On the influence of the method of approximation of an unknown function on the stability of numerical methods for solving the anomalous diffusion equation
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Cybernetics and programming

On the influence of the method of approximation of an unknown function on the stability of numerical methods for solving the anomalous diffusion equation
Litvinov Vladimir Andreevich

PhD in Physics and Mathematics

Docent, the department of Informatics and Special Technology, Barnaul Law Institute of the Ministry of Internal Affairs of Russia

656038, Russia, Altaiskii krai, g. Barnaul, ul. Chkalova, 49

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The subject of the research is numerical algorithms for solving fractional partial differential equations. The object of the study is the stability of several algorithms for the numerical solution of the anomalous diffusion equation. Algorithms based on the difference representation of the fractional Riemann-Liuville derivative and the Caputo derivative for various orders of accuracy are considered. A comparison is made of the results of numerical calculations using the analyzed algorithms for a model problem with the exact solution of the anomalous diffusion equation for various orders of the fractional derivative with respect to the spatial coordinate. The results of the work were obtained on the basis of the analysis of the constructed difference schemes for stability, the conducted numerical experiments and a comparative analysis of the data obtained. The main conclusions of the study are the advantage of using the approximation of the fractional Caputo derivative compared to using the difference scheme for the fractional Riemann-Liouville derivative in the numerical solution of the anomalous diffusion equation. The paper also indicates the importance of choosing the method of difference approximation of the second derivative, which is a derivative of the Caputo.

Keywords: derivative Of Caputo, approximation, diffusion, difference scheme, algorithm, stability, fractional derivative, numerical method, Riemann-Liouville derivative, differential equation
This article written in Russian. You can find full text of article in Russian here .

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