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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due March 18 (Friday) }}
\rhead{\thepage} \cfoot{{\sf Math 4340/6340 $|$ Advanced Engineering
    Mathematics $|$ Spring 2022 $|$ Dr.~Macauley}}


\begin{document}

$\;$ \vspace{-6mm}

\begin{center}
  \textsc{Topics: Self-adjoint operators and Sturm-Liouville theory}
\end{center}

%\noindent Read the following:
%\begin{itemize}
%\item Chapters 1 \& 2 of \emph{Visual Group Theory} (VGT). \vspace{-2mm}
%\item VGT Exercises 1.1--1.4, 1.8--1.12, 2.13--2.17. \vspace{-2mm}
%\item The article \emph{Group Think} by Steven Strogatz, which
%  appeared in the NY Times in 2010. \vspace{-2mm}
%\item Francis Su's essay on good mathematical writing.
%\end{itemize} 


\noindent For consistency, we will say that a \emph{Sturm-Liouville
  equation} is a second-order differential equation in the following
\emph{self-adjoint form}:
\begin{equation}
  \label{eq:self-adjoint}
  -\frac{d}{dx}(p(x)y')+q(x)y=\lambda w(x)y\,,
\end{equation}
where $p(x)>0$ and $w(x)>0$ is called the \emph{weight}, or
\emph{density} function. If we divide through by $w(x)$, we can write this
equation as $Ly=\lambda y$, where $L$ is a \emph{self-adjoint} linear
operator. The possible values of $\lambda$ are the \emph{eigenvalues},
and solutions are the \emph{eigenfunctions}.


\begin{enumerate}
  
\item Write the following differential equations in self-adjoint
  form. That is, put them in the above form, and find $p(x)$, $q(x)$,
  and the \emph{weight} $w(x)$. Also, write out the corresponding
  linear operator $L$.
  \begin{enumerate}
  \item Airy's equation: $y''+(\lambda-x)y=0$
  \item Laguerre's equation: $xy''+(1-x)y'+\lambda y=0$
  \item An arbitrary linear equation: $y''+P(x)y'+Q(x)y=\lambda R(x)y$.
    [\emph{Hint}: Multiply through by an \emph{integrating factor},
      $e^{\int P(x)dx}$.]
%  \item The equation $x^3y''-xy'+2y=\lambda y$. 
  \end{enumerate}
  
  
\item In this problem, we will find all solutions to the
  Sturm-Liouville problem
  \[
  -y''=\lambda y,\qquad y'(0)=y(L)=0.
  \]
  \begin{enumerate}
  \item First, suppose that $\lambda=0$. That is, solve $y''=0$,
    $y'(0)=y(L)=0$.
  \item Next, suppose $\lambda=-\omega^2\leq 0$.  That is, solve the
    boundary value problem $y''=\omega^2 y$,
    $y'(0)=y(L)=0$. [\emph{Hint}: When the domain is finite, e.g.,
      $[0,L]$, it is usually more convenient to use $\cosh$ and
      $\sinh$ instead of exponentials.]
  \item Finally, suppose $\lambda=\omega^2>0$. That is, solve
    $y''=-\omega^2 y$, $y'(0)=y(L)=0$.
  \item Summarize the results from Parts (a)--(c) in terms of the
    eigenvalues and corresponding eigenfunctions of a particular
    linear differential operator $L$. What is $L$?
  \item Sketch the first four eigenfunctions on $[0,L]$.
  \end{enumerate}



\item By the main theorem of Sturm-Liouville theory, if we define an
  inner product as
  \begin{equation}
    \label{eq:SL-innerproduct}
    \<f,g\>=\int_a^b f(x)\overline{g(x)}w(x)\,dx\,, 
  \end{equation}
  then the eigenfunctions $\{y_n(x)\}$ form an \emph{orthogonal basis}
  (Note: not necessarily \emph{orthonormal}!) for the space of
  functions, integrable on $[a,b]$ with $\<f,f\><\infty$ that satisfy
  the boundary condtions. This means that for any $f\in \mathsf{L}^2([a,b],w)$
  with the same boundary conditions, we can write
  \[
  f(x)=\sum_{n=1}^\infty c_ny_n(x)\,.
  \]
  \begin{enumerate}
  \item Consider the Sturm-Liouville problem from Part~(c) of the
    previous problem:
    \[
    -y''=\lambda y\,,\qquad y'(0)=0\,,\quad y(L)=0\,.
    \]
    What is $w(x)$?
  \item The function $f(x)=x^2-L^2$ is clearly continuous and
    satisfies $f'(0)=f(L)=0$. Compute the \emph{norm}
    $||f||:=\<f,f\>^{1/2}$ of $f$.
  \item Since the eigenfunctions form a basis for the subspace of
    $\mathsf{L}^2([0,L];w)$ that satisfy the above boundary conditions, we
    can write
    \[
    x^2-L^2=\sum_{n=1}^\infty c_ny_n(x)\,,\qquad 0\leq x\leq L\,.
    \]
    Write down a formula for the $c_n$'s. Leave your answer in terms
    of an integral -- no need to actually compute it!  [\emph{Hint}:
      Don't forget that $y_n(x)$ isn't necessarily of unit length!]
  \end{enumerate}


