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Question: u0 u jo 0 figure 11 dirichlet problem in a...

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u=0 u= Jo 0 Figure 11. Dirichlet problem in a rectangle 4 Problem 1. Consider the Dirichlet problem illustrated in Figure 11. More precisely, we look for a solution of the steady-state heat Au = 0 in the rectangle R = {(z, y) : 0 < x < π, 0 <バ1) that va equation at vanishes the vertical sides of R, and so that u(z, 0) o() and u(, 1) fi(a), and u(x, 1 temperature distribution on the where fo and fi are initial data which fix the horizontal sides of the rectangle. Use separation of variables to show that if fo and fi have Fourier expansions 0o fo(x) = Σ A fl (x) = ΣBk sin kx, k Sin kx and k-1 then (sinh k(1-y)Ag + sinh kyB u(x,y) = u(,y sinh k --Ak sinh k k= 1 We recall the definitions of the hyperbolic sine and cosine functions: sinh x = and cosh x =

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