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Unformatted text preview: MATH 220: PROBLEM SET 4 DUE THURSDAY, OCTOBER 22, 2009 Problem 1. Solve the wave equation on the line: u tt c 2 u xx = 0 , u ( x, 0) = ( x ) , u t ( x, 0) = ( x ) , with ( x ) = , x < 1 , 1 + x, 1 < x < , 1 x, < x < 1 , , x > 1 . and ( x ) = , x < 1 , 2 , 1 < x < 1 , , x > 1 . Also describe in t > 0 where the solution vanishes, and where it is C , and compare it with the general results discussed in lecture (Huygens principle and propagation of singularities). Problem 2. Consider the PDE (1) u tt ( c 2 u ) + qu = 0 , u ( x, 0) = ( x ) , u t ( x, 0) = ( x ) , where c,q 0, depend on x only, and c is bounded between positive constants, i.e. for some c 1 ,c 2 > 0, c 1 c ( x ) c 2 for all x R n . Assume that u is C 2 throughout this problem, and u is real-valued. (All calculations would go through if one wrote | u t | 2 , etc., in the complex valued case.) (i) Fix x R n and R...
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