Elements of Green's Functions and Propagation

Potentials, Diffusion, and Waves

Author: Gabriel Barton

Publisher: Oxford University Press

ISBN: 9780198519980

Category: Science

Page: 465

View: 8510

This text takes the student with a background in undergraduate physics and mathematics towards the skills and insights needed for graduate work in theoretical physics. The author uses Green's functions to explore the physics of potentials, diffusion, and waves. These are important phenomena in their own right, but this study of the partial differential equations describing them also prepares the student for more advanced applications in many-body physics and field theory. Calculations are carried through in enough detail for self-study, and case histories illustrate the interplay between physical insight and mathematical formalism. The aim is to develop the habit of dialogue with the equations and the craftsmanship this fosters in tackling the problem. The book is based on the author's extensive teaching experience.
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Green's Functions with Applications

Author: Dean G. Duffy

Publisher: CRC Press

ISBN: 1420034790

Category: Mathematics

Page: 464

View: 8455

Since its introduction in 1828, using Green's functions has become a fundamental mathematical technique for solving boundary value problems. Most treatments, however, focus on its theory and classical applications in physics rather than the practical means of finding Green's functions for applications in engineering and the sciences. Green's Functions with Applications systematically presents the various methods of deriving these useful functions. It leads readers through the process of developing Green's functions for ordinary and partial differential equations. In addition to exploring the classical problems involving the wave, heat, and Helmholtz equations, the book includes special sections on leaky modes, water waves, and absolute/convective instability. The author gives special attention to the numerical evaluation of Green's functions. By illustrating many of the functions in the text and problem sets, he helps readers develop an intuition about the behavior of Green's function in certain problems. He also considers the questions of the computational efficiency and possible methods for accelerating the process. With its wealth of examples and problems drawn from the literature, this book provides a treasure-trove of methods to construct and compute Green's functions. It is the most exhaustive source book of Green's functions yet available and the only one designed specifically for engineering and scientific applications.
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Green's Functions and Infinite Products

Bridging the Divide

Author: Yuri A. Melnikov

Publisher: Springer Science & Business Media

ISBN: 9780817682804

Category: Mathematics

Page: 165

View: 6421

Green's Functions and Infinite Products provides a thorough introduction to the classical subjects of the construction of Green's functions for the two-dimensional Laplace equation and the infinite product representation of elementary functions. Every chapter begins with a review guide, outlining the basic concepts covered. A set of carefully designed challenging exercises is available at the end of each chapter to provide the reader with the opportunity to explore the concepts in more detail. Hints, comments, and answers to most of those exercises can be found at the end of the text. In addition, several illustrative examples are offered at the end of most sections. This text is intended for an elective graduate course or seminar within the scope of either pure or applied mathematics.
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Green's Functions and Finite Elements

Author: Friedel Hartmann

Publisher: Springer Science & Business Media

ISBN: 3642295231

Category: Technology & Engineering

Page: 330

View: 6155

This book elucidates how Finite Element methods look like from the perspective of Green’s functions, and shows new insights into the mathematical theory of Finite Elements. Practically, this new view on Finite Elements enables the reader to better assess solutions of standard programs and to find better model of a given problem. The book systematically introduces the basic concepts how Finite Elements fulfill the strategy of Green’s functions and how approximating of Green’s functions. It discusses in detail the discretization error and shows that are coherent with the strategy of “goal oriented refinement”. The book also gives much attention to the dependencies of FE solutions from the parameter set of the model.
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Integral Transform Techniques for Green's Function

Author: Kazumi Watanabe

Publisher: Springer Science & Business Media

ISBN: 331900879X

Category: Mathematics

Page: 190

View: 2461

In this book mathematical techniques for integral transforms are described in detail but concisely. The techniques are applied to the standard partial differential equations, such as the Laplace equation, the wave equation and elasticity equations. The Green's functions for beams, plates and acoustic media are also shown along with their mathematical derivations. Lists of Green's functions are presented for the future use. The Cagniard's-de Hoop method for the double inversion is described in detail, and 2D and 3D elasto-dynamics problems are fully treated.
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Green's Functions

