Average Rational Approximation For Nonlinear Boundary Value Problem

In many practical problems, things appear in a nonlinear form.As an important and special form of nonlinear approximation.rational approximation has a special significance either in practice or in life.It can solve the problem of poor convergence and instability of the traditional approximation methods,the subject is an important and has a powerful vitality.The traditional method of approximation includes approximation of Tay-lor expansion, Pade approximation, interpolation approximation and polyno-mial approximation, but they all have their own drawbacks:Taylor expansion has a very high precision in the deployment point nearby, in the distant place, the effect is very poor; using Taylor expansion, Pade approximation accompany its disadvantages; generally speaking. Lagrange interpolation approximation’s accuracy is better, this method is now used more, but on a finite interval, when the change of curvature function is larger approximation, accuracy can be poor, for example the Rung phenomenon:polynomial approximation is applied frequently, approximation works well. This paper is based on paper [35],continue to explore the approximation method that better than the poly-nomial- average rational approximation.In a large range of function approximation, the average of rational approx-imation is much better than the polynomial approximation. Commonly,when the function changes sharply,the accuracy of polynomial approximation is not very high.On the basis of the reference [35], first of all,it discusses the linear and nonlinear boundary value problems.making use of variable minimization prin-ciple, obtain the corresponding energy functional, transforming the problem to solving a nonlinear equation:secondly.it discusses the nonlinear Schrodinger equation on the infinite interval and in the real domain, Rational approxima-tion form with a new base e-kx to approximate the true solution.