论文标题
二阶线性普通微分方程的相位函数方法，带有转折点
Phase function methods for second order linear ordinary differential equations with turning points
论文作者
论文摘要
众所周知，具有缓慢变化系数的二阶线性普通微分方程允许缓慢变化的相函数。该观察结果是liouville-green方法的基础，也是许多其他技术，用于渐近溶液的渐近近似。最近，作者利用它为二阶线性的普通微分方程开发高效的求解器，该方程是振荡的。在许多感兴趣的情况下，算法在时间上达到了与溶液振荡频率无关的时间的最佳精度。在这里，我们表明，经过较小的修改后，它还允许有效地解决具有转折点的二阶微分方程方程。也就是说，对于方程式，其解决方案在某些区域具有振荡性，并且表现得像是在其他区域增加和降低指数函数的线性组合。我们介绍了证明我们方法属性的数值实验的结果，其中包括一些表明它可以用来评估与其依赖参数无关的时间来评估许多经典特殊功能。
It is well known that second order linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation is the basis of the Liouville-Green method and many other techniques for the asymptotic approximation of the solutions of such equations. More recently, it was exploited by the author to develop a highly efficient solver for second order linear ordinary differential equations whose solutions are oscillatory. In many cases of interest, that algorithm achieves near optimal accuracy in time independent of the frequency of oscillation of the solutions. Here we show that, after minor modifications, it also allows for the efficient solution of second order differential equation equations which have turning points. That is, it is effective in the case of equations whose solutions are oscillatory in some regions and behave like linear combinations of increasing and decreasing exponential functions in others. We present the results of numerical experiments demonstrating the properties of our method, including some which show that it can used to evaluate many classical special functions in time independent of the parameters on which they depend.
