Nicola Bellomo

Livio Gibelli

This volume explores the complex problems that arise in the modeling and simulation of crowd dynamics in order to present the state-of-the-art of this emerging field and contribute to future research activities. Experts in various areas apply their unique perspectives to specific aspects of crowd dynamics, covering the topic from multiple angles. These include a demonstration of how virtual reality may solve dilemmas in collecting empirical data; a detailed study on pedestrian movement in smoke-filled environments; a presentation of one-dimensional conservation laws with point constraints on the flux; a collection of new ideas on the modeling of crowd dynamics at the microscopic scale; and others. Applied mathematicians interested in crowd dynamics, pedestrian movement, traffic flow modeling, urban planning, and other topics will find this volume a valuable resource. Additionally, researchers in social psychology, architecture, and engineering may find this information relevant to their work.

Theory, Models, and Safety Problems

Crowd Dynamics, Volume 1

Luigi Preziosi

Addressed to engineers, scientists, and applied mathematicians, this book explores the fundamental aspects of mathematical modelling in applied sciences and related mathematical and computational methods. After providing the general framework needed for mathematical modelling-definitions, classifications, general modelling procedures, and validation methods-the authors deal with the analysis of discrete models. This includes modelling methods and related mathematical methods. The analysis of models is defined in terms of ordinary differential equations. The analysis of continuous models, particularly models defined in terms of partial differential equations, follows. The authors then examine inverse type problems and stochastic modelling. Three appendices provide a concise guide to functional analysis, approximation theory, and probability, and a diskette included with the book includes ten scientific programs to introduce the reader to scientific computation at a practical level.

Modelling Mathematical Methods and Scientific Computation

Antonio Romano

Modeling and Applied Mathematics Modeling the behavior of real physical systems by suitable evolution equa tions is a relevant, maybe the fundamental, aspect of the interactions be tween mathematics and applied sciences. Modeling is, however, only the first step toward the mathematical description and simulation of systems belonging to real world. Indeed, once the evolution equation is proposed, one has to deal with mathematical problems and develop suitable simula tions to provide the description of the real system according to the model. Within this framework, one has an evolution equation and the re lated mathematical problems obtained by adding all necessary conditions for their solution. Then, a qualitative analysis should be developed: this means proof of existence of solutions and analysis of their qualitative be havior. Asymptotic analysis may include a detailed description of stability properties. Quantitative analysis, based upon the application ofsuitable methods and algorithms for the solution of problems, ends up with the simulation that is the representation of the dependent variable versus the independent one. The information obtained by the model has to be compared with those deriving from the experimental observation of the real system. This comparison may finally lead to the validation of the model followed by its application and, maybe, further generalization.

Mechanics and Dynamical Systems with Mathematica®

Bertrand Lods

Roberto Revelli

Luca Ridolfi

Analysis of nonlinear models and problems is crucial in the application of mathematics to real-world problems. This book approaches this important topic by focusing on collocation methods for solving nonlinear evolution equations and applying them to a variety of mathematical problems. These include wave motion models, hydrodynamic models of vehicular traffic flow, convection-diffusion models, reaction-diffusion models, and population dynamics models. The book may be used as a textbook for graduate courses on collocation methods, nonlinear modeling, and nonlinear differential equations. Examples and exercises are included in every chapter.

Solutions to Nonlinear Problems

Generalized Collocation Methods

Elena de Angelis

This collection of selected chapters offers a comprehensive overview of state-of-the-art mathematical methods and tools for modeling and analyzing cancer phenomena. Topics covered include stochastic evolutionary models of cancer initiation and progression, tumor cords and their response to anticancer agents, and immune competition in tumor progression and prevention. The complexity of modeling living matter requires the development of new mathematical methods and ideas. This volume, written by first-rate researchers in the field of mathematical biology, is one of the first steps in that direction.

Genesis, Evolution, Immune Competition, and Therapy

Selected Topics in Cancer Modeling

Pierre Degond

Eitan Tadmor

This volume collects ten surveys on the modeling, simulation, and applications of active particles using methods ranging from mathematical kinetic theory to nonequilibrium statistical mechanics. The contributing authors are leading experts working in this challenging field, and each of their chapters provides a review of the most recent results in their areas and looks ahead to future research directions. The approaches to studying active matter are presented here from many different perspectives, such as individual-based models, evolutionary games, Brownian motion, and continuum theories, as well as various combinations of these. Applications covered include biological network formation and network theory; opinion formation and social systems; control theory of sparse systems; theory and applications of mean field games; population learning; dynamics of flocking systems; vehicular traffic flow; and stochastic particles and mean field approximation. Mathematicians and other members of the scientific community interested in active matter and its many applications will find this volume to be a timely, authoritative, and valuable resource.

Advances in Theory, Models, and Applications

Active Particles, Volume 1

Develops different mathematical methods and tools to model living systems. This book presents material that can be used in such real-world applications as immunology, transportation engineering, and economics. It is of interest to those involved in modeling complex social systems and living matter in general.

A Kinetic Theory and Stochastic Game Approach

Modeling Complex Living Systems

Mario Pulvirenti

Modeling complex biological, chemical, and physical systems, in the context of spatially heterogeneous mediums, is a challenging task for scientists and engineers using traditional methods of analysis. Modeling in Applied Sciences is a comprehensive survey of modeling large systems using kinetic equations, and in particular the Boltzmann equation and its generalizations. An interdisciplinary group of leading authorities carefully develop the foundations of kinetic models and discuss the connections and interactions between model theories, qualitative and computational analysis and real-world applications. This book provides a thoroughly accessible and lucid overview of the different aspects, models, computations, and methodology for the kinetic-theory modeling process. Topics and Features: * Integrated modeling perspective utilized in all chapters * Fluid dynamics of reacting gases * Self-contained introduction to kinetic models * Becker–Doring equations * Nonlinear kinetic models with chemical reactions * Kinetic traffic-flow models * Models of granular media * Large communication networks * Thorough discussion of numerical simulations of Boltzmann equation This new book is an essential resource for all scientists and engineers who use large-scale computations for studying the dynamics of complex systems of fluids and particles. Professionals, researchers, and postgraduates will find the book a modern and authoritative guide to the topic.

Nicola Bellomo

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