Outline: Derivatives of the Laplace equations, the wave equations and diffusion equation; Methods to solve equations: separation of variables, Fourier series and integrals and characteristics; maximum principles, Green’s functions.

Here you can find homework problems and solutions. The problems are selected from
the text book (*Partial differential equations, an introduction, Walter A. Strauss, The second edition, John Wiley & Sons, Ltd.*) and are listed here for your convenience.

## Basic Info

- Course Name: Partial Differential Equations
- Lecturers: Chair Prof. Xiao-Ping Wang
- Sessions: T1A (Wednesday 16:30-17:20 at Room 1505), T1B (Monday 17:30-18:20 at Room 1511)
- Course Page: http://www.math.ust.hk/~mawang/teaching/math4052/math4052a.html
- Semester: 2016-Autumn

#### Homework Problems

Homework 1 (Due in tutorial class in the week of Sept 12):

- (Page 5, Q2(a)(b)). Which of the following operators are linear?
- $ \mathscr{ L } u = u_x + x u_y $.
- $ \mathscr{ L } u = u_x + u u_y $.

- (Page 5, Q3(c)(d)). For each of the following equations, state the order and
whether it is nonlinear, linear inhomogeneous, or linear homogeneous; provide
reasons.
- $ u_t - u_{xxt} + u u_x = 0 $.
- $ u_{tt} - u_{xx} + x^2 = 0 $.

- (Page 9, Q1). Solve the first-order equation $ 2u_t + 3u_x = 0 $ with the auxiliary condition $ u=\sin x $ when $ t=0 $.

- (Page 5, Q2(a)(b)). Which of the following operators are linear?
Homework 2 (Due in tutorial class in the week of Sept 19):

- (Page 27, Q3). Solve the boundary problem $ u’^\prime = 0 $ for $ 0<x<1 $ with $ u’(0)+k u(0) = 0 $ and $ u’(1)\pm k u(1)=0 $. Do the $+$ and $-$ cases separately. What is special about the case $k=2$?
- (Page 31, Q1). What is the type of each of the following equations?
- $ u_{xx} - u_{xy} + 2 u _y + u_{yy} - 3 u_{yx} + 4u = 0$.
- $ 9 u_{xx} + 6 u_{xy} + u_{yy} + u_{x} = 0$.

- (Page 32, Q6). Consider the equation $ 3u_y + u_{xy} = 0$.
- What is its type?
- Find the general solution. (
*Hint:*Substitute $v=u_y$.) - With the auxiliary conditions $u(x,0)= e^{-3x}$ and $u_y(x,0)=0$, does a solution exist? Is it unique?

Homework 3 (Due in tutorial class in the week of Sept 26):

- (Page 38, Q1). Solve $ u_{tt} = c^2 u_{xx}$, $u(x,0)=e^x$, $u_t(x,0) = \sin x$.
- (Page 38, Q2). Solve $ u_{tt} = c^2 u_{xx}$, $u(x,0)=\log (1+x^2)$, $u_t(x,0) = 4+x$.
- (Page 38, Q9). Solve $u_{xx} - 3 u_{xt} - 4u_{tt} = 0$,
$u(x,0)=x^2$, $u_t(x,0)=e^x$. (
*Hint:*Factor the operator as we did for the wave equation.) - (Page 41, Q4). If $u(x,t)$ satisfies the wave equation $u_{tt} = u_{xx}$, prove the identity $$u(x+h,t+k) + u(x-h,t-k) = u(x+k,t+h) + u(x-k,t-h)$$ for all $x,t,h$, and $k$. Sketch the quadrilateral $Q$ whose vertices are the arguments in the identity.

Homework 4 (Due in tutorial class in the week of Oct. 3):

- (Page 52, Q3). Use the method of Green’s function to solve the diffusion equation $u_t = k u_{xx}$, subject to the initial condition $u(x,0)=\phi(x)$, where $ \phi(x) = e^{3x} $.
- (Page 52, Q4). Solve the diffusion equation above if $\phi(x) = e^{-x}$ for $x>0$ and $\phi(x)=0$ for $x<0$.
- (Page 52, Q5). Prove properties (a)-(e) seen on Page 47 of the diffusion equation $u_t = k u_{xx}$.

