Overview of partial differential equations

M. Brio, G. M. Webb, A. R. Zakharian

Research output: Chapter in Book/Report/Conference proceedingChapter

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 languageEnglish (US)
Title of host publicationMathematics in Science and Engineering
PublisherElsevier
Pages1-57
Number of pages57
EditionC
DOIs
StatePublished - Jan 1 2010
Externally publishedYes

Publication series

NameMathematics in Science and Engineering
NumberC
Volume213
ISSN (Print)0076-5392

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)

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  • Cite this

    Brio, M., Webb, G. M., & Zakharian, A. R. (2010). Overview of partial differential equations. In Mathematics in Science and Engineering (C ed., pp. 1-57). (Mathematics in Science and Engineering; Vol. 213, No. C). Elsevier. https://doi.org/10.1016/S0076-5392(10)21306-1