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s11_mthsc208_ws7a-HeatEqn

s11_mthsc208_ws7a-HeatEqn - MthSc 208(Spring 2011 Worksheet...

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MthSc 208 (Spring 2011) Worksheet 7a MthSc 208: Differential Equations (Spring 2011) In-class Worksheet 7a: The Heat Equation NAME: We will solve for the function u ( x, t ) defined for 0 x π and t 0 which satisfies the following initial value problem of the heat equation: u t = c 2 u xx u (0 , t ) = u ( π, t ) = 0 , u ( x, 0) = x ( π - x ) , (a) Carefully descsribe (and sketch) a physical situation that this models. (b) Assume that u ( x, t ) = f ( x ) g ( t ). Compute u t and u xx , and derive boundary conditions for f ( x ). (c) Plug u = fg back into the PDE and separate variables by dividing both sides of the equation by c 2 fg . Now set this equal to a constant λ , and write down two ODEs: one for f ( x ) and one for g ( t ). Written by M. Macauley 1
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MthSc 208 (Spring 2011) Worksheet 7a (d) Solve the ODE for g ( t ). (e) Solve the ODE for f ( x ) (including boundary conditions), and determine λ . Consider separately the cases when λ = 0, λ = ω 2 > 0, and λ = - ω 2 < 0. Written by M. Macauley 2
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MthSc 208 (Spring 2011) Worksheet 7a (f) Find the general solution of the PDE. As before, it will be a superposition (infinite sum) of solutions
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