinov, Boundary value problems for the mixed type equations with spectral parameter, Fan, Tashkent, 1997 (in Russian)

[40] M.S. Salakhitdinov, T.G. Ergashev, “Properties of some operators of fractional integro-differentiation with the Bessel function in kernels”, Dokl. Akad. Nauk. Rep. Uzbekistan, 1992, no. 10–11, 3–6 (in Russian)

[41] M.B. Kapilevich, “On an equation of mixed elliptic-hyperbolic type”, Matematicheskii sbornik, 30(72):1 (1952), 11–38 (in Russian)

[42] T.D. Dzhuraev, A. Sopuev, M. Mamazhanov, Boundary value problems for equations of parabologic-hyperbolic type, Fan, Tashkent, 1986 (in Russian)

[43] T.D. Dzhuraev, Boundary value problems for equations of mixed and mixed-composite type, Fan, Tashkent, 1979 (in Russian)

[44] V.I. Smirnov, A Course of Higher Mathematics, part 2, v. IV, Nauka, M. (in Russian)

[45] S.G. Mikhlin, Lectures on linear integral equations, Fizmatgiz, M., 1959 (in Russian)