91Ó°ÊÓ

A Sturm-Liouville eigenvalue problem is called self-adjoint if $$ \left.p\left(u \frac{d v}{d x}-v \frac{d u}{d x}\right)\right|_{a} ^{b}=0 $$ (since then \(\int_{a}^{b}[u L(v)-v L(u)] d x=0\) ) for any two functions \(u\) and \(v\) satisfying the boundary conditions. Show that the following yield self-adjoint problems. (a) \(\phi(0)=0\) and \(\phi(L)=0\) (b) \(\frac{d \phi}{d x}(0)=0\) and \(\phi(L)=0\) (c) \(\frac{d \phi}{d x}(0)-h \phi(0)=0 \quad\) and \(\quad \frac{d \phi}{d x}(L)=0\) (d) \(\phi(a)=\phi(b) \quad\) and \(\quad p(a) \frac{d \phi}{d x}(a)=p(b) \frac{d \phi}{d x}(b)\) (e) \(\phi(a)=\phi(b) \quad\) and \(\quad \frac{d \phi}{d x}(a)=\frac{d \phi}{d x}(b)\) [self-adjoint only if \(p(a)=p(b)\) ] [in the situation in which \(p(0)=0\) ] (f) \(\begin{aligned} \phi(L)-0 & \text { and } \end{aligned} \quad\left[\begin{array}{l}\text { [in the situation in which } p(0)=0] \\\ \phi(0) \text { bounded and } \lim _{x \rightarrow 0} p(x) \frac{d \phi}{d x}=0\end{array}\right.\) *(g) Under what conditions is the following self-adjoint (if \(p\) is constant)? $$ \begin{aligned} \phi(L)+\alpha \phi(0)+\beta \frac{d \phi}{d x}(0) &=0 \\ \frac{d \phi}{d x}(L)+\gamma \phi(0)+\delta \frac{d \phi}{d x}(0) &=0 \end{aligned} $$

Prove that the eigenfunctions corresponding to different eigenvalues (of the following eigenvalue problem) are orthogonal: $$ \frac{d}{d x}\left[p(x) \frac{d \phi}{d x}\right]+q(x) \phi+\lambda \sigma(x) \phi=0 $$ with the boundary conditions $$ \begin{array}{r} \phi(1)=0 \\ -2 \frac{d \phi}{d x}(2)=0 \end{array} $$ What is the weighting function?

Consider heat flow with convection (see Exercise 1.5.2): $$ \frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}-V_{0} \frac{\partial u}{\partial x} $$ (a) Show that the spatial ordinary differential equation obtained by separation of variables is not in Sturm-Liouville form. *(b) Solve the initial boundary value problem $$ \begin{aligned} &u(0, t)=0 \\ &u(L, t)=0 \\ &u(x, 0)=f(x) \end{aligned} $$ (c) Solve the initial boundary value problem $$ \begin{aligned} \frac{\partial u}{\partial x}(0, t) &=0 \\ \frac{\partial u}{\partial x}(L, t) &=0 \\ u(x, 0) &=f(x) \end{aligned} $$

Give an example of an eigenvalue problem with more than one eigenfunction corresponding to an cigenvalue.

For the Sturm-Liouville eigenvalue problem, \(\frac{d^{2} \phi}{d x^{2}}+\lambda \phi=0 \quad\) with \(\quad \frac{d \phi}{d x}(0)=0 \quad\) and \(\quad \frac{d \phi}{d x}(L)=0\) verify the following general properties: (a) There are an infinite number of eigenvalues with a smallest but no largest. (b) The \(n\)th eigenfunction has \(n-1\) zeros. (c) The eigenfunctions are complete and orthogonal. (d) What does the Rayleigh quotient say concerning negative and zero eigenvalues?