Practice-Midterm-2011

# Practice-Midterm-2011 - E4200 Practice Midterm 1 Let f x be...

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Unformatted text preview: E4200 Practice Midterm 1. Let f ( x ) be the 2- periodic function defined on (- 1 , 1) by f ( x ) = ( 1 + x on (- 1 , 0) 1- x on (0 , 1) . 1.1. Sketch the graph of f ( x ). 1.2. Show that f ( x ) is even. 1.3. Show that Z 1 (1- x )cos( nπx ) dx = if n ≥ 2 and n is even 2 ( nπ ) 2 if n ≥ 1 and n is odd . 1.4. We recall that the 2- periodic function f ( x ) can be represented by f ( x ) = a + ∞ X n =1 a n cos( nπx ) + b n sin( nπx ) , where a = 1 2 Z 1- 1 f ( x ) dx, a n = Z 1- 1 f ( x )cos( nπx ) dx, b n = Z 1- 1 f ( x )sin( nπx ) dx. Calculate the coefficients a , a n and b n for n ≥ 1. 1.5 Calculate f ( x ) and its Fourier decomposition. 2. Let 0 ≤ α < 2 πc/L , where c > 0 and L > 0. We consider the problem ∂ 2 u ∂t 2 + α ∂u ∂t = c 2 ∂ 2 u ∂x 2 x ∈ (0 ,L ) , t > u ( t, 0) = 0 , u ( t,L ) = 0 , t > u (0 ,x ) = 0 , ∂u ∂t (0 ,x ) = g ( x ) , x ∈ (0 ,L ) . 2.1. Consider elementary solutions of the form u ( t,x ) = G ( t ) φ ( x ), where u ( t,x ) solves the above equation except for the initial conditions. Find the equations (including possible boundary conditions) that G ( t ) and φ ( x ) must satisfy. 2.2. For the equation ψ 00 + λψ = 0 on (0 ,L ), the Rayleigh quotient states that λ Z L ψ 2 ( x ) dx = Z L ( ψ ) 2 ( x ) dx + ψ (0) ψ (0)- ψ ( L ) ψ ( L ) . Show that the eigenvalues λ associated to the problem for φ obtained in 2.1 must satisfy λ > 0. Solve the equation for ( λ,φ ). 2.3. We recall that the solutions to y 00 + αy + β 2 y = 0 for α < 2 β are given by y = e- αt/ 2 ( A cos( ωt ) + B sin( ωt )) , ω = 1 2 q 4 β 2- α 2 . Recalling that 0 < α < 2 πc/L , solve for G ( t ) obtained in 2.1....
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Practice-Midterm-2011 - E4200 Practice Midterm 1 Let f x be...

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