## Applications of Lie's Theory of Ordinary and Partial Differential Equations

**Author**: L Dresner

**Publisher:**CRC Press

**ISBN:**9781420050783

**Category:**Science

**Page:**225

**View:**1892

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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.

## Applications of Lie Groups to Differential Equations

**Author**: Peter J. Olver

**Publisher:**Springer Science & Business Media

**ISBN:**1468402749

**Category:**Mathematics

**Page:**497

**View:**5254

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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.

## Symmetry Analysis of Differential Equations with Mathematica®

**Author**: Gerd Baumann

**Publisher:**Springer Science & Business Media

**ISBN:**1461221102

**Category:**Mathematics

**Page:**521

**View:**3915

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The first book to explicitly use Mathematica so as to allow researchers and students to more easily compute and solve almost any kind of differential equation using Lie's theory. Previously time-consuming and cumbersome calculations are now much more easily and quickly performed using the Mathematica computer algebra software. The material in this book, and on the accompanying CD-ROM, will be of interest to a broad group of scientists, mathematicians and engineers involved in dealing with symmetry analysis of differential equations. Each section of the book starts with a theoretical discussion of the material, then shows the application in connection with Mathematica. The cross-platform CD-ROM contains Mathematica (version 3.0) notebooks which allow users to directly interact with the code presented within the book. In addition, the author's proprietary "MathLie" software is included, so users can readily learn to use this powerful tool in regard to performing algebraic computations.

## Vorlesungen über partielle Differentialgleichungen

**Author**: Vladimir I. Arnold

**Publisher:**Springer-Verlag

**ISBN:**3540350314

**Category:**Mathematics

**Page:**174

**View:**1713

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Nach seinem bekannten und viel verwendeten Buch über gewöhnliche Differentialgleichungen widmet sich der berühmte Mathematiker Vladimir Arnold nun den partiellen Differentialgleichungen in einem neuen Lehrbuch. In seiner unnachahmlich eleganten Art führt er über einen geometrischen, anschaulichen Weg in das Thema ein, und ermöglicht den Lesern so ein vertieftes Verständnis der Natur der partiellen Differentialgleichungen. Für Studierende der Mathematik und Physik ist dieses Buch ein Muss. Wie alle Bücher Vladimir Arnolds ist dieses Buch voller geometrischer Erkenntnisse. Arnold illustriert jeden Grundsatz mit einer Abbildung. Das Buch behandelt die elementarsten Teile des Fachgebiets and beschränkt sich hauptsächlich auf das Cauchy-Problem und das Neumann-Problems für die klassischen Lineargleichungen der mathematischen Physik, insbesondere auf die Laplace-Gleichung und die Wellengleichung, wobei die Wärmeleitungsgleichung und die Korteweg-de-Vries-Gleichung aber ebenfalls diskutiert werden. Die physikalische Intuition wird besonders hervorgehoben. Eine große Anzahl von Problemen ist übers ganze Buch verteilt, und ein ganzer Satz von Aufgaben findet sich am Ende. Was dieses Buch so einzigartig macht, ist das besondere Talent Arnolds, ein Thema aus einer neuen, frischen Perspektive zu beleuchten. Er lüftet gerne den Schleier der Verallgemeinerung, der so viele mathematische Texte umgibt, und enthüllt die im wesentlichen einfachen, intuitiven Ideen, die dem Thema zugrunde liegen. Das kann er besser als jeder andere mathematische Autor.

## Linear Theory of Colombeau Generalized Functions

**Author**: M Nedeljkov,S Pilipovic,D Scarpalezos

**Publisher:**CRC Press

**ISBN:**9780582356832

**Category:**Mathematics

**Page:**168

**View:**8897

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Results from the now-classical distribution theory involving convolution and Fourier transformation are extended to cater for Colombeau's generalized functions. Indications are given how these particular generalized functions can be used to investigate linear equations and pseudo differential operators. Furthermore, applications are also given to problems with nonregular data.

