Geometrical Properties Of Differential Equations

Author : Ljudmila A Bordag
Publisher : World Scientific Publishing Company
Page : 340 pages
File Size : 46,6 Mb
Release : 2015-05-27
Category : Mathematics
ISBN : 9789814667265

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Geometrical Properties of Differential Equations by Ljudmila A Bordag Pdf

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.).

Author : Ljudmila A. Bordag
Publisher : Unknown
Page : 341 pages
File Size : 44,9 Mb
Release : 2015
Category : Electronic books
ISBN : 9814667250

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Geometrical Properties of Differential Equations by Ljudmila A. Bordag Pdf

Author : V V Zharinov
Publisher : World Scientific
Page : 372 pages
File Size : 46,5 Mb
Release : 1992-03-26
Category : Mathematics
ISBN : 9789814513999

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Lecture Notes on Geometrical Aspects of Partial Differential Equations by V V Zharinov Pdf

This book focuses on the properties of nonlinear systems of PDE with geometrical origin and the natural description in the language of infinite-dimensional differential geometry. The treatment is very informal and the theory is illustrated by various examples from mathematical physics. All necessary information about the infinite-dimensional geometry is given in the text. Contents:Introduction: Internal Geometry of PDE:Differential ManifoldsLie-Backlund MappingsLie-Backlund Fields and Infinitesimal SymmetriesCartan Forms, Currents and Conservation LawsC-Spectral Sequence. Further Properties of Conservation LawsTrivial Equations. The Formal Variational CalculusEvolution EquationsExternal Geometry of PDE:Differential SubmanifoldsNormal Projection. External Fields and FormsTrivial Ambient Differential ManifoldsThe Characteristic MappingThe Green's FormulaLow-Dimensional Conservation LawsBacklund CorrespondenceFurther Studies:Lagrangian FormalismHamiltonian EquationsExample: The Nambu's StringAppendix Readership: Graduate students and researchers in mathematical physics. keywords:Differential Manifolds;Lie-Bäcklund Mappings;Cartan Forms;Currents;Conservation Laws;Lagrangian Formation;Hamiltonian Equations

Author : P.H. Kersten,I.S. Krasil'shchik
Publisher : Springer Science & Business Media
Page : 346 pages
File Size : 48,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400901797

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Geometric and Algebraic Structures in Differential Equations by P.H. Kersten,I.S. Krasil'shchik Pdf

The geometrical theory of nonlinear differential equations originates from classical works by S. Lie and A. Bäcklund. It obtained a new impulse in the sixties when the complete integrability of the Korteweg-de Vries equation was found and it became clear that some basic and quite general geometrical and algebraic structures govern this property of integrability. Nowadays the geometrical and algebraic approach to partial differential equations constitutes a special branch of modern mathematics. In 1993, a workshop on algebra and geometry of differential equations took place at the University of Twente (The Netherlands), where the state-of-the-art of the main problems was fixed. This book contains a collection of invited lectures presented at this workshop. The material presented is of interest to those who work in pure and applied mathematics and especially in mathematical physics.

Author : Anonim
Publisher : Elsevier
Page : 710 pages
File Size : 55,8 Mb
Release : 2020-01-14
Category : Mathematics
ISBN : 9780444640048

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Geometric Partial Differential Equations - Part I by Anonim Pdf

Besides their intrinsic mathematical interest, geometric partial differential equations (PDEs) are ubiquitous in many scientific, engineering and industrial applications. They represent an intellectual challenge and have received a great deal of attention recently. The purpose of this volume is to provide a missing reference consisting of self-contained and comprehensive presentations. It includes basic ideas, analysis and applications of state-of-the-art fundamental algorithms for the approximation of geometric PDEs together with their impacts in a variety of fields within mathematics, science, and engineering. About every aspect of computational geometric PDEs is discussed in this and a companion volume. Topics in this volume include stationary and time-dependent surface PDEs for geometric flows, large deformations of nonlinearly geometric plates and rods, level set and phase field methods and applications, free boundary problems, discrete Riemannian calculus and morphing, fully nonlinear PDEs including Monge-Ampere equations, and PDE constrained optimization Each chapter is a complete essay at the research level but accessible to junior researchers and students. The intent is to provide a comprehensive description of algorithms and their analysis for a specific geometric PDE class, starting from basic concepts and concluding with interesting applications. Each chapter is thus useful as an introduction to a research area as well as a teaching resource, and provides numerous pointers to the literature for further reading The authors of each chapter are world leaders in their field of expertise and skillful writers. This book is thus meant to provide an invaluable, readable and enjoyable account of computational geometric PDEs

