Abstract
We consider the initial-boundary value problem for a third-order partial differential equation with highest mixed derivative. An abstract Cauchy problem for a first-order algebraic-differential equation in a Banach space with a distinguished time variable is solved. It is proved that, by equivalent replacements, the original problem is reduced to the Cauchy problem for an algebraic-differential equations. To solve the stated problem, the Fredholm property of the operator before the highest derivative is used. Conditions under which the solution of the problem exists and is unique are determined, and this solution is found in analytical form.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. I. Barenblat, Yu. P. Zheltov, and I. N. Kochina, “On the main ideas of the theory of filtration in fractured media,” Prikl. Mat. Mekh. 24 (5), 58–73 (1960).
A. F. Chudnovskii, Thermal Physics of Soils (Nauka, Moscow, 1976) [in Russian].
V. L. Ginzburg and A. A. Rukhadze, Waves in Magnetically Active Plasma (Nauka, Moscow, 1975) [in Russian].
A. G. Sveshnikov, A. B. Al’shin, M. O. Korpusov, and Yu. D. Pletner, Linear and Nonlinear Equations of Sobolev Type (Fizmatlit, Moscow, 2007) [in Russian].
A. F. Chudnovskii, “Influence of tillage on its thermal regime,” Sb. Trudov AgroFiz., No. 23, 55–63 (1969).
M. Kh. Shkhanukov, “Some boundary value problems for a third-order equation that arise in the modeling of the filtration of a fluid in porous media,” Differ. Uravn. 18 (4), 689–699 (1982).
M. Kh. Beshtokov, “The Riemann function method and the difference method for solving a nonlocal boundary-value problem for a third-order equation of hyperbolic type,” Izv. Vuzov. Severo-Kavkaz. Region, No. 5, 6–9 (2007).
S. P. Zubova and V. I. Uskov, “Solution of a problem for a Sobolev-type third-order equation by the cascade-decomposition method,” Vestn. PMM 11, 70–83 (2015).
A. A. Shcheglova, “Study and solution of degenerate systems of ordinary differential equations by means of a change of variables,” Siberian Math. J. 36 (6), 1247–1256 (1995).
G. A. Sviridyuk and V. E. Fedorov, “Semigroups of operators with kernels,” Vestn. Chelyabinsk Gos. Univ., No. 6, 42–70 (2002).
S. P. Zubova, “On the solvability of the Cauchy problem for a descriptor of pseudoregular equations in Banach space,” Vestn. Voronezh Gos. Univ. Ser. Phys. Mat., No. 2, 192–198 (2013).
S. P. Zubova, E. V. Raetskaya, and V. I. Uskov, “Degeneracy property of a matrix-differential operator and applications,” J. Math. Sci. (N. Y.) 255 (5), 640–652 (2021).
V. I. Uskov, “Initial-boundary value problem for perturbed third order partial differential equations,” J. Math. Sci. (N. Y.) 255 (6), 779–789 (2021).
S. G. Krein, Linear Differential Equations in Banach Space (Nauka, Moscow, 1967) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematicheskie Zametki, 2022, Vol. 111, pp. 895–903 https://doi.org/10.4213/mzm13350.
Rights and permissions
About this article
Cite this article
Uskov, V.I. Solution of the Mixed Problem for a Third-Order Partial Differential Equation. Math Notes 111, 932–939 (2022). https://doi.org/10.1134/S0001434622050273
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434622050273