High accurate numerical method for solving time-fractional Fisher's equation
[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 initial value to obtain the first correction.Iterating with the new initial value,the process continues until a numerical solution satisfying the required accuracy is obtained.Finally,the obtained numerical solution is transformed back using Fourier inverse transform to obtain the solution to the original problem.Additionally,an error estimate for the SDC method is provided,and the accuracy of the SDC method depends on the regularity of F with respect to t.The numerical method established in this paper achieves high-order accuracy in both time and space domains.Numerical experiments show that,when the number of time partitions is p=7 and the number of spatial partitions is N=4,8,16,24,32,errors between the numerical solution corresponding to different fractional orders α and the exact solution diminish greatly.Due to the smoothness of the exact solution,discretizing the spatial domain using spectral methods can result in exponentially convergent accuracy.When the number of time partitions is p=3,4,5,6,7 and the number of spatial partitions is N=32,the errors between the numerical solution corresponding to different fractional orders α and the exact solution can reach high-order accuracy.Therefore,using the SDC method for time,with a small p value,the solution in the time direction can also exponentially converge,quickly achieving high-order accuracy.Compared with finite difference methods,using the SDC method not only saves computational resources but also achieves high accuracy.Finally,three-dimensional surface plots of the numerical solution for α=0.2,0.5,0.7,1,p=7,N=32 are provided.[Conclusions]The nonlinearity of the time-fractional Fisher's equation brings challenges and may need to be handled with the combination of a few numerical methods.The Fourier spectral method represents the solution using Fourier series expansion,processing the nonlinear term in Fourier space to simplify operations in the frequency domain.Additionally,Fourier series secures good orthogonality properties and rapid convergence,enabling efficient solving of the equation.The spectral deferred correction method offers advantages such as high accuracy and fast computation,even with large time steps.This numerical solution method combines the advantages of the Fourier-Galerkin spectral method and spectral deferred correction method,including high accuracy,good stability,low storage requirements,and fast computation.The stability and convergence of the constructed numerical format are demonstrated through the computation of error norms.Finally,numerical experiments confirm the efficiency and feasibility of the numerical format constructed in this paper.Compared to numerical formats developed using finite difference methods,the proposed numerical format achieves high-order accuracy in both spatial and temporal directions,with faster computational speeds.
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维普
