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Problem set 12, PDE

Problem 1

$latex \displaystyle \int_{-\infty}^\infty e^{-\pi x^2} e^{-2\pi i x\xi} dx = e^{-\pi\xi^2}$

for any $latex \xi\in\R$. Hint: For each N, calculate the contour integral

$latex \displaystyle \int_\gamma e^{-\pi (z+i\xi)^2} dz$

over the contour $latex \gamma$ around the rectangle with vertices $latex -N, N, N-i\xi, -N-i\xi$. Take $latex N\to\infty$.

Problem 2

$latex \displaystyle \int_{\R^n} e^{-\pi|x|^2} e^{-2\pi i x\cdot \xi} dx = e^{-\pi|\xi|^2}$

for every $latex \xi\in\R^n$.

Problem 3

  1. $latex K(x,t) = t^{-n/2} e^{-|x|^2/4t}$ satisfies the heat equation.

  2. $latex G(x,t) = (1 - 4\alpha t)^{-n/2} e^{\alpha|x|^2/(1-4\alpha t)}$, for any $latex \alpha>0$, also satisfies the heat equation.

Problem 4

Let $latex u_0:[0,\infty)\to\R$ be bounded and continuous, with $latex u_0(0)=0$. Find an integral representation for the solution of the problem

$latex \displaystyle \begin{cases} \partial_t u - \partial^2_x u = 0 & x>0, t>0 \\ u(x,0) = u_0(x) & x>0 \\ u(0,t) = 0 & t>0. \end{cases}$