
% Problems for 3150, S2013
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\begin{document}
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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 %{Class Time~}\underline{~~~~~~~~}
 \newline


\begin{center}
\bf Math 3150 Problems \\
{\small\bf Chapter 4}
\end{center}

{\small\sf  {\bf Due date}: See the internet due date. Problems are
collected once a week. Records are locked when the stack is
returned. Records are only corrected, never appended.

{\bf Submitted work}. Please submit one package per problem.
 Label each problem with its corresponding problem number,
e.g., \fbox{Prob3.1-4} or \fbox{Xc1.2-4}. Kindly label extra credit
problems with label \fbox{\bf Extra Credit}. You may attach this
printed sheet to simplify your work.

{\bf Labeling}. The label \fbox{Probx.y-z} means the problem is for
chapter \fbox{x}, section \fbox{y}, problem \fbox{z}. When $y=0$,
then the problem does not have a textbook analog, it is a
\BF{background problem}. Otherwise, the problem number should match
a corresponding problem in the textbook. The same labeling applies
to extra credit problems, e.g., \fbox{Xc1.0-4}, \fbox{Xc1.1-2}.}

\section*{Chapter 4: 4.1-4.2 -- Laplacian and Symmetric Drumhead Vibration}

  \Prob{Prob4.1-5}{(Laplacian in Spherical Coordinates)} Represent
 $u(x,y,z)=\left(x^2+y^2+z^2\right)^{3/2}$ in spherical coordinates
 $x=r\cos\phi\sin\theta$, $y=r\sin\phi\sin\theta$, $z=r\cos\theta$
 and then decide if $u$ satisfies Laplace's equation in
 spherical coordinates,
 $$u_{rr}+\frac{2}{r}u_r+\frac{1}{r^2}\left(
 u_{\theta\theta}+\cot\theta \, u_\theta+\csc^2\theta
 \, u_{\phi\phi}\right)=0.$$

\Prob{Xc4.1-9}{(Spherical Laplacian Symmetric Case)} Supply details
when $u(r,\theta,\phi)$ is independent of $\theta$ and $\phi$ to
verify that Laplace's equation
$$u_{rr}+\frac{2}{r}u_r+\frac{1}{r^2}\left(
 u_{\theta\theta}+\cot\theta \, u_\theta+\csc^2\theta\,
 u_{\phi\phi}\right)=0$$ reduces to the
simpler symmetric case equation
$$u_{rr}+\frac{2}{r}u_r=0.$$

\Prob{Prob4.0-1}{(Power Series Method)} Solve $y''+y=x+1$, $y(0)=0$,
$y'(0)=1$ by the power series method to obtain a series solution in
the form
$$y(x) = \sum_{n=0}^\infty a_nx^n.$$
Answer: $\dd y(x)=x+1-\cos x=x+1-\sum_{n=0}^\infty
\frac{(-1)^n}{(2n)!}x^{2n}$. To justify your answer, which may have
a different form, compare the first five nonzero terms of the two
series answers.  REF: Example 3, appendix A.5, page A44

\Prob{Prob4.0-2}{(Euler Differential Equation)} The transformation
pair $x=e^t$, $y(x)=u(t)$ changes the Euler differential equation
$Ax^2y''+Bxy'+Cy=0$ into the constant-coefficient equation
$$A\left(\frac{d^2u}{dt^2}-\frac{du}{dt}\right)+B\frac{du}{dt}+Cu=0,$$
with corresponding characteristic equation $Ar(r-1)+Br+C=0$.
REF: Example 4, appendix A.3, page A24


\begin{quote}\begin{tabular}{ll}
 (a) Solve $x^2y''+4xy'+2y=0$.  &  (c) Solve $x^2y''+xy'+4y=0$.
\\[2ex]
 (b) Solve $2x^2y''+6xy'+2y=0$. &  (d) Solve $x^2y''-xy'+5y=0$.
\end{tabular}\end{quote}

\Prob{Prob4.0-3}{(Frobenius Method)} Solve by the Frobenius method
for $y(x)=\sum_{n=0}^\infty a_nx^{n+r}$, $x>0$, where $r$ is the
largest root of the indicial equation. Find only the first three
nonzero terms of the Frobenius series. Check the answer using a
computer algebra system.
REF: Example 2, appendix A.6, page A55

 (a) $4xy''+6y'+y=0$, $4r(r-1)+6r+0=0$

 (b) $4xy''+2y'+y=0$ $4r(r-1)+2r+0=0$

 (c) $2y''-\frac{1}{x}y'+\frac{2}{x}y=0$, $2r(r-1) - r + 0=0$


\Prob{Prob4.0-4}{(Frobenius Method Case 3)} Solve for two
independent solutions of $xy''-(2+x)y'+2y=0$ using the method of
Frobenius and optionally a computer algebra system.

