Partial Differential Equations and Their Classification Into Types

Hackbusch, Wolfgang

doi:10.1007/978-3-642-11490-8_1

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2004

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Cited by 3 publications

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“…Uniqueness has been proved in Lemma 3.1. Because of Lemma 3.2 we know that the problem satisfies Fredholm's alternative (see for instance Theorem 6.5.15 of [10]). Then, existence is a consequence of uniqueness.…”

Section: State Equationmentioning

confidence: 99%

Finite Element Methods in Local Active Control of Sound

Bermúdez¹,

Gamallo²,

Rodríguez³

2004

SIAM J. Control Optim.

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Abstract. The active control of sound is analyzed in the framework of the mathematical theory of optimal control. After setting the problem in the frequency domain, we deal with the state equation, which is a Helmholtz partial differential equation. We show existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure. Two optimization problems are successively considered. The first one concerns the choice of phases and amplitudes of the actuators to minimize the noise at the sensors location. The second one consists in determining the optimal actuators placement. Both problems are then numerically solved. Error estimates are settled and numerical results for some tests are reported.

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Section: State Equationmentioning

confidence: 99%