## Geometrical Properties of Differential Equations

*Applications of the Lie Group Analysis in Financial Mathematics*

**Author**: Ljudmila A Bordag

**Publisher:**World Scientific Publishing Company

**ISBN:**9814667269

**Category:**Mathematics

**Page:**340

**View:**7572

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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.).

## Amphora

*Festschrift für Hans Wussing zu seinem 65. Geburtstag Festschrift for Hans Wussing on the Occasion of his 65th Birthday*

**Author**: S.S Demidov,M. Folkerts,D.E. Rowe,C.J. Scriba

**Publisher:**Springer-Verlag

**ISBN:**3034885997

**Category:**Mathematics

**Page:**782

**View:**4391

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eine Assistentenstelle bei GERHARD HARIG am bereits 1906 gegründeten Karl-Sudhoff-Institut für Geschichte der Medizin und Naturwissenschaften in Leipzig, die er anderen Angeboten (z. B. beim Flugzeugbau) vorzog. Nach dem Tode von Professor HARIG bekam HANS WUSSING 1967 (als einziger habilitierter Wissenschaftshistoriker in der DDR) eine Dozentur für Geschichte der Mathematik und der Naturwissenschaften und wurde zum kommissarischen Direktor des Sudhoff-Instituts eingesetzt. Ein Jahr später wurde er zum a. o. Professor für Geschichte der Mathematik und der Naturwissenschaften berufen, 1970 erfolgte die Ernennung zum ordent lichen Professor. Von 1977 bis 1982 war er Direktor des Sudhoff-Instituts und ist seit 1982 Leiter der Abteilung für Geschichte der Mathematik und der Naturwissenschaften. Die Reihe von WUSSINGs Publikationen ist lang. Eine Liste seiner Veröffentlichungen bis 1985 findet sich in der Zeitschrift NTM, Bd. 24 (1987), S. 1-5. Es ist hier nicht der Ort, all seine Arbeiten im einzelnen zu würdigen. Erwähnt seien nur die wichtigsten Buchpublikationen: 1962 erschien bei B. G. Teubner Leipzig die Mathematik in der Antike. WUSSING verfaßte Biographien von COPERNICUS, GAUSS, NEWTON und ADAM RIES. Auch seine neueste Publikation hat mit dem bekannten deutschen Rechenmeister zu tun: Die Goß von ADAM RIES konnte er trotz schwie rigster Umstände zusammen mit WOLFGANG KAUNZNER noch rechtzeitig im Jubiläumsjahr 1992 herausgeben. WUSSING ist auch ein erfolgreicher Hochschullehrer.

## Solution Sets of Differential Equations in Abstract Spaces

**Author**: Robert Dragoni,Paolo Nistri,Pietro Zecca,Jack W Macki

**Publisher:**CRC Press

**ISBN:**9780582294509

**Category:**Mathematics

**Page:**120

**View:**6081

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This book presents results on the geometric/topological structure of the solution set S of an initial-value problem x(t) = f(t, x(t)), x(0) =xo, when f is a continuous function with values in an infinite-dimensional space. A comprehensive survey of existence results and the properties of S, e.g. when S is a connected set, a retract, an acyclic set, is presented. The authors also survey results onthe properties of S for initial-value problems involving differential inclusions, and for boundary-value problems. This book will be of particular interest to researchers in ordinary and partial differential equations and some workers in control theory.

## ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

*THEORY AND APPLICATIONS*

**Author**: NITA H. SHAH

**Publisher:**PHI Learning Pvt. Ltd.

**ISBN:**8120350871

**Category:**Mathematics

**Page:**528

**View:**5319

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This revised and updated text, now in its second edition, continues to present the theoretical concepts of methods of solutions of ordinary and partial differential equations. It equips students with the various tools and techniques to model different physical problems using such equations. The book discusses the basic concepts of ordinary and partial differential equations. It contains different methods of solving ordinary differential equations of first order and higher degree. It gives the solution methodology for linear differential equations with constant and variable coefficients and linear differential equations of second order. The text elaborates simultaneous linear differential equations, total differential equations, and partial differential equations along with the series solution of second order linear differential equations. It also covers Bessel’s and Legendre’s equations and functions, and the Laplace transform. Finally, the book revisits partial differential equations to solve the Laplace equation, wave equation and diffusion equation, and discusses the methods to solve partial differential equations using the Fourier transform. A large number of solved examples as well as exercises at the end of chapters help the students comprehend and strengthen the underlying concepts. The book is intended for undergraduate and postgraduate students of Mathematics (B.A./B.Sc., M.A./M.Sc.), and undergraduate students of all branches of engineering (B.E./B.Tech.), as part of their course in Engineering Mathematics. New to the SECOND Edition • Includes new sections and subsections such as applications of differential equations, special substitution (Lagrange and Riccati), solutions of non-linear equations which are exact, method of variation of parameters for linear equations of order higher than two, and method of undetermined coefficients • Incorporates several worked-out examples and exercises with their answers • Contains a new Chapter 19 on ‘Z-Transforms and its Applications’.

## Singularities of Solutions of Second-Order Quasilinear Equations

**Author**: Laurent Veron

**Publisher:**CRC Press

**ISBN:**9780582035393

**Category:**Mathematics

**Page:**392

**View:**8442

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## Theory of Differential Equations

*Partial differential equations*

**Author**: Andrew Russell Forsyth

**Publisher:**CUP Archive

**ISBN:**N.A

**Category:**

**Page:**478

**View:**8522

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## Progress in Partial Differential Equations The Metz Surveys 2

**Author**: Michel Chipot

**Publisher:**CRC Press

**ISBN:**9780582227699

**Category:**Mathematics

**Page:**248

**View:**4104

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This volume presents papers from the conferences given at the University of Metz in 1992, and presents some recent advances in various important domains of partial differential equations and applied mathematics. A special attempt has been made to make this work accessible to young researchers and non-specialists.

## Analysis of Weakly Compressible Turbulence Using Symmetry Methods and Direct Numerical Simulation

**Author**: Raphael Gotthard Harald Arlitt

**Publisher:**Cuvillier Verlag

**ISBN:**3865373461

**Category:**

**Page:**201

**View:**3892

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## Monotone Dynamical Systems

*An Introduction to the Theory of Competitive and Cooperative Systems*

**Author**: Hal L. Smith

**Publisher:**American Mathematical Soc.

**ISBN:**0821844873

**Category:**Mathematics

**Page:**174

**View:**9086

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This book presents comprehensive treatment of a rapidly developing area with many potential applications: the theory of monotone dynamical systems and the theory of competitive and cooperative differential equations. The primary aim is to provide potential users of the theory with techniques, results, and ideas useful in applications, while at the same time providing rigorous proofs. Among the topics discussed in the book are continuous-time monotone dynamical systems, and quasimonotone and nonquasimonotone delay differential equations. The book closes with a discussion of applications to quasimonotone systems of reaction-diffusion type. Throughout the book, applications of the theory to many mathematical models arising in biology are discussed. Requiring a background in dynamical systems at the level of a first graduate course, this book is useful to graduate students and researchers working in the theory of dynamical systems and its applications.

## Elementary Lie group analysis and ordinary differential equations

**Author**: Nailʹ Khaĭrullovich Ibragimov

**Publisher:**John Wiley & Sons Inc

**ISBN:**9780471974307

**Category:**Mathematics

**Page:**347

**View:**8400

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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.

## Symmetries and Integrability of Difference Equations

**Author**: Decio Levi,Peter Olver,Zora Thomova,Pavel Winternitz

**Publisher:**Cambridge University Press

**ISBN:**1139493841

**Category:**Mathematics

**Page:**N.A

**View:**5639

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Difference equations are playing an increasingly important role in the natural sciences. Indeed many phenomena are inherently discrete and are naturally described by difference equations. Phenomena described by differential equations are therefore approximations of more basic discrete ones. Moreover, in their study it is very often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference equations. This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference ones. Each of the eleven chapters is a self-contained treatment of a topic, containing introductory material as well as the latest research results. The book will be welcomed by graduate students and researchers seeking an introduction to the field. As a survey of the current state of the art it will also serve as a valuable reference.

