斯图姆-刘维尔问题特征函数第二类正交性

斯图姆—刘维尔型特征值问题中特征函数存在另外一类正交关系,就是:ba∫[p(x)y′m(x)y′n(x)+q(x)ym(x)yn(x)]dx+hym(a)yn(a)+Hym(b)yn(b)=0(m≠n),称此正交关系为第二类正交关系。采用直接积分法和利用特征值的变分表达式并应用变分原理给出了第二类正交关系的两种不同的证明。以杆的纵振动问题为例,阐明了斯图姆—刘维尔问题特征函数第二类正交关系的物理意义。

The Second Type of Orthogonality of Eigenfunctions of Sturm-Liouville Problem

There is another type of orthogonality of eigenfunctions of the Sturm-Liouville problem as follows:∫ba[p(x)y′m(x)y′n(x)+q(x)ym(x)yn(x)]dx+hym(a)yn(a)+Hym(b)yn(b)=0(m≠n).We give two kinds of methods,that is the directly integral and applied variational expression of the eigenvalues & variational principle,to prove the orthogonality of the second one.Taking the problem of vertical vibration of a rod as an example,the physical meaning of the second type of orthogonality of eigenfunctions of the Sturm-Liouville problem is clarified.