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1737 results in Nonlinear Science and Fluid Dynamics

Page 24 of 87

Singularities: Formation, Structure, and Propagation
  • J. Eggers, M. A. Fontelos
  • Published online:
    05 September 2015
    Print publication:
    10 September 2015
  • Many key phenomena in physics and engineering are described as singularities in the solutions to the differential equations describing them. Examples covered thoroughly in this book include the formation of drops and bubbles, the propagation of a crack and the formation of a shock in a gas. Aimed at a broad audience, this book provides the mathematical tools for understanding singularities and explains the many common features in their mathematical structure. Part I introduces the main concepts and techniques, using the most elementary mathematics possible so that it can be followed by readers with only a general background in differential equations. Parts II and III require more specialised methods of partial differential equations, complex analysis and asymptotic techniques. The book may be used for advanced fluid mechanics courses and as a complement to a general course on applied partial differential equations.

Chaos, Dynamics, and Fractals
  • An Algorithmic Approach to Deterministic Chaos
  • Joseph L. McCauley
  • Published online:
    05 October 2013
    Print publication:
    07 January 1993
  • This book develops deterministic chaos and fractals from the standpoint of iterated maps, but the emphasis makes it very different from all other books in the field. It provides the reader with an introduction to more recent developments, such as weak universality, multifractals, and shadowing, as well as to older subjects like universal critical exponents, devil's staircases and the Farey tree. The author uses a fully discrete method, a 'theoretical computer arithmetic', because finite (but not fixed) precision cannot be avoided in computation or experiment. This leads to a more general formulation in terms of symbolic dynamics and to the idea of weak universality. The connection is made with Turing's ideas of computable numbers and it is explained why the continuum approach leads to predictions that are not necessarily realized in computation or in nature, whereas the discrete approach yields all possible histograms that can be observed or computed.

Page 24 of 87