# (2016 Autumn) Math 4052 Partial Differential Equations

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

#### Homework Problems

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

1. (Page 5, Q2(a)(b)). Which of the following operators are linear?
• $\mathscr{ L } u = u_x + x u_y$.
• $\mathscr{ L } u = u_x + u u_y$.
2. (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.
• $u_t - u_{xxt} + u u_x = 0$.
• $u_{tt} - u_{xx} + x^2 = 0$.
3. (Page 9, Q1). Solve the first-order equation $2u_t + 3u_x = 0$ with the auxiliary condition $u=\sin x$ when $t=0$.
• Homework 2 (Due in tutorial class in the week of Sept 19):

1. (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 $k=2$?
2. (Page 31, Q1). What is the type of each of the following equations?
• $u_{xx} - u_{xy} + 2 u _y + u_{yy} - 3 u_{yx} + 4u = 0$.
• $9 u_{xx} + 6 u_{xy} + u_{yy} + u_{x} = 0$.
3. (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):

1. (Page 38, Q1). Solve $u_{tt} = c^2 u_{xx}$, $u(x,0)=e^x$, $u_t(x,0) = \sin x$.
2. (Page 38, Q2). Solve $u_{tt} = c^2 u_{xx}$, $u(x,0)=\log (1+x^2)$, $u_t(x,0) = 4+x$.
3. (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.)
4. (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):

1. (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 $\phi(x) = e^{3x}$.
2. (Page 52, Q4). Solve the diffusion equation above if $\phi(x) = e^{-x}$ for $x>0$ and $\phi(x)=0$ for $x<0$.
3. (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):

1. (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 $$1 = \frac{4}{\pi} \left( \sin \frac{\pi x}{l} + \frac{1}{3}\sin\frac{3 \pi x}{l} + \frac{1}{5}\sin\frac{5 \pi x}{l}+ \cdots \right)$$
2. (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.
3. (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).
1. Show that the eigenfunctions are $\cos\left[ \left( n+\frac{1}{2}\right)\pi x/l\right]$.
2. Write the series expansion for a solution $u(x,t)$.
4. (Page 92, Q3). Solve the Schrödinger equation $u_t = i k u_{xx}$ for real $k$ in the interval $0<x<l$ with the boundary conditions $u_x(0,t) = 0,u(l,t) = 0$.
• Homework 6 (Due in tutorial class in the week of Oct. 16):

1. (Page 45, Q2). Consider a solution of the diffusion equation $u_t = u_{xx}$ in $\{ 0 \leq x \leq l, 0 \leq t < \infty \}$.
1. 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 $T$?
2. 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 $T$?
2. (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 $u(x,0)=4x(1-x)$.
1. Show that $0 < u(x,t) < 1$ for all $t>0$ and $0<x<1$.
2. Show that $u(x,t) = u(1-x,t)$ for all $t\geq 0$ and $0 \leq x \leq 1$.
3. Use the energy method to show that $\int _0 ^1 u^2 dx$ is a strictly decreasing function of $t$.
• Homework 7 (Due in tutorial class in the week of Oct. 31):

1. (Page 111, Q2). Let $\phi(x) \equiv x^2$ for $0 \leq x \leq 1 = l$.
1. Calculate its Fourier sine series.
2. Calculate its Fourier cosine series.
2. (Page 111, Q4). Find the Fourier cosine series of the function $| \sin x |$ in the interval $(-\pi,\pi)$. Use it to find the sums $$\sum_{n=1}^\infty \frac{1}{4 n^2 - 1} \quad \text{and} \quad \sum_{n=1}^\infty \frac{(-1)^n}{4n^2 - 1}.$$
3. (Page 134, Q1). $\sum_{n=0}^\infty (-1)^n x^{2n}$ is a geometric series.
1. Does it converge pointwise in the interval $-1<x<1$ ?
2. Does it converge uniformly in the interval $-1<x<1$ ?
3. Does it converge in the $L^2$ sense in the interval $-1<x<1$ ?
(Hint: You can compute its partial sums explicitly.)
4. (Page 134, Q5). Let $\phi(x)=0$ for $0<x<1$ and $\phi(x)=1$ for $1<x<3$.
1. Find the first four nonzero terms of its Fourier cosine series explicitly.
2. For each $x$ $(0\leq x \leq 3)$, what is the sum of this series?
3. Does it converge to $\phi(x)$ in the $L^2$ sense? Why?
4. Put $x=0$ to find the sum
$$1 + \frac{1}{2} - \frac{1}{4} - \frac{1}{5} + \frac{1}{7} + \frac{1}{8} - \frac{1}{10} - \frac{1}{11} + \dots .$$
5. (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.
1. Find the full Fourier series of $\phi(x)$ in the interval $(-1,1)$.
2. Find the first three nonzero terms explicitly.
3. Does it converge in the mean square sense?
4. Does it converge pointwise?
5. Does it converge uniformly to $\phi(x)$ in the interval $(-1,1)$?
• Homework 8 (Due in tutorial class in the week of Nov. 7):

1. (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 $r=b$.
2. (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.
1. Find the temperature. (Hint: The temperature depends only on the radius.)
2. What are the hottest and coldest temperatures?
3. Can you choose $\gamma$ so that the temperature on its outer boundary is $20\unicode{x2103}$?
3. (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 $y$.)
4. (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):

1. (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:
1. Find the maximum value of $u$ in $\bar{D}$.
2. Calculate the value of $u$ at the origin.
2. (Page 172, Q2). Solve $u_{xx} + u_{yy}=0$ in the disk $D=\{r<a\}$ with the boundary condition $$u = 1 + 3 \sin \theta \quad \text{on } r=a.$$
3. (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 $r\longrightarrow\infty$.
4. (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.
• Homework 10 (Due in tutorial class in the week of Nov. 21):

1. (Page 184, Q2). Prove the uniqueness up to constants of the Neumann problem using the energy method.
2. (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.
3. (Page 187, Q1). Derive the representation formula for harmonic functions in two dimensions: $$u(\mathbf{x}_0) = \frac{1}{2\pi} \int_{\text{bdy} D} \left[ u(\mathbf{x}) \frac{\partial}{\partial n} ( \log |\mathbf{x} - \mathbf{x}_0| ) - \frac{\partial u}{\partial n} \log |\mathbf{x} - \mathbf{x}_0| \right] ds.$$
4. (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):

1. (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.
1. It solves $G’^\prime (x) = 0$ for $x\neq x_0$ (“harmonic”).
2. $G(0) = G(l) = 0$.
3. $G(x)$ is continuous at $x_0$ and $G(x) + \frac{1}{2} | x - x_0 |$ is harmonic at $x_0$.
2. (Page 196, Q6).
1. Find the Green’s function for the half-plane $\{ (x,y): y>0 \}$.
2. Use it to solve the Dirichlet problem in the half-plane with boundary values $h(x)$.
3. Calculate the solution with $u(x,0) = 1$.
3. (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.)
4. (Page 197, Q17).
1. 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 $$\begin{eqnarray} u_{xx} + u_{yy} = 0 \text{ in } Q, \quad u(0,y) = g(y) \text{ for } y>0, \\ u(x,0) = h(x) \text{ for } x>0. \end{eqnarray}$$