A Lattice Boltzmann Method for Spatial Fractional-Order Telegraph Equations

LI Mengjun; DAI Houping; WEI Xuedan; ZHENG Zhoushun

doi:10.21656/1000-0887.410311

Volume 42 Issue 5

May 2021

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Article Navigation > Applied Mathematics and Mechanics > 2021 > 42(5): 522-530

LI Mengjun, DAI Houping, WEI Xuedan, ZHENG Zhoushun. A Lattice Boltzmann Method for Spatial Fractional-Order Telegraph Equations[J]. Applied Mathematics and Mechanics, 2021, 42(5): 522-530. doi: 10.21656/1000-0887.410311

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A Lattice Boltzmann Method for Spatial Fractional-Order Telegraph Equations

doi: 10.21656/1000-0887.410311

¹College of Mathematics and Statistics, Jishou University, Jishou, Hunan 416000, P.R.China
²School of Mathematics and Statistics, Central South University, Changsha 410083, P.R.China

Funds: The National Natural Science Foundation of China（51974377）

Received Date: 2020-10-15
Rev Recd Date: 2021-04-06
Publish Date: 2021-05-01

Abstract

Abstract

The lattice Boltzmann method (LBM) was applied to numerically solve Riemann-Liouville spatial fractional-order telegraph equations. Firstly, the integral term of the fractional-order operator was discretized and the order of convergence was analyzed. Then, a 1D and 3-velocity (D1Q3) LBM evolution model with modified functions was established. The expressions of equilibrium distribution functions and correction functions were deduced by means of the Chapman-Enskog multi-scale analysis and the Taylor expansion technique. Therefore, the macroscopic equation was exactly recovered from the established evolution model. Numerical results show the stability and effectiveness of the model.

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References

[1]	杨云冲, 徐忠昌. 带阻尼项的时间空间分数阶电报方程的差分格式及其稳定性分析[J]. 计算机与数字工程, 2015,43(12): 2027-2129, 2144.(YANG Yunchong, XU Zhongchang. Difference approximation and stability analysis of the time-space fractional telegraph equation with damp[J]. Computer and Digital Engineering,2015,〖STHZ〗 43(12): 2027-2129, 2144.(in Chinese))
[2]	SEVIMLICAN A. An approximation to solution of space and time fractional telegraph equations by He’s variational iteration method[J]. Mathematical Problems in Engineering,2010,2010: 290631.
[3]	MOMANI S. Analytic and approximate solutions of the space- and time-fractional telegraph equations[J]. Applied Mathematics and Computation,2005,170(2): 1126-1134.
[4]	ORSINGHER E, BEGHIN L. Time-fractional telegraph equations and telegraph processes with Brownian time[J]. Probability Theory and Related Fields,2004,128(1): 141-160.
[5]	SRIVASTAVA V K, AWASTHI M K, TAMSIR M. RDTM solution of Caputo time fractional-order hyperbolic telegraph equation[J]. AIP Advances,2013,3(3): 61-72.
[6]	ZHAO Z G, LI C P. Fractional difference/finite element approximations for the time-space fractional telegraph equation[J]. Applied Mathematics and Computation,2012,219(6): 2975-2988.
[7]	LI H, JIANG W, LI W Y. Space-time spectral method for the Cattaneo equation with time fractional derivative[J]. Applied Mathematics and Computation,2019,349(6): 325-336.
[8]	马亮亮, 刘冬兵. 二维变系数空间分数阶电报方程数值解[J]. 辽宁工程技术大学学报（自然科学版）, 2014,33(3): 429-432.(MA Liangliang, LIU Dongbing. Numerical solution to the two-dimensional space fractional order telegraph equation with variable coefficients[J]. Journal of Liaoning Technical University(Natural Science),2014,33(3): 429-432.(in Chinese))
[9]	杨文洁. 时间分数阶电报方程的差分方法[D]. 硕士学位论文. 济南: 山东师范大学, 2020.(YANG Wenjie. Difference methods for time fractional telegraph equation[D]. Master Thesis. Jinan: Shandong Normal University, 2020.(in Chinese))
[10]	MODANLI M, AKGUL A. On solutions of fractional order telegraph partial differential equation by Crank-Nicolson finite difference method[J]. Applied Mathematics and Nonlinear Sciences,2020,5(2): 163-170.
[11]	BENZI R, SUCCI S, VERGASSOLA M. The lattice Boltzmann equation: theory and applications[J]. Physics Reports,1992,223(3): 145-197.
[12]	CHEN S Y, DOOLEN G D. Lattice Boltzmann method for fluid flows[J]. Annual Review of Fluid Mechanics,1998,〖STHZ〗 30(1): 329-364.
[13]	QIAN Y H, SUCCI S, ORSZAG S A. Recent advances in lattice Boltzmann computing[J]. Annual Review of Computational Physics,1995,3(1): 195-242.
[14]	郭照立, 郑楚光. 格子Boltzmann方法的原理及其应用[M]. 北京: 科学出版社, 2008.(GUO Zhaoli, ZHENG Chuguang. Theory and Application of Lattice Boltzmann Method [M]. Beijing: Science Press, 2008.(in Chinese))
[15]	郭照立, 王能超, 李青, 等. 流体动力学的格子Boltzmann方法[M]. 武汉: 湖北科学技术出版社, 2002.(GUO Zhaoli, WANG Nengchao, LI Qing, et al. Lattice Boltzmann Method for Hydrodynamics [M]. Wuhan: Hubei Science and Technology Press, 2002.(in Chinese))
[16]	SUN Y X, TIAN Z F. High-order upwind compact finite-difference lattice Boltzmann method for viscous incompressible flows[J]. Computers and Mathematics With Applications,2020,80(7): 1858-1872.
[17]	BENHAMOU J, JAMI M, MEZRHAB A, et al. Numerical study of natural convection and acoustic waves using the lattice Boltzmann method[J]. Heat Transfer,2020,49(6): 3779-3796.
[18]	LI Q H, CHAI Z H, SHI B C. Lattice Boltzmann model for a class of convection-diffusion equations with variable coefficients[J]. Computers and Mathematics With Applications,2015,70(4): 548-561.
[19]	DUAN Y L, KONG L H, GUO M. Numerical simulation of a class of nonlinear wave equations by lattice Boltzmann method[J]. Communications in Mathematics and Statistics,2017,5(1): 13-35.
[20]	ZHU J Q, LU S H, GAO D Y, et al. Numerical analysis on supercritical natural convection by lattice Boltzmann method[J]. Numerical Heat Transfer(Part B): Fundamentals,2020,77(6): 461-473.