REF: Example 4, appendix A.6, page A59. 

DETAILS: Indicial equation $r(r-1)-2r+0=0$,
with roots $r=0$, $r=3$. Then $y_1(x)=x^3\sum_{n=0}^\infty a_m x^m$
and $y_2(x)=k y_1(x)\ln|x|+x^0\sum_{n=0}^\infty b_nx^n$.

MAPLE: There is no log term. The coefficient of $\_C1$ is $y_1(x)=
1+x/4+x^2/20+\cdots$
 and
the coefficient of $\_C2$ is $y_2(x)=12+12x+6x^2+\cdots$.
\begin{verbatim}                                    
 de:=x^2*diff(y(x),x,x)-x*(2+x)*diff(y(x),x)+2*x*y(x)=0;
 dsolve({de},y(x),series);
\end{verbatim}

{\small\sf \BF{References}: Asmar {\em PDE and BVP}, Appendix A.5 or A.6,
and Edwards-Penney {\em DE and BVP}, section 8.4, or {\em DE and Linear
Algebra}, chapter 11.}

\Prob{Prob4.2-1}{(Radially Symmetric Drumhead)} Solve the radially
symmetric drumhead problem for $u(r,t)$ on the domain $0<r<2$,
$t>0$:
$$
 \begin{array}{rcl}
\dd u_{tt}(r,t) &=& \dd u_{rr}(r,t)+\frac{1}{r}u_{r}(r,t), \\
 u(2,t) &=& 0,\\
 u(r,0) &=& 0,\\
 \dd u_t(r,0) &=& 1.
 \end{array}
$$

\Prob{Xc4.2-12a}{(Series Identity for $J_0(x)$)} Using Bessel's
equation of order zero,
$$ x^2y''+xy'+x^2y=0,$$
derive from the Frobenius method the series formula
$$J_0(x)=\sum_{n=0}^\infty \frac{(-1)^n}{4^n(n!)^2}x^{2n}.$$
Observe that this is a special solution obtained from the Frobenius
series $y=\sum_{n=0}^\infty a_nx^n$ by taking $a_0=1$.

\Prob{Xc4.2-12b}{(Bessel Function Identities)} Establish the
identities
$$
 \begin{array}{lcl}
   \dd \int J_1(x)dx &=& -J_0(x)+c, \\
   \dd \int xJ_0(x)dx &=& xJ_1(x)+c.
 \end{array}
$$

\newpage
\section*{Chapter 4: 4.3-4.4 -- Non-Symmetric Drumhead and 2D Wave Equation}



 \Prob{Prob4.3-3}{(Non-Symmetric
Drumhead)} Solve the drumhead problem for $u(r,\theta,t)$. Plot the
drumhead for $t=0,1,3$ using a truncated series of four nonzero
numerically approximated coefficients. Assume in the problem
statement $ 0<r<2$, $0<\theta<2\pi$, $t>0$.
$$
 \begin{array}{rcl}
 u_{tt}(r,\theta,t) &=& \dd u_{rr}(r,\theta,t)+\frac{1}{r}u_{r}(r,\theta,t)+\frac{1}{r^2}u_{\theta\theta}(r,\theta,t), \\
 u(2,\theta,t) &=& 0,\\
 u(r,0,t) &=& u(r,2\pi,t),\\
 \dd u_\theta(r,0,t) &=& \dd u_\theta(r,2\pi,t), \\
 u(r,\theta,0) &=& \dd (4-r^2)r\sin\theta,\\
 \dd u_t(r,\theta,0) &=& 1.
 \end{array}
$$
%$$
% \begin{array}{rcl}
% u_{tt} &=& c^2\left(u_{rr}+\frac{1}{r}u_{rr}+\frac{1}{r^2}u_{\theta\theta}\right),\,\, 0<r<a, \, 0<\theta<2\pi,\, t>0, \\
% u(a,\theta,t) &=& 0,\,\, 0 < \theta < 2\pi,\, t > 0,\\
% u(r,0,t) &=& u(r,2\pi,t,\,\, 0<r<a,\, t>0,\\
% u_\theta(r,0,t) &=& u_\theta(r,2\pi,t),\,\, 0<r<2,\, t>0, \\
% u(r,\theta,0) &=& f(r,\theta),\\
% u_t(r,\theta,0) &=& g(r,\theta).
% \end{array}
%$$

