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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1996 • Question 12

Question

Math 6420 • PDE Fall 1996 • Question 12
A uniform spherical potato at temperature \(T_0\) is placed in the center of a large oven that is kept at a uniform temperature \(T_1\). Estimate how long it takes for the temperature at the center of the potato to reach \(\frac{1}{2}(T_0+T_1)\). Hint: The Laplacian in three dimensions with spherical symmetry is \[ \nabla^2u=\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{du}{dr}\right). \]

Raw LaTeX

A uniform spherical potato at temperature \(T_0\) is placed in
the center of a large oven that is kept at a uniform temperature \(T_1\). Estimate how long
it takes for the temperature at the center of the potato to reach \(\frac{1}{2}(T_0+T_1)\).

Hint: The Laplacian in three dimensions with spherical symmetry is
\[
  \nabla^2u=\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{du}{dr}\right).
\]
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Topic Tags

Elliptic Equations Heat Equation and Parabolic Equations Laplace Equation Linear Programming and Optimization