Construction and Applications

Author: Yuri A. Melnikov,Max Y. Melnikov

Publisher: Walter de Gruyter

ISBN: 3110253399

Category: Mathematics

Page: 447

View: 2279

This monograph is looking at applied elliptic and parabolic type partial differential equations in two variables. The elliptic type includes the Laplace, static Klein-Gordon and biharmonic equation. The parabolic type is represented by the classical heat equation and the Black-Scholes equation which has emerged as a mathematical model in financial mathematics. This book is a useful source for everyone who is studying or working in the fields of science, finance, or engineering that involve practical solution of partial differential equations.
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Green's Function and Boundary Elements of Multifield Materials

Author: Qing-Hua Qin

Publisher: Elsevier

ISBN: 9780080478067

Category: Technology & Engineering

Page: 266

View: 6230

Green's Function and Boundary Elements of Multifield Materials contains a comprehensive treatment of multifield materials under coupled thermal, magnetic, electric, and mechanical loads. Its easy-to-understand text clarifies some of the most advanced techniques for deriving Green's function and the related boundary element formulation of magnetoelectroelastic materials: Radon transform, potential function approach, Fourier transform. Our hope in preparing this book is to attract interested readers and researchers to a new field that continues to provide fascinating and technologically important challenges. You will benefit from the authors' thorough coverage of general principles for each topic, followed by detailed mathematical derivation and worked examples as well as tables and figures where appropriate. In-depth explanations of the concept of Green's function Coupled thermo-magneto-electro-elastic analysis Detailed mathematical derivation for Green's functions
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Green's Functions in Quantum Physics

Author: Eleftherios N. Economou

Publisher: Springer Science & Business Media

ISBN: 3540288414

Category: Science

Page: 480

View: 7765

The new edition of a standard reference will be of interest to advanced students wishing to become familiar with the method of Green's functions for obtaining simple and general solutions to basic problems in quantum physics. The main part is devoted to the simplest kind of Green's functions, namely the solutions of linear differential equations with a -function source. It is shown that these familiar Green's functions are a powerful tool for obtaining relatively simple and general solutions of basic problems such as scattering and boundlevel information. The bound-level treatment gives a clear physical understanding of "difficult" questions such as superconductivity, the Kondo effect, and, to a lesser degree, disorder-induced localization. The more advanced subject of many-body Green's functions is presented in the last part of the book. This third edition is 50% longer than the previou and offers end-of-chapter problems and solutions (40% are solved) and additional appendices to helpit is to serve as an effective self-tutorial and self-sufficient reference. Throughout, it demonstrates the powerful and unifying formalism of Green's functions across many applications, including transport properties, carbon nanotubes, and photonics and photonic crystals.
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Green’s Functions and Linear Differential Equations

Theory, Applications, and Computation

Author: Prem K. Kythe

Publisher: CRC Press

ISBN: 1439840091

Category: Mathematics

Page: 382

View: 6971

Green’s Functions and Linear Differential Equations: Theory, Applications, and Computation presents a variety of methods to solve linear ordinary differential equations (ODEs) and partial differential equations (PDEs). The text provides a sufficient theoretical basis to understand Green’s function method, which is used to solve initial and boundary value problems involving linear ODEs and PDEs. It also contains a large number of examples and exercises from diverse areas of mathematics, applied science, and engineering. Taking a direct approach, the book first unravels the mystery of the Dirac delta function and then explains its relationship to Green’s functions. The remainder of the text explores the development of Green’s functions and their use in solving linear ODEs and PDEs. The author discusses how to apply various approaches to solve initial and boundary value problems, including classical and general variations of parameters, Wronskian method, Bernoulli’s separation method, integral transform method, method of images, conformal mapping method, and interpolation method. He also covers applications of Green’s functions, including spherical and surface harmonics. Filled with worked examples and exercises, this robust, self-contained text fully explains the differential equation problems, includes graphical representations where necessary, and provides relevant background material. It is mathematically rigorous yet accessible enough for readers to grasp the beauty and power of the subject.
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Green’s Functions in Classical Physics

