一类变系数非线性奇摄动方程的变形坐标法

The Method of Strained Coordinates for a Class of Variable Coefficient Nonlinear Equation

利用变形坐标法,讨论了一类变系数的非线性奇摄动问题:(xn+εym)dy/dx+nxn-1y=1,y(1)=a>1,x∈[0,1],0<ε<<1,m,n为自然数,a为常数.通过与L-P方法的对比和对参数几种不同取值的分类探讨,得到了该变系数非线性奇摄动方程的一致有效的渐近解.并且通过数值模拟,证实了方程的精确解和用变形坐标法得到的渐近解的一致性,从而说明用变形坐标法解此类奇摄动方程的渐近解的有效性.

This paper discusses the following variable coefficient nonlinear singular perturbation problem by using the method of strained coordinates：（x^n＋εy^m）dy/dx＋nx^n-1y=1,y（1）=a〉1,x∈[0,1].0〈ε〈〈1, m,n is natural number, a is a constant. Contrasting the method of L-P and the method of strained coordinates and analyzing the different value of parameter, we get the uniform effective asymptotic solution of the nonlinear singularly perturbed equation. The numerical simulations confirm the correctness by comparing the exact solution and the uniformly valid asymptotic solution. Finally it can be concluded that it is effective to solve this kind of equation by using deformation coordinate method.