Wave equations &amp; energy

William O. Bray; Ellen Hunter; William O. Bray; Ellen Hunter

doi:10.3934/math.2019.3.463

AIMS Mathematics

2019, Volume 4, Issue 3: 463-481. doi: 10.3934/math.2019.3.463

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Research article

Wave equations & energy

William O. Bray ^{1
,
,},
Ellen Hunter ²

1.
Department of Mathematics, Missouri State University, 901 S. National, Springfield, MO 65897, USA
2.
Department of Mathematics, Missouri State University, 901 S. National, Springfield, MO 65897, USA

Received: 01 February 2019 Accepted: 29 April 2019 Published: 10 May 2019
MSC : 35L05, 34B24

The focus of this work is apply Fourier analytic methods based on Parseval's equality to the computation of kinetic and potential energy of solutions of initial boundary value problems for general wave type equations on a finite interval. As a consequence, an energy equipartion principle for the solution is obtained. Within our methods are some new results regarding eigenfunction expansions arising from regular Sturm-Liouville problems in Sobolev spaces.
- wave equation,
- Sturm-Liouville problem,
- Sobolev space,
- energy conservation,
- energy equipartition
Citation: William O. Bray, Ellen Hunter. Wave equations & energy[J]. AIMS Mathematics, 2019, 4(3): 463-481. doi: 10.3934/math.2019.3.463

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Abstract

The focus of this work is apply Fourier analytic methods based on Parseval's equality to the computation of kinetic and potential energy of solutions of initial boundary value problems for general wave type equations on a finite interval. As a consequence, an energy equipartion principle for the solution is obtained. Within our methods are some new results regarding eigenfunction expansions arising from regular Sturm-Liouville problems in Sobolev spaces.

References

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[5]	W. O. Bray, A Journey into Partial Differential Equations, Jones & Bartlett Learning, 2012.
[6]	G. Birkhoff, G. Carlo-Rota, Ordinary Differential Equations, 4th edition, J. Wiley, 1989.
[7]	C. T. Fulton, S. A. Pruess, Eigenvalue and eigenfunction asypmtotics for regular Sturm-Liouville problems, Journal of Mathematical Analysis Applications, 188 (1994), 297-340. doi: 10.1006/jmaa.1994.1429
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[9]	G. Leoni, A First Course in Sobolev Spaces, Vol. 105, American Mathematical Soc., 2009.
[10]	R. L. Wheeden, Measure and Integral, An Introduction to Real Analysis, 2 ed., CRC Press, 2015.

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