数学代写|Math372 Partial Differential Equations

The problem numbers refer to problems from your text book second edition. I will often assign problems which are not in the text book. Keep in mind that there is a firm ‘no late homework’ policy.
Assignment 1: Assigned Wed 01/18. Due Wed 01/25

1. Sec. 1.1. 3,4
2. Sec. 1.2. 2, 3, 11, 12 You don’t need the ‘coordinate method’ to do 11 , as the hint suggests. It can be done directly using the method of characteristics.
3. Find the general solutions of the PDEs
   (a) $\left(1+x^2\right) \partial_x u+\partial_y u=y u^2$.
   (b) $(1-x y) \partial_x u+\partial_y u+\partial_z u=(y-z) u$.

Assignment 2: Assigned Wed $01 / 25$. Due Wed $02 / 01$

1. Sec. 1.3. $1,2,6,10$
2. Newtons law of cooling says that a body loses heat to it’s surroundings at a rate proportional to the temperature difference. Consider a thin (1D) wire immersed in a medium of constant temperature $\theta_0$, which exchanges heat with the surroundings according to Newtons law. Find a PDE satisfied by the temperature in the wire.
3. Let $\rho(x, y, t)$ be the density of a fluid at time $t$ and position $x, y$. Let $u(x, y, t)$ be the instantaneous velocity of the fluid at time $t$ and position $x, y$, and derive a PDE satisfied by $\rho$. HINT: Assume that mass is conserved, and compute the rate of change of mass in some region $D$.
4. Sec. 1.4. $3,4,6$

Assignment 3: Assigned Wed 02/01. Due Wed 02/08

1. Sec. 2.1. $5,10,11$ [See problem 9 for a hint on 10.]
2. Sec. 2.2. $2,3$.
3. (a) Suppose $u$ satisfies $u_{t t}-c^2 u_{x x}=0$ on the interval $(a, b)$, for $t>0$. Under what boundary conditions on $u$ is energy is conserved? Prove it, and provide some physical explanation.
   (b) Let $D$ be some region in (2 or 3 ) dimensional space. Suppose $u$ satisfies $u_{t t}-c^2 \triangle u=0$ in the region $D$. Define the energy to be the area/volume integral $E=\int_D\left(u_t^2+c^2|\nabla u|^2\right) d v$. Under what boundary conditions on $u$ is energy conserved? Prove it. Hint: From the previous part you should be able to guess the boundary conditions. For the proof, it requires replacing ‘integration by parts’ from your previous proof by a clever application of the divergence theorem.

Let $D$ be a region in $\mathbb{R}^3$, and $c, r>0$ be constants, and $a, f$ be functions depending only on the spatial variables $x_1, x_2$ and $x_3$. Show that solutions to the PDE
$$
\partial_t^2 u-c^2 \triangle u+a u+r \partial_t u=f
$$
with Dirichlet boundary conditions
$$
u=0 \text { on } \partial D
$$
and initial data
$$
u(x, 0)=\varphi(x) \quad \& \quad \partial_t u(x, 0)=\psi(x)
$$
are unique. That is, if $u_1$ and $u_2$ are two solutions to the above PDE, with the same boundary conditions and initial data, show that they are equal. [Hint: Suppose $u_1$ and $u_2$ are two solutions. Set $v=u_1-u_2$. Now try and cook up some ‘energy’ which will help you show $v$ is 0 . Note, the energy you cook up (if you do it right) won’t be conserved! It will however decrease with time.]