abstract-wave-Eq

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

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

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 10/11/2010 for the course MATH 11 taught by Professor Smith during the Three '10 term at ADFA.

### Page1 / 4

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

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

View Full Document
Ask a homework question - tutors are online