# Quiz2 - Quiz 2 Consider the heat equation u 2u = k 2 , 0 <...

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Quiz 2 Consider the heat equation ∂u ∂t = k 2 u ∂x 2 , 0 < x < π, t > 0 ∂u ∂x (0 ,t ) = 0 , ∂u ∂x ( π,t ) = 0 , t > 0 , u ( x, 0) = 1 for 0 < x < π 2 , and u ( x, 0) = - 1 for π 2 < x < π. (1) 1. Using the method of separation of variables u ( x,t ) = G ( t ) φ ( x ), write the ordinary diﬀerential equations that G and φ must satisfy (with their boundary conditions). 2. Solve for G ( t ) knowing G (0). 3. Solve the eigenvalue problem for φ (we assume to save time that φ 00 + λφ = 0 with the proper boundary conditions has no non-trivial solution when λ < 0). 4. Write down the most general solution u ( x,t ) that can be constructed from the above method of separation of variables. Justify the formula. 5. Extending the initial condition by evenness on ( - π,π ) use the Fourier theorem to obtain the solution to (1). Calculate explicitly the coeﬃcients that enter the formula. We recall that for an even function f ( x ), f ( x ) = a 0 + X n =1 a n cos( nx ) , a 0 = 1

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## This note was uploaded on 02/05/2012 for the course APMAE 4200 taught by Professor R during the Fall '11 term at Columbia.

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Quiz2 - Quiz 2 Consider the heat equation u 2u = k 2 , 0 <...

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