WS03 - functions. (b) Now suppose that h ( x ) is also...

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APPM 4/5350 Worksheet Week 3 Figure 1: www.xkcd.com 1. Eigenvalues/Eigenvectors (a) Solve Av = λ v where A = ± 3 2 2 3 ² Identify the eigenvalues and eigenvectors. What relationship do the eigenvectors have? (b) now solve Av = λ v where A = 2 ∂x 2 + 2 ∂y 2 Once again, identify the eigenvalues and eigenfunctions. How are the eigenvectors re- lated now? 2. Laplace’s Equation in a Rectangle Consider 2 w = 0 in a rectangle, 0 x a , 0 y b . The boundary conditions are: w (0 ,y ) = f ( y ) w ( a,y ) = g ( y ) w ( x, 0) = h ( x ) w ( x,b ) = k ( x ) (a) Separate variables. Suppose that f ( y ) = g ( y ) = h ( x ) = 0 . How does this affect your choice of sign for the separation constant? Now solve the problem in terms of given
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Unformatted text preview: functions. (b) Now suppose that h ( x ) is also non-zero. Outline how you would solve the problem now (you do not need to solve it again). (c) Now suppose that the boundary conditions are now: w (0 ,y ) = 0 w x ( a,y ) = 0 w ( x, 0) = 0 w ( x,b ) = k ( x ) How will the solution differ from your solution to part a ? Solve it if you need to! 3. Boundary Conditions What are periodic boundary conditions? For what kind of domain do you use them, and why? Written by Anil Damle and Anna Lieb 1 revised August 2010...
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