@inbook{c4622e79b8ec40238c31e7d7fef7af7e,

title = "Overview of partial differential equations",

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 Schr{\"o}dinger 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.",

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, Schr{\"o}dinger, Shocks and traveling waves, Solution operator, Telegrapher equations, Wave, Wave breaking, Wave packets, Well-posedness",

author = "M. Brio and Webb, {G. M.} and Zakharian, {A. R.}",

year = "2010",

doi = "10.1016/S0076-5392(10)21306-1",

language = "English (US)",

series = "Mathematics in Science and Engineering",

publisher = "Elsevier",

number = "C",

pages = "1--57",

booktitle = "Mathematics in Science and Engineering",

edition = "C",

}