 \Prob{Xc4.3-13}{(Two-Dimensional Heat Conduction)} Solve
the heat conduction problem in a circular plate for $u(r,\theta,t)$.
Assume in the problem statement $0<r<1$, $0 < \theta < 2\pi$, $t>0$.
$$
 \begin{array}{rcl}
 u_{t}(r,\theta,t) &=& \dd u_{rr}(r,\theta,t)+\frac{1}{r}u_{r}(r,\theta,t)+\frac{1}{r^2}u_{\theta\theta}(r,\theta,t), \\
 u(1,\theta,t) &=& \sin 3\theta,\\
 u(r,\theta,0) &=& 0.
 \end{array}
$$

\section*{Chapter 4: 4.4 --     Steady-State Temperature in a Disk}
 % Section 4.4: 5, 15, 17;

 \Prob{Prob4.4-5}{(Dirichlet Problem on a Disk)} Solve the
 unit disk steady-state heat problem on $0<r<1$, $0\le \theta < 2\pi$ for $u(r,\theta)$:
% $$
% \begin{array}{rcl}
% \dd u_{rr}+\frac{1}{r}u_{rr}+\frac{1}{r^2}u_{\theta\theta} &=& 0, \\
% \dd u(a,\theta) &=& = f(\theta).
% \end{array}
%$$
 $$
  \begin{array}{ll}
 \begin{array}{rcl}
 \dd u_{rr}(r,\theta)+\frac{1}{r}u_{r}(r,\theta)+\frac{1}{r^2}u_{\theta\theta}(r,\theta) &=&
 0,
 \end{array}
 \\
 \begin{array}{rcl}
 \dd u(1,\theta) &=&
 \left\{
   \begin{array}{rl}
       100 &    0 \le \theta \le 0.25\pi, \\
         0 & 0.25\pi<\theta < 2\pi .
   \end{array}\right.
 \end{array}
 \end{array}
$$

 \Prob{Prob4.4-15a}{(Exterior Dirichlet Problem on a Disk)}
Solve the
 unit disk exterior steady-state heat problem on $r>1$, $0\le \theta < 2\pi$ for $u(r,\theta)$:
% $$
% \begin{array}{rcl}
% \dd u_{rr}+\frac{1}{r}u_{rr}+\frac{1}{r^2}u_{\theta\theta} &=& 0, \\
% \dd u(a,\theta) &=& = f(\theta).
% \end{array}
%$$
 $$
  \begin{array}{ll}
 \begin{array}{rcl}
 \dd u_{rr}(r,\theta)+\frac{1}{r}u_{r}(r,\theta)+\frac{1}{r^2}u_{\theta\theta}(r,\theta) &=&
 0,
 \end{array}
 \\
 \begin{array}{rcl}
 \dd u(1,\theta) &=&
 \left\{
   \begin{array}{rl}
       100 &    0 \le \theta \le 0.25\pi, \\
         0 & 0.25\pi<\theta < 2\pi .
   \end{array}\right.
 \end{array}
 \end{array}
$$

 \Prob{Xc4.4-11}{(Dirichlet Series Formula)} Establish the identity
 $$ \sum_{n=1}^\infty \frac{u^n}{n}\sin(n\theta) =
 \arctan\left(\frac{u\sin\theta}{1-u\cos\theta}\right).$$

 {\small\sf \BF{Hint}: 
Use the Taylor expansion (1)  $\sum_{n=1}^\infty \frac{z^n}{n}= -ln(1-z)$.
Replace $u$ by $r$ to derive the identity. 
Let $z=r e^{i\theta}$ and expand the Taylor series into real and complex
series. Then the imaginary part of equation (1) is the equality desired.
Decoding the RHS involves writing $1-z=(1-x)+(-y)i= r_1 e^{i \theta_1} $ 
followed by using the formula $-\tan(\theta_1) = -(-y)/(1-x)$.}

\newpage
 \Prob{Prob4.4-15b}{(Cartesian Coordinates)}
 Use the {\em Dirichlet series formula} with $u=1/r$ to convert to rectangular $xy$-coordinates the exterior
 Dirichlet problem solution in polar form
 $$
 u(r,\theta) = \frac{25}{2} + \frac{100}{\pi}\sum_{n=1}^\infty
 \frac{r^{-n}(1-(-1)^n)}{n} \sin (n\theta).$$

{\small\sf \BF{Reference}: Asmar {\em PDE and BVP}, section 4.4,
where you will find several useful
 trigonometric identities. Use $(-1)^n=\cos(n\pi)$, $x=r\cos\theta$,
 $y=r\sin\theta$ for the conversion.}

\BF{Answer}: $u(r,\theta)=\frac{25}{2}+\frac{100}{\pi}\left(
\arctan(v_1)+\arctan(v_2) \right)$ where $\dd
v_1=\frac{\sin\theta}{r-\cos\theta}$ and $\dd
v_2=\frac{\sin\theta}{r+\cos\theta}$. In terms of $x$ and $y$, $\dd v_1=\frac{y}{x^2+y^2-x}$, $\dd v_2=\frac{y}{x^2+y^2+x}$.

