分数阶反应扩散模型在图灵斑图中的应用及数值模拟
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摘要
斑图是在空间或时间上具有某些规律性的非均匀宏观结构,是可以用反应扩散系统描述其图案形成的数学模型之一。反应扩散系统中,稳定状态会在某些条件下失稳,产生空间定态图纹,即图灵斑图。分数阶反应扩散系统可以用来描述反常扩散运动。通过分数阶拉普拉斯算子的谱分解进行线性稳定性分析,研究系统模型的图灵不稳定性,详细阐述分数阶图灵斑图的数学机制和二维分数阶Gierer-Meinhardt模型下斑图的形成机理。在数值计算中,采用了高效、高精度的数值格式,空间离散采用傅里叶谱方法,离散结果具有谱精度。时间离散采用四阶龙格库塔指数时间差分方法。在数值模拟方面,以分数阶Gierer-Meinhardt模型为例,发现系统可以通过控制分数阶阶数的变化生成斑图,并验证了之前的理论结果。
Patterns are non-uniform macrostructures with some regularity in space or time, which can describe the pattern formation by a reaction diffusion system. The stable state will be unstable under certain conditions and spontaneously produce the spatial stationary pattern, namely Turing pattern. We can describe anomalous diffusion motion through the fractional reaction diffusion system. The paper obtains the Turing instability of two-dimensional fractional order Gierer-Meinhardt model by spectral decomposition of fractional Laplacian operator. An efficient high-precision numerical scheme is used in the numerical simulation. The Fourier spectral method is used in the spatial discretization, which has spectral accuracy. The Runge-Kutta exponential time difference method is applied to the time discretization. Numerical simulations in Gierer-Meinhardt model show that the system can generate patterns by controlling the value of fractional order in the system, and verify the theoretical results in the previous stability analysis.
Patterns are non-uniform macrostructures with some regularity in space or time, which can describe the pattern formation by a reaction diffusion system. The stable state will be unstable under certain conditions and spontaneously produce the spatial stationary pattern, namely Turing pattern. We can describe anomalous diffusion motion through the fractional reaction diffusion system. The paper obtains the Turing instability of two-dimensional fractional order Gierer-Meinhardt model by spectral decomposition of fractional Laplacian operator. An efficient high-precision numerical scheme is used in the numerical simulation. The Fourier spectral method is used in the spatial discretization, which has spectral accuracy. The Runge-Kutta exponential time difference method is applied to the time discretization. Numerical simulations in Gierer-Meinhardt model show that the system can generate patterns by controlling the value of fractional order in the system, and verify the theoretical results in the previous stability analysis.
引文
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[ 2 ]HARRISON L G.Kinetic theory of living pattern[M].London:Cambridge University Press,2005.
[ 3 ]KEMER B S,OSIPOV V V.Static,traveling and pulsating auto solitons[M].New York:Springer Publishing Company,1994:58-74.
[ 4 ]ASTROV Y,AMMELT E,TEPERICK S,et al.Hexagon and stripe turing structures in a gas discharge system[J].Phys Lett A,1996,211(3):184-190.
[ 5 ]张荣培,刘佳,王语.Chebyshev谱配置方法求解Allen-Cahn方程[J].沈阳师范大学学报(自然科学版),2017,35(4):435-440.
[ 6 ]LIANG P.Neurocomputation by reaction diffusion[J].Phys Rev Lett,1995,75(9):1863-1866.
[ 7 ]STALIUNAS K,SANCHDZ-MORCILO V J.Turing patterns in nonlinear optics[J].Opt Commun,2000,177(1):389-395.
[ 8 ]CASTETS V,DULOS E,BOISSONADE J,et al.Experimental evidence of a sustained standing turing-type nonequilibrium chemical pattern[J].Phys Rev Lett,1990,64(24):2953.
[ 9 ]LHEUREUX I.Self-organized rhythmic patterns in geochemical systems[J].Philos Trans,2013,371(2004):20120356.
[10]YIN H W,WEN X.Pattern formation through temporal fractional derivatives[J].Sci Rep,2018,8(1):5070.
[11]Li X J,Xu C.Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation[J].Commun Comp Phys,2010,8(5):1016.
[12]KHADER M M.On the numerical solutions for the fractional diffusion equation[J].Commun Nonlinear Sci & Numer Sim,2011,16(6):2535-2542.
[13]KHADER M M,SWEILAM N H.Approximate solutions for the fractional advection-dispersion equation using Legendre pseudo-spectral method[J].Comp Appl Mathe,2014,33(3):739-750.
[14]LI X,XU C.A space-time spectral method for the time fractional diffusion equation[J].SIAM J Numer Anal,2009,47(3):2108-2131.
[15]HANERT E.A comparison of three eulerian numerical methods for fractional-order transport models[J].Environ Fluid Mech,2010,10(1/2):7-20.
[16]ZAYERNOURI M,KARNIADAKIS G E.Fractional spectral collocation method[J].Siam J Sci Comp,2014,36(1):A40-A62.