### 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 |

### Fingerprint

### 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

*Mathematics in Science and Engineering*,

*213*(C), 1-57. https://doi.org/10.1016/S0076-5392(10)21306-1

**Overview of partial differential equations.** / Brio, Moysey; Webb, G. M.; Zakharian, A. R.

Research output: Contribution to journal › Article

*Mathematics in Science and Engineering*, vol. 213, no. C, pp. 1-57. https://doi.org/10.1016/S0076-5392(10)21306-1

}

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 -