6.5: Problems
- Page ID
- 106236
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)6.1. Find the adjoint operator and its domain for \(L u=u^{\prime \prime}+4 u^{\prime}-3 u, u^{\prime}(0)+ 4 u(0)=0, u^{\prime}(1)+4 u(1)=0\)
6.2. 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 \(u^{\prime}(a)=u^{\prime}(b)\).]
6.3. 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.
6.4. Find the eigenvalues and eigenfunctions of the given Sturm-Liouville problems.
a. \(y^{\prime \prime}+\lambda y=0, y^{\prime}(0)=0=y^{\prime}(\pi)\).
b. \(\left(x y^{\prime}\right)^{\prime}+\dfrac{\lambda}{x} y=0, y(1)=y\left(e^{2}\right)=0\).
6.5. 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.
6.6. In Example 6.10 we found a bound on the lowest eigenvalue for the given eigenvalue problem.
a. Verify the computation in the example.
b. Apply the method using
\(y(x)=\left\{\begin{array}{cc}
x, & 0<x<\dfrac{1}{2} \\[4pt]
1-x, & \dfrac{1}{2}<x<1
\end{array}\right.\)
Is this an upper bound on \(\lambda_{1}\)
c. Use the Rayleigh quotient to obtain a good upper bound for the lowest eigenvalue of the eigenvalue problem: \(\phi^{\prime \prime}+\left(\lambda-x^{2}\right) \phi=0, \phi(0)=0\), \(\phi^{\prime}(1)=0\)
6.7. Use the method of eigenfunction expansions to solve the problem:
\[y^{\prime \prime}+4 y=x^{2}, \quad y(0)=y(1)=0 \nonumber \]
6.8. Determine the solvability conditions for the nonhomogeneous boundary value problem: \(u^{\prime \prime}+4 u=f(x), u(0)=\alpha, u^{\prime}(1)=\beta\)