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Method Of Lines Analysis Of Turing Models (Hardcover)
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Method Of Lines Analysis Of Turing Models (Hardcover)
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This book is directed toward the numerical integration (solution)
of a system of partial differential equations (PDEs) that describes
a combination of chemical reaction and diffusion, that is,
reaction-diffusion PDEs. The particular form of the PDEs
corresponds to a system discussed by Alan Turing and is therefore
termed a Turing model.Specifically, Turing considered how a
reaction-diffusion system can be formulated that does not have the
usual smoothing properties of a diffusion (dispersion) system, and
can, in fact, develop a spatial variation that might be interpreted
as a form of morphogenesis, so he termed the chemicals as
morphogens.Turing alluded to the important impact computers would
have in the study of a morphogenic PDE system, but at the time
(1952), computers were still not readily available. Therefore, his
paper is based on analytical methods. Although computers have since
been applied to Turing models, computer-based analysis is still not
facilitated by a discussion of numerical algorithms and a readily
available system of computer routines.The intent of this book is to
provide a basic discussion of numerical methods and associated
computer routines for reaction-diffusion systems of varying form.
The presentation has a minimum of formal mathematics. Rather, the
presentation is in terms of detailed examples, presented at an
introductory level. This format should assist readers who are
interested in developing computer-based analysis for
reaction-diffusion PDE systems without having to first study
numerical methods and computer programming (coding).The numerical
examples are discussed in terms of: (1) numerical integration of
the PDEs to demonstrate the spatiotemporal features of the
solutions and (2) a numerical eigenvalue analysis that corroborates
the observed temporal variation of the solutions. The resulting
temporal variation of the 2D and 3D plots demonstrates how the
solutions evolve dynamically, including oscillatory long-term
behavior.In all of the examples, routines in R are presented and
discussed in detail. The routines are available through this link
so that the reader can execute the PDE models to reproduce the
reported solutions, then experiment with the models, including
extensions and application to alternative models.
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