Rational Approximation To Solve The Two Point Boundary Value

Rational function approximation theory approximation is an important research topic. In the early19th century and20th century, Chebyshev and C.dela Valle Poussin began to study the real axis has the best rational function approximation problem bounded interval, studied the presence of rational functions of best approximation sex. Mathematicians put a lot of energy and enthusiasm, but without success. Mainly rational approximation calculation instability, the solution is not unique, without a good initial value.in a wide range of function approximation, rational polynomial approx-imation much better results than the polynomial approximation effects when comparing dramatic changes in the function, commonly used polynomial ap-proximation is not very good, for example, the use of the function power series Taylor expand can get rational polynomial approximation, this is only a partial result, it only launched a high accuracy in the vicinity of the point, in distant places is poor; rational Pade approximation, like Taylor series expansion as, away from the expansion point, the same accuracy is not high; Lagrange interpolation approximation, its accuracy is better in general, which is now the most used method, but on a finite interval, when the curvature changes in function approximation accuracy may be very large poor, such as Rung example; interpolation can be used in a wide range of approximation, but this interpolation approach to meet the very demanding solvability condi-tion, its overall accuracy is difficult to ensure.This article first best average rational approximation are discussed and put forward three important tips to solve rational approximation:the reg-ularization of thought, Newton flow line method to construct a good initial comparison algorithm through several approximation calculations, the results show that the calculation is simple and stable, good function of the nature, the effect is still very good approximation. Secondly, piecewise continuous graphics, we constructed a high-order polynomial are segmented, and is not the best approach. Therefore, the best average of the provisions proposed boundary rational approximation, can reduce the number of times, and is the best approximation. Finally, the use of rational approximations to solve the two point boundary value. Variational principle by minimizing the defor-mation, and then use an alternative approximation rational approximation we require u (x), the solution on a, a very complex problem b nonlinear equations, using Newtonian flow lines method to solve nonlinear equations. ultimate idea is to use rational approximations to solutions of nonlinear boundary val-ue problems and boundary value problems for unbounded regions, research solitary wave problems.