Homework 5 (Due in tutorial class in the week of Oct. 10):

- (Page 89, Q2). Consider a metal rod ($0<x<l$), insulated along its sides but not at its ends, which is initially at temperature $=1$. Suddenly both ends are plunged into a bath of temperature $=0$. Write the differential equation, boundary conditions, and initial condition. Write the formula for the temperature $u(x,t)$ at later times. In this problem,
*assume*the infinite series expansion $$ 1 = \frac{4}{\pi} \left( \sin \frac{\pi x}{l} + \frac{1}{3}\sin\frac{3 \pi x}{l} + \frac{1}{5}\sin\frac{5 \pi x}{l}+ \cdots \right) $$ - (Page 89, Q4). Consider waves in a resistant medium that satisfy the problem $$ u_{tt} = c^2 u_{xx} - r u_t \quad \text{for} \quad 0<x<l $$ $$ u= 0 \quad \text{at both ends} $$ $$ u(x,0)= \phi(x) \quad u_{t}(x,0) = \psi(x),$$ where $r$ is a constant, $0<r<2\pi c/l$. Write down the series expansion of the solution.
- (Page 92, Q2). Consider the equation $u_{tt}=c^2 u_{xx}$ for $0<x<l$, with the boundary conditions $u_x(0,t) = 0,u(l,t)=0$ (Neumann at the left, Dirichlet at the right).
- Show that the eigenfunctions are $\cos\left[ \left( n+\frac{1}{2}\right)\pi x/l\right]$.
- Write the series expansion for a solution $u(x,t)$.

- (Page 92, Q3). Solve the SchrÃ¶dinger equation $u_t = i k u_{xx}$ for real $k$ in the interval $0<x<l$ with the boundary conditions $u_x(0,t) = 0,u(l,t) = 0$.

- (Page 89, Q2). Consider a metal rod ($0<x<l$), insulated along its sides but not at its ends, which is initially at temperature $=1$. Suddenly both ends are plunged into a bath of temperature $=0$. Write the differential equation, boundary conditions, and initial condition. Write the formula for the temperature $u(x,t)$ at later times. In this problem,
Homework 6 (Due in tutorial class in the week of Oct. 16):

- (Page 45, Q2). Consider a solution of the diffusion equation
$u_t = u_{xx}$ in $\{ 0 \leq x \leq l, 0 \leq t < \infty \}$.
- Let $M(T) = $ the maximum of $u(x,t)$ in the closed rectangle $\{ 0 \leq x \leq l, 0 \leq t \leq T \}$. Does $M(T)$ increase or decrease as a function of $T$?
- Let $m(T) = $ the minimum of $u(x,t)$ in the closed rectangle $\{ 0 \leq x \leq l, 0 \leq t \leq T \}$. Does $m(T)$ increase or decrease as a function of $T$?

- (Page 46, Q4). Consider the diffusion equation $u_t = u_{xx}$ in $\{ 0<x<1,0<t< \infty \}$
with $u(0,t)=u(1,t)=0$ and $u(x,0)=4x(1-x)$.
- Show that $ 0 < u(x,t) < 1 $ for all $t>0$ and $0<x<1$.
- Show that $ u(x,t) = u(1-x,t) $ for all $t\geq 0$ and $0 \leq x \leq 1$.
- Use the energy method to show that $ \int _0 ^1 u^2 dx$ is a strictly decreasing function of $t$.

- (Page 45, Q2). Consider a solution of the diffusion equation
$u_t = u_{xx}$ in $\{ 0 \leq x \leq l, 0 \leq t < \infty \}$.
Homework 7 (Due in tutorial class in the week of Oct. 31):

- (Page 111, Q2). Let $\phi(x) \equiv x^2$ for $0 \leq x \leq 1 = l$.
- Calculate its Fourier sine series.
- Calculate its Fourier cosine series.

- (Page 111, Q4). Find the Fourier cosine series of the function $ | \sin x | $ in the interval $(-\pi,\pi)$. Use it to find the sums $$ \sum_{n=1}^\infty \frac{1}{4 n^2 - 1} \quad \text{and} \quad \sum_{n=1}^\infty \frac{(-1)^n}{4n^2 - 1}. $$
- (Page 134, Q1). $\sum_{n=0}^\infty (-1)^n x^{2n}$ is a
geometric series.
- Does it converge pointwise in the interval $-1<x<1$ ?
- Does it converge uniformly in the interval $-1<x<1$ ?
- Does it converge in the $L^2$ sense in the interval $-1<x<1$ ?

