Abstract
Chapter 1 begins with some examples of partial differential equations in science and engineering and their linearization and dispersion equations. The concepts of well-posedness, regularity, and solution operator for systems of partial differential equations (PDE's) are discussed. Instabilities can arise from both numerical methods and from real physical instabilities. Some physical instabilities are described, including: (a) the distinction between convective and absolute instabilities, (b) the Rayleigh-Taylor and Kelvin-Helmholtz instabilities in fluids, (c) wave breaking and gradient catastrophe in gas dynamics and in conservation laws, (d) modulational or Benjamin Feir instabilities and nonlinear Schrödinger related equations, (e) three-wave resonant interactions and explosive instabilities associated with negative energy waves. Basic wave concepts are described (e.g. wave-number surfaces, group velocity, wave action, wave diffraction, and wave energy equations). A project from semiconductor transport modeling is described.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1-57 |
| Number of pages | 57 |
| Journal | Mathematics in Science and Engineering |
| Volume | 213 |
| Issue number | C |
| DOIs | |
| State | Published - 2008 |
| Externally published | Yes |
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Keywords
- Absolute and convective instabilities
- Advection
- Airy
- Diffraction
- Dispersion relation
- Group velocity
- Heat
- Linear and nonlinear resonant wave interaction
- Modulational instability
- Partial differential equations
- Rayleigh-Taylor and Kevin-Helmholtz instabilities
- Regularity
- Schrödinger
- Shocks and traveling waves
- Solution operator
- Telegrapher equations
- Wave
- Wave breaking
- Wave packets
- Well-posedness
ASJC Scopus subject areas
- Mathematics(all)
- Engineering(all)
Cite this
Overview of partial differential equations. / Brio, Moysey; Webb, G. M.; Zakharian, A. R.
In: Mathematics in Science and Engineering, Vol. 213, No. C, 2008, p. 1-57.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Overview of partial differential equations
AU - Brio, Moysey
AU - Webb, G. M.
AU - Zakharian, A. R.
PY - 2008
Y1 - 2008
N2 - Chapter 1 begins with some examples of partial differential equations in science and engineering and their linearization and dispersion equations. The concepts of well-posedness, regularity, and solution operator for systems of partial differential equations (PDE's) are discussed. Instabilities can arise from both numerical methods and from real physical instabilities. Some physical instabilities are described, including: (a) the distinction between convective and absolute instabilities, (b) the Rayleigh-Taylor and Kelvin-Helmholtz instabilities in fluids, (c) wave breaking and gradient catastrophe in gas dynamics and in conservation laws, (d) modulational or Benjamin Feir instabilities and nonlinear Schrödinger related equations, (e) three-wave resonant interactions and explosive instabilities associated with negative energy waves. Basic wave concepts are described (e.g. wave-number surfaces, group velocity, wave action, wave diffraction, and wave energy equations). A project from semiconductor transport modeling is described.
AB - Chapter 1 begins with some examples of partial differential equations in science and engineering and their linearization and dispersion equations. The concepts of well-posedness, regularity, and solution operator for systems of partial differential equations (PDE's) are discussed. Instabilities can arise from both numerical methods and from real physical instabilities. Some physical instabilities are described, including: (a) the distinction between convective and absolute instabilities, (b) the Rayleigh-Taylor and Kelvin-Helmholtz instabilities in fluids, (c) wave breaking and gradient catastrophe in gas dynamics and in conservation laws, (d) modulational or Benjamin Feir instabilities and nonlinear Schrödinger related equations, (e) three-wave resonant interactions and explosive instabilities associated with negative energy waves. Basic wave concepts are described (e.g. wave-number surfaces, group velocity, wave action, wave diffraction, and wave energy equations). A project from semiconductor transport modeling is described.
KW - Absolute and convective instabilities
KW - Advection
KW - Airy
KW - Diffraction
KW - Dispersion relation
KW - Group velocity
KW - Heat
KW - Linear and nonlinear resonant wave interaction
KW - Modulational instability
KW - Partial differential equations
KW - Rayleigh-Taylor and Kevin-Helmholtz instabilities
KW - Regularity
KW - Schrödinger
KW - Shocks and traveling waves
KW - Solution operator
KW - Telegrapher equations
KW - Wave
KW - Wave breaking
KW - Wave packets
KW - Well-posedness
UR - http://www.scopus.com/inward/record.url?scp=77955235717&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77955235717&partnerID=8YFLogxK
U2 - 10.1016/S0076-5392(10)21306-1
DO - 10.1016/S0076-5392(10)21306-1
M3 - Article
AN - SCOPUS:77955235717
VL - 213
SP - 1
EP - 57
JO - Mathematics in Science and Engineering
JF - Mathematics in Science and Engineering
SN - 0076-5392
IS - C
ER -