Chapter

From linear to nonlinear equations, fundamental results

Klaus Böhmer

in Numerical Methods for Nonlinear Elliptic Differential Equations

Published in print October 2010 | ISBN: 9780199577040
Published online January 2011 | e-ISBN: 9780191595172 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199577040.003.0001

Series: Numerical Mathematics and Scientific Computation

From linear to nonlinear equations, fundamental results

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Chapter 1 demonstrates, for the simple mechanical example of a bent rod, the change in character from linear to nonlinear regimes. This is followed by several examples for different types of nonlinear elliptic differential equations in mathematics, science and in engineering, e.g. the Monge‐Ampère, the reaction‐diffusion, the von Kármán and the Navièr‐Stokes equations. These problems require appropriate analytical results for (space‐) discretization methods. The necessary tools from functional analysis and calculus in Banach spaces are summarized.

Keywords: linear regimes; nonlinear regimes; Monge‐Ampère; reaction‐diffusion; von Kármán equations; Navièr-Stokes equations; basic functional analysis; Banach spaces

Chapter.  8208 words.  Illustrated.

Subjects: Mathematical and Statistical Physics

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