This book is devoted to explaining a wide range of applications of con tinuous symmetry groups to physically important systems of differential equations.

Author: Peter J. Olver

Publisher: Springer Science & Business Media

ISBN: 9781468402742

Category: Mathematics

Page: 497

View: 465

This book is devoted to explaining a wide range of applications of con tinuous symmetry groups to physically important systems of differential equations. Emphasis is placed on significant applications of group-theoretic methods, organized so that the applied reader can readily learn the basic computational techniques required for genuine physical problems. The first chapter collects together (but does not prove) those aspects of Lie group theory which are of importance to differential equations. Applications covered in the body of the book include calculation of symmetry groups of differential equations, integration of ordinary differential equations, including special techniques for Euler-Lagrange equations or Hamiltonian systems, differential invariants and construction of equations with pre scribed symmetry groups, group-invariant solutions of partial differential equations, dimensional analysis, and the connections between conservation laws and symmetry groups. Generalizations of the basic symmetry group concept, and applications to conservation laws, integrability conditions, completely integrable systems and soliton equations, and bi-Hamiltonian systems are covered in detail. The exposition is reasonably self-contained, and supplemented by numerous examples of direct physical importance, chosen from classical mechanics, fluid mechanics, elasticity and other applied areas.

In physics groups emerge as transformations. For applications of Lie groups to differential equations we consider the transformation of coordinates and fields. Such transformations form a Lie transformation group.

Author: Willi-hans Steeb

Publisher: World Scientific Publishing Company

ISBN: 9789813107014

Category: Science

Page: 472

View: 868

This textbook comprehensively introduces students and researchers to the application of continuous symmetries and their Lie algebras to ordinary and partial differential equations. Covering all the modern techniques in detail, it relates applications to cutting-edge research fields such as Yang-Mills theory and string theory.Aimed at readers in applied mathematics and physics rather than pure mathematics, the material is ideally suited to students and researchers whose main interest lies in finding solutions to differential equations and invariants of maps.A large number of worked examples and challenging exercises help readers to work independently of teachers, and by including SymbolicC++ implementations of the techniques in each chapter, the book takes full advantage of the advancements in algebraic computation.Twelve new sections have been added in this edition, including: Haar measure, Sato's theory and sigma functions, universal algebra, anti-self dual Yang-Mills equation, and discrete Painlevé equations.

Author: Vladimir DorodnitsynPublish On: 2010-12-01

Intended for researchers, numerical analysts, and graduate students in various fields of applied mathematics, physics, mechanics, and engineering sciences, Applications of Lie Groups to Difference Equations is the first book to provide a ...

Author: Vladimir Dorodnitsyn

Publisher: CRC Press

ISBN: 1420083104

Category: Mathematics

Page: 344

View: 889

Intended for researchers, numerical analysts, and graduate students in various fields of applied mathematics, physics, mechanics, and engineering sciences, Applications of Lie Groups to Difference Equations is the first book to provide a systematic construction of invariant difference schemes for nonlinear differential equations. A guide to methods

In comparison with the known beginner guides to Lie group analysis, the book is oriented toward students who are interested in financial mathematics, mathematical finance and economics.We provide the results of the Lie group analysis of ...

Author: Ljudmila A Bordag

Publisher: World Scientific Publishing Company

ISBN: 9789814667265

Category: Mathematics

Page: 340

View: 409

This textbook is a short comprehensive and intuitive introduction to Lie group analysis of ordinary and partial differential equations. This practical-oriented material contains a large number of examples and problems accompanied by detailed solutions and figures. In comparison with the known beginner guides to Lie group analysis, the book is oriented toward students who are interested in financial mathematics, mathematical finance and economics. We provide the results of the Lie group analysis of actual models in Financial Mathematics using recent publications. These models are usually formulated as nonlinear partial differential equations and are rather difficult to make use of. With the help of Lie group analysis it is possible to describe some important properties of these models and to obtain interesting reductions in a clear and understandable algorithmic way. The book can serve as a short introduction for a further study of modern geometrical analysis applied to models in financial mathematics. It can also be used as textbook in a master's program, in an intensive compact course, or for self study. The textbook with a large number of examples will be useful not only for students who are interested in Financial Mathematics but also for people who are working in other areas of research that are not directly connected with Physics (for instance in such areas of Applied Mathematics like mathematical economy, bio systems, coding theory, etc.).

This enables readers to quickly grasp the essentials and start applying the methods to find solutions. The book includes worked examples and problems from a wide range of scientific and engineering fields.

Author: L Dresner

Publisher: CRC Press

ISBN: 1420050788

Category: Science

Page: 242

View: 782

Lie's group theory of differential equations unifies the many ad hoc methods known for solving differential equations and provides powerful new ways to find solutions. The theory has applications to both ordinary and partial differential equations and is not restricted to linear equations. Applications of Lie's Theory of Ordinary and Partial Differential Equations provides a concise, simple introduction to the application of Lie's theory to the solution of differential equations. The author emphasizes clarity and immediacy of understanding rather than encyclopedic completeness, rigor, and generality. This enables readers to quickly grasp the essentials and start applying the methods to find solutions. The book includes worked examples and problems from a wide range of scientific and engineering fields.

