Study Of The Numerical Methods And Properties For Several Nonlinear Partial Differential Equations

Whether in theory or in practical application,nonlinear partial differential equations is an important branch of modern mathematics,they are used to describe the mechanical,process control,ecological and economic system,chemical circulation system and the problems in the field of epidemiology and so on.For space,time,time delay effects are fully considered when using nonlinear partial differential equations describing the above problem,thus more accurately reflecting the actual.In this paper,using finite difference method for a class of nonlinear partial differential equations,main works are as follows:First,the generalized regularized long wave(GRLW)equation is studied by finite difference method.We design three types of fourth-order compact finite difference schemes.It is proved by the discrete energy method that the compact scheme is uniquely solvable,the convergence and unconditional stability of the difference schemes are obtained,and its numerical convergence order is O(?2+h4)in the L?-norm.Further,three types of the compact schemes are conservative.Numerical experiment results show that the theory is accurate and the method is efficient and reliable.Secondly,a new compact finite difference scheme is proposed and analyzed for the extended Fisher-Kolmogorov(EFK)equation in two space dimension with Dirichlet boundary conditions.A priori bounds are proved using Lyapunov functional.Further,existence,uniqueness and convergence of the difference solutions with order O(?2 ?h4)in the L?-norm are proved.Numerical results are given to support the theoretical analysis.Finally,the long-time behavior of the finite difference solution to the generalized BBM(GBBM)equation in two space dimensions with Dirichlet boundary conditions.The unique solvability of numerical solution is shown.It is proved that there exists a global attractor of the discrete dynamical system.Finally,we obtain the long-time stability and convergence of the difference scheme.Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems.Numerical experiment results show that the theory is accurate and the schemes are efficient and reliable.