(*Hint*: You can compute its partial sums explicitly.)

- (Page 134, Q5). Let $\phi(x)=0$ for $0<x<1$ and $\phi(x)=1$
for $1<x<3$.
- Find the first four nonzero terms of its Fourier cosine series explicitly.
- For each $x$ $(0\leq x \leq 3)$, what is the sum of this series?
- Does it converge to $\phi(x)$ in the $L^2$ sense? Why?
- Put $x=0$ to find the sum

$$ 1 + \frac{1}{2} - \frac{1}{4} - \frac{1}{5} + \frac{1}{7} + \frac{1}{8} - \frac{1}{10} - \frac{1}{11} + \dots .$$

- (Page 134~Page 135, Q7). Let
$$ \phi(x) = \left\{ \begin{align} -1-x \quad & \text{for}\quad -1 < x < 0 \\ +1-x \quad & \text{for}\quad 0 < x < 1. \end{align} \right.$$
- Find the full Fourier series of $\phi(x)$ in the interval $(-1,1)$.
- Find the first three nonzero terms explicitly.
- Does it converge in the mean square sense?
- Does it converge pointwise?
- Does it converge uniformly to $\phi(x)$ in the interval $(-1,1)$?

- (Page 111, Q2). Let $\phi(x) \equiv x^2$ for $0 \leq x \leq 1 = l$.
Homework 8 (Due in tutorial class in the week of Nov. 7):

- (Page 160, Q6). Solve $u_{xx} + u_{yy}=1$ in the annulus $a<r<b$ with $u(x,y)$ vanishing on both parts of the boundary $r=a$ and $r=b$.
- (Page 160, Q9). A spherical shell with inner radius $1$
and outer radius $2$ has a steady-state temperature distribution. Its
inner boundary is held at $100\unicode{x2103}$. Its outer boundary satisfies
$\partial u / \partial r = - \gamma < 0 $, where $\gamma$ is a constant.
- Find the temperature. (
*Hint*: The temperature depends only on the radius.) - What are the hottest and coldest temperatures?
- Can you choose $\gamma$ so that the temperature on its outer boundary is $20\unicode{x2103}$?

- Find the temperature. (
- (Page 164~Page 165, Q1). Solve $u_{xx} + u_{yy}=0$
in the rectangle $0<x<a$, $0<y<b$ with the following boundary conditions:
$$ \begin{align} & u_x = -a \quad \text{on } x=0 \qquad & u_x = 0 \quad \text{on } x=a \\ & u_y = b \quad \text{on } y=0 \qquad & u_y = 0 \quad \text{on } y=b. \end{align}
$$
(
*Hint*: Note that the necessary condition of Exercise 6.1.11 is satisfied. A shortcut is to guess that the solution might be a quadratic polynomial in $x$ and $y$.) - (Page 165, Q4). Find the harmonic function in the square $\{ 0<x<1, 0<y<1 \}$ with the boundary conditions $u(x,0)=x$, $u(x,1)=0$, $u_x(0,y)=0$, $u_x( 1,y )=y^2$.

Homework 9 (Due in tutorial class in the week of Nov. 14):

- (Page 172, Q1). Suppose that $u$ is a harmonic function in the disk $D=\{r<2\}$ and that $u=3 \sin 2 \theta + 1$
for $r=2$. Without finding the solution, answer the following questions:
- Find the maximum value of $u$ in $\bar{D}$.
- Calculate the value of $u$ at the origin.

- (Page 172, Q2). Solve $u_{xx} + u_{yy}=0$ in the disk $D=\{r<a\}$ with the boundary condition $$ u = 1 + 3 \sin \theta \quad \text{on } r=a. $$
- (Page 175, Q1). Solve $u_{xx} + u_{yy}=0$ in the
*exterior*$\{r>a\}$ of a disk, with the boundary condition $u=1+3\sin\theta$ on $r=a$, and the condition at infinity that $u$ be bounded as $r\longrightarrow\infty$. - (Page 176, Q10).Solve $u_{xx} + u_{yy}=0$ in the quarter-disk $\{ x^2 + y^2 < a^2, x>0, y>0 \}$ with the following BCs: $$ u=0 \quad \text{on } x=0 \text{ and on } y=0 \quad \text{and} \quad \frac{\partial u}{\partial r} = 1 \quad \text{on } r=a.$$ Write the answer as an infinite series and write the first two nonzero terms explicitly.

