4.5: Problems
- Page ID
- 90940
Prove the if \(u(x)\) and \(v(x)\) satisfy the general homogeneous boundary conditions \[\begin{array}{r} \alpha_{1} u(a)+\beta_{1} u^{\prime}(a)=0, \\ \alpha_{2} u(b)+\beta_{2} u^{\prime}(b)=0 \end{array}\label{eq:1}\] at \(x=a\) and \(x=b\), then \[p(x)\left[u(x) v^{\prime}(x)-v(x) u^{\prime}(x)\right]_{x=a}^{x=b}=0 .\nonumber \]
Prove Green’s Identity \(\int_{a}^{b}(u \mathcal{L} v-v \mathcal{L} u) d x=\left.\left[p\left(u v^{\prime}-v u^{\prime}\right)\right]\right|_{a} ^{b}\) for the general Sturm-Liouville operator \(\mathcal{L}\).
Find the adjoint operator and its domain for \(L u=u^{\prime \prime}+4 u^{\prime}-3 u, u^{\prime}(0)+\) \(4 u(0)=0, u^{\prime}(1)+4 u(1)=0\).
Show that a Sturm-Liouville operator with periodic boundary conditions on \([a, b]\) is self-adjoint if and only if \(p(a)=p(b)\). [Recall, periodic boundary conditions are given as \(u(a)=u(b)\) and \(u^{\prime}(a)=u^{\prime}(b)\).]
The Hermite differential equation is given by \(y^{\prime \prime}-2 x y^{\prime}+\lambda y=0\). Rewrite this equation in self-adjoint form. From the Sturm-Liouville form obtained, verify that the differential operator is self adjoint on \((-\infty, \infty)\). Give the integral form for the orthogonality of the eigenfunctions.
Find the eigenvalues and eigenfunctions of the given Sturm-Liouville problems.
- \(y^{\prime \prime}+\lambda y=0, y^{\prime}(0)=0=y^{\prime}(\pi)\).
- \(\left(x y^{\prime}\right)^{\prime}+\frac{\lambda}{x} y=0, y(1)=y\left(e^{2}\right)=0\).
The eigenvalue problem \(x^{2} y^{\prime \prime}-\lambda x y^{\prime}+\lambda y=0\) with \(y(1)=y(2)=0\) is not a Sturm-Liouville eigenvalue problem. Show that none of the eigenvalues are real by solving this eigenvalue problem.
In Example 4.2.7 we found a bound on the lowest eigenvalue for the given eigenvalue problem.
- Verify the computation in the example.
- Apply the method using \[y(x)=\left\{\begin{array}{cc} x, & 0<x<\frac{1}{2} \\ 1-x, & \frac{1}{2}<x<1 \end{array}\right.\nonumber \] Is this an upper bound on \(\lambda_{1}\)
- Use the Rayleigh quotient to obtain a good upper bound for the lowest eigenvalue of the eigenvalue problem: \(\phi^{\prime \prime}+\left(\lambda-x^{2}\right) \phi=0\), \(\phi(0)=0, \phi^{\prime}(1)=0\).
Use the method of eigenfunction expansions to solve the problems:
- \(y^{\prime \prime}=x^{2}, \quad y(0)=y(1)=0\).
- \(y^{\prime \prime}+4 y=x^{2}, \quad y^{\prime}(0)=y^{\prime}(1)=0\).
Determine the solvability conditions for the nonhomogeneous boundary value problem: \(u^{\prime \prime}+4 u=f(x), u(0)=\alpha, u^{\prime}(\pi / 4)=\beta\).