Stochastic Ordinary and Stochastic Partial Differential Equations

Peter Kotelenez

ISBN 10: 1489986588 / ISBN 13: 9781489986580

Published by Springer New York, Springer New York Sep 2014, 2014

About this Item

This item is printed on demand - Print on Demand Titel. Neuware -The present volume analyzes mathematical models of time-dependent physical p- nomena on three levels: microscopic, mesoscopic, and macroscopic. We provide a rigorous derivation of each level from the preceding level and the resulting me- scopic equations are analyzed in detail. Following Haken (1983, Sect. 1. 11. 6) we deal, ¿at the microscopic level, with individual atoms or molecules, described by their positions, velocities, and mutual interactions. At the mesoscopic level, we describe the liquid by means of ensembles of many atoms or molecules. The - tension of such an ensemble is assumed large compared to interatomic distances but small compared to the evolving macroscopic pattern. . . . At the macroscopic level we wish to study the corresponding spatial patterns. ¿ Typically, at the mac- scopic level, the systems under consideration are treated as spatially continuous systems such as uids or a continuous distribution of some chemical reactants, etc. Incontrast,onthemicroscopiclevel,Newtonianmechanicsgovernstheequationsof 1 motion of the individual atoms or molecules. These equations are cast in the form 2 of systems of deterministic coupled nonlinear oscillators. The mesoscopic level is probabilistic in nature and many models may be faithfully described by stochastic 3 ordinary and stochastic partial differential equations (SODEs and SPDEs), where the latter are de ned on a continuum. The macroscopic level is described by ti- dependent partial differential equations (PDE¿s) and its generalization and simpl- cations. In our mathematical framework we talk of particles instead of atoms and mo- cules. The transition from the microscopic description to a mesoscopic (i. e.Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 472 pp. Englisch.

Seller Inventory # 9781489986580

Bibliographic Details

• Title: Stochastic Ordinary and Stochastic Partial Differential Equations
• Publisher: Springer New York, Springer New York Sep 2014
• Publication Date: 2014
• Language: English
• ISBN 10: 1489986588
• ISBN 13: 9781489986580
• Binding: Taschenbuch
• Condition: Neu

About this title

Synopsis

Stochastic Partial Differential Equations analyzes mathematical models of time-dependent physical phenomena on microscopic, macroscopic and mesoscopic levels. It provides a rigorous derivation of each level from the preceding one and examines the resulting mesoscopic equations in detail. Coverage first describes the transition from the microscopic equations to the mesoscopic equations. It then covers a general system for the positions of the large particles.

From the Back Cover

This book provides the first rigorous derivation of mesoscopic and macroscopic equations from a deterministic system of microscopic equations. The microscopic equations are cast in the form of a deterministic (Newtonian) system of coupled nonlinear oscillators for N large particles and infinitely many small particles. The mesoscopic equations are stochastic ordinary differential equations (SODEs) and stochastic partial differential equatuions (SPDEs), and the macroscopic limit is described by a parabolic partial differential equation.

 A detailed analysis of the SODEs and (quasi-linear) SPDEs is presented. Semi-linear (parabolic) SPDEs are represented as first order stochastic transport equations driven by Stratonovich differentials. The time evolution of correlated Brownian motions is shown to be consistent with the depletion phenomena experimentally observed in colloids. A covariance analysis of the random processes and random fields as well as a review section of various approaches to SPDEs are also provided.

An extensive appendix makes the book accessible to both scientists and graduate students who may not be specialized in stochastic analysis.

 Probabilists, mathematical and theoretical physicists as well as mathematical biologists and their graduate students will find this book useful.

 Peter Kotelenez is a professor of mathematics at Case Western Reserve University in Cleveland, Ohio.