\item Consider the following Sturm-Liouville problem:
  \[
  -y''-y'=\lambda y\,,\qquad y(0)=0\,\quad y(2)=0\,.
  \]
  \begin{enumerate}
  \item Find the eigenvalues and eigenfunctions. [\emph{Hint}: You
    will encounter a \emph{discriminant} of $D=1-4\lambda$. As before,
    there will be three cases: $D=0$, $D>0$, and $D<0$.]
  \item Write this differential equation in standard form, as in
    Eq.~\eqref{eq:self-adjoint}. [\emph{Hint}: First, multiply through
      by an \emph{integrating factor}, $e^x$.]
  \item Write a formula for $\<y_n,y_m\>$ in terms of an
    integral. What is this integral equal to when $n\neq m$?
  \end{enumerate}


%\newpage

\item Consider the following Sturm-Liouville equation on $[-1,1]$, called
  \emph{Legendre's differential equation}:
  \begin{equation}
    \label{eq:legendre}
    (1-x^2)y''-2xy'+\lambda y=0
  \end{equation}
  In this problem, you will find the eigenvalue and eigenfunctions,
  which have already come up several times in this class in different
  settings.
  \begin{enumerate}
  \item Write Legendre's equation into \emph{self-adjoint form}, as in
    Equation~\ref{eq:self-adjoint}. That is, find $p(x)$, $q(x)$, and
    $w(x)$, and the self-adjoint operator $L$. This is called a
    \emph{singular Sturm-Liouville problem} on the inteveral
    $[a,b]=[-1,1]$ because the function $p(x)$ satisfies
    $p(-1)=p(1)=0$, and so boundary conditions on $y(x)$ are not
    needed.
  \item Assume that there is a power series solution of the form
    $\displaystyle\sum_{n=0}^\infty a_nx^n$. Plug this back into
    Eq.~\eqref{eq:legendre} and find the recurrence relation for the
    coefficients.
  \item Recall from HW 5 that a generalized power series solution will
    have radius of convergence $R=1$, i.e., it will be defined on the
    open interval $(-1,1)$, but \emph{not} on its endpoints, $a=-1$ or
    $b=1$. However, if we have a \emph{polynomial} solution (that is,
    only finitely many non-zero terms, which happens when $a_{n+2}=0$
    for some $n$), then this will certainly be defined on all of
    $[-1,1]$. What values of $\lambda$ lead to a polynomial solution?
    (These are the \emph{eigenvalues} of $L$.)
  \item The \emph{eigenfunction} for eigenvalue $\lambda_k$ is a
    polynomial $P_k(x)$ called the \emph{Legendre polynomial} of
    degree $k$. (These arose on HW 2 and HW 5.) By Sturm-Liouville
    theory, they form an \emph{orthogonal basis} of $\mathsf{L}^2([-1,1])$,
    meaning that
    \[
    \<P_n,P_m\>:=\int_{-1}^1 P_n(x)P_m(x)\,dx=0\,,\qquad n\neq m\,.
    \]
    Use the recurrence relation to write out the first five Legendre
    polynomials, $P_k(x)$, for $k=0,\dots,4$. Normalize each one so
    they form an \emph{orthonormal} set. 
  \item Write the polynomial $f(x)=3x^3-2x^2+4$ using the first four
    Legendre polynomials. That is, find $C_0$, $C_1$, $C_2$, and $C_3$
    such that
    \[
    3x^3-2x^2+4=C_0P_0(x)+C_1P_1(x)+C_2P_2(x)+C_3P_3(x)\,.
    \]
    \emph{Hint}: This is \emph{very} similar to a problem you did on HW 2!
  \end{enumerate}


\end{enumerate}

\end{document}
