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{ \large \textbf{\textsc{Math 5587 -- Homework 7 (Due Thursday Oct 27)}}}
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\begin{enumerate}
\item Consider the full Fourier series on the interval $-\ell \leq x \leq \ell$ for a function $f(x)$
\[f(x) = \frac{1}{2}A_0 + \sum_{n=1}^\infty A_n \cos\left(\frac{n\pi x}{\ell}\right) + B_n\sin\left(\frac{n\pi x}{\ell}\right),\]
where
\[A_n=\frac{1}{\ell}\int_{-\ell}^\ell f(x) \cos\left(\frac{n\pi x}{\ell}\right) \, dx \  \ \text{ and } \ \ B_n =\frac{1}{\ell}\int_{-\ell}^\ell f(x)\sin\left(\frac{n\pi x}{\ell}\right) \, dx.\]
\begin{enumerate}
\item Assume that $f$ is an even function. Show that 
\[f(x) = \frac{1}{2}A_0 + \sum_{n=1}^\infty A_n \cos\left(\frac{n\pi x}{\ell}\right),\]
where
\[A_n=\frac{2}{\ell}\int_{0}^\ell f(x) \cos\left(\frac{n\pi x}{\ell}\right) \, dx.\]
\item Assume that $f$ is an odd function. Show that 
\[f(x) = \sum_{n=1}^\infty B_n \sin\left(\frac{n\pi x}{\ell}\right),\]
where
\[B_n=\frac{2}{\ell}\int_{0}^\ell f(x) \sin\left(\frac{n\pi x}{\ell}\right) \, dx.\]
\end{enumerate}
\item Let $f$ be a $2\pi$-periodic $k$-times continuously differentiable function and consider the complex version of the Fourier series
\[f(x) = \sum_{n=-\infty}^\infty c_n e^{inx} \ \ \text{ where } \ \ c_n = \frac{1}{2\pi}\int_{-\pi}^\pi f(x) e^{-inx}\, dx.\]
Recall that for complex-valued functions $f$ and $g$ on the interval $[-\pi,\pi]$ we define
\[(f,g) = \int_{-\pi}^\pi f(x)\bar{g(x)} \, dx \ \ \text{ and } \ \ \|f\| = (f,f)^\frac{1}{2}.\]
\begin{enumerate}
\item Let $g$ be any $2\pi$-periodic complex-valued differentiable function. Use integration by parts to show that
\[(f,g')  = -(f',g)\]
[Hint: Differentiation and complex conjugation commute, so $\frac{d}{dx} \left(\bar{g(x)}\right)= \bar{g'(x)}$.]
\item Use part (a) to show that
\[c_n = \frac{1}{2\pi(in)^k} \left(f^{(k)},e^{inx}\right),\]
where $f^{(k)}$ denotes the $k^{\rm th}$ derivative of $f$. [Hint: Start by using part (a) to show that 
\[c_n = \frac{1}{2\pi in} \left(f',e^{inx}\right).\]
Then repeat the argument $k$ times (or use induction).]
\item Use part (b) and the Cauchy-Schwarz inequality (HW 6) to show that for any $k\geq 1$
\[|c_n| \leq  \frac{L_k}{n^k}  \ \ \text{ where } \ \ L_k = \frac{1}{\sqrt{2\pi}} \|f^{(k)}\|.\]
That is, if $f$ is $k$-times continuously differentiable then the Fourier coefficients $c_n$ of $f$ decay to zero at least as fast as $1/n^k$.
\end{enumerate}
\item Let $X(x)$ be a solution of the eigenvalue problem
\[X'' + \lambda X(x) = 0 \ \ \text{ for } a \leq x \leq b,\]
where $\lambda$ is a real number.
\begin{enumerate}
\item Assume that
\begin{equation}\label{eq:nonneg}
XX'\Big\vert_a^b = X(b)X'(b) - X(a)X'(a) \leq 0.
\end{equation} 
Show that if $X$ is not the zero function, then $\lambda \geq 0$. Hence \eqref{eq:nonneg} guarantees there are no negative eigenvalues. [Hint: Show that $\lambda \|X\|^2 = \lambda(X,X) = -(X'',X)$. Then integrate by parts and use \eqref{eq:nonneg} to show that $\lambda\|X\|^2 \geq 0$.]
\item Show that \eqref{eq:nonneg} holds when $X$ satisfies 
\begin{enumerate}
\item Homogeneous Dirichlet boundary conditions $X(a)=X(b)=0$,
\item Homogeneous Neumann boundary conditions $X'(a)=X'(b)=0$,
\item Mixed boundary conditions $X'(a)=X(b)=0$ or $X(a)=X'(b)=0$,
\item Periodic boundary conditions $X(a)=X(b)$ and $X'(a)=X'(b)$.
\end{enumerate}
\end{enumerate}
\item Let $X_1(x)$ and $X_2(x)$ be solutions of the eigenvalue problems
\[X_1'' + \lambda_1 X_1(x) = 0 \ \ \text{ for } a \leq x \leq b,\]
and
\[X_2'' + \lambda_2 X_2(x) = 0 \ \ \text{ for } a \leq x \leq b,\]
for different eigenvalues $\lambda_1 \neq \lambda_2$.
\begin{enumerate}
\item Show that
\[-X_1''X_2 + X_1X_2'' = \frac{d}{dx} (-X_1'X_2 + X_1X_2').\]
[Hint: Use product rule to expand the right hand side.]
\item Use part (a) and the fundamental theorem of calculus to show that
\[(\lambda_1-\lambda_2)(X_1,X_2) = (-X_1'X_2 + X_1X_2')\Big\vert_a^b.\]
\item Conclude from part (b) that $X_1$ and $X_2$ are orthogonal (i.e., $(X_1,X_2)=0$) if and only if
\begin{equation}\label{eq:orth}
(-X_1'X_2 + X_1X_2')\Big\vert_a^b = 0.
\end{equation}
\item Boundary conditions are called \emph{symmetric} whenever \eqref{eq:orth} holds. By part (b), whenever we have symmetric boundary conditions, eigenfunctions corresponding to different eigenvalues are always orthogonal. Show that each of the boundary conditions listed below are symmetric.
\begin{enumerate}
\item Homogeneous Dirichlet boundary conditions $X_i(a)=X_i(b)=0$,
\item Homogeneous Neumann boundary conditions $X_i'(a)=X_i'(b)=0$,
\item Periodic boundary conditions $X_i(a)=X_i(b)$ and $X_i'(a)=X_i'(b)$,
\item Robin boundary conditions $X_i(a) - AX'_i(a) = X_i(b) + AX'_i(b) = 0$, where $A$ is a real number.
\end{enumerate}
In each case, the boundary conditions are assumed to hold for both $i=1$ and $i=2$.
\end{enumerate}
\item Solve the heat equation 
\[\left\{\begin{aligned}
u_t - u_{xx} &=0,&&\text{ for } 0 < x < \pi \text{ and } t>0\\
u(0,t) &=0,&&\text{ for } t>0\\
u(\pi,t)&=\pi,&&\text{ for } t>0\\
u(x,0) &= \cos(x),&&\text{ for } 0 \leq x \leq \pi.
\end{aligned}\right.\]
Give a complete answer, including the values of the coefficients $B_n$ of the series. After solving for $u$, compute
\[\lim_{t\to \infty} u(x,t).\]
[Hint: First define $v(x,t) = u(x,t) - x$ to convert the problem into one with homogenous Dirichlet conditions $v(0,t)=v(\pi,t)=0$. Then solve for $v$ instead.]
\end{enumerate}


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