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Both of these are nonlinear second-order diffusion equations. One finds a large amount of exercises and open problems that can serve as a starting point for further research. The authors have produced a well-written book, which gives a good picture of what is known about the canonical equation.

There are a large number of exercises, which fill in details of proofs or provide illuminating examples or straightforward generalisations as well as a good number of open problems. There are also a large number of numerically computed graphs of branching curves and bifurcation curves throughout the book, which provide insights into the mathematically formulated results. The book is a valuable contribution to the literature, for both the specialist and the nonspecialist reader.

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JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser. Buy eBook. Buy Hardcover. Buy Softcover. FAQ Policy. About this Textbook The study of spatial patterns in extended systems, and their evolution with time, poses challenging questions for physicists and mathematicians alike.

Spatial Patterns: Higher Order Models in Physics and Mechanics

Show all. Real Eigenvalues Pages Peletier, L. Estimates Pages Peletier, L.

Periodic Solutions Pages Peletier, L. Kinks and Pulses Pages Peletier, L.

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A comprehensive overview is given of these active areas of research. The study of dynamic equations on a measure chain time scale is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on measure chains can build bridges between continuous and discrete mathematics.

This work is an introduction to measure chain theory with particular emphasis on its usefulness in allowing for the simultaneous development of differential equations and difference equations.

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The study of measure chain theory has also led to several important applications, e. With numerous examples throughout, the book may be used in a special topics seminar at the senior undergraduate or beginning graduate levels. May also serve as a good reference to stimulate the development of new kinds of equations with potentially new applications. Spatial Patterns offers a study of nonlinear higher order model equations that are central to the description and analysis of spatio-temporal pattern formation in the natural sciences.

Through a unique combination of results obtained by rigorous mathematical analysis and computational studies, the text exhibits the principal families of solutions, such as kinks, pulses and periodic solutions, and their dependence on critical eigenvalue parameters, and points to a rich structure, much of which still awaits exploration. The exposition unfolds systematically, first focusing on a single equation to achieve optimal transparency, and then branching out to wider classes of equations.

The presentation is based on results from real analysis and the theory of ordinary differential equations. The book is intended for mathematicians who wish to become acquainted with this new area of partial and ordinary differential equations, for mathematical physicists who wish to learn about the theory developed for a class of well-known higher order pattern-forming model equations, and for graduate students who are looking for an exciting and promising field of research.

As the population exceeds the six billion mark, questions of population explosion, of how many people the earth can support and under which conditions, become pressing.

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Some of the questions and challenges raised can be addressed through the use of mathematical models, but not all. The goal of this book is to search for a balance between simple and analyzable models and unsolvable models which are capable of addressing important questions such as these. Part I focuses on single species simple models including those which have been used to predict the growth of human and animal population in the past.

6. Time Evolution and the Schrödinger Equation

Single population models are, in some sense, the building blocks of more realistic models - the subject of Part II. Their role is fundamental to the study of ecological and demographic processes including the role of population structure and spatial heterogeneity - the subject of Part III. This book, which will include both examples and exercises, will be useful to practitioners, graduate students, and scientists working in the field.