Travelling Waves in Nonlinear Diffusion-Convection Reaction

Brian H. Gilding, Robert Kersner (Autoren)

Buch | Softcover

IX, 210 Seiten

2012 | Softcover reprint of the original 1st ed. 2004
Springer Basel (Verlag)
978-3-0348-9638-2 (ISBN)

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This monograph has grown out of research we started in 1987, although the foun dations were laid in the 1970's when both of us were working on our doctoral theses, trying to generalize the now classic paper of Oleinik, Kalashnikov and Chzhou on nonlinear degenerate diffusion. Brian worked under the guidance of Bert Peletier at the University of Sussex in Brighton, England, and, later at Delft University of Technology in the Netherlands on extending the earlier mathematics to include nonlinear convection; while Robert worked at Lomonosov State Univer sity in Moscow under the supervision of Anatolii Kalashnikov on generalizing the earlier mathematics to include nonlinear absorption. We first met at a conference held in Rome in 1985. In 1987 we met again in Madrid at the invitation of Ildefonso Diaz, where we were both staying at 'La Residencia'. As providence would have it, the University 'Complutense' closed down during this visit in response to student demonstra tions, and, we were very much left to our own devices. It was natural that we should gravitate to a research topic of common interest. This turned out to be the characterization of the phenomenon of finite speed of propagation for nonlin ear reaction-convection-diffusion equations. Brian had just completed some work on this topic for nonlinear diffusion-convection, while Robert had earlier done the same for nonlinear diffusion-absorption. There was no question but that we bundle our efforts on the general situation.

Travelling waves are observed in many natural processes, ranging from the spread of diseases to the combustion of fuels. A common feature of a multitude of these phenomena is that they can be similarly described in terms of a dispersive or diffusive mechanism, an advective or convective process, some kind of reaction, sorption, source or sink mechanism, or, an intricate balance between several of these factors. This book presents an overview of the occurence of travelling waves when these underlying phenomena can be mathematically modelled by a partial differential equation. The contents provide an up-to-date review of the state of the art in this field of research, touching on many different areas of application and including a large number of new results.

1 Introduction.- 2 General theory.- 2.1 Basic hypotheses.- 2.2 Integral equation theory.- 2.3 Proof of equivalence.- 2.4 Illustration of difficulties.- 2.5 Further properties.- 2.6 Classical results.- Bibliographical notes.- 3 Transformations.- Bibliographical notes.- 4 Travelling waves.- 4.1 Admissible wave speeds.- 4.2 Number of solutions.- Bibliographical notes.- 5 Convection-diffusion.- Bibliographical notes.- 6 Reaction-diffusion.- 6.1 Sink term.- 6.2 Source term.- 6.3 Smooth coefficients.- Bibliographical notes.- 7 Power-law equations.- Bibliographical notes.- 8 Wavefronts.- 8.1 Admissible wave speeds.- 8.2 Number of wavefronts.- 8.3 Illustrations.- 8.4 Multiple equilibria.- Bibliographical notes.- 9 Wavefronts for convection-diffusion.- Bibliographical notes.- 10 Wavefronts for reaction-diffusion.- 10.1 Fixed sign.- 10.2 One sign change.- 10.3 Smooth coefficients.- Bibliographical notes.- 11 Unbounded waves.- 12 Wavefronts and unbounded waves for power-law equations.- 12.1 Convection-diffusion.- 12.2 Reaction-diffusion with linear convection.- 12.3 Reaction-convection-diffusion.- Bibliographical notes.- 13 Explicit travelling-wave solutions.- 13.1 Power-law equations.- 13.2 Generalizations of the Fisher equation.- 13.3 Generating further explicit solutions.- Bibliographical notes.

Erscheint lt. Verlag	24.10.2012
Reihe/Serie	Progress in Nonlinear Differential Equations and Their Applications
Zusatzinfo	IX, 210 p.
Verlagsort	Basel
Sprache	englisch
Maße	155 x 235 mm
Gewicht	349 g
Themenwelt	Sachbuch/Ratgeber ► Natur / Technik ► Garten
	Informatik ► Weitere Themen ► Bioinformatik
	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
	Naturwissenschaften ► Biologie ► Evolution
Schlagworte	Evolution • Genetics • partial differential equation • Partial differential equations • population dynamics • Travelling Waves
ISBN-10	3-0348-9638-7 / 3034896387
ISBN-13	978-3-0348-9638-2 / 9783034896382
Zustand	Neuware