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PRACTICE-Final-2011

# PRACTICE-Final-2011 - Practice Final APMA 4200 2011 PROBLEM...

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Practice Final, APMA 4200, 2011. PROBLEM 1. Consider the function g ( x ) = x for x (0 , π ) , - x for x ( - π, 0) . 1.1. Show that the function is even. 1.2. Calculate its Fourier coefficients. We recall that a 2 π periodic function can be represented as f ( x ) = a 0 + X n =1 a n cos( nx ) + b n sin( nx ) , b n = 1 π Z π - π f ( y ) sin( ny ) dy, a 0 = 1 2 π Z π - π f ( y ) dy, a n = 1 π Z π - π f ( y ) cos( ny ) dy. 1.3. Is the function g ( x ) extended by periodicity a continuous function on the whole line x ( -∞ , )? 1.4. Justify that you can differentiate term by term in the Fourier series expansion of g ( x ) and deduce the Fourier series expansion of the function h ( x ) = 1 for x (0 , π ) , - 1 for x ( - π, 0) . PROBLEM 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) = x for 0 < x < π. (1) 2.1. 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 (including boundary conditions for φ ). 2.2. Solve for G ( t ) knowing G (0). 2.3. 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). 2.4. Write down the most general solution u ( x, t ) that can be constructed from the above method of separation of variables. Justify the formula. 2.5. Solve for u ( x, t ) in (1) as a superposition of sines and cosines in the x variable.

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