Section
|
Mathematics
|
Title
|
A numerical method for solving the second initial-boundary value problem for a multidimensional third-order pseudoparabolic equation
|
Author(-s)
|
Beshtokov M.Kh.a
|
Affiliations
|
Institute of Applied Mathematics and Automation, Kabardino-Balkarian Scientific Center of the Russian Academy of Sciencesa
|
Abstract
|
The work is devoted to the study of the second initial-boundary value problem for a general-form third-order differential equation of pseudoparabolic type with variable coefficients in a multidimensional domain with an arbitrary boundary. In this paper, a multidimensional pseudoparabolic equation is reduced to an integro-differential equation with a small parameter, and a locally one-dimensional difference scheme by A.A. Samarskii is used. Using the maximum principle, an a priori estimate is obtained for the solution of a locally one-dimensional difference scheme in the uniform metric in the $C$ norm. The stability and convergence of the locally one-dimensional difference scheme are proved.
|
Keywords
|
pseudoparabolic equation, moisture transfer equation, locally one-dimensional scheme, stability, convergence of the difference scheme, additivity of the scheme
|
UDC
|
35L35
|
MSC
|
519.63
|
DOI
|
10.35634/vm210303
|
Received
|
11 May 2021
|
Language
|
Russian
|
Citation
|
Beshtokov M.Kh. A numerical method for solving the second initial-boundary value problem for a multidimensional third-order pseudoparabolic equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 3, pp. 384-408.
|
References
|
- Barenblatt G.I., Entov V.M., Ryzhik V.M. Theory of fluid flows through natural rocks, Springer, 1990. https://www.springer.com/gp/book/9780792301677
- Shkhanukov M.Kh. Some boundary value problems for a third-order equation that arise in the modeling of the filtration of a fluid in porous media, Differentsial'nye Uravneniya, 1982, vol. 18, no. 4, pp. 689-699 (in Russian). http://mi.mathnet.ru/eng/de4523
- Cuesta C., van Duijn C.J., Hulshof J. Infiltration in porous media with dynamic capillary pressure: travelling waves, European Journal of Applied Mathematics, 2000, vol. 11, issue 4, pp. 381-397. https://doi.org/10.1017/S0956792599004210
- Chudnovskii A.F. Teplofizika pochv (The thermophysics of soils), Moscow: Nauka, 1976.
- Hallaire M. Le potentiel efficace de l'eau dans le sol en régime de dessèchement, L'Eau et la Production Végétale, Paris: Institut national de la recherche agronomique, 1964, no. 9, pp. 27-62.
- Colton D.L. On the analytic theory of pseudoparabolic equations, The Quarterly Journal of Mathematics, 1972, vol. 23, issue 2, pp. 179-192. https://doi.org/10.1093/qmath/23.2.179
- Dzekcer E.S. Equation of motion of underground water with a free surface in multilayer media, Sov. Phys., Dokl., 1975, vol. 20, pp. 24-26. https://zbmath.org/?q=an:0331.76056
- Chen P.J., Gurtin M.E. On a theory of heat conduction involving two temperatures, Zeitschrift für angewandte Mathematik und Physik ZAMP, 1968, vol. 19, issue 4, pp. 614-627. https://doi.org/10.1007/BF01594969
- Ting T.W. Certain non-steady flows of second-order fluids, Archive for Rational Mechanics and Analysis, 1963, vol. 14, issue 1, pp. 1-26. https://doi.org/10.1007/BF00250690
- Barenblatt G.I., Bertsch M., Dal Passo R., Ughi M. A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM Journal on Mathematical Analysis, 1993, vol. 24, no. 6, pp. 1414-1439. https://doi.org/10.1137/0524082
- Novick-Cohen A., Pego R.L. Stable patterns in a viscous diffusion equation, Transactions of the American Mathematical Society, 1991, vol. 324, no. 1, pp. 331-351. https://doi.org/10.1090/S0002-9947-1991-1015926-7
- Sveshnikov A.A., Al'shin A.B., Korpusov M.O., Pletner Yu.D. Lineinye i nelineinye uravneniya sobolevskogo tipa (Linear and nonlinear Sobolev type equations), Moscow: Fizmatlit, 2007.
