Abstract: In this work, we study the pseudoparabolic equation in the three dimensional space. The equation of this form implies the presence of cylindrical or spherical symmetry that enables one to move from a three-dimensional problem to one-dimensional problem, but with degeneration. In this regard, we study the solvability and stability of solutions to boundary value problems for degenerate pseudoparabolic equation of the third order of general form with variable coefficients and third kind condition, as well as difference schemes approximating this problem on uniform grids. The main result consists in proving a priori estimates for a solution to both the differential and difference problems by means of the method of energy inequalities. The obtained inequalities imply stability of the solution relative to initial data and right side. Because of the linearity of the considered problems these inequalities allow us to state the convergence of the approximate solution to the exact solution of the considered differential problem under the assumption of the existence of the solutions in the class of sufficiently smooth functions. On the test examples the numerical experiments are performed confirming the theoretical results obtained in the work.

Keywords: equation with degeneration, boundary value problem, condition of the third kind, a priori estimate, difference scheme, stability and convergence of a difference scheme, moisture transfer equation, pseudo-parabolic equation

For citation: Beshtokov M. KH., Kanchukoyev V. Z., Erzhibova F. A. A Boundary Value Problem for a degenerate
Moisture Transfer Equation with a Condition of the Third Kind // Vladikavkazskii matematicheskii
zhurnal [Vladikavkaz Math. J.], 2017, vol. 19, no. 1, pp. 13-26. DOI 10.23671/VNC.2018.4.9164

1. Dzektser E. S. Equations of motion of free-surface underground
water in layered media. Doklady Akademii Nauk SSSR [Doklady
Mathematics], 1975, vol. 220, no. 3, pp. 540-543 (in Russian).

2. Rubinshtein L. I. On heat propagation in heterogeneous media.
Izv. Akad. Nauk SSSR, Ser. Geogr. , 1948, vol. 12, no. 1,
pp. 27-45 (in Russian).

3. Ting T. W. A cooling process according to two-temperature theory
of heat conduction. J. Math. Anal. Appl. , 1974, vol. 45, no. 1,
pp. 23-31.

4. Hallaire M. L'eau et la production vegetable. Inst. National de
la Recherche Agronomique , 1964, no. 9.

5. Chudnovskii A. F. Teplofizika pochv [Thermal Physics of Soils].
Moscow, Nauka, 1976, 352 p. (in Russian).

6. Barenblat G. I., Zheltov Yu. P., and Kochina I. N. Basic concept in
the theory of seepage of homogeneous liquids in fissured rocks.
Prikladnaya matematika i mechanika [J. Appl. Math.
Mech.], 1960, vol. 25, no. 5, pp. 852-864 (in Russian).

7. Beshtokov M. Kh. Riemann function method and finite difference
method for solving a nonlocal boundary value problem for a
third-order hyperbolic equation. Izvestiya Vysshikh Uchebnykh
Zavedenii. Severo-Kavkaz. Reg. , 2007, no. 5, pp. 6-9 (in Russian).

8. Beshtokov M. Kh. Finite-difference method for a nonlocal boundary
value problem for a third-order pseudoparabolic equation.
Differ. Equations , 2013, vol. 49, no. 9, pp. 1134-141.
DOI: 10.1134/S0012266113090085.

9. Beshtokov M. Kh. On a boundary value problem for a third-order
pseudoparabolic equation with a nonlocal condition. I Izvestiya
Vysshikh Uchebnykh Zavedenii. Severo-Kavkaz. Reg. , 2013, no. 1,
pp. 5-10 (in Russian).

10. Beshtokov M. Kh. A numerical method for solving one nonlocal
boundary value problem for a third-order hyperbolic equation.
Comput. Math. Math. Phys. , 2014, vol. 54, no. 9, pp.
1441-1458. DOI: 10.7868/S0044466914090051.

11. Samarskii A. A. Teoriya raznostnih shem [Theory of Difference
Schemes]. Moscow, Nauka, 1983, 616 p. (in Russian).

12. Olisaev E. G. Raznostnye Metody Reshenija Nelokal'Nyh Kraevyh
Zadach Dlja Uravnenija Parabolicheskogo Tipa s Vyrozhdeniem .
Candidate's Dis. In Math. And Physics. Moscow, Russ. State Library,
2003 (In Russian).

13. Ladyzhenskaya O. A. Kraevie zadachi matematicheskoi fiziki
[Boundary Value Problems of Mathematical Physics]. Moscow, Nauka,
1973, 408 p. (in Russian).

14. Andreev V. B. On the convergence of difference schemes
approximating the second and third boundary value problems for
elliptic equations. USSR Comput. Math. Math. Phys. , 1968, vol. 8,
no. 6, pp. 44-62. DOI: 10.1016/0041-5553(68)90092-X.

15. Samarskii A. A., Gulin A. V. Ustoichivost raznostnih shem
[Stability of Finite Difference Schemes]. Moscow, Nauka, 1973, 416
p. (in Russian).