wave-eq - 2 t 2-( x-x ) 2-( y-y ) 2 ] 1 / 2 dxdy + t [ same...

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Math 425, Spring 2011 Jerry L. Kazdan The Wave Equation in R 2 and R 3 To Solve u tt = c 2 Δ u , x R n with u ( x , 0 ) = φ ( x ) , u t ( x , 0 ) = ψ ( x ) In R 3 [Poisson: Kirchoff’s formula ] u ( x 0 , t 0 ) = 1 4 π c 2 t 0 ZZ S ψ ( x ) dS + t 0 ± 1 4 π c 2 t 0 ZZ S φ ( x ) dS ² , where S is the sphere centered at x 0 with radius ct 0 . In R 2 . Here we write ( x , y ) R 2 . u ( x 0 , y 0 , t 0 ) = 1 2 π c ZZ D ψ ( x , y ) [ c
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Unformatted text preview: 2 t 2-( x-x ) 2-( y-y ) 2 ] 1 / 2 dxdy + t [ same expression with ] , where D is the disk ( x-x ) 2 +( y-y ) 2 c 2 t 2 ....
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This document was uploaded on 03/06/2012.

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