- (Page 172, Q1). Suppose that $u$ is a harmonic function in the disk $D=\{r<2\}$ and that $u=3 \sin 2 \theta + 1$
for $r=2$. Without finding the solution, answer the following questions:
Homework 10 (Due in tutorial class in the week of Nov. 21):

- (Page 184, Q2). Prove the uniqueness up to constants of the Neumann problem using the energy method.
- (Page 184, Q3). Prove the uniqueness of the Robin problem $ \partial u / \partial n + a(\mathbf{x}) u(\mathbf{x}) = h(\mathbf{x})$ provided that $a(\mathbf{x})>0$ on the boundary.
- (Page 187, Q1). Derive the representation formula for harmonic functions in two dimensions: $$ u(\mathbf{x}_0) = \frac{1}{2\pi} \int_{\text{bdy} D} \left[ u(\mathbf{x}) \frac{\partial}{\partial n} ( \log |\mathbf{x} - \mathbf{x}_0| ) - \frac{\partial u}{\partial n} \log |\mathbf{x} - \mathbf{x}_0| \right] ds.$$
- (Page 187, Q2). Let $\phi(\mathbf{x})$ be any $\mathbf{C}^2$ function defined on all of three-dimensional space that vanishes outside some sphere. Show that $$ \phi(\mathbf{0}) = - \iiint \frac{1}{|\mathbf{x}|} \Delta \phi(\mathbf{x}) \frac{d\mathbf{x}}{4\pi}. $$ The integration is taken over the region where $\phi(\mathbf{x})$ is not zero.

Homework 11 (Due in tutorial class in the week of Nov. 28):

- (Page 196, Q1). Find the one-dimensional Green’s function for the interval $(0,l)$. The three properties defining it can be restated as follows.
- It solves $G’^\prime (x) = 0$ for $x\neq x_0$ (“harmonic”).
- $G(0) = G(l) = 0$.
- $G(x)$ is continuous at $x_0$ and $G(x) + \frac{1}{2} | x - x_0 |$ is harmonic at $x_0$.

- (Page 196, Q6).
- Find the Green’s function for the half-plane $ \{ (x,y): y>0 \} $.
- Use it to solve the Dirichlet problem in the half-plane with boundary values $h(x)$.
- Calculate the solution with $u(x,0) = 1$.

- (Page 197, Q9). Find the Green’s function for the tilted half-space $ \{ (x,y,z): ax+by+cz>0 \} $. (
*Hint*: Either do it from scratch by reflecting across the tilted plane, or change variables in the double integral (3) $$ 0 = \iint_{\text{bdy }D} \left( u \frac{\partial H}{\partial n} - \frac{\partial u}{\partial n} H \right) dS $$ using a linear transformation.) - (Page 197, Q17).
- Find the Green’s function for the quadrant
$$ Q = \{ (x,y) : x>0,y>0 \}.$$
(
*Hint*: Either use the method of reflection or reduce to the half-plane problem by the transformation $(x,y) \mapsto (x^2-y^2,2xy)$.) 2, Use your answer in the previous part to solve the Dirichlet problem $$ \begin{eqnarray} u_{xx} + u_{yy} = 0 \text{ in } Q, \quad u(0,y) = g(y) \text{ for } y>0, \\ u(x,0) = h(x) \text{ for } x>0. \end{eqnarray} $$

- Find the Green’s function for the quadrant
$$ Q = \{ (x,y) : x>0,y>0 \}.$$
(

- (Page 196, Q1). Find the one-dimensional Green’s function for the interval $(0,l)$. The three properties defining it can be restated as follows.

#### Answers and Hints

- Homework 1: Answers and Hints.
- Homework 2: Answers and Hints.
- Homework 3: Answers and Hints.
- Homework 4: Answers and Hints.
- Homework 5: Answers and Hints.
- Homework 6: Answers and Hints.
- Homework 7: Answers and Hints.
- Homework 8: Answers and Hints.
- Homework 9: Answers and Hints.
- Homework 10: Answers and Hints.
- Homework 11: Answers and Hints.