Multiple Time Scales

Multiple Time Scales
Author: Jeremiah U. Brackbill
Publisher: Academic Press
Total Pages: 457
Release: 2014-05-10
Genre: Mathematics
ISBN: 1483257568

Multiple Time Scales presents various numerical methods for solving multiple-time-scale problems. The selection first elaborates on considerations on solving problems with multiple scales; problems with different time scales; and nonlinear normal-mode initialization of numerical weather prediction models. Discussions focus on analysis of observations, nonlinear analysis, systems of ordinary differential equations, and numerical methods for problems with multiple scales. The text then examines the diffusion-synthetic acceleration of transport iterations, with application to a radiation hydrodynamics problem and implicit methods in combustion and chemical kinetics modeling. The publication ponders on molecular dynamics and Monte Carlo simulations of rare events; direct implicit plasma simulation; orbit averaging and subcycling in particle simulation of plasmas; and hybrid and collisional implicit plasma simulation models. Topics include basic moment method, electron subcycling, gyroaveraged particle simulation, and the electromagnetic direct implicit method. The selection is a valuable reference for researchers interested in pursuing further research on the use of numerical methods in solving multiple-time-scale problems.

Multiple Time Scale Dynamics

Multiple Time Scale Dynamics
Author: Christian Kuehn
Publisher: Springer
Total Pages: 816
Release: 2015-02-25
Genre: Mathematics
ISBN: 3319123165

This book provides an introduction to dynamical systems with multiple time scales. The approach it takes is to provide an overview of key areas, particularly topics that are less available in the introductory form. The broad range of topics included makes it accessible for students and researchers new to the field to gain a quick and thorough overview. The first of its kind, this book merges a wide variety of different mathematical techniques into a more unified framework. The book is highly illustrated with many examples and exercises and an extensive bibliography. The target audience of this book are senior undergraduates, graduate students as well as researchers interested in using the multiple time scale dynamics theory in nonlinear science, either from a theoretical or a mathematical modeling perspective.

Multiple-Time-Scale Dynamical Systems

Multiple-Time-Scale Dynamical Systems
Author: Christopher K.R.T. Jones
Publisher: Springer Science & Business Media
Total Pages: 278
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461301173

Systems with sub-processes evolving on many different time scales are ubiquitous in applications: chemical reactions, electro-optical and neuro-biological systems, to name just a few. This volume contains papers that expose the state of the art in mathematical techniques for analyzing such systems. Recently developed geometric ideas are highlighted in this work that includes a theory of relaxation-oscillation phenomena in higher dimensional phase spaces. Subtle exponentially small effects result from singular perturbations implicit in certain multiple time scale systems. Their role in the slow motion of fronts, bifurcations, and jumping between invariant tori are all explored here. Neurobiology has played a particularly stimulating role in the development of these techniques and one paper is directed specifically at applying geometric singular perturbation theory to reveal the synchrony in networks of neural oscillators.

Dynamic Inequalities On Time Scales

Dynamic Inequalities On Time Scales
Author: Ravi Agarwal
Publisher: Springer
Total Pages: 264
Release: 2014-10-30
Genre: Mathematics
ISBN: 3319110020

This is a monograph devoted to recent research and results on dynamic inequalities on time scales. The study of dynamic inequalities on time scales has been covered extensively in the literature in recent years and has now become a major sub-field in pure and applied mathematics. In particular, this book will cover recent results on integral inequalities, including Young's inequality, Jensen's inequality, Holder's inequality, Minkowski's inequality, Steffensen's inequality, Hermite-Hadamard inequality and Čebyšv's inequality. Opial type inequalities on time scales and their extensions with weighted functions, Lyapunov type inequalities, Halanay type inequalities for dynamic equations on time scales, and Wirtinger type inequalities on time scales and their extensions will also be discussed here in detail.

