hw11Marsh2008

hw11Marsh2008 - u ( x, t ). (d) Write a MATLAB program to...

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CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 11 Posted Wednesday 26 March 2008. Due Wednesday 2 April 2008 in class. 1. [50 points] Consider the wave equation 2 u ∂t 2 = c 2 2 u ∂x 2 for 0 x 1 and t 0 subject to the mixed boundary conditions u (0 , t ) = 0 , ∂u ∂x (1 , t ) = 0 for all t 0 and initial conditions u ( x, 0) = ψ ( x ) = s n =1 b n φ n ( x ) , ∂u ∂t ( x, 0) = γ ( x ) = s n =1 d n φ n ( x ) , where the functions φ n are the eigenfunctions φ n ( x ) = 2 sin( r λ n x ) of the operator Lu = - d 2 u dx 2 with initial conditions u (0) = ( du/dx )(1) = 0 and eigenvalues λ n = ( n - 1 / 2) 2 π 2 for n = 1 , 2 , . . . . (Recall that you computed these eigenvalues and eigenfunctions on Problem Set 6. You may Fnd it helpful to review your solution to that problem.) (a) We wish to write the solution to this wave equation in the form u ( x, t ) = s j =1 a j ( t ) φ j ( x ) . Show that the coe±cients a j ( t ) obey the ordinary di²erential equation d 2 a j dt 2 ( t ) = - c 2 λ j a j ( t ) with initial conditions a j (0) = b j , da j dt (0) = d j . (b) Write down the solution to the di²erential equation in part (a). (c) Use your solution to part (b) to write out a formula for the solution
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Unformatted text preview: u ( x, t ). (d) Write a MATLAB program to compute solutions to this di²erential equation with c = 1 and boundary conditions u ( x, 0) = 0 , ∂u ∂t ( x, 0) = x + sin( πx ) . Submit plots of the solution at times t = 0 , . 5 , 1 . , 1 . 5 , 2 . 0. 2. [50 points] This question concerns the homogeneous wave equation on an unbounded spatial domain: ∂ 2 u ∂t 2 = ∂ 2 u ∂x 2 ,-∞ < x < ∞ , t > . Find the solution u ( x, t ) to this equation with the following initial conditions: (a) u ( x, 0) = 2 sin( x ) e-x 2 , ∂u ∂t ( x, 0) = 0; (b) u ( x, 0) = 0, ∂u ∂t ( x, 0) =-2 x (1 + x 2 ) 2 ; (c) u ( x, 0) = 2 sin( x ) e-x 2 , ∂u ∂t ( x, 0) =-2 x (1 + x 2 ) 2 . (d) Produce a plot (or plots) showing your solution to part (c) over-10 ≤ x ≤ 10 at times t = 0 , 1 , 2 , 3 , 4 , 5....
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This note was uploaded on 01/21/2012 for the course CAAM 330 taught by Professor Tompson during the Fall '09 term at UVA.

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hw11Marsh2008 - u ( x, t ). (d) Write a MATLAB program to...

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