Author: Tom Rother

Publisher: Springer

ISBN: 3319524372

Category: Science

Page: 267

View: 7934

This book presents the Green’s function formalism in a basic way and demonstrates its usefulness for applications to several well-known problems in classical physics which are usually solved not by this formalism but other approaches. The book bridges the gap between applications of the Green’s function formalism in quantum physics and classical physics. This book is written as an introduction for graduate students and researchers who want to become more familiar with the Green’s function formalism. In 1828 George Green has published an essay that was unfortunately sunken into oblivion shortly after its publication. It was rediscovered only after several years by the later Lord Kelvin. But since this time, using Green’s functions for solving partial differential equations in physics has become an important mathematical tool. While the conceptual and epistemological importance of these functions were essentially discovered and discussed in modern physics - especially in quantum field theory and quantum statistics - these aspects are rarely touched in classical physics. In doing it, this book provides an interesting and sometimes new point of view on several aspects and problems in classical physics, like the Kepler motion or the description of certain classical probability experiments in finite event spaces. A short outlook on quantum mechanical problems concludes this book.
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Heat Conduction Using Green’s Functions, 2nd Edition

Author: Kevin D. Cole,James V. Beck,A. Haji-Sheikh,Bahman Litkouhi

Publisher: Taylor & Francis

ISBN: 143989521X

Category: Science

Page: 663

View: 1277

Since its publication more than 15 years ago, Heat Conduction Using Green’s Functions has become the consummate heat conduction treatise from the perspective of Green’s functions—and the newly revised Second Edition is poised to take its place. Based on the authors’ own research and classroom experience with the material, this book organizes the solution of heat conduction and diffusion problems through the use of Green’s functions, making these valuable principles more accessible. As in the first edition, this book applies extensive tables of Green’s functions and related integrals, and all chapters have been updated and revised for the second edition, many extensively. Details how to access the accompanying Green’s Function Library site, a useful web-searchable collection of GFs based on the appendices in this book The book reflects the authors’ conviction that although Green’s functions were discovered in the nineteenth century, they remain directly relevant to 21st-century engineers and scientists. It chronicles the authors’ continued search for new GFs and novel ways to apply them to heat conduction. New features of this latest edition— Expands the introduction to Green’s functions, both steady and unsteady Adds a section on the Dirac Delta Function Includes a discussion of the eigenfunction expansion method, as well as sections on the convergence speed of series solutions, and the importance of alternate GF Adds a section on intrinsic verification, an important new tool for obtaining correct numerical values from analytical solutions A main goal of the first edition was to make GFs more accessible. To facilitate this objective, one of the authors has created a companion Internet site called the Green’s Function Library, a web-searchable collection of GFs. Based on the appendices in this book, this library is organized by differential equation, geometry, and boundary condition. Each GF is also identified and cataloged according to a GF numbering system. The library also contains explanatory material, references, and links to related sites, all of which supplement the value of Heat Conduction Using Green’s Functions, Second Edition as a powerful tool for understanding.
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Progress in Nonequilibrium Green's Functions

Proceedings of the Conference "Kadanoff-Baym Equations: Progress and Perspectives for Many-body Physics", Rostock, Germany, 20-24 September 1999

Author: Michael Bonitz,Renate Nareyka,Dirk Semkat

Publisher: World Scientific

ISBN: 9789810242183

Category: Science

Page: 564

View: 521

Equilibrium and nonequilibrium properties of correlated many-body systems are of growing interest in many fields of physics, including condensed matter, dense plasmas, nuclear matter and particles. The most powerful and general method which applies equally to all these areas is given by quantum field theory.Written by the leading experts and understandable to non-specialists, this book provides an overview on the basic ideas and concepts of the method of nonequilibrium Green's functions. It is complemented by modern applications of the method to a variety of topics, such as optics and transport in dense plasmas and semiconductors; correlations, bound states and coherence; strong field effects and short-pulse lasers; nuclear matter and QCD.Authors include: Gordon Bayan, Pawel Danielewicz, Don DuBois, Hartmut Haug, Klaus Henneberger, Antti-Pekka Jauho, J”rn Kuoll, Dietrich Kremp, Pavel Lipavsky and Paul C Martin.
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Proceedings of the Conference, Progress in Nonequilibrium Green's Functions, Dresden, Germany, 19-23 August 2002

