Application Of Reaction Diffusion Model In Turing Pattern And Numerical Simulation

A pattern is a non-uniform macroscopic structure with some regularity in space or time,which can be described by a reaction diffusion system.In a reaction-diffusion system,the steady state will be unstable under certain conditions and spontaneously produce a spatial stationary pattern,that is,a Turing pattern.The generation of the Turing pattern corresponds to the coupling of a nonlinear reaction dynamics process to the diffusion process.Turing instability has attracted the attention of scientists in many fields.In this paper,from the two aspects of the integer order and fractional order reaction diffusion equations,the reaction diffusion system in the Turing pattern is explored and numerical simulation is carried out.The linear stability analysis can explain the pattern formation mechanism of the Gierer-Meinhardt model,which can explain the process of generating the Turing pattern after the diffusion term is added to the diffusion term in the nonlinear ordinary differential equation system.The Chebyshev spectral method and the compact implicit integration factor method are used for spatial and temporal discretization,featuring high precision,good stability,and small storage.In terms of fractional order,the linear stability analysis is carried out by spectral decomposition of fractional Laplacian.The mathematical mechanism of fractional order graph and the formation mechanism of two-dimensional fractional Gierer-Meinhardt model are elaborated.The numerical method uses the Fourier spectrum method and the fourth-order Runge-Kutta exponential time difference method(ETD4RK method).Numerical simulations show that under certain conditions,the system can generate patterns by controlling the variation of fractional order and verify the theoretical results.The first chapter introduces the basic concept of the Turing pattern and the reaction diffusion system in the Turing pattern,and describes the research significance and research progress of the Turing pattern.The second chapter explores the derivation process and characteristics of Chebyshev spectral method,compact implicit integral factor method,Fourier spectrum method and ETD4 RK method.In the thirdchapter,the application and numerical simulation of reaction-diffusion model in Turing pattern are discussed.Firstly,the Turing pattern is described by dimensionless reaction diffusion equations.Then the mathematical mechanism of pattern formation is deduced.Chebyshev spectral method is used for spatial discretization,and compact implicit integral factor method is used for temporal discretization.Finally,numerical experiments are carried out.In chapter 4,on the basis of Chapter 3,fractional Laplace operator is adopted,which is more widely used.The equations are discretized by combining Fourier spectral method and ETD4 RK method.The variation and trend of the graphics are analyzed by numerical experiments.These numerical results verify the theoretical result.