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s11_mthsc208_hw20

# s11_mthsc208_hw20 - MthSc 208 Fall 2011(Differential...

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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 ( π,t ) + γ u ( π,t ) = 0 , u ( x, 0) = h ( x ) , where γ is a non-negative constant, and h ( x ) and arbitrary function on [0 ,π ] (a) Describe a physical situation that this models. Be sure to describe the impact of the initial condition, both boundary conditions and the constant γ . (b) What is the steady-state solution, and why? (Use your physical intuition)....
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s11_mthsc208_hw20 - MthSc 208 Fall 2011(Differential...

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