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Volume 14, Issue 4
Stability and Convergence Analyses of the FDM Based on Some L-Type Formulae for Solving the Subdiffusion Equation

Reza Mokhtari, Mohadese Ramezani & Gundolf Haase

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 945-971.

Published online: 2021-09

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  • Abstract

Some well-known L-type formulae, i.e., L1, L1-2, and L1-2-3 formulae, are usually employed to approximate the Caputo fractional derivative of order α ∈ (0, 1). In this paper, we aim to elaborate on the stability and convergence analyses of some finite difference methods (FDMs) for solving the subdiffusion equation, i.e., a diffusion equation which exploits the Caputo time-fractional derivative of order $α$. In fact, the FDMs considered here are based on the usual central difference scheme for the spatial derivative, and the Caputo derivative is approximated by using methods such as the L1, L1-2, and L1-2-3 formulae. Thanks to a specific type of the discrete version of the Gronwall inequality, we show that the FDMs are unconditionally stable in the maximum norm and also discrete $H^1$ norm. Then, we prove that the finite difference method which uses the L1, L1-2, and L1-2-3 formulae has the global order of convergence $2−α$, $3−α$, and 3, respectively. Finally, some numerical tests confirm the theoretical results. A brief conclusion finishes the paper.

  • AMS Subject Headings

65M12, 65M06, 26A33

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-945, author = {Mokhtari , RezaRamezani , Mohadese and Haase , Gundolf}, title = {Stability and Convergence Analyses of the FDM Based on Some L-Type Formulae for Solving the Subdiffusion Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {4}, pages = {945--971}, abstract = {

Some well-known L-type formulae, i.e., L1, L1-2, and L1-2-3 formulae, are usually employed to approximate the Caputo fractional derivative of order α ∈ (0, 1). In this paper, we aim to elaborate on the stability and convergence analyses of some finite difference methods (FDMs) for solving the subdiffusion equation, i.e., a diffusion equation which exploits the Caputo time-fractional derivative of order $α$. In fact, the FDMs considered here are based on the usual central difference scheme for the spatial derivative, and the Caputo derivative is approximated by using methods such as the L1, L1-2, and L1-2-3 formulae. Thanks to a specific type of the discrete version of the Gronwall inequality, we show that the FDMs are unconditionally stable in the maximum norm and also discrete $H^1$ norm. Then, we prove that the finite difference method which uses the L1, L1-2, and L1-2-3 formulae has the global order of convergence $2−α$, $3−α$, and 3, respectively. Finally, some numerical tests confirm the theoretical results. A brief conclusion finishes the paper.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2021-0020}, url = {http://global-sci.org/intro/article_detail/nmtma/19525.html} }
TY - JOUR T1 - Stability and Convergence Analyses of the FDM Based on Some L-Type Formulae for Solving the Subdiffusion Equation AU - Mokhtari , Reza AU - Ramezani , Mohadese AU - Haase , Gundolf JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 945 EP - 971 PY - 2021 DA - 2021/09 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2021-0020 UR - https://global-sci.org/intro/article_detail/nmtma/19525.html KW - Stability analysis, order of convergence, Caputo derivative, L1 formula, L1-2 formula, L1-2-3 formula, subdiffusion equation, Gronwall inequality. AB -

Some well-known L-type formulae, i.e., L1, L1-2, and L1-2-3 formulae, are usually employed to approximate the Caputo fractional derivative of order α ∈ (0, 1). In this paper, we aim to elaborate on the stability and convergence analyses of some finite difference methods (FDMs) for solving the subdiffusion equation, i.e., a diffusion equation which exploits the Caputo time-fractional derivative of order $α$. In fact, the FDMs considered here are based on the usual central difference scheme for the spatial derivative, and the Caputo derivative is approximated by using methods such as the L1, L1-2, and L1-2-3 formulae. Thanks to a specific type of the discrete version of the Gronwall inequality, we show that the FDMs are unconditionally stable in the maximum norm and also discrete $H^1$ norm. Then, we prove that the finite difference method which uses the L1, L1-2, and L1-2-3 formulae has the global order of convergence $2−α$, $3−α$, and 3, respectively. Finally, some numerical tests confirm the theoretical results. A brief conclusion finishes the paper.

Reza Mokhtari, Mohadese Ramezani & Gundolf Haase. (2021). Stability and Convergence Analyses of the FDM Based on Some L-Type Formulae for Solving the Subdiffusion Equation. Numerical Mathematics: Theory, Methods and Applications. 14 (4). 945-971. doi:10.4208/nmtma.OA-2021-0020
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