WS04 - ∂x(0,t = 1 ∂u ∂x L,t = β Make sure you can...

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APPM 4/5350 Worksheet Week 4 Figure 1: www.xkcd.com 1. Consider 2 u = 0 inside a disk of radius R , with u ( R,θ ) = 0 (i) What other boundary conditions should you enforce? (ii) Write down the steps you would use to solve the boundary value problem (you don’t need to solve it here). (iii) What condition applies at u (0 ) in terms of the variables given in the problem? Hint: You still don’t need to solve the whole problem here. Use one of the proper- ties of solutions to Laplace’s Equation. (iv) Where on the disk does u achieve a maximum value? A minimum value? Explain why this must be so. 2. When is the following problem well-posed (if ever)? ∂u ∂t = 2 u ∂x 2 u ( x, 0) = f ( x ) ∂u
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Unformatted text preview: ∂x (0 ,t ) = 1 ∂u ∂x ( L,t ) = β Make sure you can explain what it means for a problem to be well-posed. 3. What is the compatibility condition for Laplace’s equation in three dimensions with heat flow given on the boundaries (note that Ω denotes the domain, and δ Ω denotes the bound-ary of the domain): ∇ 2 u ( x ) = 0 ∀ x ∈ Ω-c ∇ u · ˆn | δ Ω = g ( x ) BONUS! (don’t spend too much time on this © ) Prove that all positive even integers ≥ 4 can be expressed as the sum of two primes (Strong Goldbach Conjecture) Written by Anil Damle and Anna Lieb 1 revised September 2010...
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