220-HW4 - MATH 220: PROBLEM SET 4 DUE THURSDAY, OCTOBER 22,...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 2

220-HW4 - MATH 220: PROBLEM SET 4 DUE THURSDAY, OCTOBER 22,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online