Partial Differential Equations

Assignment 5

1. [5 points] Use Fourier transform to derive the Poisson formula for the upper half plane in 2-dimension.

2. Fundamental Solution

   1. [5 points] Define the fundamental solution:

      $$\begin{equation*} \phi(x,t) = \begin{cases} \frac{1}{(4\pi t)^{n/2}} e^{-\frac{|x|^2}{4t}}, & x \in \mathbb{R}^n, t > 0, \\ 0, & x \in \mathbb{R}^n, t = 0. \end{cases} \end{equation*}$$

      Show that$\phi$satisfies$\phi_t - \Delta \phi = 0$for$x \in \mathbb{R}^n, t > 0$and that

      $$\begin{equation*} \lim\limits_{(x,t)\to(x_0,0)} \phi(x,t) = 0, \quad \text{for } x_0 \neq 0. \end{equation*}$$

   2. [5 points] Show that

      $$\begin{equation*} \int_{\mathbb{R}^n} \phi(x,t)dx = 1 \quad \forall t > 0. \end{equation*}$$

   3. [5 points] For$\delta > 0$, show that

      $$\begin{equation*} \lim\limits_{t\to0} \int_{|x - y| > \delta} \phi(x - y, t)dy = 0. \end{equation*}$$

3. [5 points] Solve the heat equation:

   $$\begin{equation*} \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0, \quad x \in \mathbb{R}, t > 0, \end{equation*}$$

   with initial condition$u(x,0) = f(x)$using Fourier transform (assume appropriate conditions on$f$).

4. [5 points] Consider the equation:

   $$\begin{equation*} \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0, \quad x \in \mathbb{R}, t > 0. \end{equation*}$$

   Find all solutions of the form$u(x,t) = \frac{1}{\sqrt{t}} v\left(\frac{x}{2\sqrt{t}}\right)$.

5. [10 points] Let$E(x,t,r)$be the heat ball and$E(1) = E(0,0,1)$. Show that:

   $$\begin{equation*} \int\int_{E(1)} \frac{|y|^2}{s^2} dyds = 4. \end{equation*}$$

   Use an appropriate transformation to evaluate:

   $$\begin{equation*} \int\int_{E(r)} \frac{|y|^2}{|s|^2} dyds, \end{equation*}$$

   where$E(r) = E(0,0,r)$.

6. [5 points] Define:

   $$\begin{equation*} g(t) = \begin{cases} e^{-\frac{1}{t^\alpha}}, & t > 0, \\ 0, & t \leq 0, \end{cases} \end{equation*}$$

   with$\alpha > 1$, and consider:

   $$\begin{equation*} u(x,t) = \sum\limits_{k=0}^{\infty} \frac{g^{(k)}(t)}{(2k)!} x^{2k}. \end{equation*}$$

   Show that this provides infinitely many solutions to the heat equation with zero boundary conditions.

7. [5 points] Find a sequence of solutions$u_n$of the one-dimensional heat equation:

   $$\begin{equation*} \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0, \quad x \in (0, 2\pi), t > 0, \end{equation*}$$

   with boundary conditions$u(0,t) = u(2\pi,t) = 0$. Using separation of variables, construct a series solution and derive the condition ensuring that$u(x,0) = f(x)$.

8. Show that if$u$satisfies the heat equation$u_t - \Delta u = 0$in$\Omega \times (0,T)$, then the following maximum principle holds:

   $$\begin{equation*} \sup\limits_{\Omega \times (0,T)} u(x,t) = \sup\limits_{\Gamma_T} u(x,t). \end{equation*}$$

9. [5 points] Show that if$u$satisfies:

   $$\begin{equation*} u_t - \Delta u = u, \quad \text{in } \Omega \times (0,T), \end{equation*}$$

   with$u(x,0) = 0$for$x \in \Omega$and$u(x,t) = 0$on$\partial\Omega \times [0,T]$, then$u(x,t) = 0$in$\Omega \times (0,T)$.

10. [5 points] Solve the heat equation:

    $$\begin{equation*} u_t = u_{xx}, \quad x > 0, t > 0, \end{equation*}$$
    with initial and boundary conditions:
    $$\begin{equation*} u(x,0) = g(x), \quad x > 0, \quad u(0,t) = 0, \quad t > 0. \end{equation*}$$
    (Hint: Use an odd extension to rewrite the equation in$\mathbb{R} \times (0,\infty)$.)