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Unformatted text preview: E4200 2011 Midterm Problem 1. Consider the wave equation for u ( x,t ): ∂ 2 u ∂t 2 = c 2 ∂ 2 u ∂x 2 , < x < π, t > ∂u ∂x (0 ,t ) = 0 , ∂u ∂x ( π,t ) = 0 , t > , u ( x, 0) = x < x < π ∂u ∂t ( x, 0) = 1 < x < π 2- 1 π 2 < x < π. (1) 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 including the boundary conditions for φ . You are asked to write an equation for φ that does not depend on the constant velocity c . 1.2. Assuming that the eigenvalues in the eigenvalue problem for φ are non negative, i.e., λ ≥ 0, solve for G ( t ) knowing G (0) and G (0) (you need to separate the cases λ = 0 and λ > 0). 1.3. Use the Rayleigh quotient to show that the eigenvalues λ in the eigenvalue problem for φ are non- negative, i.e., λ ≥ 0. We recall that the Rayleigh quotient is given in this context by λ =- φφ π + Z π ( φ ) 2 dx Z π φ 2 dx . Can we have λ = 0? 1.4. Solve the eigenvalue problem for φ . 1.5. Write down the elementary solutions u n ( x,t ) and the most general solution u ( x,t ) of the above PDE (except for the initial conditions) that can be constructed by the method of separation of variables. 1.6. 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 (you may want to use integrations by parts). We recall that for an even function f ( x ), f ( x ) = a + ∞ X n =1 a n cos( nx ) , a = 1 π Z π f ( y ) dy, a n = 2 π Z π f ( y )cos( ny ) dy. Problem 2. Consider the problem for u ( x,t ): √ 2 + x 4 ∂u ∂t = ∂ ∂x √ 2 + x 4 ∂u ∂x ,- 1 < x < 1 , t > ∂u ∂x (- 1 ,t ) = ∂u ∂x (1 ,t ) = 0 , t > u ( x, 0) = f ( x ) ,- 1 < x < 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.
- Fall '11