{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

PRACTICE-Final-2011

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}