The numerical solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative

Kaouther Bouchama; Yacine Arioua; Abdelkrim Merzougui

doi:10.3934/naco.2021026

Article Contents

2022, Volume 12, Issue 3: 621-636. Doi: 10.3934/naco.2021026

This issue Previous Article Adaptive controllability of microscopic chaos generated in chemical reactor system using anti-synchronization strategy Next Article A mesh adaptation algorithm using new monitor and estimator function for discontinuous and layered solution

The numerical solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative

1.
Laboratory for Pure and Applied Mathematics, University of M'sila, Bp 166 M'sila, 28000, Algeria
2.
Department of Mathematics, Laboratory for Pure and Applied Mathematics, University of M'sila, Bp 166 M'sila, 28000, Algeria

^*Corresponding author: Yacine Arioua
^*Corresponding author: Yacine Arioua

Received: May 2021

Revised: June 2021

Early access: July 2021

Published: September 2022

This work was financially supported by the General Direction of Scientific Research and Technological Development (DGRSDT)-Algeria, PRFU(Grant No. C00L03UN280120180010)

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Abstract

In this paper, a numerical approximation solution of a space-time fractional diffusion equation (FDE), involving Caputo-Katugampola fractional derivative is considered. Stability and convergence of the proposed scheme are discussed using mathematical induction. Finally, the proposed method is validated through numerical simulation results of different examples.

Keywords:

Fractional diffusion equation,
Caputo-Katugampola fractional derivative,
Finite difference scheme,
Convergence,
Stability.

Mathematics Subject Classification: Primary: 35R11, 65M06, 65M12.

Citation:

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Figure 1. Graphical comparison of the numerical and the exact solution with $ h = 0.001 $, $ k = 0.1 $, $ \rho = 2 $, $ \alpha = 0.7, $ $ n = 20 $ and $ m = 25 $

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Figure 2. Graphical comparison of the numerical and the exact solution with $ k = 0.1 $, $ \rho = 2 $, $ \alpha = 0.6, $ $ \beta = 1.8, $ $ n = 30 $ and $ m = 25 $

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Figure 3. Graphical comparison of the numerical and the exact solution with $ k = 0.1 $, $ \rho = 2, $ $ \alpha = 0.9 $, $ (a)\ \beta = 1 $, $ (b)\ \beta = 2 $ and $ m = 25 $

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Figure 4. Graphical comparison of the numerical and the exact solution with $ h = 0.005 $, $ k = 0.1 $, $ \rho = 3 $, $ \alpha = 0.7, $ $ n = 20 $ and $ m = 15 $

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Figure 5. Graphical comparison of the numerical and the exact solution with $ k = 0.1 $, $ \rho = 3 $, $ \alpha = 0.8, $ $ \beta = 1.8, $ $ n = 40 $ and $ m = 15 $

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Figure 6. Graphical comparison of the numerical and the exact solution with $ k = 0.1 $, $ \rho = 3 $, $ \alpha = 0.9, $ $ (a)\ \beta = 1 $, $ (b)\ \beta = 2 $ and $ m = 15 $

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