abstract-wave-Eq

abstract-wave-Eq - P DE LECTURE NOTES, M ATH 237A-B 181 13....

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PDE LECTURE NOTES, MATH 237A-B 181 13. Abstract Wave Equation In the next section we consider (13.1) u tt 4 u =0 with u ( x, 0) = f ( x ) and u t ( x, 0) = g ( x ) for x R n . Before working with this explicit equation we will work out an abstract Hilbert space theory f rst. Theorem 13.1 (Existence) . Suppose A : H H is a self-adjoint non-positive operator, i.e. A = A and A 0 and f D ( A ) and g g D ¡ A ¢ are given. Then (13.2) u ( t )=cos( t A ) f + sin( t A ) A g satis f es: (1) ˙ u ( t )=cos( t A ) Af +sin( t A ) g exists and is continuous. (2) ¨ u ( t ) exists and is continuous (13.3) ¨ u ( t )= Au with u (0) = f and ˙ u (0) = g. (3) d dt Au ( t )= cos( t A ) Af +sin( t A ) Ag exists and is contin- uous. Eq. (13.3) is Newton’s equation of motion for an in f nite dimensional harmonic oscillation. Given any solution u to Eq. (13.3) it is natural to de f ne its energy by E ( t, u ):= 1 2 [ k ˙ u ( t ) k 2 + k ωu ( t ) k 2 ]= K.E . + P.E. where ω := A. Notice that Eq. (13.3) becomes ¨ u + ω 2 u =0 with this de f nition of ω. Lemma 13.2 (Conservation of Energy) . Suppose u is a solution to Eq. (13.3) such that d dt Au ( t ) exists and is continuous. Then ˙ E ( t )=0 . Proof. ˙ E ( t )=Re( ˙ u, ¨ u )+Re( ωu,ω ˙ u )=Re( ˙ u, ω 2 u ) Re( ω 2 u, ˙ u )=0 . Theorem 13.3
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This note was uploaded on 10/11/2010 for the course MATH 11 taught by Professor Smith during the Three '10 term at ADFA.

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abstract-wave-Eq - P DE LECTURE NOTES, M ATH 237A-B 181 13....

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