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Unformatted text preview: MAE105 Final Exam (open book, closed notes) Name:_________________________ Time: 3:10 to 6:10pm Date: December 12, 2003 Problem 1 (a) (2 Points) Find a general expression x = x ( t ), for the characteristics of the following PDE: ∂ t ∂ u + x sin t ∂ x ∂ u = . [Note that your expression must include a constant of integration, say, x (0) = x .] (b) (0.5 Point) In the x , t-plane, t > 0, sketch a typical characteristic curve. (c) (2 Points) Find the general solution of this PDE. (d) (1.5 Points) Specialize the solution in (b) such that at t = 0, we have u ( x ,0) = u = ( x + 1). Problem 2 Consider the wave equation ∂ t 2 ∂ 2 u- ∂ x 2 ∂ 2 u = (1) in a finite domain, 0 < x < 1, t > 0, with the boundary conditions u (0, t ) = u (1, t ) = 0 , and initial conditions u ( x , 0) = f ( x ) = 1- x x for 1 / 2 < x < 1 . for < x < 1 / 2 ∂ t ∂ u ( x , 0) = g ( x ) = sin( π x ) , < x < 1 . (a) (1 Point) Draw the ( x , t )-plane, and below the x-axis, draw the initial conditions in two graphs, as discussed in the class and in your book....
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- Spring '07
- Fourier Series, Partial differential equation, general solution, Boundary conditions