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Author: Richard D. Tatum Publisher: ISBN: Category : Nonlinear chemical kinetics Languages : en Pages : 0
Book Description
Reaction-diffusion describes the process in which multiple participating chemicals or agents react with each other, while simultaneously diffusing or spreading through a liquid or gaseous medium. Typically, these processes are studied for their ability to produce nontrivial patterns that evolve over time. These patterns, often referred to as Turing structures or Turing patterns, are diffusion driven. In the presence of diffusion, the Turing patterns are observable, but are not present in the absence of diffusion. It is important for reactiondiffusion models to replicate the behavior that is experimentally observed. That is to say that the models must be able to produce solutions with traits, such as pattern type, that are similar to experimentally observed traits. Mathematically, we seek to explain certain aspects of the models such as pattern selection in the hope of broadening our understanding of the underlying process for which the model represents. I analyze a mixed reaction-diffusion system containing an instability that results in nontrivial Turing structures. This system uses a homotopy parameter [beta] to vary the effect of both local ([beta] = 1) and nonlocal ([beta] = 0) diffusion. Furthermore, I consider [element-of symbol]-scaled kernels J such that [th]J is -independent for [th] [element-of symbol] R. For [th]
Author: Richard D. Tatum Publisher: ISBN: Category : Nonlinear chemical kinetics Languages : en Pages : 0
Book Description
Reaction-diffusion describes the process in which multiple participating chemicals or agents react with each other, while simultaneously diffusing or spreading through a liquid or gaseous medium. Typically, these processes are studied for their ability to produce nontrivial patterns that evolve over time. These patterns, often referred to as Turing structures or Turing patterns, are diffusion driven. In the presence of diffusion, the Turing patterns are observable, but are not present in the absence of diffusion. It is important for reactiondiffusion models to replicate the behavior that is experimentally observed. That is to say that the models must be able to produce solutions with traits, such as pattern type, that are similar to experimentally observed traits. Mathematically, we seek to explain certain aspects of the models such as pattern selection in the hope of broadening our understanding of the underlying process for which the model represents. I analyze a mixed reaction-diffusion system containing an instability that results in nontrivial Turing structures. This system uses a homotopy parameter [beta] to vary the effect of both local ([beta] = 1) and nonlocal ([beta] = 0) diffusion. Furthermore, I consider [element-of symbol]-scaled kernels J such that [th]J is -independent for [th] [element-of symbol] R. For [th]
Author: Fuensanta Andreu-Vaillo Publisher: American Mathematical Soc. ISBN: 0821852302 Category : Mathematics Languages : en Pages : 274
Book Description
Nonlocal diffusion problems arise in a wide variety of applications, including biology, image processing, particle systems, coagulation models, and mathematical finance. These types of problems are also of great interest for their purely mathematical content. This book presents recent results on nonlocal evolution equations with different boundary conditions, starting with the linear theory and moving to nonlinear cases, including two nonlocal models for the evolution of sandpiles. Both existence and uniqueness of solutions are considered, as well as their asymptotic behaviour. Moreover, the authors present results concerning limits of solutions of the nonlocal equations as a rescaling parameter tends to zero. With these limit procedures the most frequently used diffusion models are recovered: the heat equation, the $p$-Laplacian evolution equation, the porous media equation, the total variation flow, a convection-diffusion equation and the local models for the evolution of sandpiles due to Aronsson-Evans-Wu and Prigozhin. Readers are assumed to be familiar with the basic concepts and techniques of functional analysis and partial differential equations. The text is otherwise self-contained, with the exposition emphasizing an intuitive understanding and results given with full proofs. It is suitable for graduate students or researchers. The authors cover a subject that has received a great deal of attention in recent years. The book is intended as a reference tool for a general audience in analysis and PDEs, including mathematicians, engineers, physicists, biologists, and others interested in nonlocal diffusion problems.
Author: Gabriela Caristi Publisher: CRC Press ISBN: 1000117197 Category : Mathematics Languages : en Pages : 428
Book Description
"Based on the proceedings of the International Conference on Reaction Diffusion Systems held recently at the University of Trieste, Italy. Presents new research papers and state-of-the-art surveys on the theory of elliptic, parabolic, and hyperbolic problems, and their related applications. Furnishes incisive contribution by over 40 mathematicians representing renowned institutions in North and South America, Europe, and the Middle East."
