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.
翻译:众所周知,具有缓慢变化系数的第二顺序单线性普通差分方程式承认缓慢变化的阶段功能。这一观察是Liouville-Green法和此类方程式解决方案无症状近似的其他许多技术的基础。它也是最近开发的数字算法的基础,在许多有兴趣的情况下,该算法在时间上独立于方程式系数的大小,并达到与其条件号所预测的相等的精确度。我们在这里指出,通过将第二顺序同源线性普通差分方程式的阶段功能法与适应性Levin方法的变异法结合起来,评估骨骼组成部分,可以有效和准确地解决一大批二等同的二等同线性普通差方程式。