Author : Frantisek Neuman
Publisher : Springer
Page : 344 pages
File Size : 42,7 Mb
Release : 1991
Category : Mathematics
ISBN : UOM:39015029206920

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Global Properties of Linear Ordinary Differential Equations by Frantisek Neuman Pdf

This volume presents an authoritative, unified overview of the methods and results concerning the global properties of linear differential equations of order n (n>=2). It does not, however, seek to be comprehensive. Rather, it contains a selection of results which richly illustrate the unified approach presented. By making use of recent methods and results from many different areas of mathematics and by introducing several original methods, global solutions of problems previously studied only locally are given. The structure of global transformations is described algebraically, and a new geometrical approach is introduced which leads to global canonical forms suitable for Cartan's moving frame-of-reference method. The theory discussed also provides effective tools for solving some open problems, especially relating to the distribution of zeros of solutions. In addition, the theory of functional equations plays an important role in studying the asymptotic behaviour of solutions. Applications to differential geometry and functional equations are also described. The volume is largely self-contained. This book is for mathematicians, computer scientists, physicists, chemists, engineers, and others whose work involves the use of linear differential equations.

Author : Solomon Lefschetz
Publisher : Unknown
Page : 408 pages
File Size : 46,7 Mb
Release : 1963
Category : Differential equations
ISBN : UOM:39015000978588

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Differential Equations: Geometric Theory by Solomon Lefschetz Pdf

Author : Isabel Maddison
Publisher : Unknown
Page : 82 pages
File Size : 52,8 Mb
Release : 1896
Category : Differential equations
ISBN : UOM:39015064582433

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On Singular Solutions of Differential Equations in Two Variables, and the Geometrical Properties of Certain Invariants Andcovariants of Their Complete Primitives ... by Isabel Maddison Pdf

Author : Ernst Hairer,Christian Lubich,Gerhard Wanner
Publisher : Springer Science & Business Media
Page : 526 pages
File Size : 52,5 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662050187

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Geometric Numerical Integration by Ernst Hairer,Christian Lubich,Gerhard Wanner Pdf

This book deals with numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by numerous figures, treats applications from physics and astronomy, and contains many numerical experiments and comparisons of different approaches.

Author : Antonin Chambolle,Matteo Novaga,Enrico Valdinoci
Publisher : Springer Science & Business Media
Page : 400 pages
File Size : 41,5 Mb
Release : 2014-01-17
Category : Mathematics
ISBN : 9788876424731

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Geometric Partial Differential Equations by Antonin Chambolle,Matteo Novaga,Enrico Valdinoci Pdf

This book is the outcome of a conference held at the Centro De Giorgi of the Scuola Normale of Pisa in September 2012. The aim of the conference was to discuss recent results on nonlinear partial differential equations, and more specifically geometric evolutions and reaction-diffusion equations. Particular attention was paid to self-similar solutions, such as solitons and travelling waves, asymptotic behaviour, formation of singularities and qualitative properties of solutions. These problems arise in many models from Physics, Biology, Image Processing and Applied Mathematics in general, and have attracted a lot of attention in recent years.

Author : Luigi Ambrosio,Norman Dancer
Publisher : Springer Science & Business Media
Page : 347 pages
File Size : 46,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642571862

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Calculus of Variations and Partial Differential Equations by Luigi Ambrosio,Norman Dancer Pdf

At the summer school in Pisa in September 1996, Luigi Ambrosio and Norman Dancer each gave a course on the geometric problem of evolution of a surface by mean curvature, and degree theory with applications to PDEs respectively. This self-contained presentation accessible to PhD students bridged the gap between standard courses and advanced research on these topics. The resulting book is divided accordingly into 2 parts, and neatly illustrates the 2-way interaction of problems and methods. Each of the courses is augmented and complemented by additional short chapters by other authors describing current research problems and results.

Author : Robert Everist Greene,Shing-Tung Yau
Publisher : American Mathematical Soc.
Page : 560 pages
File Size : 49,5 Mb
Release : 1993
Category : Mathematics
ISBN : 9780821814949

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Differential Geometry: Partial Differential Equations on Manifolds by Robert Everist Greene,Shing-Tung Yau Pdf

The first of three parts comprising Volume 54, the proceedings of the Summer Research Institute on Differential Geometry, held at the University of California, Los Angeles, July 1990 (ISBN for the set is 0-8218-1493-1). Part 1 begins with a problem list by S.T. Yau, successor to his 1980 list ( Sem

