Numerical method for solving the subdiffusion differential equation with nonlocal boundary conditions

Murat A. Sultanov; Vladimir E. Misilov; Makhmud A. Sadybekov; Murat A. Sultanov; Vladimir E. Misilov; Makhmud A. Sadybekov

doi:10.3934/math.20241726

AIMS Mathematics

2024, Volume 9, Issue 12: 36385-36404. doi: 10.3934/math.20241726

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Research article

Numerical method for solving the subdiffusion differential equation with nonlocal boundary conditions

1.
Department of Mathematics, Faculty of Natural Science, Khoja Akhmet Yassawi International Kazakh-Turkish University, B. Sattarhanov Street 29, Turkistan 160200, Kazakhstan
2.
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of RAS, S. Kovalevskaya Street 16, Ekaterinburg 620108, Russia
3.
Department of High Performance Computing Technologies, Institute of Natural Sciences and Mathematics, Ural Federal University, Mira Street 19, Ekaterinburg 620002, Russia
4.
Institute of Mathematics and Mathematical Modeling, 125 Pushkin street, 050010 Almaty, Kazakhstan

Received: 09 September 2024 Revised: 23 December 2024 Accepted: 26 December 2024 Published: 31 December 2024
MSC : 35K20, 35R11, 65M06

This work was devoted to the construction of a numerical algorithm for solving the initial boundary value problem for the subdiffusion equation with nonlocal boundary conditions. For the case of not strongly regular boundary conditions, the well-known methods cannot be used. We applied an algorithm that consists of reducing the nonlocal problem to a sequential solution of two subproblems with local boundary conditions. The solution to the original problem was summed up from the solutions of the subproblems. To solve the subproblems, we constructed implicit difference schemes on the basis of the L1 formula for approximating the Caputo fractional derivative and central difference for approximating the space derivatives. Stability and convergence of the schemes were established. The Thomas algorithm was used to solve systems of linear algebraic equations. Numerical experiments were conducted to study the constructed algorithm. In terms of accuracy and stability, the algorithm performs well. The results of experiments confirmed that the convergence order of the method coincides with the theoretical one, $ O(\tau^{2-\alpha}+h^2) $.
- differential equations,
- fractional derivative,
- subdiffusion equation,
- nonlocal problems,
- not strongly regular boundary conditions,
- boundary value problems,
- numerical algorithms
Citation: Murat A. Sultanov, Vladimir E. Misilov, Makhmud A. Sadybekov. Numerical method for solving the subdiffusion differential equation with nonlocal boundary conditions[J]. AIMS Mathematics, 2024, 9(12): 36385-36404. doi: 10.3934/math.20241726

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Abstract

This work was devoted to the construction of a numerical algorithm for solving the initial boundary value problem for the subdiffusion equation with nonlocal boundary conditions. For the case of not strongly regular boundary conditions, the well-known methods cannot be used. We applied an algorithm that consists of reducing the nonlocal problem to a sequential solution of two subproblems with local boundary conditions. The solution to the original problem was summed up from the solutions of the subproblems. To solve the subproblems, we constructed implicit difference schemes on the basis of the L1 formula for approximating the Caputo fractional derivative and central difference for approximating the space derivatives. Stability and convergence of the schemes were established. The Thomas algorithm was used to solve systems of linear algebraic equations. Numerical experiments were conducted to study the constructed algorithm. In terms of accuracy and stability, the algorithm performs well. The results of experiments confirmed that the convergence order of the method coincides with the theoretical one, $ O(\tau^{2-\alpha}+h^2) $.

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