Abstract
In this chapter, we present a short and even more far from exhaustive theoretical study of the wave equation. We establish the existence and uniqueness of the solution, as well as the energy estimates. We describe the qualitative behavior of solutions, which is very different from that of the heat equation. Again, we will mostly work in one dimension of space. In the same chapter, we introduce finite difference methods for the numerical approximation of the wave equation. Here again, stability issues are prominent, and significantly more delicate than for the heat equation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Admittedly, there is a third case, \(\varOmega ={\mathbb R}_+^*\), but we will not consider it here.
- 2.
We admit here that both formulations coincide in the smooth case.
- 3.
Boundary conditions are a delicate question for such systems.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Le Dret, H., Lucquin, B. (2016). The Wave Equation. In: Partial Differential Equations: Modeling, Analysis and Numerical Approximation. International Series of Numerical Mathematics, vol 168. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-27067-8_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-27067-8_9
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-27065-4
Online ISBN: 978-3-319-27067-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)