Skip to main content
SpringerLink
Log in

On the numerical solution of a nonlocal boundary value problem for a degenerating pseudoparabolic equation

  • Numerical Methods
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We study a nonlocal boundary value problem for a degenerating pseudoparabolic third-order equation of the general form. For the solution of the problem, we obtain a priori estimates in differential and difference form, which imply the stability of the solution with respect to the initial data and right-hand side on a layer as well as the convergence of the solution of the difference problem to the solution of the differential problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Samarskii, A.A., Some Problems of the Theory of Differential Equations, Differ. Uravn., 1980, vol. 16, no. 11, pp. 1925–1935.

    MathSciNet  Google Scholar 

  2. Canon, J.R., The Solution of the Heat Equation Subject to the Specification of Energy, Quart. Appl. Math., 1963, vol. 21, no. 2, pp. 155–160.

    MathSciNet  Google Scholar 

  3. Kamynin, L.I., A Boundary Value Problem in the Theory of Heat Conduction with Nonclassical Boundary Conditions, Zh. Vychisl. Mat. Mat. Fiz., 1964, vol. 4, no. 6, pp. 1006–1024.

    MathSciNet  Google Scholar 

  4. Chudnovskii, A.F., Some Corrections in Statement and Solutions of Problems of Heat and Moisture Transport in Soil, in Sb. tr. po agronomicheskoi fizike (Collection of Works on Agronomic Physics), 1969, vol. 23, pp. 41–54.

    Google Scholar 

  5. Dzektser, E.S., Equations of Motion of Soil Water with Free Surface in Many-Layer Media, Dokl. Akad. Nauk SSSR, 1975, vol. 220, no. 3, pp. 540–543.

    Google Scholar 

  6. Rubinshtein, L.I., On the Question about the Propagation of Heat in Heterogeneous Media, Izv. Akad. Nauk SSSR Ser. Geograf. Geofiz., 1948, vol. 12, no. 1, pp. 27–45.

    MathSciNet  Google Scholar 

  7. Ting, T.W., A Cooling Process According to Two-Temperature Theory of Heat Conduction, J. Math. Anal. Appl., 1974, vol. 45, no. 9, pp. 23–31.

    Article  MathSciNet  MATH  Google Scholar 

  8. Hallaire, M., L’eau et la production vegetable, Inst. Natl. Rech. Agronomique, 1964, no. 9, p. 455.

    Google Scholar 

  9. Chudnovskii, A.F., Teplofizika pochv (Heat Physics of Soils), Moscow, 1976.

    Google Scholar 

  10. Barenblat, G.I., Zheltov, Yu.P., and Kochina, I.N., On Basic Representations of Filtration Theory of Fluids in Cracked Solids, Prikl. Mat. Mekh., 1960, vol. 25, no. 5, pp. 852–864.

    Google Scholar 

  11. Sveshnikov, A.A., Al’shin, A.B., Korpusov, M.O., and Pletner, Yu.D., Lineinye i nelineinye uravneniya sobolevskogo tipa (Linear and Nonlinear Equations of the Sobolev Type), Moscow, 2007.

    Google Scholar 

  12. Colton, D.L., Pseudoparabolic Equations in One Space Variable, J. Differential Equations, 1972, vol. 12, pp. 559–565.

    Article  MathSciNet  MATH  Google Scholar 

  13. Shkhanukov, M.X., 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., 1982, vol. 18, no. 4, pp. 689–699.

    MathSciNet  MATH  Google Scholar 

  14. Showalter, R.E. and Ting, T.W., Pseudoparabolic Partial Differential Equations, SIAM J. Math. Anal., 1970, vol. 1, no. 1, pp. 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  15. Ting, T.W., Certain Nonsteady Flows of Second-Order Fluids, Arch. Ration. Mech. Anal., 1963, vol. 14, no. 1, pp. 1–26.

    Article  MATH  Google Scholar 

  16. Vodakhova, V.A., A Boundary Value Problem with A. M. Nakhushev’s Nonlocal Condition for a Pseudoparabolic Equation in a Moisture Transfer, Differ. Uravn., 1982, vol. 18, no. 2, pp. 280–285.

    MathSciNet  MATH  Google Scholar 

  17. Kozhanov, A.I., On a Nonlocal Boundary Value Problem with Variable Coefficients for the Heat Equation and the Aller Equation, Differ. Uravn., 2004, vol. 40, no. 6, pp. 763–774.

    MathSciNet  MATH  Google Scholar 

  18. Beshtokov, M.Kh., Method of Riemann Function and Difference Method for the Solution of a Nonlocal Boundary Value Problem for Third-Order Equation of the Hyperbolic Type, Izv. Vyssh. Uchebn. Zaved. Severo-Kavkaz. Region, Rostov, 2007, no. 5, pp. 6–9.

    Google Scholar 

  19. Beshtokov, M.Kh., Finite-Difference Method for a Nonlocal Boundary Value Problem for a Third-Order Pseudoparabolic Equation, Differ. Uravn., 2013, vol. 49, no. 9, pp. 1170–1177.

    MathSciNet  MATH  Google Scholar 

  20. Beshtokov, M.Kh., On a Certain Boundary Value Problem for a Pseudoparabolic Third-Order Equation with Nonlocal Condition, Izv. Vyssh. Uchebn. Zaved. Severo-Kavkaz. Region, Rostov, 2013, no. 1, pp. 5–10.

    MATH  Google Scholar 

  21. Beshtokov, M.Kh., A Numerical Method for Solving One Nonlocal Boundary Value Problem for a Third- Order Hyperbolic Equation, Zh. Vychisl. Mat. Mat. Fiz., 2014, vol. 54, no. 9, pp. 1497–1514.

    MathSciNet  MATH  Google Scholar 

  22. Samarskii, A.A., Teoriya raznostnykh skhem (Theory of Difference Schemes), Moscow: Nauka, 1983.

    Google Scholar 

  23. Olisaev, E.G., Difference Methods for Nonlocal Boundary Value Problems for an Equation of the Parabolic Type with Degeneration, Cand. Sci. (Phys.–Math.) Dissertation, Moscow, 2003.

    Google Scholar 

  24. Ladyzhenskaya, O.A., Kraevye zadachi matematicheskoi fiziki (Boundary Value Problems of Mathematical Physics), Moscow: Nauka, 1973.

    Google Scholar 

  25. Andreev, V.B., The Convergence of Difference Schemes Which Approximate the Second and Third Boundary Value Problems for Elliptic Equations, Zh. Vychisl. Mat. Mat. Fiz., 1968, vol. 8, no. 6, pp. 1218–1231.

    MathSciNet  Google Scholar 

  26. Samarskii, A.A. and Gulin, A.V., Ustoichivost’ raznostnykh skhem (Stability of Difference Schemes), Moscow, 1973.

    Google Scholar 

  27. Faddeev, D.K. and Faddeeva, V.N., Vychislitel’nye metody lineinoi algebry (Computational Methods of Linear Algebra), Moscow, 1960.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kabardino-Balkar State University, Nalchik, Russia

    M. Kh. Beshtokov

Authors
  1. M. Kh. Beshtokov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Kh. Beshtokov.

Additional information

Original Russian Text © M.Kh. Beshtokov, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 10, pp. 1393–1406.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beshtokov, M.K. On the numerical solution of a nonlocal boundary value problem for a degenerating pseudoparabolic equation. Diff Equat 52, 1341–1354 (2016). https://doi.org/10.1134/S0012266116100104

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266116100104

Search

Navigation