TY - JOUR
T1 - Finite difference/Hermite–Galerkin spectral method for multi-dimensional time-fractional nonlinear reaction–diffusion equation in unbounded domains
AU - Guo, Shimin
AU - Mei, Liquan
AU - Zhang, Zhengqiang
AU - Chen, Jie
AU - He, Yuan
AU - Li, Ying
N1 - Publisher Copyright:
© 2019 Elsevier Inc.
PY - 2019/6
Y1 - 2019/6
N2 - The aim of this paper is to develop an efficient finite difference/Hermite–Galerkin spectral method for the time-fractional nonlinear reaction–diffusion equation in unbounded domains with one, two, and three spatial dimensions. For this purpose, we employ the L2−1 σ formula to discretize the temporal Caputo derivative. Additionally, we apply the Hermite–Galerkin spectral method with scaling factor for the approximation in space. The stability of the fully discrete scheme is established to show that our method is unconditionally stable. Numerical experiments including one-, two-, and three-dimensional cases of the problem are carried out to verify the accuracy of our scheme. The scheme is show-cased by solving two problems of practical interest, including the fractional Allen–Cahn and Gray–Scott models, together with an analysis of the properties of the fractional orders.
AB - The aim of this paper is to develop an efficient finite difference/Hermite–Galerkin spectral method for the time-fractional nonlinear reaction–diffusion equation in unbounded domains with one, two, and three spatial dimensions. For this purpose, we employ the L2−1 σ formula to discretize the temporal Caputo derivative. Additionally, we apply the Hermite–Galerkin spectral method with scaling factor for the approximation in space. The stability of the fully discrete scheme is established to show that our method is unconditionally stable. Numerical experiments including one-, two-, and three-dimensional cases of the problem are carried out to verify the accuracy of our scheme. The scheme is show-cased by solving two problems of practical interest, including the fractional Allen–Cahn and Gray–Scott models, together with an analysis of the properties of the fractional orders.
KW - Finite difference
KW - Fractional calculus
KW - Hermite polynomial/function
KW - Nonlinear reaction–diffusion equation
KW - Unbounded domain
UR - http://www.scopus.com/inward/record.url?scp=85060695193&partnerID=8YFLogxK
U2 - 10.1016/j.apm.2019.01.018
DO - 10.1016/j.apm.2019.01.018
M3 - Article
AN - SCOPUS:85060695193
SN - 0307-904X
VL - 70
SP - 246
EP - 263
JO - Applied Mathematical Modelling
JF - Applied Mathematical Modelling
ER -