ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## Numerical Methods for Solving the Second Boundary Value Problem for a Multidimensional Sobolev Type Equation

### Author(s):

M. KH. Beshtokov

Department of Computational Methods,
Institute of Applied Mathematics and Automation, KBSC RAS
Address: Shortanova 89A, Nalchik, 360000, Russia
North-Caucasus Federal University,
North-Caucasus Center for Mathematical Research,
Address: Pushkin 1, Stavropol, 355017, Russia

beshtokov-murat@yandex.ru

### Abstract:

The second boundary value problem is investigated for a multidimensional Sobolev-type differential equation with variable coefficients. The considered equation is reduced to an integro-differential equation of parabolic type with a small parameter. For an approximate solution of the obtained problem, a locally one-dimensional difference scheme is constructed. Using the method of energy inequalities, an a priori estimate is obtained for the solution of a locally one-dimensional difference scheme, which implies its stability and convergence. For a two-dimensional problem, an algorithm is constructed for the numerical solution of the second boundary value problem for a partial differential equation of Sobolev type.

### Keywords

• a priori estimate
• Boundary value problems
• convergence
• integro-differential equation
• locally one-dimensional scheme
• Sobolev type differential equation
• stability

### References:

1. Barenblatt G. I., Yentov V. M., Ryzhik V. M. Dvizheniye zhidkostey i gazov v prirodnykh plastakh [The movement of liquids and gases in natural formations], Moscow: Nedra, 1984. 211 p
2. Shkhanukov M. KH. [On some third-order boundary value problems arising in the simulation of fluid filtration in porous media] Differents. uravnen, 18:4 (1982), 689-700. (In Russian)
3. van Duijn C. J., Cuesta C., Hulshof J. Infiltration in Porous Media with Dynamic Capillary Pressure: Travelling Waves // European Journal of Anaesthesiology, 11 (2000), 381-397
4. Chudnovskiy A. F. Teplofizika pochv [Thermal physics of soils], Moscow: Nauka, 1976. 353 p
5. Hallaire M. Le potentiel efficace de l'eau dans le sol en regime de dessechement // L 'Eau et la Production Vegetale. Paris: Institut National de la Recherche Agronomique, 9 (1964), 27-62
6. Colton D. L. On the analytic theory of pseudoparabolic equations // Quart. J. Math., 23 (1972), 179-192
7. Dzektser Ye. S. [The equations of motion of groundwater with a free surface in multilayer environments] DAN SSSR, 220:3 (1975), 540-543. (In Russian)
8. Chen P. J., Curtin M. E. On a theory of heat conduction involving two temperatures // Journal of Applied Mathematics and Physics (ZAMP). 19 (1968), 614-627
9. Ting T. W. Certain non-steady flows of second-order fluids // Archive for Rational Mechanics and Analysis, 14 (1963), 1-26
10. Ivanova M. V., Ushakov V. I. The second boundary-value problem for pseudoparabolic equations in noncylindrical domains // Mathematical Notes, 72:1 (2002), 43-47
11. Ablabekov B. S., Mukanbetova A. T. [On the solvability of solutions of the second initial-boundary value problem for a pseudoparabolic equation with a small parameter] // Nauka, novyye tekhnologii i innovatsii Kyrgyzstana, 3 (2019), 41-47. (In Russian)
12. Fedorenko R. P. The speed of convergence of one iterative process // USSR Computational Mathematics and Mathematical Physics, 4:3 (1964), 227-235
13. Bakhvalov N. S. On the convergence of a relaxation method with natural constraints on the elliptic operator // USSR Computational Mathematics and Mathematical Physics, 6:5 (1966), 101-135
14. Brandt A. Multi-level adaptive solutions to boundary value problems // Math. Comput. 31 (1977), 333-390
15. Zenkevich O. Metod konechnykh elementov v tekhnike [Finite element method in engineering]. Moscow: Mir, 1975, 271 p
16. Gallagher R. Metod konechnykh elementov. Osnovy. [Finite Element Method. Basics]. Moscow: Mir, 1984, 428 p
17. Olshanskii M. A., An analysis of the multigtrid method for the convection-diffusion equations with the Dirichlet boundary conditions // Comput. Math. Math. Phys., 44:8 (2004), 1374-1403
18. Olshanskii M. A., Reusken A. Convergence analysis of a multigrid solver for a finite element method applied to convection-dominated model problem // SIAM J. Num. Anal. 43 (2004), 1261-1291
19. Beshtokov M. Kh. Finite-difference method for a nonlocal boundary value problem for a third-order pseudoparabolic equation // Diff. Eq., 49:9 (2013), 1170-1177
20. Beshtokov M. Kh. A numerical method for solving one nonlocal boundary value problem for a third-order hyperbolic equation // Computational Mathematics and Mathematical Physics, 54:9 (2014), 1441-1458
21. Beshtokov M. Kh. Difference method for solving a nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coefficients // Comput. Math. Math. Phys., 56:10 (2016), 1763-1777
22. Beshtokov M. KH Differential and difference boundary value problem for loaded third-order pseudo-parabolic differential equations and difference methods for their numerical solution // Comput. Math. Math. Phys., 57:12 (2017), 1973-1993
23. Beshtokov M. KH. Krayevyye zadachi dlya nagruzhennykh psevdoparabolicheskikh uravneniy drobnogo poryadka i raznostnyye metody ikh resheniya [Boundary value problems for loaded pseudoparabolic equations of fractional order and difference methods for their solution] // Izvestiya vuzov. Matematika, 2 (2019), 3-12
24. Vishik M. I., Lyusternik L. A. [Regular degeneration and boundary layer for linear differential equations with small parameter] // Uspekhi Mat. Sciences, 1967, Т. 12, №5, s. 3-122. (In Russ. )
25. Godunov S. K., Ryaben'kiy V. S. Raznostnyye skhemy [Difference schemes], Moscow: Nauka, 1977. 440 p
26. Samarskii A. A. Teoriya raznostnykh skhem [Theory of difference schemes], Moscow: Nauka, 1983. 656 p
27. Andreev V. B. On the convergence of difference schemes approximating the second and third boundary value problems for elliptic equations // USSR Computational Mathematics and Mathematical Physics, 8:6 (1968), 44-62