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solh2 - Homework Set 2 Math 716 Due Friday January 26th 1...

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Homework Set 2: Math 716, Due: Friday, January 26th 1. Consider the wave equation u tt = c 2 u xx , u ( x, 0) = φ ( x ) , u t ( x, 0) = ψ ( x ) for x R with c negationslash = 0. Assume φ C 2 and ψ C 1 have compact support ( i.e. there is a number R> 0 such that φ ( x ) = ψ ( x ) = 0 for | x | >R ). Denote d’Alembert’s solution of (1) by u ( x,t ). Using d’Alembert’s solution, show that the solutions to (1) depend in a stable fashion on the initial data φ and ψ . More precisely: if u 1 and u 2 satisfy (1) with initial data ( φ 1 1 ) and ( φ 2 2 ), respectively, show that sup x R | u 1 ( x,t ) u 2 ( x,t ) | ≤ C ( t ) bracketleftbigg sup x R | φ 1 ( x ) φ 2 ( x ) | + sup x R | ψ 1 ( x ) ψ 2 ( x ) | bracketrightbigg for some constant C ( t ) that may depend on t , but not on the initial data. What is a good estimate for C ( t )? Can you find similar estimates for x ( u 1 u 2 ) and xx ( u 1 u 2 )? Solution: Consider v = u 1 u 2 . It satisfies v tt c 2 v xx = 0 , v ( x, 0) = φ 1 ( x ) φ 2 ( x ) φ ( x ) , v t ( x, 0) = ψ 1 ( x ) ψ 2 ( x ) ψ ( x ) We know from D’Alembert solution v ( x,t ) = 1 2 [ φ ( x + ct ) + φ ( x ct )] + 1 2 c integraldisplay x + ct x - ct ψ ( y ) dy So, | v ( x,t ) | ≤ bardbl φ bardbl + bardbl ψ bardbl 1 2 c integraldisplay x + ct x - ct dy (1 + t ) [ bardbl φ
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