论文标题
二阶不均匀线性普通微分方程的相位函数方法
Phase function methods for second order inhomogeneous linear ordinary differential equations
论文作者
论文摘要
众所周知，具有缓慢变化的系数的二阶均匀线性差分方程允许缓慢变化的相函数。该观察结果是liouville-green方法和许多其他方程溶液渐近近似的技术。它也是最近开发的数值算法的基础，在许多感兴趣的情况下，该算法与方程系数的幅度无关，并且与其条件数量所预测的相当的准确性。在这里，我们指出，通过将二阶均质线性普通微分方程与自适应Levin方法的变体相结合以评估振荡积分的变体，可以通过将相位函数方法与评估振荡性积分的变体相结合，从而有效，准确地求解了一大批二阶二阶线性线性差分方程。
It is well known that second order homogeneous linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation underlies the Liouville-Green method and many other techniques for the asymptotic approximation of the solutions of such equations. It is also the basis of a recently developed numerical algorithm that, in many cases of interest, runs in time independent of the magnitude of the equation's coefficients and achieves accuracy on par with that predicted by its condition number. Here we point out that a large class of second order inhomogeneous linear ordinary differential equations can be efficiently and accurately solved by combining phase function methods for second order homogeneous linear ordinary differential equations with a variant of the adaptive Levin method for evaluating oscillatory integrals.