Author: Michael Bonitz,Dirk Semkat

Publisher: World Scientific

ISBN: 9789812705129

Category: Science

Page: 549

View: 8944

Equilibrium and nonequilibrium properties of correlated many-body systems are of growing interest in many areas of physics, including condensed matter, dense plasmas, nuclear matter and particles. The most powerful and general method which is equally applied to all these areas is given by quantum field theory. This book provides an overview of the basic ideas and concepts of the method of nonequilibrium Green''s functions, written by the leading experts and presented in a way accessible to non-specialists and graduate students. It is complemented by invited review papers on modern applications of the method to a variety of topics, such as optics and quantum transport in semiconductors; superconductivity; strong field effects, QCD, and state-of-the-art computational concepts OCo from Green''s functions to quantum Monte Carlo and time-dependent density functional theory.The proceedings have been selected for coverage in: OCo Index to Scientific & Technical Proceedings (ISTP CDROM version / ISI Proceedings)"
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Green's Functions and Boundary Element Analysis for Modeling of Mechanical Behavior of Advanced Materials

Author: J. R. Berger,V. K. Tewary

Publisher: DIANE Publishing

ISBN: 9780788148187

Category:

Page: 166

View: 2887

Demonstrates the potential of Green's functions & boundary element methods in solving a broad range of practical materials science problems. Papers include: Accurate Discretization of Integral Operators, Boundary Element Analysis of Bimaterials Using Anisotropic Elastic Green's Functions, Mechanical Properties of Metal-Matrix Composites, Approximate Operators for Boundary Integral Equations in Transient Elastodynamics, Simulation of the Electrochemical Machining Process Using a 2D Fundamental Singular Solution, Elastic Green's Functions for Anisotropic Solids, & more. Charts & tables.
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Applications of Green's Functions in Science and Engineering

Author: Michael D. Greenberg

Publisher: Courier Dover Publications

ISBN: 0486805840

Category: Mathematics

Page: 160

View: 6193

Concise and highly regarded, this treatment of Green's functions and their applications in science and engineering is geared toward undergraduate and graduate students with only a moderate background in ordinary differential equations and partial differential equations. The text also includes a wealth of information of a more general nature on boundary value problems, generalized functions, eigenfunction expansions, partial differential equations, and acoustics. The two-part treatment begins with an overview of applications to ordinary differential equations. Topics include the adjoint operator, delta function, the Green's function method, and the eigenfunction method. The second part, which explores applications to partial differential equations, covers functions for the Laplace, Helmholtz, diffusion, and wave operators. A full index, exercises, suggested reading list, a new preface, and a new brief errata list round out the text.
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Green’s Functions in the Theory of Ordinary Differential Equations

Author: Alberto Cabada

Publisher: Springer Science & Business Media

ISBN: 1461495067

Category: Mathematics

Page: 168

View: 1573

This book provides a complete and exhaustive study of the Green’s functions. Professor Cabada first proves the basic properties of Green's functions and discusses the study of nonlinear boundary value problems. Classic methods of lower and upper solutions are explored, with a particular focus on monotone iterative techniques that flow from them. In addition, Cabada proves the existence of positive solutions by constructing operators defined in cones. The book will be of interest to graduate students and researchers interested in the theoretical underpinnings of boundary value problem solutions.
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Many-Particle Physics

Author: Gerald D. Mahan

Publisher: Springer Science & Business Media

ISBN: 9780306434235

Category: Science

Page: 1032

View: 7699

This textbook is for a course in advanced solid-state theory. It is aimed at graduate students in their third or fourth year of study who wish to learn the advanced techniques of solid-state theoretical physics. The method of Green's functions is introduced at the beginning and used throughout. Indeed, it could be considered a book on practical applications of Green's functions, although I prefer to call it a book on physics. The method of Green's functions has been used by many theorists to derive equations which, when solved, provide an accurate numerical description of many processes in solids and quantum fluids. In this book I attempt to summarize many of these theories in order to show how Green's functions are used to solve real problems. My goal, in writing each section, is to describe calculations which can be compared with experiments and to provide these comparisons whenever available. The student is expected to have a background in quantum mechanics at the level acquired from a graduate course using the textbook by either L. I. Schiff, A. S. Davydov, or I. Landau and E. M. Lifshiftz. Similarly, a prior course in solid-state physics is expected, since the reader is assumed to know concepts such as Brillouin zones and energy band theory. Each chapter has problems which are an important part of the lesson; the problems often provide physical insights which are not in the text. Sometimes the answers to the problems are provided, but usually not.
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