- Ivanova M.V., Ushakov V.I. The second boundary-value problem for pseudoparabolic equations in noncylindrical domains, Mathematical Notes, 2002, vol. 72, no. 1, pp. 43-47. https://doi.org/10.1023/A:1019812920385
- Vabishchevich P.N. On a new class of additive (splitting) operator-difference schemes, Mathematics of Computation, 2012, vol. 81, no. 277, pp. 267-276. https://www.jstor.org/stable/23075227
- Vabishchevich P.N., Grigor'ev A.V. Splitting schemes for pseudoparabolic equations, Differential Equations, 2013, vol. 49, no. 7, pp. 807-814. https://doi.org/10.1134/S0012266113070033
- Čiegis R., Tumanova N. On construction and analysis of finite difference schemes for pseudoparabolic problems with nonlocal boundary conditions, Mathematical Modelling and Analysis, 2014, vol. 19, no. 2, pp. 281-297. https://doi.org/10.3846/13926292.2014.910562
- Čiegis R., Suboč O., Bugajev A. Parallel algorithms for three-dimensional parabolic and pseudoparabolic problems with different boundary conditions, Nonlinear Analysis: Modelling and Control, 2014, vol. 19, no. 3, pp. 382-395. https://doi.org/10.15388/NA.2014.3.5
- Ablabekov B.S., Baiserkeeva A.B. Explicit solution Cauchy problem for two-dimensional pseudo-parabolic equations, Izvestiya Vuzov Kyrgyzstana, 2015, no. 10, pp. 3-7 (in Russian). https://www.elibrary.ru/item.asp?id=28766897
- Ablabekov B.S., Mukanbetova A.T. On solvability of solutions of the second initial-boundary problem for pseudoparabolic equations with a small parameter, Nauka, Novye Tekhnologii i Innovatsii Kyrgyzstana, 2019, no. 3, pp. 41-47 (in Russian). http://science-journal.kg/en/journal/1/archive/12271
- Shivanian E., Aslefallah M. Stability and convergence of spectral radial point interpolation method locally applied on two-dimensional pseudoparabolic equation, Numerical Methods for Partial Differential Equations, 2017, vol. 33, no. 3, pp. 724-741. https://doi.org/10.1002/num.22119
- Aslefallah M., Abbasbandy S., Shivanian E. Meshless singular boundary method for two-dimensional pseudo-parabolic equation: analysis of stability and convergence, Journal of Applied Mathematics and Computing, 2020, vol. 63, issues 1-2, pp. 585-606. https://doi.org/10.1007/s12190-020-01330-x
- Hussain M., Haq S., Ghafoor A. Meshless RBFs method for numerical solutions of two-dimensional high order fractional Sobolev equations, Computers and Mathematics with Applications, 2020, vol. 79, no. 3, pp. 802-816. https://doi.org/10.1016/j.camwa.2019.07.033
- Amiraliyev G.M., Cimen E., Amirali I., Cakir M. High-order finite difference technique for delay pseudo-parabolic equations, Journal of Computational and Applied Mathematics, 2017, vol. 321, pp. 1-7. https://doi.org/10.1016/j.cam.2017.02.017
- Jachimavičiené J., Sapagovas M., Štikonas A., Štikoniené O. On the stability of explicit finite difference schemes for a pseudoparabolic equation with nonlocal conditions, Nonlinear Analysis: Modelling and Control, 2014, vol. 19, no. 2, pp. 225-240. https://doi.org/10.15388/NA.2014.2.6
- Beshtokov M.Kh. Finite-difference method for a nonlocal boundary value problem for a third-order pseudoparabolic equation, Differential Equations, 2013, vol. 49, no. 9, pp. 1134-1141. https://doi.org/10.1134/S0012266113090085
- Beshtokov M.Kh. Difference method for solving a nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coefficients, Computational Mathematics and Mathematical Physics, 2016, vol. 56, no. 10, pp. 1763-1777. https://doi.org/10.1134/S0965542516100043
- Beshtokov M.Kh. Differential and difference boundary value problem for loaded third-order pseudo-parabolic differential equations and difference methods for their numerical solution, Computational Mathematics and Mathematical Physics, 2017, vol. 57, no. 12, pp. 1973-1993. https://doi.org/10.1134/S0965542517120089
- Beshtokov M.Kh. Boundary value problems for degenerating and nondegenerating Sobolev-type equations with a nonlocal source in differential and difference forms, Differential Equations, 2018, vol. 54, no. 2, pp. 250-267. https://doi.org/10.1134/S0012266118020118
- Beshtokov M.Kh. Boundary-value problems for loaded pseudoparabolic equations of fractional order and difference methods of their solving, Russian Mathematics, 2019, vol. 63, no. 2, pp. 1-10. https://doi.org/10.3103/S1066369X19020014
- Beshtokov M.Kh. Boundary value problems for a loaded modified fractional-order moisture transfer equation with the Bessel operator and difference methods for their solution, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 2, pp. 158-175. https://doi.org/10.35634/vm200202
- Mesloub S., Bachar I. On a nonlocal 1-D initial value problem for a singular fractional-order parabolic equation with Bessel operator, Advances in Difference Equations, 2019, vol. 2019, issue 1, article number: 254. https://doi.org/10.1186/s13662-019-2196-z
- Luc N.H., Jafari H., Kumam P., Tuan N.H. On an initial value problem for time fractional pseudo-parabolic equation with Caputo derivative, Mathematical Methods in the Applied Sciences, 2021. https://doi.org/10.1002/mma.7204
- Samarskii A.A. The theory of difference schemes, New York: CRC Press, 2001. https://doi.org/10.1201/9780203908518
- Vishik M.I., Lyusternik L.A. Regular degeneration and boundary layer for linear differential equations with small parameter, Uspekhi Matematicheskikh Nauk, 1957, vol. 12, issue 5 (77), pp. 3-122 (in Russian). http://mi.mathnet.ru/eng/umn7705
- Godunov S.K., Ryaben'kii V.S. Raznostnye skhemy (Difference schemes), Moscow: Nauka, 1977.
- Samarskii A.A., Gulin A.V. Ustoichivost' raznostnykh skhem (Stability of difference schemes), Moscow: Nauka, 1973.
|
Full text
|
|