DG Method Of Variable Coefficient Singular Perturbed ODE’s

Ordinary differential equations with Dirichlet boundary arise widely in many fields,like science and technology and economic and so on.Many other methods had been used to solve this problem, such as difference methods, spectral methods and continuous finite element methods. The DG method developed in recent years is widely used,people are increasingly keen on this method for solving various kinds of equations.The best advantage is the ap-proximate solution of the problem is stable, and there were no oscillation phe-nomena;secondly,the solution of the problem is high accuracy and there is a super convergence; Another advantage is that it looses demand of the smooth-ness of the solution to the equations.This is also the tremendous impetus to apply it to solve various kinds of problem in mathematical and physics.This paper mainly discusses how to solve the problem of the singularly per-turbed equations with variable coefficients and Dirichlet boundary condition by DG method eux+b(x)u=f(x), u(0)=u0, b(x)=1-x, x∈J=[0,1]., There are new features of this problem:on the one hand,with various co-efficients b(x)=1-x and it turns to zero at the point x=1; on the other hand,with double singularity.In this problem, x=0is singular point while x=1is a turn point with inner boundary layer where the solution is u=0(e-1/2). the area near the singular point x=0is the boundary layer, the error line is Radau type by DG method as the constant coefficient situation when τ=(m+1)e|lne/b(0). but we meet the essential difficulty when x>τ.The work and the innovations we mainly do in this paper as follows:1:The interval we discussed is divided into three pieces,the singular interval J0=(0, τ),τ=(m+1)e|lne|/6(0); the smooth interval J1=(τ,1-T), T=c(?) and the inner boundary layer J2=(1-T,1). Based on the solution of the problem, we research the regular properties in each interval.2:The thickness of the inner boundary layer is found accurately τ (?) and N is the right number of the subdivision.c can be determined according to the derivation.According to the theoretical analysis,we used the variablestep in J1interval hj=hb(xj), and keep the error line on Gauss type in J1interval by DG method and the super convergence order O(hm+2).3:After determined the thickness of the inner boundary layer,we used uniform grid in J2interval,and keep the error line on Radau type.