WS07 - APPM 4/5350 Worksheet Week 7 Figure 1: www.xkcd.com...

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APPM 4/5350 Worksheet Week 7 Figure 1: www.xkcd.com 1. What are the requirements to differentiate a Fourier sine series on [0 ,L ] term by term? 2. What are the requirements to differentiate a Fourier series on [ - L,L ] term by term? 3. Draw a function that satisfies 2 u = 0 on [0 ,a ] when u (0) = 0 and u ( a ) = 8 . 4. Draw a function that does not satisfy 2 u = 0 on [0 ,a ] but does satisfy u (0) = 0 and u ( a ) = 8 . 5. What is the solution to 2 u = 0 in a circle of radius R with f ( R,θ ) = 0 ? 6. Draw the Fourier cosine and sine series, and their periodic extensions, of f ( x ) = x 2 on [0 ,L ] 7. Prove that the solution to Laplace’s equation with f ( r,θ ) = g ( θ ) on the boundary is unique. 8. Explain why f ( x ) = g ( y ) x,y implies f ( x ) = c and g ( y ) = c where c is a constant. 9. Write down the coefficients that solve f ( x ) = n =0 a n φ n ( x ) if the set { φ n ( x ) } n =0 form an orthogonal basis for the space that contains f ( x ) . 10. Where would you need boundary conditions to solve
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This note was uploaded on 10/24/2010 for the course APPM 4350 taught by Professor Ablowitz during the Fall '08 term at Colorado.

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