Once the temperature in an object reaches a steady state, the heat equation becomes the Laplace equation. Use separation of variables to derive the steady-state solution to the heat equation on the rectangle R = [0, 1] × [0, 1] with the following Dirichlet boundary conditions: u = 0 on the left and right sides; u = f(x) on the bottom; u = g(x) on the top. That is, solve uxx + uyy = 0 subject to u(0, y) = u(1, y) = 0, u(x, 0) = f(x), and u(x, 1) = g(x). You may assume the separation constant is negative: F''/F = −k, for k > 0. Extra credit: Plot u(x, y) when f(x) = sin (πx) and g(x) = sin (2πx).
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