Final-04

# Final-04 - MAE105 Final Exam(open book closed notes no...

This preview shows pages 1–3. Sign up to view the full content.

MAE105 Final Exam (open book, closed notes, no phone, calculators or computers) Name:_________________________ Time: 3:00 to 6:00pm Date: December 10, 2004 Problem 1 (a) (1 Point) Find a general expression x = x ( t ), for the characteristics of the following PDE: t u + x cos t x u = t u . (1) [Note that your expression must include a constant of integration, say, x (0) = x 0 .] (b) (0.5 Point) In the x , t -plane, t > 0, sketch a typical characteristic curve for 0 t < 2 π . (c) (1 Point) Show that on the characteristics, ODE (1) reduces to dt du = t u . (2) Find the general solution of this PDE. (d) (1 Point) Specialize the solution in (c) such that at t = 0, we have u ( x 0 ,0) = u 0 = (ln( x 0 ) + 1). Problem 2 Consider the wave equation t 2 2 u - x 2 2 u = 0 (3) in a finite domain, 0 < x < 1, t > 0, with the boundary conditions u (0, t ) = u (1, t ) = 0 , (4) and initial conditions u ( x , 0) = f ( x ) = 1 - x x sin( π x ) for 1 / 2 < x < 1 . for 0 < x < 1 / 2 (5) t u ( x , 0) = g ( x ) = sin( π x ) , 0 < 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. (b) (1 Point) Extend the initial conditions such that the boundary conditions are satisfied for all t > 0, if we use the general solution for the infinite domain.

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

View Full Document
- 2 - (c) (1 Point) In the x , t -plane, draw the characteristics that pass through the following points: [ x = 1 / 4, t = 4 / 3] .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

Final-04 - MAE105 Final Exam(open book closed notes no...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online