Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems

  • 48 Accesses

Abstract

Boundary-value problems for the heat conduction equation are considered in the case where the boundary conditions contain a fractional derivative. Problems of this type arise when the heat processes are simulated by a nonstationary heat flow by using the one-dimensional thermal model of a two-layer system (coating — base). It is proved that the problem under consideration is correct. A one-parameter family of difference schemes is constructed; it is shown that these schemes are stable and convergent in the uniform metric.

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

References

  1. 1.

    É. M. Kartashov,Analytic Methods in the Theory of Heat Conduction in the Solid State [in Russian], Vysshaya Shkola, Moscow (1985).

  2. 2.

    A. N. Tikhonov and A. A. Samarskii,Equations in Mathematical Physics [in Russian], Nauka, Moscow (1966).

  3. 3.

    É. Goursat,A Course of Mathematical Analysis. Vol. 3 [Russian translation], Gostekhizdat, Moscow (1933).

  4. 4.

    S. G. Samko, A. A. Kilbas, and O. I. Marichev,Integrals and Derivatives of Fractional Order and Some Applications [in Russian]. Nauka i Tekhnika, Minsk (1987).

  5. 5.

    A. A. Samarskii and A. V. Gulin,Stability of Difference Schemes [in Russian], Nauka, Moscow (1973).

  6. 6.

    R. Richtmyer and K. Morton,Difference Methods for Initial-Value Problems, Wiley-Interscience, New York (1967).

Download references

Author information

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1289–1398, September, 1993.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Shkhanukov, M.K., Kerefov, A.A. & Berezovskii, A.A. Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems. Ukr Math J 45, 1445–1455 (1993). https://doi.org/10.1007/BF01058643

Download citation

Keywords

  • Boundary Condition
  • Heat Conduction
  • Heat Flow
  • Difference Scheme
  • Difference Method