 \Prob{Prob4.4-15c}{(Isotherms)} Consider the exterior Dirichlet problem on the unit
 disk, $r>1$, $0\le \theta < 2\pi$,
 $$
  \begin{array}{ll}
 \begin{array}{rcl}
 \dd u_{rr}(r,\theta)+\frac{1}{r}u_{r}(r,\theta)+\frac{1}{r^2}u_{\theta\theta}(r,\theta) &=&
 0,
 \end{array}
 \\
 \begin{array}{rcl}
 \dd u(1,\theta) &=&
 \left\{
   \begin{array}{rl}
       100 &    0 \le \theta \le 0.25\pi, \\
         0 & 0.25\pi<\theta < 2\pi .
   \end{array}\right.
 \end{array}
 \end{array}
$$
The isotherms are the $xy$-plane curves of constant temperature
described by $u(x,y)=T$, where $0\le T \le 100$. They are circles.
Find their equations and plot a representative set of isotherms.

\section*{Chapter 4: 4.5  --     Steady-State Temperature in a Cylinder}
 % Section 4.5: 1


 \Prob{Prob4.5-1}{(Steady-State Temperature in a Cylinder)} 
The radially symmetric case of a Dirchlet problem on a cylinder uses
Laplace's equation in cylindrical coordinates. Solve the boundary value problem.
$$
\begin{array}{l} 
\dd\frac{\partial^2 u}{\partial \rho^2} +
\frac{1}{\rho} \frac{\partial u}{\partial \rho} + \frac{\partial^2
u}{\partial z^2} = 0, \\[2ex]
u(\rho,z)=0 \mbox{~on the bottom ($z=0$)}, \\
u(\rho,z)=0 \mbox{~on lateral surface ($\rho=1$, $0<z<2$)}, \\
u(\rho,z)=100 \mbox{~on the top $z=2$, $0 < \rho < 1$}.
\end{array}
$$

\BF{Answer}: 
$\dd u(\rho,z)=200\sum_{n=1}^\infty A_n J_0(\lambda_n
\rho)\sinh(\lambda_n z)$ where $\lambda_n$ is the $n$th
positive root $x$ of $J_0(x)=0$. The coefficient $A_n$ equals the
reciprocal of $\lambda_n J_1(\lambda_n)\sinh(2\lambda_n)$. Suggestion:
use the formulas developed in section 4.5 of Asmar.

\section*{Chapter 4: 4.6  --     Poisson's Equation on a Disk}
 % Section 4.6: 5

\Prob{Prob4.6-5}{(Poisson's Equation on a Disk)} 
Solve the Poisson disk problem by decompsition into two problems: (1)
Poisson problem with zero boundary conditions, and (2) Dirichlet problem
with non-homogenous boundary condition.
$$
\begin{array}{l} 
\dd\frac{\partial^2 u}{\partial x^2} +
 \frac{\partial^2 u}{\partial y^2} = -u+1, \\[2ex]
u(1,\theta)=0,\quad 0\le\theta\le 2\pi.
\end{array}
$$



\BF{Answer}: The problem is first converted to polar coordinates
$(r,\theta)$, using $u(x,y)=u(r,\theta)$ and $x=r\cos\theta,
y=r\sin\theta$. Then
$ u(r,\theta)=2\sum_{n=1}^\infty A_n J_0(\lambda_n r)$ where
$\lambda_n$ is the $n$th positive root $x$ of $J_0(x)=0$.
The coefficient $A_n$ equals the reciprocal of $\lambda_n
(1-\lambda_n^2)J_1(\lambda_n)$. Suggestion: use the formulas developed
in sections 4.4 and 4.6 of Asmar.



\end{document}
\endinput
 Zhu's problems
 • Section 2.1: 20;
 • Section 2.2: 8, 15;
 • Section 2.3: 5;
 • Section 2.4: 7(a), 15;
 • Section 3.3: 9, 12(a,b);
 • Section 3.4: 15;
 • Section 3.5: 13;
 • Section 3.6: 3;
 • Section 3.7: 5, 13;
 • Section 3.9: 3;
 • Section 4.4: 5, 15, 17;
 • Section 7.1: 8;
 • Section 7.2: 20, 31;
 • Section 7.3: 19.
