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Unformatted text preview: MthSc 208, Fall 2011 (Differential Equations) Dr. Macauley HW 20 Due Monday April 25th, 2011 (1) Let u ( x,t ) be the temperature of a bar of length 10, that is insulated so that no heat can enter or leave. Suppose that initially, the temperature increases linearly from 70 at one endpoint, to 80 at the other endpoint. (a) Sketch the initial heat distribution on the bar, and express it as a function of x . (b) Write down an initial/boundary value problem to which u ( x,t ) is a solution (Let the constant from the heat equation be c 2 ). (c) What will the steady-state solution be? (2) Consider the following PDE: u t = c 2 2 u x 2 , u (0 ,t ) = 0 , u x ( ,t ) + u ( ,t ) = 0 , u ( x, 0) = h ( x ) , where is a non-negative constant, and h ( x ) and arbitrary function on [0 , ] (a) Describe a physical situation that this models. Be sure to describe the impact of the initial condition, both boundary conditions and the constant . (b) What is the steady-state solution, and why? (Use your physical intuition)....
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