Nonlinear Symmetries and Nonlinear Equations
Nonlinear Symmetries and Nonlinear Equations

Author: G. Gaeta

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 275

ISBN-13: 9401110182

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The study of (nonlinear) dift"erential equations was S. Lie's motivation when he created what is now known as Lie groups and Lie algebras; nevertheless, although Lie group and algebra theory flourished and was applied to a number of dift"erent physical situations -up to the point that a lot, if not most, of current fun damental elementary particles physics is actually (physical interpretation of) group theory -the application of symmetry methods to dift"erential equations remained a sleeping beauty for many, many years. The main reason for this lies probably in a fact that is quite clear to any beginner in the field. Namely, the formidable comple:rity ofthe (algebraic, not numerical!) computations involved in Lie method. I think this does not account completely for this oblivion: in other fields of Physics very hard analytical computations have been worked through; anyway, one easily understands that systems of dOlens of coupled PDEs do not seem very attractive, nor a very practical computational tool.

Nonlinear Reaction-Diffusion Systems
Nonlinear Reaction-Diffusion Systems

Author: Roman Cherniha

Publisher: Springer

Published: 2017-09-18

Total Pages: 173

ISBN-13: 3319654675

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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.

Symmetry and Perturbation Theory in Nonlinear Dynamics
Symmetry and Perturbation Theory in Nonlinear Dynamics

Author: Giampaolo Cicogna

Publisher: Springer Science & Business Media

Published: 2003-07-01

Total Pages: 218

ISBN-13: 354048874X

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has been in the of a Symmetry major ingredient development quantum perturba tion and it is a basic of the of theory, ingredient theory integrable (Hamiltonian and of the the use in context of non Hamiltonian) systems; yet, symmetry gen eral is rather recent. From the of view of nonlinear perturbation theory point the use of has become dynamics, widespread only through equivariant symmetry bifurcation in this attention has been confined to linear even theory; case, mostly symmetries. in recent the and of methods for dif Also, theory practice symmetry years ferential has become and has been to a equations increasingly popular applied of the of the book Olver This by variety problems (following appearance [2621). with is and deals of nature theory deeply geometrical symmetries general (pro vided that described i.e. in this context there is are vector no they by fields), to limit attention to linear reason symmetries. In this look the basic tools of i.e. normal book we at perturbation theory, introduced Poincar6 about and their inter a forms (first by century ago) study action with with no limitation to linear ones. We focus on the most symmetries, basic fixed the and i.e. a setting, systems having point (at origin) perturbative around thus is local.

Symmetry and Integration Methods for Differential Equations
Symmetry and Integration Methods for Differential Equations

Author: George Bluman

Publisher: Springer Science & Business Media

Published: 2008-01-10

Total Pages: 425

ISBN-13: 0387216499

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This text discusses Lie groups of transformations and basic symmetry methods for solving ordinary and partial differential equations. It places emphasis on explicit computational algorithms to discover symmetries admitted by differential equations and to construct solutions resulting from symmetries. This new edition covers contact transformations, Lie-B cklund transformations, and adjoints and integrating factors for ODEs of arbitrary order.

Separation of Variables and Exact Solutions to Nonlinear PDEs
Separation of Variables and Exact Solutions to Nonlinear PDEs

Author: Andrei D. Polyanin

Publisher: CRC Press

Published: 2021-09-20

Total Pages: 349

ISBN-13: 1000463664

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Separation of Variables and Exact Solutions to Nonlinear PDEs is devoted to describing and applying methods of generalized and functional separation of variables used to find exact solutions of nonlinear partial differential equations (PDEs). It also presents the direct method of symmetry reductions and its more general version. In addition, the authors describe the differential constraint method, which generalizes many other exact methods. The presentation involves numerous examples of utilizing the methods to find exact solutions to specific nonlinear equations of mathematical physics. The equations of heat and mass transfer, wave theory, hydrodynamics, nonlinear optics, combustion theory, chemical technology, biology, and other disciplines are studied. Particular attention is paid to nonlinear equations of a reasonably general form that depend on one or several arbitrary functions. Such equations are the most difficult to analyze. Their exact solutions are of significant practical interest, as they are suitable to assess the accuracy of various approximate analytical and numerical methods. The book contains new material previously unpublished in monographs. It is intended for a broad audience of scientists, engineers, instructors, and students specializing in applied and computational mathematics, theoretical physics, mechanics, control theory, chemical engineering science, and other disciplines. Individual sections of the book and examples are suitable for lecture courses on partial differential equations, equations of mathematical physics, and methods of mathematical physics, for delivering special courses and for practical training.

Symmetry Methods for Differential Equations
Symmetry Methods for Differential Equations

Author: Peter Ellsworth Hydon

Publisher: Cambridge University Press

Published: 2000-01-28

Total Pages: 230

ISBN-13: 9780521497862

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This book is a straightforward introduction to the subject of symmetry methods for solving differential equations, and is aimed at applied mathematicians, physicists, and engineers. The presentation is informal, using many worked examples to illustrate the main symmetry methods. It is written at a level suitable for postgraduates and advanced undergraduates, and is designed to enable the reader to master the main techniques quickly and easily.The book contains some methods that have not previously appeared in a text. These include methods for obtaining discrete symmetries and integrating factors.

Nonlinear Reaction-Diffusion-Convection Equations
Nonlinear Reaction-Diffusion-Convection Equations

Author: Roman Cherniha

Publisher: CRC Press

Published: 2017-11-02

Total Pages: 261

ISBN-13: 1498776191

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It is well known that symmetry-based methods are very powerful tools for investigating nonlinear partial differential equations (PDEs), notably for their reduction to those of lower dimensionality (e.g. to ODEs) and constructing exact solutions. This book is devoted to (1) search Lie and conditional (non-classical) symmetries of nonlinear RDC equations, (2) constructing exact solutions using the symmetries obtained, and (3) their applications for solving some biologically and physically motivated problems. The book summarises the results derived by the authors during the last 10 years and those obtained by some other authors.

Applications of Symmetry Methods to Partial Differential Equations
Applications of Symmetry Methods to Partial Differential Equations

Author: George W. Bluman

Publisher: Springer Science & Business Media

Published: 2009-10-30

Total Pages: 415

ISBN-13: 0387680284

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This is an acessible book on the advanced symmetry methods for differential equations, including such subjects as conservation laws, Lie-Bäcklund symmetries, contact transformations, adjoint symmetries, Nöther's Theorem, mappings with some modification, potential symmetries, nonlocal symmetries, nonlocal mappings, and non-classical method. Of use to graduate students and researchers in mathematics and physics.

Symmetries of Maxwell’s Equations
Symmetries of Maxwell’s Equations

Author: W.I. Fushchich

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 228

ISBN-13: 9400937296

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Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the fina\ question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gu\ik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.