Author: Masaharu Taniguchi Publisher: ISBN: 9784864970976 Category : Mathematics Languages : en Pages : 0
Book Description
The study on traveling fronts in reaction-diffusion equations is the first step to understand various kinds of propagation phenomena in reaction-diffusion models in natural science. One dimensional traveling fronts have been studied from the 1970s, and multidimensional ones have been studied from around 2005. This volume is a text book for graduate students to start their studies on traveling fronts. Using the phase plane analysis, we study the existence of traveling fronts in several kinds of reaction-diffusion equations. For a nonlinear reaction term, a bistable one is a typical one. For a bistable reaction-diffusion equation, we study the existence and stability of two-dimensional V-form fronts, and we also study pyramidal traveling fronts in three or higher space dimensions. The cross section of a pyramidal traveling front forms a convex polygon. It is known that the limit of a pyramidal traveling front gives a new multidimensional traveling front. For the study the multidimensional traveling front, studying properties of pyramidal traveling fronts plays an important role. In this volume, we study the existence, uniqueness and stability of a pyramidal traveling front as clearly as possible for further studies by graduate students. For a help of their studies, we briefly explain and prove the well-posedness of reaction-diffusion equations and the Schauder estimates and the maximum principles of solutions.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets
Author: Nikos I. Kavallaris Publisher: Springer ISBN: 3319679449 Category : Technology & Engineering Languages : en Pages : 310
Book Description
This book presents new developments in non-local mathematical modeling and mathematical analysis on the behavior of solutions with novel technical tools. Theoretical backgrounds in mechanics, thermo-dynamics, game theory, and theoretical biology are examined in details. It starts off with a review and summary of the basic ideas of mathematical modeling frequently used in the sciences and engineering. The authors then employ a number of models in bio-science and material science to demonstrate applications, and provide recent advanced studies, both on deterministic non-local partial differential equations and on some of their stochastic counterparts used in engineering. Mathematical models applied in engineering, chemistry, and biology are subject to conservation laws. For instance, decrease or increase in thermodynamic quantities and non-local partial differential equations, associated with the conserved physical quantities as parameters. These present novel mathematical objects are engaged with rich mathematical structures, in accordance with the interactions between species or individuals, self-organization, pattern formation, hysteresis. These models are based on various laws of physics, such as mechanics of continuum, electro-magnetic theory, and thermodynamics. This is why many areas of mathematics, calculus of variation, dynamical systems, integrable systems, blow-up analysis, and energy methods are indispensable in understanding and analyzing these phenomena. This book aims for researchers and upper grade students in mathematics, engineering, physics, economics, and biology.
Author: Markus Kirkilionis Publisher: Springer Science & Business Media ISBN: 9783540441984 Category : Mathematics Languages : en Pages : 446
Book Description
Applied mathematics is a central connecting link between scientific observations and their theoretical interpretation. Nonlinear analysis has surely contributed major developments which nowadays shape the face of applied mathematics. At the beginning of the millennium, all sciences are expanding at increased speed. Technological, ecological, economical and medical problem solving is a central issue of every modern society. Mathematical models help to expose fundamental structures hidden in these problems and serve as unifying tools to deepen our understanding. What are the new challenges applied mathematics has to face with the increased diversity of scientific problems? In which direction should the classical tools of nonlinear analysis be developed further? How do new available technologies influence the development of the field? How can problems be solved which have been beyond reach in former times? It is the aim of this book to explore new developments in the field by way of discussion of selected topics from nonlinear analysis.
Author: Yang Kuang Publisher: Academic Press ISBN: 0080960022 Category : Mathematics Languages : en Pages : 413
Book Description
Delay Differential Equations emphasizes the global analysis of full nonlinear equations or systems. The book treats both autonomous and nonautonomous systems with various delays. Key topics addressed are the possible delay influence on the dynamics of the system, such as stability switching as time delay increases, the long time coexistence of populations, and the oscillatory aspects of the dynamics. The book also includes coverage of the interplay of spatial diffusion and time delays in some diffusive delay population models. The treatment presented in this monograph will be of great value in the study of various classes of DDEs and their multidisciplinary applications.
Author: Richard D. Tatum
Author: Richard D. Tatum
Author: Fuensanta Andreu-Vaillo
Author: Gabriela Caristi
Author: Masaharu Taniguchi
Author: Arnaud Ducrot
Author: Joseph Boyd Raquepas
Author: Nikos I. Kavallaris
Author: Markus Kirkilionis
Author: Yang Kuang
Author: Franz Rothe