## Nonlinear Boundary Value Problems in Science and Engineering

**Author**: C. Rogers,W. F. Ames

**Publisher:**Academic Press

**ISBN:**0080958702

**Category:**Computers

**Page:**415

**View:**2976

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Overall, our object has been to provide an applications-oriented text that is reasonably self-contained. It has been used as the basis for a graduate-level course both at the University of Waterloo and at the Centro Studie Applicazioni in Tecnologie Avante, Bari, Italy. The text is aimed, in the main, at applied mathematicians with a strong interest in physical applications or at engineers working in theoretical mechanics.

## Conjugate Gradient Type Methods for Ill-Posed Problems

**Author**: Martin Hanke

**Publisher:**CRC Press

**ISBN:**9780582273702

**Category:**Mathematics

**Page:**144

**View:**891

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The conjugate gradient method is a powerful tool for the iterative solution of self-adjoint operator equations in Hilbert space.This volume summarizes and extends the developments of the past decade concerning the applicability of the conjugate gradient method (and some of its variants) to ill posed problems and their regularization. Such problems occur in applications from almost all natural and technical sciences, including astronomical and geophysical imaging, signal analysis, computerized tomography, inverse heat transfer problems, and many more This Research Note presents a unifying analysis of an entire family of conjugate gradient type methods. Most of the results are as yet unpublished, or obscured in the Russian literature. Beginning with the original results by Nemirovskii and others for minimal residual type methods, equally sharp convergence results are then derived with a different technique for the classical Hestenes-Stiefel algorithm. In the final chapter some of these results are extended to selfadjoint indefinite operator equations. The main tool for the analysis is the connection of conjugate gradient type methods to real orthogonal polynomials, and elementary properties of these polynomials. These prerequisites are provided in a first chapter. Applications to image reconstruction and inverse heat transfer problems are pointed out, and exemplarily numerical results are shown for these applications.

## Lie-Theoretic Ode Numerical Analysis, Mechanics and Differential Systems

**Author**: Robert Hermann

**Publisher:**Math-Sci Press

**ISBN:**9780915692453

**Category:**Mathematics

**Page:**286

**View:**3822

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## Symmetries of Partial Differential Equations

*Conservation Laws - Applications - Algorithms*

**Author**: A.M. Vinogradov

**Publisher:**Springer Science & Business Media

**ISBN:**9780792305941

**Category:**Mathematics

**Page:**242

**View:**2273

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2 The authors of these issues involve not only mathematicians, but also speci alists in (mathematical) physics and computer sciences. So here the reader will find different points of view and approaches to the considered field. A. M. VINOGRADOV 3 Acta Applicandae Mathematicae 15: 3-21, 1989. © 1989 Kluwer Academic Publishers. Symmetries and Conservation Laws of Partial Differential Equations: Basic Notions and Results A. M. VINOORADOV Department of Mathematics, Moscow State University, 117234, Moscow, U. S. S. R. (Received: 22 August 1988) Abstract. The main notions and results which are necessary for finding higher symmetries and conservation laws for general systems of partial differential equations are given. These constitute the starting point for the subsequent papers of this volume. Some problems are also discussed. AMS subject classifications (1980). 35A30, 58005, 58035, 58H05. Key words. Higher symmetries, conservation laws, partial differential equations, infinitely prolonged equations, generating functions. o. Introduction In this paper we present the basic notions and results from the general theory of local symmetries and conservation laws of partial differential equations. More exactly, we will focus our attention on the main conceptual points as well as on the problem of how to find all higher symmetries and conservation laws for a given system of partial differential equations. Also, some general views and perspectives will be discussed.