Problem set 12, PDE

Problem 1

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

for any \xi\in\RHint: For each N, calculate the contour integral

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

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

Problem 2

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

for every \xi\in\R^n.

Problem 3

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

Problem 4

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

\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}

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