The text provides an ideal introduction to the modern group analysis and addresses issues to both beginners and experienced researchers in the application of Lie group methods.

Author: Nail H. Ibragimov

Publisher: CRC Press

ISBN: 0849344883

Category: Mathematics

Page: 452

View: 435

Today Lie group theoretical approach to differential equations has been extended to new situations and has become applicable to the majority of equations that frequently occur in applied sciences. Newly developed theoretical and computational methods are awaiting application. Students and applied scientists are expected to understand these methods. Volume 3 and the accompanying software allow readers to extend their knowledge of computational algebra. Written by the world's leading experts in the field, this up-to-date sourcebook covers topics such as Lie-Bäcklund, conditional and non-classical symmetries, approximate symmetry groups for equations with a small parameter, group analysis of differential equations with distributions, integro-differential equations, recursions, and symbolic software packages. The text provides an ideal introduction to the modern group analysis and addresses issues to both beginners and experienced researchers in the application of Lie group methods.

Moreover, this theory provides a universal tool for tackling considerable numbers of differential equations when other means of integration fail. * This is the first modern text on ordinary differential equations where the basic integration ...

Author: N. H. Ibragimov

Publisher: John Wiley & Sons

ISBN: STANFORD:36105026109822

Category: Mathematics

Page: 376

View: 899

Lie group analysis, based on symmetry and invariance principles, is the only systematic method for solving nonlinear differential equations analytically. One of Lie's striking achievements was the discovery that the majority of classical devices for integration of special types of ordinary differential equations could be explained and deduced by his theory. Moreover, this theory provides a universal tool for tackling considerable numbers of differential equations when other means of integration fail. * This is the first modern text on ordinary differential equations where the basic integration methods are derived from Lie group theory * Includes a concise and self contained introduction to differential equations * Easy to follow and comprehensive introduction to Lie group analysis * The methods described in this book have many applications The author provides students and their teachers with a flexible text for undergraduate and postgraduate courses, spanning a variety of topics from the basic theory through to its many applications. The philosophy of Lie groups has become an essential part of the mathematical culture for anyone investigating mathematical models of physical, engineering and natural problems.

Volume 2 offers a unique blend of classical results of Sophus Lie with new, modern developments and numerous applications which span a period of more than 100 years.

Author: Nail H. Ibragimov

Publisher: CRC Press

ISBN: 0849328640

Category: Mathematics

Page: 570

View: 663

Volume 2 offers a unique blend of classical results of Sophus Lie with new, modern developments and numerous applications which span a period of more than 100 years. As a result, this reference is up to date, with the latest information on the group theoretic methods used frequently in mathematical physics and engineering. Volume 2 is divided into three parts. Part A focuses on relevant definitions, main algorithms, group classification schemes for partial differential equations, and multifaceted possibilities offered by Lie group theoretic philosophy. Part B contains the group analysis of a variety of mathematical models for diverse natural phenomena. It tabulates symmetry groups and solutions for linear equations of mathematical physics, classical field theory, viscous and non-Newtonian fluids, boundary layer problems, Earth sciences, elasticity, plasticity, plasma theory (Vlasov-Maxwell equations), and nonlinear optics and acoustics. Part C offers an English translation of Sophus Lie's fundamental paper on the group classification and invariant solutions of linear second-order equations with two independent variables. This will serve as a concise, practical guide to the group analysis of partial differential equations.

36(3), 545–548 (1988) L.V. Sabinin, Smooth Quasigroups and Loops (Kluwer Academic Publishers, Dordrecht, 1999) P.J. Olver, Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, vol.

Author: Giovanni Falcone

Publisher: Springer

ISBN: 9783319621814

Category: Mathematics

Page: 361

View: 349

This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters. From representation theory and Gerstenhaber algebras to control theory, from differential equations to Finsler geometry and Lepage manifolds, the book introduces young researchers in Mathematics to a wealth of different topics, encouraging a multidisciplinary approach to research. As such, it is suitable for students in doctoral courses, and will also benefit researchers who want to expand their field of interest.

This book is a comprehensive introduction to the application of continuous symmetries and their Lie algebras to ordinary and partial differential equations.

Author: W-H Steeb

Publisher: World Scientific Publishing Company

ISBN: 9789813105034

Category: Science

Page: 372

View: 864

This book is a comprehensive introduction to the application of continuous symmetries and their Lie algebras to ordinary and partial differential equations. It is suitable for students and research workers whose main interest lies in finding solutions to differential equations. It therefore caters for readers primarily interested in applied mathematics and physics rather than pure mathematics. The book provides an application-orientated text that is reasonably self-contained. A large number of worked examples have been included to help readers working independently of a teacher. The advance of algebraic computation has made it possible to write programs for the tedious calculations in this research field, and thus the book also makes a survey of computer algebra packages.