第一种推广为，将方程解的展开式扩展到负指数，可以求得方程的负指数的椭圆函数解。
The first generalization is that the expansion of solutions of equations is generalized to negative exponent, and negative exponential elliptic function solutions are obtained.
重正化群方法已成为获得这类问题精确解的一致有效渐近展开式的有用工具。
Renormalization group method is an effective tool to obtain the uniformly valid asymptotic expansion exact solutions of this kind of problems.
我们利用边界层校正法以及微分不等式理论证明了解的存在定理，并构造出其解的一致有效渐近展开式。
Using the method of boundary layer correction and the differential inequality theory, we prove the existence theorem of solutions and construct the uniformly valid asymptotic expansions of.
利用同伦分析方法，得到了该模型解的近似展开式。
Using the homotopy analysis method, the approximation of the solution for the model was obtained.
对地图投影学中经常遇到的等量纬度正反解展开式进行了新的研究。
The expansions of conformal latitude in map projection are studied in this paper.
然后，运用微分不等式理论，证明了形式渐近解的一致有效性，并得出了解得任意阶的一致有效展开式。
And then, the uniform validity of solution is proved and the uniform valid asymptotic expansions of arbitrary order are obtained by using the theories of differential inequalities.
其收敛性的证明是依据其渐近扩散展开式，在边界层上得到的误差估计逼近其离散纵标方法的解。
Our proof of the convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the discrete-ordinate method.
在一般的条件下，证明了解的存在性，而且得到解及其各导数的高阶一致有效渐近展开式。
Under the general conditions, we prove the existence of the solution and get the asymptotic expansions of the solution and its derivatives, which are uniformly valid for the higher orders.
利用变分迭代理论，简捷地得到了该模型解的近似展开式。
Using the variational iteration theory, the approximation of the solution for its model is obtained for short cut calculation.
通过求解由一阶泰勒展开式得到的线性方程组，避免了为求解此平面而求解非线性方程组最小二乘解的过程，使算法简化。
The first order Taylor series expansion replaces the non-linear equation used in solving this plane, and thus simplifies the algorithm.
在适当条件下，获得了配置解的导数在结点集上成立的精细误差展开式。
It is shown that the first derivative of the corresponding collocation solution admits, under suitable assumptions, a precise error expansion at the knots.
在适当条件下，获得了配置解的导数在结点集上成立的精细误差展开式。
It is shown that the first derivative of the corresponding collocation solution admits, under suitable assumptions, a precise error expansion at the knots.
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