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A Numerical Method for Solving the Third Boundary Value Problem for the Convection-Diffusion Equation with a Fractional Time Derivative in a Multidimensional Domain

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Abstract

In this paper, we study the third boundary value problem for the convection-diffusion equation with a time fractional derivative and variable coefficients in a multidimensional domain. For an approximate solution in a rectangular parallelepiped of the problem set, in the same area, the third boundary value problem for a differential equation with a small parameter is considered. An a priori estimate is obtained from which it follows the convergence of the solution of the differential problem with a small parameter to the solution of the original problem for small values of the parameter. For a problem with a small parameter, a locally one-dimensional difference scheme by A. A. Samarski is constructed. Using the maximum principle for solving the difference problem, an a priori estimate is obtained in the grid norm \(C\), which expresses the stability of the locally one-dimensional difference scheme. The uniform convergence of the locally one-dimensional scheme is proved for \(0<\alpha<1\).

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REFERENCES

  1. V. E. Tarasov, Theoretical Physics Models with Integro-Differentiation of Fractional Order (Inst. Komp’yut. Issled., Izhevsk, 2011) [in Russian].

    Google Scholar 

  2. V. V. Uchaikin, The Method of Fractional Derivatives (Artishok, Ulyanovsk, 2008) [in Russian].

    Google Scholar 

  3. A. A. Alikhanov, ‘‘Boundary value problems for the diffusion equation of the variable order in differential and difference settings,’’ Appl. Math. 219, 3938–3946 (2012).

    MathSciNet  MATH  Google Scholar 

  4. A. A. Alikhanov, ‘‘A new difference scheme for the time fractional diffusion equation,’’ J. Comput. Phys. 280, 424–438 (2015).

    Article  MathSciNet  Google Scholar 

  5. K. Diethelm and G. Walz, ‘‘Numerical solution of fractional order differential equations by extrapolation,’’ Numer. Algorithms 16, 231–253 (2016).

    Article  MathSciNet  Google Scholar 

  6. Y. N. Zhang, Z. Z. Sun, and H. L. Liao, ‘‘Finite difference methods for the time fractional diffusion equation on non-uniform meshs,’’ J. Comput. Phys. 265, 195–210 (2014).

    Article  MathSciNet  Google Scholar 

  7. M. Kh. Beshtokov, ‘‘Local and nonlocal boundary value problems for degenerating and nondegenerating pseudoparabolic equations with a Riemann–Liouville fractional derivative,’’ Differ. Equat. 54, 758–774 (2018).

    Article  MathSciNet  Google Scholar 

  8. M. Kh. Beshtokov, ‘‘To boundary-value problems for degenerating pseudoparabolic equations With Gerasimov–Caputo fractional derivative,’’ Russ. Math. 62 (10), 1–14 (2018).

    Article  MathSciNet  Google Scholar 

  9. M. Kh. Beshtokov, ‘‘Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative,’’ Differ. Equat. 55, 919–928 (2019).

    MathSciNet  MATH  Google Scholar 

  10. M. Kh. Beshtokov, ‘‘Boundary-value problems for loaded pseudoparabolic equations of fractional order and difference methods of their solving,’’ Russ. Math. 63 (2), 1–10 (2019).

    Article  MathSciNet  Google Scholar 

  11. M. M. Lafisheva and M. Kh. Shkhanukov-Lafishev, ‘‘Locally one-dimensional difference schemes for the fractional order diffusion equation,’’ Comput. Math. Math. Phys. 48, 1875–1884 (2008).

    Article  MathSciNet  Google Scholar 

  12. B. A. Ashabokov, Z. V. Beshtokova and M. Kh. Shkhanukov-Lafishev, ‘‘Locally one-dimensional difference scheme for a fractional tracer transport equation, ‘‘Comput. Math. Math. Phys. 57, 1498–1510 (2017).

    Article  MathSciNet  Google Scholar 

  13. F. I. Taukenova and M. Kh. Shkhanukov-Lafishev, ‘‘Difference methods for solving boundary value problems for fractional differential equations,’’ Comput. Math. Math. Phys. 46, 1785–1795 (2006).

    Article  MathSciNet  Google Scholar 

  14. A. K. Bazzaev and M. Kh. Shkhanukov-Lafishev, ‘‘Locally one-dimensional scheme for fractional diffusion equations with robin boundary conditions,’’ Comput. Math. Math. Phys. 350, 1141–1149 (2010).

    Article  Google Scholar 

  15. M. I. Vishik and L. A. Lyusternik, ‘‘Regular degeneration and boundary layer for linear differential equations with small parameter,’’ Usp. Mat. Nauk 12 (5), 3–122 (1967).

    MathSciNet  MATH  Google Scholar 

  16. S. K. Godunov and V. S. Ryben’kiy, Difference Schemes (Nauka, Moscow, 1977) [in Russian]

    Google Scholar 

  17. N. H. Abel, Solution de quelques problemes al’aide d’integrales definies (Dover, New York, 1965).

    Google Scholar 

  18. S. K. Kumykova, ‘‘A certain boundary value problem with shift for the equation \(signy^{m}u_{xx}+u_{yy}=0\),’’ Differ. Uravn. 12, 79–88 (1976).

    MathSciNet  Google Scholar 

  19. O. A. Ladyzhenskaya, Boundary Value Problems in Mathematical Physics (Nauka, Moscow, 1973) [in Russian].

    Google Scholar 

  20. A. A. Samarskii, The Theory of Difference Schemes (Marcel Dekker, New York, 2001).

    Book  Google Scholar 

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

    MATH  Google Scholar 

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Funding

The reported study was funded by RFBR and NSFC according to the research project no. 20-51-53007.

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Correspondence to M. Kh. Beshtokov.

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(Submitted by A. V. Lapin)

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Beshtokov, M.K. A Numerical Method for Solving the Third Boundary Value Problem for the Convection-Diffusion Equation with a Fractional Time Derivative in a Multidimensional Domain. Lobachevskii J Math 42, 1630–1642 (2021). https://doi.org/10.1134/S1995080221070052

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  • DOI: https://doi.org/10.1134/S1995080221070052

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