Applications Of Nonstandard Finite Difference Method And Parametrized Perturbation Method To Differential Equations

In this paper, we construct the nonstandard finite diference schemes on a few kindsof partial diferential equations, and discuss the global properties of the discrete schemes.Then, we use parametrized perturbation method to solve several kinds of nolinear ordinarydiferential equations. We compare our numerical solutions with others by numericaltests.Firstly, a nonstandard finite diference scheme is constructed for one dimensionalparabolic partial diferential equations——FitzHugh-Nagumo equations. Three propertiesof this scheme are proved. The first property is the fixed points of the associated spaceindependent equation of FitzHugh-Nagumo equations are elementary stable. The secondproperty is that the scheme preserves the principle of energy conservation for the corre-sponding stationary equation of FitzHugh-Nagumo equations. Third, combining the non-standard finite diference methods for the space independent equation and the stationaryequation, we could derive the nonstandard finite diference scheme for FitzHugh-Nagumoequations. The positivity and the boundedness of the solution are proved with a certainfunctional relation between time step size and space step size.Next, Hopf-Cole transformation is used to linearize the Burgers’equation which isa one-dimensional quasi-linear parabolic partial diferential equation. The nonstandardfinite diference scheme of the resulting equation is obtained by the nonlocal approxima-tion. Then, the consistency, stability and convergence of the scheme are analyzed. Thepositivity and the boundedness are also proved with a certain condition of the time stepsize and space step size.And then, we consider a hyperbolic partial diferential equation——Telegraph equa-tion. We also adopt the nonlocal approximation method and construct its nonstandardfinite diference scheme. We show that this scheme is stable and consistent, The condi-tions of positivity and boundedness are also given.The last part of paper is about some applications of parametrized perturbation methodto several ordinary diferential equations. We apply the parametrized perturbation methodto Prey-Predator equation, Volterra equation and mKdv equation respectively. After com-paring our numerical solutions with the exact solutions and other numerical solutions ob- tained by other numerical methods, we show that the parametrized perturbation methodis efective for linear and nonlinear diferential equations.