Abstract
In this article, a non-classical problem with an integral condition for a parabolic-hyperbolic equation of the third order, has been formulated and investigated. By the method of integral equations, the unique solvability of the considered problem has been proved. The problem is equivalently reduced to a problem for a second order parabolic-hyperbolic equation with unknown right-hand side. The formulas of the Cauchy problem for telegraph equation and the solution of the first boundary value problem for an inhomogeneous parabolic equation were used.
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REFERENCES
P. Ya. Polubarinova-Kochina, Theory of Groundwater Movement (Nauka, Moscow, 1977) [in Russian].
O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of Nonlinear Acoustics (Nauka, Moscow, 1975) [in Russian].
T. D. Dzhuraev, Boundary Value Problems for Equations of Mixed and Mixed-Composite Type (Fan, Tashkent, 1979) [in Russian].
M. S. Salakhitdinov, Equations of Mixed-Composite Type (Fan, Tashkent, 1974) [in Russian].
N. I. Ionkin, ‘‘The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition,’’ Differ. Uravn. 13, 294–304 (1977).
A. A. Samarskiy, ‘‘On some problems of the theory of differential equations,’’ Differ. Uravn. 16, 1925–1935 (1980).
A. M. Nakhushev, ‘‘An approximate method for solving boundary value problems for differential equations and its application to the dynamics of soil moisture and groundwater,’’ Differ. Uravn. 18, 72–81 (1982).
A. B. Golovanchikov, I. E. Simonova, and B. V. Simonov, ‘‘The solution of diffusion problem with integral boundary condition,’’ Fundam. Prikl. Mat. 7, 339–349 (2001).
T. K. Yuldashev, ‘‘On an optimal control of inverse thermal processes with an integral condition of redefinition,’’ Vestn. TvGU, Ser.: Prikl. Mat., No. 4, 65–87 (2019).
T. K. Yuldashev, ‘‘Nonlinear optimal control of thermal processes in nonlinear inverse problem,’’ Lobachevskii J. Math. 41 (1), 124–136 (2020).
L. S. Pulkina and A. E. Savenkova, ‘‘A problem with second kind integral conditions for hyperbolic equation,’’ Vestn. Samar. Univ., Estestvennonauch. Ser., Nos. 1–2, 33–45 (2016).
Z. A. Nakhusheva, Nonlocal Boundary Value Problems for Basic and Mixed Types of Differential Equations (Kab.-Balk. Nauch. Tsentr RAN, Nalchik, 2011) [in Russian].
K. B. Sabitov, ‘‘Boundary value problem for equation of parabolic-hyperbolic type with nonlocal integral condition,’’ Differ. Equat. 46, 1472–1481 (2010).
Yu. K. Sabitova, ‘‘Boundary-value problem with nonlocal integral condition for mixed-type equations with degeneracy on the transition line,’’ Math. Notes 98, 454–465 (2015).
A. K. Urinov and Sh. T. Nishonova, ‘‘A problem with integral conditions for an elliptic-parabolic equation,’’ Math. Notes 102, 68–80 (2017).
A. K. Urinov and A. O. Mamanazarov, ‘‘Problems with an integral condition for a parabolic-hyperbolic equation with a non-characteristic line of change of type,’’ Vestn. Nats. Univ. Uzbek., No. 2/2, 227–238 (2017).
A. K. Urinov and K. S. Khalilov, ‘‘On a nonlocal problem for parabolic-hyperbolic equations,’’ Dokl. Adyg. (Cherkessk.) Mezhdun. Akad. Nauk 15 (1), 24–30 (2013).
A. K. Urinov and K. S. Khalilov, ‘‘Some non-classical problems for one class of parabolic-hyperbolic equation,’’ Dokl. Adyg. (Cherkessk.) Mezhdun. Akad. Nauk 16 (4), 42–49 (2014).
A. K. Urinov and K. S. Khalilov, ‘‘Non-local problems for of parabolic-hyperbolic equation with a integral boundary condition,’’ Dokl. Akad. Nauk Resp. Uzb., No. 2, 6–9 (2014).
O. S. Zikirov, ‘‘On a problem with integral conditions for a third-order equation,’’ Uzb. Mat. Zh., No. 2, 26–31 (2006).
G. A. Lukina, ‘‘Boundary value problems with integral conditions for the linearized Korteweg - de Vries equation,’’ Vestn. Yuzh.-Ural. Univ., Ser.: Mat. Model. Program. 8 (17), 52–61 (2011).
O. S. Zikirov and D. K. Kholikov, ‘‘Mixed problem with an integral condition for third-order equations,’’ Mat. Zam. SVFU 21 (2), 22–30 (2014).
T. K. Yuldashev, ‘‘Nonlinear integro-differential equation of pseudoparabolic type with nonlocal integral condition,’’ Vestn. Volgogr. Univ., Ser. 1: Fiz. Mat. 32 (1), 11–23 (2016).
T. K. Yuldashev, ‘‘On Fredholm partial integro-differential equation of the third order,’’ Russ. Math. 59 (9), 62–66 (2015).
T. K. Yuldashev, ‘‘An inverse problem for a third-order Fredholm integro-differential equation with a degenerate kernel,’’ Vladik. Mat. Zh. 18 (2), 76–85 (2016).
T. K. Yuldashev, ‘‘Mixed problem for pseudoparabolic integrodifferential equation with degenerate kernel,’’ Differ. Equat. 53, 99–108 (2017).
T. K. Yuldashev, B. I. Islomov, and E. K. Alikulov, ‘‘Boundary-value problems for loaded third-order parabolic-hyperbolic equations in infinite three-dimensional domains,’’ Lobachevskii J. Math. 41 (5), 926–944 (2020).
Ya. T. Mehraliev and U. S. Alizade, ‘‘On the problem of identifying a linear source for the third-order hyperbolic equation with integral condition,’’ Probl. Fiz., Mat. Tekh. 40 (3), 80–87 (2019).
A. I. Kozhanov and A. V. Dyuzheva, ‘‘Non-local problems with an integral condition for third-order differential equations,’’ Vestn. Samar. Tekh. Univ., Ser.: Fiz.-Mat. Nauki 24, 607–620 (2020).
O. S. Zikirov and M. M. Sagdullayeva, ‘‘Solvability of a non-local problem for a third-order equation with the heat operator in the main part,’’ Vestn. KRAUNTS, Fiz.-Mat. Nauki 30 (1), 20–30 (2020).
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1977) [in Russian].
G. N. Watson, Theory of Bessel functions (Cambridge, London, 1966).
M. S. Salakhitdinov and A. K. Urinov, Boundary Value Problems for the Mixed Type Equations with Spectral Parameter (Fan, Tashkent, 1997) [in Russian].
V. I. Smirnov, Mixed Type Equation (Vysshaya Shkola, Moscow, 1981), Vol. 4 [in Russian].
A. M. Nakhushev, Equations of Mathematical Biology (Vysshaya Shkola, Moscow, 1995) [in Russian].
S. G. Mikhlin, Lectures on Linear Integral Equations (Fizmatgiz, Moscow, 1959) [in Russian].
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Khalilov, Q.S. A Nonlocal Problem for a Third Order Parabolic-Hyperbolic Equation with a Spectral Parameter. Lobachevskii J Math 42, 1274–1285 (2021). https://doi.org/10.1134/S1995080221060123
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DOI: https://doi.org/10.1134/S1995080221060123