Abstract
Boundary-value problems for a third-order hyperbolic equation that is degenerate inside a domain and contains the Aller operator in the principal part, are examined. The existence and uniqueness theorem for solutions of the problem is proved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Zh. A. Balkizov, “The first boundary-value problem for a hyperbolic equation degenerate inside a domain,” Vladikavkaz. Mat. Zh., 18, No. 2, 19–30 (2016).
Zh. A. Balkizov, “Boundary-value problem for a hyperbolic equation that is degenerate inside a domain,” Izv. Vyssh. Ucheb. Zaved. Sev.-Kav. Region. Estestv. Nauki, No. 1, 5–10 (2016).
G. I. Barenblatt, Iu. P. Zheltov, and I. N. Kochina, “Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata],” Prikl. Mat. Mekh., 24, No. 5, 852–864 (1960).
A. F. Chudnovsky, Soil Thermophysics [in Russian], Nauka, Moscow (1976).
B. D. Coleman, R. J. Duffin, V. J. Mizel, “Instability, uniqueness, and nonexistence theorems for the equation on a strip,” Arch. Rat. Mech. Anal., 19, 100–116 (1965).
D. Colton, “Pseudoparabolic equations in one space variable,” J. Differ. Equ., 12, No. 3, 559–565 (1972).
G. M. Fichtenholz, Course of Differential and Integral Calculus [in Russian], Vol. 1, Fizmatlit. Moscow (2003).
M. Hallaire, L’eauet la Productions Vegetable, Inst. Natl. Recherche Agronom. (1964).
T. Sh. Kalmenov, “Criterion of the uniqueness of a solution to the Darboux problem for a degenerate hyperbolic equation,” Differ. Uravn., 7, No. 1, 178–181 (1971).
T. Sh. Kalmenov, “On the Darboux problem for a degenerate hyperbolic equation,” Differ. Uravn., 10, No. 1, 59–68 (1974).
R. Kh. Makaova, “The second boundary-value problem for the generalized Aller equation with the fractional Riemann–Liouville derivative,” Dokl. Adyg. Akad. Nauk, 17, No. 3, 35–38 (2015).
A. M. Nakhushev, Fractional Calculus and Its Applications [in Russian], Fizmatlit, Moscow (2003).
A. M. Nakhushev, Loaded Equations and Their Applications [in Russian], Nauka, Moscow (2012).
A. M. Nakhushev, Equations of Mathematical Biology [in Russian], Vysshaya Shkola, Moscow (1995).
A. V. Pskhu, “Initial-value problem for a linear fractional ordinary differential equation,” Mat. Sb., 202, No. 4, 111–122 (2011).
M. Kh. Shkhanukov, “On some boundary-value problems for third-order equations arising from the simulation of fluid filtration in porous media,” Differ. Uravn., 18, No. 4, 689–699 (1982).
R. E. Showalter and T. W. Ting, “Pseudoparabolic partial differential equations,” SIAM J. Math. Anal., 1, No. 1, 1–26 (1970).
M. M. Smirnov, Degenerate Hyperbolic Equations [in Russian], Vysheishaya Shkola, Minsk (1977).
V. A. Vodakhova, “Boundary-value problem with the Nakhushev nonlocal condition for a pseudoparabolic moisture transfer equation,” Differ. Uravn., 18, No. 2, 280–285 (1982).
E. M. Wright, “The generalized Bessel function of order greater than one,” Quart. J. Math. Oxford Ser., 11, 36–48 (1940).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 149, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, 2018.
Rights and permissions
About this article
Cite this article
Makaova, R.K. Boundary-Value Problem for a Third-Order Hyperbolic Equation that is Degenerate Inside a Domain and Contains the Aller Operator in the Principal Part. J Math Sci 250, 780–787 (2020). https://doi.org/10.1007/s10958-020-05043-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-05043-1