Second-order linear differential equations with two turning points
Second-order linear differential equations with two turning points
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DOI：
10.1098/rsta.1975.0023
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发表时间：
1975-03
期刊：
影响因子：
--
通讯作者：
F. W. J. Olver
中科院分区：
文献类型：
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作者：
F. W. J. Olver
对于大值的实参数u，考虑形式为d2w/dx2={u2f(u，a，x)+g(u，a，x)}w的微分方程。其中，x是在开放的，可能无限的区间(x1，x2)上的实变量，a是有界实参数。假设f{u，a，x)和g{u，a，x)在(x1，x2)内无奇异性，且f(u，a，x)恰好有两个零点，这两个零点连续地依赖于a并且与a的某一值重合。除了零点的邻域外，g(u，a，x)的绝对值比U2F(u，a，x)小。通过应用Liouville变换，将微分方程转化为四种标准形式之一，具有连续的系数。然后用抛物线柱面函数构造了解的渐近逼近。这些近似对大的u是有效的，对xε(x1，x2)是一致的，对a也是一致的。每个近似都伴随着一个严格的和现实的误差界。文中还给出了抛物柱面函数的一些新性质。
Differential equations of the form d2w/dx2={u2f(u,a,x)+g(u,a,x)}w are considered for large values of the real parameter u. Here x is a real variable ranging over an open, possibly infinite, interval (x1, x2), and a is a bounded real parameter. It is assumed that f {u, a, x) and g{u,a, x) are free from singularity within (x1, x2), and f (u, a, x) has exactly two zeros, which depend continuously on a and coincide for a certain value of a. Except in the neighbourhoods of the zeros, g(u,a,x) is small in absolute value compared with u2f(u, a, x). By application of the Liouville transformation, the differential equation is converted into one of four standard forms, with continuous coefficients. Asymptotic approximations for the solutions are then constructed in terms of parabolic cylinder functions. These approximations are valid for large u, uniformly with respect to x ε (x1, x2) and also uniformly with respect to a. Each approximation is accompanied by a strict and realistic error bound. The paper also includes some new properties of parabolic cylinder functions.