Author : Galina Filipuk,Yoshishige Haraoka,Sławomir Michalik
Publisher : Birkhäuser
Page : 471 pages
File Size : 48,7 Mb
Release : 2017-06-23
Category : Mathematics
ISBN : 9783319528427

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Analytic, Algebraic and Geometric Aspects of Differential Equations by Galina Filipuk,Yoshishige Haraoka,Sławomir Michalik Pdf

This volume consists of invited lecture notes, survey papers and original research papers from the AAGADE school and conference held in Będlewo, Poland in September 2015. The contributions provide an overview of the current level of interaction between algebra, geometry and analysis and demonstrate the manifold aspects of the theory of ordinary and partial differential equations, while also pointing out the highly fruitful interrelations between those aspects. These interactions continue to yield new developments, not only in the theory of differential equations but also in several related areas of mathematics and physics such as differential geometry, representation theory, number theory and mathematical physics. The main goal of the volume is to introduce basic concepts, techniques, detailed and illustrative examples and theorems (in a manner suitable for non-specialists), and to present recent developments in the field, together with open problems for more advanced and experienced readers. It will be of interest to graduate students, early-career researchers and specialists in analysis, geometry, algebra and related areas, as well as anyone interested in learning new methods and techniques.

Author : Alex Kasman
Publisher : American Mathematical Society
Page : 366 pages
File Size : 43,6 Mb
Release : 2023-03-30
Category : Mathematics
ISBN : 9781470472627

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Glimpses of Soliton Theory by Alex Kasman Pdf

This book challenges and intrigues from beginning to end. It would be a treat to use for a capstone course or senior seminar. —William J. Satzer, MAA Reviews on Glimpses of Soliton Theory (First Edition) Solitons are nonlinear waves which behave like interacting particles. When first proposed in the 19th century, leading mathematical physicists denied that such a thing could exist. Now they are regularly observed in nature, shedding light on phenomena like rogue waves and DNA transcription. Solitons of light are even used by engineers for data transmission and optical switches. Furthermore, unlike most nonlinear partial differential equations, soliton equations have the remarkable property of being exactly solvable. Explicit solutions to those equations provide a rare window into what is possible in the realm of nonlinearity. Glimpses of Soliton Theory reveals the hidden connections discovered over the last half-century that explain the existence of these mysterious mathematical objects. It aims to convince the reader that, like the mirrors and hidden pockets used by magicians, the underlying algebro-geometric structure of soliton equations provides an elegant explanation of something seemingly miraculous. Assuming only multivariable calculus and linear algebra, the book introduces the reader to the KdV Equation and its multisoliton solutions, elliptic curves and Weierstrass $wp$-functions, the algebra of differential operators, Lax Pairs and their use in discovering other soliton equations, wedge products and decomposability, the KP Hierarchy, and Sato's theory relating the Bilinear KP Equation to the geometry of Grassmannians. Notable features of the book include: careful selection of topics and detailed explanations to make the subject accessible to undergraduates, numerous worked examples and thought-provoking exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of Mathematica® to facilitate computation and animate solutions. The second edition refines the exposition in every chapter, adds more homework exercises and projects, updates references, and includes new examples involving non-commutative integrable systems. Moreover, the chapter on KdV multisolitons has been greatly expanded with new theorems providing a thorough analysis of their behavior and decomposition.

Author : Shing-tung Yau
Publisher : Princeton University Press
Page : 720 pages
File Size : 51,7 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400881918

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Seminar on Differential Geometry. (AM-102), Volume 102 by Shing-tung Yau Pdf

This collection of papers constitutes a wide-ranging survey of recent developments in differential geometry and its interactions with other fields, especially partial differential equations and mathematical physics. This area of mathematics was the subject of a special program at the Institute for Advanced Study in Princeton during the academic year 1979-1980; the papers in this volume were contributed by the speakers in the sequence of seminars organized by Shing-Tung Yau for this program. Both survey articles and articles presenting new results are included. The articles on differential geometry and partial differential equations include a general survey article by the editor on the relationship of the two fields and more specialized articles on topics including harmonic mappings, isoperimetric and Poincaré inequalities, metrics with specified curvature properties, the Monge-Arnpere equation, L2 harmonic forms and cohomology, manifolds of positive curvature, isometric embedding, and Kraumlhler manifolds and metrics. The articles on differential geometry and mathematical physics cover such topics as renormalization, instantons, gauge fields and the Yang-Mills equation, nonlinear evolution equations, incompleteness of space-times, black holes, and quantum gravity. A feature of special interest is the inclusion of a list of more than one hundred unsolved research problems compiled by the editor with comments and bibliographical information.