Methods and Applications of Singular Perturbations

Methods and Applications of Singular Perturbations
Author: Ferdinand Verhulst
Publisher: Springer Science & Business Media
Total Pages: 332
Release: 2006-06-04
Genre: Mathematics
ISBN: 0387283137

Contains well-chosen examples and exercises A student-friendly introduction that follows a workbook type approach

Principles of Neural Coding

Principles of Neural Coding
Author: Rodrigo Quian Quiroga
Publisher: CRC Press
Total Pages: 625
Release: 2013-05-06
Genre: Medical
ISBN: 1439853312

Understanding how populations of neurons encode information is the challenge faced by researchers in the field of neural coding. Focusing on the many mysteries and marvels of the mind has prompted a prominent team of experts in the field to put their heads together and fire up a book on the subject. Simply titled Principles of Neural Coding, this b

Nonlinear Time Scale Systems in Standard and Nonstandard Forms

Nonlinear Time Scale Systems in Standard and Nonstandard Forms
Author: Anshu Narang-Siddarth
Publisher: SIAM
Total Pages: 231
Release: 2014-01-01
Genre: Mathematics
ISBN: 1611973341

This book introduces key concepts for systematically controlling engineering systems that possess interacting phenomena occurring at widely different speeds. The aim is to present the reader with control techniques that extend the benefits of model reduction of singular perturbation theory to a larger class of nonlinear dynamical systems. New results and relevant background are presented through insightful examples that cover a wide range of applications from different branches of engineering. This book is unique because it: presents a new perspective on existing control methods and thus broadens their application to a larger class of nonlinear dynamical systems; discusses general rather than problem-specific developments to certain applications or disciplines in order to provide control engineers with useful analytical tools ; addresses new control problems using singular perturbation methods, including closed-form results for control of nonminimum phase systems.

Multiple Scale and Singular Perturbation Methods

Multiple Scale and Singular Perturbation Methods
Author: J.K. Kevorkian
Publisher: Springer
Total Pages: 634
Release: 1996-05-15
Genre: Mathematics
ISBN: 0387942025

This book is a revised and updated version, including a substantial portion of new material, of our text Perturbation Methods in Applied Mathematics (Springer Verlag, 1981). We present the material at a level that assumes some familiarity with the basics of ordinary and partial differential equations. Some of the more advanced ideas are reviewed as needed; therefore this book can serve as a text in either an advanced undergraduate course or a graduate-level course on the subject. Perturbation methods, first used by astronomers to predict the effects of small disturbances on the nominal motions of celestial bodies, have now become widely used analytical tools in virtually all branches of science. A problem lends itself to perturbation analysis if it is "close" to a simpler problem that can be solved exactly. Typically, this closeness is measured by the occurrence of a small dimensionless parameter, E, in the governing system (consisting of differential equations and boundary conditions) so that for E = 0 the resulting system is exactly solvable. The main mathematical tool used is asymptotic expansion with respect to a suitable asymptotic sequence of functions of E. In a regular perturbation problem, a straightforward procedure leads to a system of differential equations and boundary conditions for each term in the asymptotic expansion. This system can be solved recursively, and the accuracy of the result improves as E gets smaller, for all values of the independent variables throughout the domain of interest. We discuss regular perturbation problems in the first chapter.

Conformable Dynamic Equations on Time Scales

Conformable Dynamic Equations on Time Scales
Author: Douglas R. Anderson
Publisher: CRC Press
Total Pages: 347
Release: 2020-08-29
Genre: Mathematics
ISBN: 100009393X

The concept of derivatives of non-integer order, known as fractional derivatives, first appeared in the letter between L’Hopital and Leibniz in which the question of a half-order derivative was posed. Since then, many formulations of fractional derivatives have appeared. Recently, a new definition of fractional derivative, called the "fractional conformable derivative," has been introduced. This new fractional derivative is compatible with the classical derivative and it has attracted attention in areas as diverse as mechanics, electronics, and anomalous diffusion. Conformable Dynamic Equations on Time Scales is devoted to the qualitative theory of conformable dynamic equations on time scales. This book summarizes the most recent contributions in this area, and vastly expands on them to conceive of a comprehensive theory developed exclusively for this book. Except for a few sections in Chapter 1, the results here are presented for the first time. As a result, the book is intended for researchers who work on dynamic calculus on time scales and its applications. Features Can be used as a textbook at the graduate level as well as a reference book for several disciplines Suitable for an audience of specialists such as mathematicians, physicists, engineers, and biologists Contains a new definition of fractional derivative About the Authors Douglas R. Anderson is professor and chair of the mathematics department at Concordia College, Moorhead. His research areas of interest include dynamic equations on time scales and Ulam-type stability of difference and dynamic equations. He is also active in investigating the existence of solutions for boundary value problems. Svetlin G. Georgiev is currently professor at Sorbonne University, Paris, France and works in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, dynamic calculus on time scales, and integral equations.