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This book presents several fundamental results in solving nonlinear reaction-diffusion equations and systems using symmetry-based methods. Reaction-diffusion systems are fundamental modeling tools for mathematical biology with applications to ecology, population dynamics, pattern formation, morphogenesis, enzymatic reactions and chemotaxis. The book discusses the properties of nonlinear reaction-diffusion systems, which are relevant for biological applications, from the symmetry point of view, providing rigorous definitions and constructive algorithms to search for conditional symmetry (a nontrivial generalization of the well-known Lie symmetry) of nonlinear reaction-diffusion systems. In order to present applications to population dynamics, it focuses mainly on two- and three-component diffusive Lotka-Volterra systems. While it is primarily a valuable guide for researchers working with reaction-diffusion systems and those developing the theoretical aspects of conditional symmetry conception, parts of the book can also be used in master's level mathematical biology courses.
Table des matières
1 Scalar reaction-diffusion equations - conditional symmetry, exact solutions and applications.- 2 Q-conditional symmetries of reaction-diffusion systems.- 3 Conditional symmetries and exact solutions of diffusive Lotka-Volterra systems.- 4 Q-conditional symmetries of the first type and exact solutions of nonlinear reaction-diffusion systems.- A List of reaction-diffusion systems and exact solutions.- Index.
A propos de l'auteur
Roman Cherniha graduated in mathematics from the Taras Shevchenko Kyiv State University (1981), and defended his PhD dissertation (1987) and habilitation (2003) at the Institute of Mathematics, NAS of Ukraine. During his early career, he gained substantial experience on the field of applied mathematics and physics at the Institute of Technical Heat Physics (Kyiv). Since 1992, he has held a permanent position at the Institute of Mathematics. He spent a few years abroad working at the Henri Poincaré Unniversity (a temporary CNRS position) and the University of Nottingham (Marie Curie Research Fellow). He has a wide range of research interests including: non-linear partial differential equations (especially reaction-diffusion equations): Lie and conditional symmetries, exact solutions and their properties; development of new methods for analytically solving non-linear PDEs; application of modern methods for analytically solving nonlinear boundary-value problems arising in real world application; analytically and numerically solving boundary-value problems with free boundaries; development of mathematical models describing the specific processes arising in physics, biology and medicine.
Vasyl’ Davydovych graduated in mathematics from the Lesya Ukrainka Volyn National University (2009), and defended his PhD dissertation (2014) at the Institute of Mathematics, NAS of Ukraine. At present, he is a junior researcher at the Institute of Mathematics at the NAS of Ukraine. He is currently investigating nonlinear PDEs using symmetry-based methods. His primary aim is the study of nonlinear reaction-diffusion systems arising in real-world applications (such as the diffusive Lotka-Volterra type systems).
Résumé
This book presents several fundamental results in solving nonlinear reaction-diffusion equations and systems using symmetry-based methods. Reaction-diffusion systems are fundamental modeling tools for mathematical biology with applications to ecology, population dynamics, pattern formation, morphogenesis, enzymatic reactions and chemotaxis. The book discusses the properties of nonlinear reaction-diffusion systems, which are relevant for biological applications, from the symmetry point of view, providing rigorous definitions and constructive algorithms to search for conditional symmetry (a nontrivial generalization of the well-known Lie symmetry) of nonlinear reaction-diffusion systems. In order to present applications to population dynamics, it focuses mainly on two- and three-component diffusive Lotka-Volterra systems. While it is primarily a valuable guide for researchers working with reaction-diffusion systems and those developing the theoretical aspects of conditional symmetry conception, parts of the book can also be used in master’s level mathematical biology courses.
Texte suppl.
“The aim of this book is to identify particular solutions for a wide range of reaction-diffusion Systems … . This book is primarily addressed to mathematicians working in the field of reaction-diffusion systems. Biological mathematicians can readily use the particular solutions that are listed in this book (rather convenient summary tables are given).” (Thomas Giletti, zbMATH 1391.35003, 2018)
Commentaire
"The aim of this book is to identify particular solutions for a wide range of reaction-diffusion Systems ... . This book is primarily addressed to mathematicians working in the field of reaction-diffusion systems. Biological mathematicians can readily use the particular solutions that are listed in this book (rather convenient summary tables are given)." (Thomas Giletti, zbMATH 1391.35003, 2018)
