hw8.pdf - APM346 HOMEWORK 8 1 Solve the following heat equations In each case identify the equilibrium state(the term in the series expansion of u which

# hw8.pdf - APM346 HOMEWORK 8 1 Solve the following heat...

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APM346 HOMEWORK 8 1. Solve the following heat equations. In each case identify the equilibrium state (the term in the series expansion of u which does not decay as t → ∞ ). (a) ( t - 2 x ) u = 0 , x (0 , 1) , u ( t, 0) = 0 , u ( t, 1) = 1 , u (0 , x ) = x 2 . (b) ( t - 2 x ) u = 0 , x (0 , 1) , x u ( t, 0) = 0 , x u ( t, 1) = 1 , u (0 , x ) = 1 2 x 2 . Note that your ansatz for u should be modeled after the form of solutions to the corresponding homogeneous initial-boundary value problem. 2. Solve the forced wave equation

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