时间分数阶Fisher方程的高精度数值解法  

High accurate numerical method for solving time-fractional Fisher’s equation

作　　者：王晶 陈雪娟[1] 朱小娟 WANG Jing;CHEN Xuejuan;ZHU Xiaojuan(School of Science,Jimei University,Xiamen 361021,China)

机构地区：[1]集美大学理学院,福建厦门361021

出　　处：《厦门大学学报（自然科学版）》2024年第6期1124-1131,共8页Journal of Xiamen University：Natural Science

基　　金：福建省自然科学基金(2022J01338,2020J01703);福建省教育厅项目(JAT190326);集美大学基金(ZP2020062,ZP2020054);集美大学数字福建大数据建模与智能计算研究所开放基金。

摘　　要：[目的]时间分数阶Fisher方程可以描述流体力学、热核反应、等离子体物理和传染病传播等问题中的非线性现象.但关于该方程高效的数值格式研究成果较少,且大多采用差分法对方程进行离散.为了使分数阶Fisher方程得到更广泛的应用,本文给出一种求解非线性时间分数阶Fisher方程的高精度数值解法.[方法]在空间上,采用Fourier-Galerkin谱方法进行离散得到一组关于时间的非线性常微分方程组;在时间上,采用谱延迟校正法对时间常微分方程组进行迭代校正,得到高精度的数值解.[结果]该数值解法结合了Fourier-Galerkin谱方法和谱延迟校正法的特点,具有精度高、稳定性好、储存量小及计算时间快等优点.最后通过数值算例验证了所构造的数值格式在时间和空间方向上都能达到高阶精度.[结论]将Fourier-Galerkin谱方法与谱延迟校正法相结合,计算时间分数阶Fisher方程的数值解.通过计算误差范数,验证了所构造的数值格式的稳定性和收敛性.对比差分法所构造的数值格式,本文构造的数值格式在时空方向上都能够达到高阶精度,并且运行速度更快.[Objective] The time-fractional Fisher's equation can describe nonlinear phenomena in fluid mechanics,thermal nuclear reactions,plasma physics,and the spread of infectious diseases.With the development of fractional calculus theory,the time-fractional Fisher's equation has garnered widespread attention for its ability to effectively handle nonlinear dynamic characteristics.Due to the limited research on efficient numerical schemes for the time-fractional Fisher's equation,finite difference techniques are used for discretization in most existing method.To facilitate the broader application of the fractional Fisher's equation,we propose the usage of a high-accuracy numerical method.[Methods] For the discretization of the space,the Fourier-Galerkin spectral method is used,thus resulting in a set of nonlinear ordinary differential equations(ODEs) with respect to time.For the discretization of the time,the spectral deferred correction(SDC) method iteratively corrects the ODEs to obtain high-accuracy numerical solutions.The core idea of the SDC method lies in transforming ODEs into corresponding Picard integral equations,discretized in time using Gauss-Legendre grids.By means of either forward or backward Euler methods to solve the integral system,and by solving a series of correction equations on the same Gauss-type grid,the solution is refined to higher-order accuracy.[Results] Herein the Fourier-Galerkin spectral method and the SDC method are jointly used to compute the time-fractional Fisher's equation.By discretizing the spatial domain of the time-fractional Fisher's equation using the Fourier-Galerkin spectral method,a set of nonlinear ordinary differential equations(ODEs) with respect to time is obtained.Also,our theoretical analyses confirm the convergence and the unconditional stability of this semi-discretization formulation.By transforming the ODEs into corresponding Picard integral equations,defining residual and error functions,and deriving correction equations,an error can be computed and added to the ini

关 键 词：时间分数阶Fisher方程 谱延迟校正法 Fourier-Galerkin谱方法 稳定性 收敛性 

分 类 号：O241.82[理学—计算数学]