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Author : W. I. Fushchich,W. M. Shtelen,N. I. Serov
Publisher : Unknown
Page : 464 pages
File Size : 42,8 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 9401731993
Author : Vilʹgelʹm Ilʹich Fushchich,Vladimir Mikhaĭlovich Shtelenʹ,Nikolaĭ Ivanovich Serov
Publisher : Springer
Page : 472 pages
File Size : 40,6 Mb
Release : 1993-02-28
Category : Mathematics
ISBN : UCAL:B4513652
This volume presents an account of the current state of algebraic-theoretic methods as applied to linear and nonlinear multidimensional equations of mathematical and theoretical physics. Equations are considered that are invariant under Euclid, Galilei, Schrödinger, Poincaré, conformal, and some other Lie groups, with special emphasis being given to the construction of wide classes of exact solutions of concrete nonlinear partial differential equations, such as d'Alembert, Liouville, Monge-Ampère, Hamilton-Jacobi, eikonal, Schrödinger, Navier-Stokes, gas dynamics, Dirac, Maxwell-Dirac, Yang-Mills, etc. Ansätze for spinor, as well as scalar and vector fields are described and formulae for generating solutions via conformal transformations are found explicitly for scalar, spinor, vector, and tensor fields with arbitrary conformal degree. The classical three-body problem is considered for the group-theoretic point of view. The symmetry of integro-differential equations is also studied, and the method of finding final nonlocal transformations is described. Furthermore, the concept of conditional symmetry is introduced and is used to obtain new non-Lie Ansätze for nonlinear heat and acoustic equations. The volume comprises an Introduction, which presents a brief account of the main ideas, followed by five chapters, appendices, and a comprehensive bibliography. This book will be of interest to researchers, and graduate students in physics and mathematics interested in algebraic-theoretic methods in mathematical and theoretical physics.
Author : W.I. Fushchich,W.M. Shtelen,N.I. Serov
Publisher : Springer Science & Business Media
Page : 456 pages
File Size : 52,7 Mb
Release : 2013-03-14
Category : Science
ISBN : 9789401731980
by spin or (spin s = 1/2) field equations is emphasized because their solutions can be used for constructing solutions of other field equations insofar as fields with any spin may be constructed from spin s = 1/2 fields. A brief account of the main ideas of the book is presented in the Introduction. The book is largely based on the authors' works [55-109, 176-189, 13-16, 7*-14*,23*, 24*] carried out in the Institute of Mathematics, Academy of Sciences of the Ukraine. References to other sources is not intended to imply completeness. As a rule, only those works used directly are cited. The authors wish to express their gratitude to Academician Yu.A. Mitropoi sky, and to Academician of Academy of Sciences of the Ukraine O.S. Parasyuk, for basic support and stimulation over the course of many years; to our cowork ers in the Department of Applied Studies, LA. Egorchenko, R.Z. Zhdanov, A.G. Nikitin, LV. Revenko, V.L Lagno, and I.M. Tsifra for assistance with the manuscript.
Author : Andrei D. Polyanin,Alexei I. Zhurov
Publisher : CRC Press
Page : 402 pages
File Size : 47,7 Mb
Release : 2021-09-19
Category : Mathematics
ISBN : 9781000463637
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.
Author : Sergey V. Meleshko
Publisher : Springer Science & Business Media
Page : 367 pages
File Size : 41,5 Mb
Release : 2006-06-18
Category : Technology & Engineering
ISBN : 9780387252650
Differential equations, especially nonlinear, present the most effective way for describing complex physical processes. Methods for constructing exact solutions of differential equations play an important role in applied mathematics and mechanics. This book aims to provide scientists, engineers and students with an easy-to-follow, but comprehensive, description of the methods for constructing exact solutions of differential equations.
Author : Victor A. Galaktionov,Sergey R. Svirshchevskii
Publisher : CRC Press
Page : 530 pages
File Size : 45,5 Mb
Release : 2006-11-02
Category : Mathematics
ISBN : 9781420011623
Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics is the first book to provide a systematic construction of exact solutions via linear invariant subspaces for nonlinear differential operators. Acting as a guide to nonlinear evolution equations and models from physics and mechanics, the book
Author : Daniel J. Arrigo
Publisher : John Wiley & Sons
Page : 192 pages
File Size : 42,8 Mb
Release : 2015-01-07
Category : Mathematics
ISBN : 9781118721445
A self-contained introduction to the methods and techniques of symmetry analysis used to solve ODEs and PDEs Symmetry Analysis of Differential Equations: An Introduction presents an accessible approach to the uses of symmetry methods in solving both ordinary differential equations (ODEs) and partial differential equations (PDEs). Providing comprehensive coverage, the book fills a gap in the literature by discussing elementary symmetry concepts and invariance, including methods for reducing the complexity of ODEs and PDEs in an effort to solve the associated problems. Thoroughly class-tested, the author presents classical methods in a systematic, logical, and well-balanced manner. As the book progresses, the chapters graduate from elementary symmetries and the invariance of algebraic equations, to ODEs and PDEs, followed by coverage of the nonclassical method and compatibility. Symmetry Analysis of Differential Equations: An Introduction also features: Detailed, step-by-step examples to guide readers through the methods of symmetry analysis End-of-chapter exercises, varying from elementary to advanced, with select solutions to aid in the calculation of the presented algorithmic methods Symmetry Analysis of Differential Equations: An Introduction is an ideal textbook for upper-undergraduate and graduate-level courses in symmetry methods and applied mathematics. The book is also a useful reference for professionals in science, physics, and engineering, as well as anyone wishing to learn about the use of symmetry methods in solving differential equations.
Author : Roman Cherniha,Mykola Serov,Oleksii Pliukhin
Publisher : CRC Press
Page : 244 pages
File Size : 48,9 Mb
Release : 2017-11-02
Category : Mathematics
ISBN : 9781351650878
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.
Author : Nail H. Ibragimov
Publisher : CRC Press
Page : 572 pages
File Size : 52,9 Mb
Release : 1995-10-24
Category : Mathematics
ISBN : 0849394198
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 modern group analysis and addresses issues to both beginners and experienced researchers in the application of Lie group methods.
Author : Roman Cherniha,Vasyl' Davydovych
Publisher : Springer
Page : 160 pages
File Size : 47,7 Mb
Release : 2017-09-18
Category : Mathematics
ISBN : 9783319654676
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.
Author : Solomon Manukure
Publisher : Springer Nature
Page : 389 pages
File Size : 50,8 Mb
Release : 2024-07-01
Category : Electronic
ISBN : 9783031595394
Author : George W. Bluman,Alexei F. Cheviakov,Stephen Anco
Publisher : Springer Science & Business Media
Page : 415 pages
File Size : 48,5 Mb
Release : 2009-10-30
Category : Mathematics
ISBN : 9780387680286
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.
Author : Roman M. Cherniha
Publisher : MDPI
Page : 427 pages
File Size : 49,5 Mb
Release : 2018-07-06
Category : Electronic book
ISBN : 9783038425267
This book is a printed edition of the Special Issue "Lie Theory and Its Applications" that was published in Symmetry
Author : Nail H. Ibragimov
Publisher : CRC Press
Page : 452 pages
File Size : 44,6 Mb
Release : 1993-10-20
Category : Mathematics
ISBN : 0849344883
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.
Author : Roman Cherniha,Mykola Serov,Oleksii Pliukhin
Publisher : CRC Press
Page : 261 pages
File Size : 49,9 Mb
Release : 2017-11-02
Category : Mathematics
ISBN : 9781498776196
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.