arrow
Volume 15, Issue 2
Artificial Boundary Conditions for Time-Fractional Telegraph Equation

Wang Kong & Zhongyi Huang

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 360-386.

Published online: 2022-03

Export citation
  • Abstract

In this paper, we study the numerical solution of the time-fractional telegraph equation on the unbounded domain. We first introduce the artificial boundaries $Γ_±$ to get a finite computational domain. On the artificial boundaries $Γ_±$, we use the Laplace transform to construct the exact artificial boundary conditions (ABCs) to reduce the original problem to an initial-boundary value problem on a bounded domain. In addition, we propose a finite difference scheme based on the $\mathcal{L}1$_$2$ formule for the Caputo fractional derivative in time direction and the central difference scheme for the spatial directional derivative to solve the reduced problem. In order to reduce the effect of unsmoothness of the solution at the initial moment, we use a fine mesh and low-order interpolation to discretize the solution near $t = 0$. Finally, some numerical results show the efficiency and reliability of the ABCs and validate our theoretical results.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{NMTMA-15-360, author = {Kong , Wang and Huang , Zhongyi}, title = {Artificial Boundary Conditions for Time-Fractional Telegraph Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {2}, pages = {360--386}, abstract = {

In this paper, we study the numerical solution of the time-fractional telegraph equation on the unbounded domain. We first introduce the artificial boundaries $Γ_±$ to get a finite computational domain. On the artificial boundaries $Γ_±$, we use the Laplace transform to construct the exact artificial boundary conditions (ABCs) to reduce the original problem to an initial-boundary value problem on a bounded domain. In addition, we propose a finite difference scheme based on the $\mathcal{L}1$_$2$ formule for the Caputo fractional derivative in time direction and the central difference scheme for the spatial directional derivative to solve the reduced problem. In order to reduce the effect of unsmoothness of the solution at the initial moment, we use a fine mesh and low-order interpolation to discretize the solution near $t = 0$. Finally, some numerical results show the efficiency and reliability of the ABCs and validate our theoretical results.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2021-0067}, url = {http://global-sci.org/intro/article_detail/nmtma/20356.html} }
TY - JOUR T1 - Artificial Boundary Conditions for Time-Fractional Telegraph Equation AU - Kong , Wang AU - Huang , Zhongyi JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 360 EP - 386 PY - 2022 DA - 2022/03 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2021-0067 UR - https://global-sci.org/intro/article_detail/nmtma/20356.html KW - Artificial boundary conditions, time-fractional telegraph equation, finite difference scheme, fractional Cattaneo heat conduction law. AB -

In this paper, we study the numerical solution of the time-fractional telegraph equation on the unbounded domain. We first introduce the artificial boundaries $Γ_±$ to get a finite computational domain. On the artificial boundaries $Γ_±$, we use the Laplace transform to construct the exact artificial boundary conditions (ABCs) to reduce the original problem to an initial-boundary value problem on a bounded domain. In addition, we propose a finite difference scheme based on the $\mathcal{L}1$_$2$ formule for the Caputo fractional derivative in time direction and the central difference scheme for the spatial directional derivative to solve the reduced problem. In order to reduce the effect of unsmoothness of the solution at the initial moment, we use a fine mesh and low-order interpolation to discretize the solution near $t = 0$. Finally, some numerical results show the efficiency and reliability of the ABCs and validate our theoretical results.

Wang Kong & Zhongyi Huang. (2022). Artificial Boundary Conditions for Time-Fractional Telegraph Equation. Numerical Mathematics: Theory, Methods and Applications. 15 (2). 360-386. doi:10.4208/nmtma.OA-2021-0067
Copy to clipboard
The citation has been copied to your clipboard