A numerical method for solving the second initial-boundary value problem for a multidimensional third-order pseudoparabolic equation

M. Kh. Beshtokov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 31 (2021) no. 3, pp. 384-408

Résumé

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.

Détail

Mots-clés : pseudoparabolic equation
Keywords: moisture transfer equation, locally one-dimensional scheme, stability, convergence of the difference scheme, additivity of the scheme.

Bibliographie

[1] 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

[2] 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, 18:4 (1982), 689–699 (in Russian) | Zbl

[3] Cuesta C., van Duijn C. J., Hulshof J., “Infiltration in porous media with dynamic capillary pressure: travelling waves”, European Journal of Applied Mathematics, 11:4 (2000), 381–397 | DOI | Zbl

[4] Chudnovskii A. F., The thermophysics of soils, Nauka, M., 1976

[5] Hallaire M., “Le potentiel efficace de l'eau dans le sol en régime de dessèchement”, L'Eau et la Production Végétale, 9, Institut national de la recherche agronomique, Paris, 1964, 27–62

[6] Colton D. L., “On the analytic theory of pseudoparabolic equations”, The Quarterly Journal of Mathematics, 23:2 (1972), 179–192 | DOI | Zbl

[7] Dzekcer E. S., “Equation of motion of underground water with a free surface in multilayer media”, Sov. Phys., Dokl., 20 (1975), 24–26 | Zbl

[8] Chen P. J., Gurtin M. E., “On a theory of heat conduction involving two temperatures”, Zeitschrift f{ü}r angewandte Mathematik und Physik ZAMP, 19:4 (1968), 614–627 | DOI | Zbl

[9] Ting T. W., “Certain non-steady flows of second-order fluids”, Archive for Rational Mechanics and Analysis, 14:1 (1963), 1–26 | DOI | Zbl

[10] 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, 24:6 (1993), 1414–1439 | DOI | Zbl

[11] Novick-Cohen A., Pego R. L., “Stable patterns in a viscous diffusion equation”, Transactions of the American Mathematical Society, 324:1 (1991), 331–351 | DOI | Zbl

[12] Sveshnikov A. A., Al'shin A. B., Korpusov M. O., Pletner Yu. D., Linear and nonlinear Sobolev type equations, Fizmatlit, M., 2007

[13] Ivanova M. V., Ushakov V. I., “The second boundary-value problem for pseudoparabolic equations in noncylindrical domains”, Mathematical Notes, 72:1 (2002), 43–47 | DOI | DOI | Zbl

[14] Vabishchevich P. N., “On a new class of additive (splitting) operator-difference schemes”, Mathematics of Computation, 81:277 (2012), 267–276 https://www.jstor.org/stable/23075227 | DOI | Zbl

[15] Vabishchevich P. N., Grigor'ev A. V., “Splitting schemes for pseudoparabolic equations”, Differential Equations, 49:7 (2013), 807–814 | DOI | Zbl

[16] Čiegis R., Tumanova N., “On construction and analysis of finite difference schemes for pseudoparabolic problems with nonlocal boundary conditions”, Mathematical Modelling and Analysis, 19:2 (2014), 281–297 | DOI

[17] Čiegis R., Suboč O., Bugajev A., “Parallel algorithms for three-dimensional parabolic and pseudoparabolic problems with different boundary conditions”, Nonlinear Analysis: Modelling and Control, 19:3 (2014), 382–395 | DOI

[18] Ablabekov B. S., Baiserkeeva A. B., “Explicit solution Cauchy problem for two-dimensional pseudo-parabolic equations”, Izvestiya Vuzov Kyrgyzstana, 2015, no. 10, 3–7 (in Russian)

[19] 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, 41–47 (in Russian)

[20] 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, 33:3 (2017), 724–741 | DOI | Zbl

[21] 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, 63:1–2 (2020), 585–606 | DOI

[22] 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, 79:3 (2020), 802–816 | DOI | Zbl

[23] 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, 321 (2017), 1–7 | DOI | Zbl

[24] 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, 19:2 (2014), 225–240 | DOI | Zbl

[25] Beshtokov M. Kh., “Finite-difference method for a nonlocal boundary value problem for a third-order pseudoparabolic equation”, Differential Equations, 49:9 (2013), 1134–1141 | DOI | Zbl

[26] 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, 56:10 (2016), 1763–1777 | DOI | DOI | Zbl

[27] 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, 57:12 (2017), 1973–1993 | DOI | DOI | Zbl

[28] Beshtokov M. Kh., “Boundary value problems for degenerating and nondegenerating Sobolev-type equations with a nonlocal source in differential and difference forms”, Differential Equations, 54:2 (2018), 250–267 | DOI | DOI | Zbl

[29] Beshtokov M. Kh., “Boundary-value problems for loaded pseudoparabolic equations of fractional order and difference methods of their solving”, Russian Mathematics, 63:2 (2019), 1–10 | DOI | DOI | Zbl

[30] 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, 30:2 (2020), 158–175 | DOI

[31] 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, 2019:1 (2019), 254 | DOI | Zbl

[32] 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 | DOI

[33] Samarskii A. A., The theory of difference schemes, CRC Press, New York, 2001 | DOI

[34] Vishik M. I., Lyusternik L. A., “Regular degeneration and boundary layer for linear differential equations with small parameter”, Uspekhi Matematicheskikh Nauk, 12:5(77) (1957), 3–122 (in Russian) | Zbl

[35] Godunov S. K., Ryaben'kii V. S., Difference schemes, Nauka, M., 1977

[36] Samarskii A. A., Gulin A. V., Stability of difference schemes, Nauka, M., 1973