Based On Spectral Method To Solve Three Classes Of Typical Nonlinear Partial Differential Equations

Spectral methods are important numerical methods for solving partial differential equations(PDEs),they are relatively mature numerical methods developed rapidly in recent 40 years.Compared with finite difference methods and finite element methods,they have the advantages of fast solving speed,high accuracy and infinite order convergence.Since the 1970 s and 1980 s,with the rapid development of modern computer technology,the spectral methods have reached an unprecedented height.They are widely used to solve differential equations involving physics,ocean,atmospheric science and engineering technology.Their basic idea is to use the global smooth trial function to globally approximate the exact solutions of the problems.Therefore,as long as the differential equations are smooth enough and the numerical algorithms are designed properly,the spectral methods can be used to solve the selected differential equations efficiently.In this paper,the spectral methods are used to solve three typical nonlinear PDEs.Solving nonlinear PDEs is a hot topic in the current numerical analysis.In this paper,the spectral methods are combined with the variable-step Runge-Kutta time step method and the iterative idea to solve three types of typical nonlinear PDEs,namely,nonlinear hyperbolic,nonlinear parabolic and nonlinear elliptic PDEs.For nonlinear hyperbolic differential equations,two types of one-dimensional and two-dimensional Klein-Gordon equations with periodic boundary conditions are numerically solved;for nonlinear parabolic differential equations,one type of one-dimensional Fisher equation is selected and two types of one-dimensional and two-dimensional viscous Burgers equations are numerically solved;for nonlinear elliptic differential equations,one type of one-dimensional Poisson-Boltzmann equation is numerically solved.Error estimates are made for some of differential equations with exact solutions,and some equations are selected and compared with the latest numerical solution results.The numerical results in this article are better than those obtained in the references.