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{\Large Homework \#2} \\
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{\large Math 427/527}
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{\bf Note : } Math 427 students may do the Math 527 questions for extra credit. You may work in pairs on this assignment, but pairs can only be two 427 students or two 527 students but not mixed pairs.
All plots must have axes labels, and a title. Also, be sure to use appropriate axis limits for each plot. Make your plots interesting!
\begin{enumerate}
\item Show that for non-integer values of $\nu$, the functions $J_\nu(x)$ and
$J_{-\nu}(x)$ are independent.
\item This problem will show you an interesting property of the function $\Gamma(x)$. Create a plot over integer values $0,1,2,\hdots,10$ of the function f(n) = $n!$, where $n!$ is "n-factorial". Now, on the same graph, plot the continuous function $\Gamma(x+1)$ over the domain $x\in [0,10]$. You should see that the gamma function interpolates the factorial values. To demonstrate this property, print a table of values comparing
$\Gamma(n+1)$ and $n!$ for integer values $n=0,1,2,\hdots,10$, and
conclude that $\Gamma(n) = (n-1)!$ (at least for small $n$). Hint : For the plot, use the \code{semilogy} function. For example, to plot
the points $(0,0!), (5,5!)$ and $(10,10!)$, you could use:
\begin{verbatim}
semilogy([0:5:10],[factorial(0:5:10)],'.','markersize',30);
\end{verbatim}
Try "\code{help colon}" at the \Matlab prompt if you are unfamilar with the colon (":") operator.
\item Bessel functions come in two pairs of independent functions
$(J_\nu(x),Y_\nu(x))$ and $(I_\nu(x),K_\nu(x))$, where $J_\nu(x)$ and $I_\nu(x)$ are "first kind" functions and $Y_\nu(x)$ and $K_\nu(x)$ are "second kind" functions. If you do a search on Bessel functions in Wikipedia, you will come across some intimidating formulas. We will de-mystify a few here. {\bf Note} : We use $\nu$ in place of the
$\alpha$ used by Wikipedia.
\begin{enumerate}
\item \label{prob:I} The Modified Bessel function of the first kind of order $\nu$ is defined by
$I_\nu(x) = i^{-\nu} J_\nu(ix)$, where $i=\sqrt{-1}$. Show that $I_v$ satisfies the ODE
\begin{equation*}
x^2 y'' + xy' - (x^2 + \nu^2)y = 0
\end{equation*}
\item Show that for $\nu=0$, the following is exactly the first independent solution
to \Prob{I} that we found in class.
\begin{equation*}
I_\nu(x) = \sum_{m=0}^\infty \frac{1}{m!\Gamma(m+\nu+1)}
\left(\frac{x}{2}\right)^{2m+\nu}
\end{equation*}
For $a_0$, use the choice
\begin{equation*}
a_0 = \frac{1}{2^n n!}
\end{equation*}
\item The conventional choice of a second independent solution to the ODE from \Prob{I} is expressed using the formula
\begin{equation*}
K_\nu(x) = \frac{\pi}{2}\frac{I_{-\nu}(x) - I_\nu(x)}{\sin(\nu \pi)}
\end{equation*}
for any $\nu$. Show that $K_\nu(x)$ satisfies the ODE in \Prob{I}.
\ignore{
You will need to become familiar with a few of the recursion relations relating derivatives of $I_\nu(x)$ to first kind modified Bessel functions of orders
$\nu-1$ and $\nu+1$.
}
\item \label{prob:wiki} Show that the formula in \Fig{wiki}, taken from Wikipedia is equivalent to the series solution for $Y_n(x)$ given by expression (8) in Section 5.5 of the class textbook.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.9\textwidth]{eqn_wiki.png}
\end{center}
\caption{Formula from Wikipedia entry "Bessel function", used in \Prob{wiki}.}
\label{fig:wiki}
\end{figure}
\end{enumerate}
\item \label{prob:probK}
In class, we found the first solution to the differential equation.
\begin{equation*}
xy'' + y' - x y = 0
\end{equation*}
Find a second independent solution as a series solution using the Method of Frobenius.
This solution is related to the Modified Bessel function of the second kind. Show all of your work.
\item Solve the ODE in \Prob{probK} numerically, using \code{ode45} over the domain $x \in[0.5,5]$. Take as your boundary conditions $y(0.5) = y'(0.5) = 1$. Plot your solution.
On the same graph, plot the exact solution (obtained using WolframAlpha, for example) over the domain $[0,5]$. Describe an advantage of having the analytic solution, rather than just a numerical solution for this problem.
\item ({\bf Math 527}) The classic text "Conduction of Heat in Solids", by H. S. Carslaw and J. C. Jaeger (Oxford University Press, 1959) proposes a model for heat flow in a wire. The model geometry is a cylinder of radius $a$ and the distribution of
temperature $T(r)$ in the wire is a function of radius $r$ only. Assuming constant thermal resistivity $R=1/K$, a simple model of heat flow in the wire is given by
\begin{equation}
\frac{1}{r}\frac{d}{dr}\left(r \frac{dT}{dr}\right) + \frac{A_0}{K} = 0, \qquad 0 \le r \le a
\label{eqn:constant}
\end{equation}
where $K$ is the thermal conductivity of the wire and $A_0$ is constant rate of heat production due to Joule heating.
\begin{enumerate}
\item A more realistic model allows the thermal resistivity $R$ to vary linearly with temperature as $R = R_0(1+\alpha(T-T_0)$, where $R_0$ is the resistance at a reference temperature $T_0$ and $\alpha$ is the temperature coefficient of resistivity. Show that the model in \eqn{constant} becomes
\begin{equation}
\frac{1}{r}\frac{d}{dr}\left(r \frac{dT}{dr}\right) + \beta^2 T = -\frac{A_0}{K_0}\left(1 - \alpha T_0\right), \qquad
\beta^2 = \alpha A_0/K_0,
\end{equation}
where $K_0 = 1/R_0$.
\item Find the general solution to this model. {\bf Hint 1} : First solve the homogeneous problem, then find a very simple solution to the non-homogeneous problem. Use the principle of superposition to find the general solution. {\bf Hint 2} : Be sure your solution is physical over the domain $0 \le r \le a$.
\item Suppose the temperature on the surface of the wire is held fixed at $T(a)= T_0$. Find the particular solution to model equation.
\item Find physical values for resistivity $R_0$, rate of heat production, temperature coefficient $\alpha$ at a reference temperature $T_0$, diameter, and a surface temperature $T_0$ for copper wiring. Verify that the units you use are consistent. Plot your solution $T(r)$ using these values. Cite the sources you used.
{\bf Hint} : For the units part of this question, convince yourself that the equation makes sense when $K_0$ is a thermal conductivity. Then to convert between thermal conductivity and electrical conductivity of a metal, use the Wiedemann-Franz Law, which states that
\begin{equation*}
K_0 = LT_0\sigma
\end{equation*}
where $\sigma$ is the electical conductivity ($\si{\per\ohm\per\meter}$) and
$L = \num{2.44e-8}$ is the Lorenz number ($\si{\watt\ohm\per\kelvin\tothe{2}}$).
\end{enumerate}
\end{enumerate}
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