Riemann-Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions

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Author: Thomas Trogdon,Sheehan Olver

Publisher: SIAM

ISBN: 1611974208

Category: Mathematics

Page: 373

View: 1676

Riemann?Hilbert problems are fundamental objects of study within complex analysis. Many problems in differential equations and integrable systems, probability and random matrix theory, and asymptotic analysis can be solved by reformulation as a Riemann?Hilbert problem.This book, the most comprehensive one to date on the applied and computational theory of Riemann?Hilbert problems, includes an introduction to computational complex analysis, an introduction to the applied theory of Riemann?Hilbert problems from an analytical and numerical perspective, and a discussion of applications to integrable systems, differential equations, and special function theory. It also includes six fundamental examples and five more sophisticated examples of the analytical and numerical Riemann?Hilbert method, each of mathematical or physical significance or both.

Unified Transform for Boundary Value Problems

Applications and Advances

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Author: Athanasios S. Fokas,Beatrice Pelloni

Publisher: SIAM

ISBN: 1611973813

Category: Mathematics

Page: 310

View: 8660

This book describes state-of-the-art advances and applications of the unified transform and its relation to the boundary element method. The authors present the solution of boundary value problems from several different perspectives, in particular the type of problems modeled by partial differential equations (PDEs). They discuss recent applications of the unified transform to the analysis and numerical modeling of boundary value problems for linear and integrable nonlinear PDEs and the closely related boundary element method, a well-established numerical approach for solving linear elliptic PDEs.÷ The text is divided into three parts. Part I contains new theoretical results on linear and nonlinear evolutionary and elliptic problems. New explicit solution representations for several classes of boundary value problems are constructed and rigorously analyzed. Part II is a detailed overview of variational formulations for elliptic problems. It places the unified transform approach in a classic context alongside the boundary element method and stresses its novelty. Part III presents recent numerical applications based on the boundary element method and on the unified transform.

Symmetries and Integrability of Difference Equations

Lecture Notes of the Abecederian School of SIDE 12, Montreal 2016

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Author: Decio Levi,Raphaël Rebelo,Pavel Winternitz

Publisher: Springer

ISBN: 3319566660

Category: Science

Page: 435

View: 4271

This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations. Difference equations are playing an increasingly important role in the natural sciences. Indeed, many phenomena are inherently discrete and thus naturally described by difference equations. More fundamentally, in subatomic physics, space-time may actually be discrete. Differential equations would then just be approximations of more basic discrete ones. Moreover, when using differential equations to analyze continuous processes, it is often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference ones. Each of the nine peer-reviewed chapters in this volume serves as a self-contained treatment of a topic, containing introductory material as well as the latest research results and exercises. Each chapter is presented by one or more early career researchers in the specific field of their expertise and, in turn, written for early career researchers. As a survey of the current state of the art, this book will serve as a valuable reference and is particularly well suited as an introduction to the field of symmetries and integrability of difference equations. Therefore, the book will be welcomed by advanced undergraduate and graduate students as well as by more advanced researchers.

Variational Methods for the Numerical Solution of Nonlinear Elliptic Problem

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Author: Roland Glowinski

Publisher: SIAM

ISBN: 1611973783

Category: Mathematics

Page: 462

View: 4527

Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems÷addresses computational methods that have proven efficient for the solution of a large variety of nonlinear elliptic problems. These methods can be applied to many problems in science and engineering, but this book focuses on their application to problems in continuum mechanics and physics. This book differs from others on the topic by presenting examples of the power and versatility of operator-splitting methods; providing a detailed introduction to alternating direction methods of multipliers and their applicability to the solution of nonlinear (possibly nonsmooth) problems from science and engineering; and showing that nonlinear least-squares methods, combined with operator-splitting and conjugate gradient algorithms, provide efficient tools for the solution of highly nonlinear problems. The book provides useful insights suitable for advanced graduate students, faculty, and researchers in applied and computational mathematics as well as research engineers, mathematical physicists, and systems engineers.

Solitons and the Inverse Scattering Transform

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Author: Mark J. Ablowitz,Harvey Segur

Publisher: SIAM

ISBN: 089871477X

Category: Mathematics

Page: 425

View: 7522

A study, by two of the major contributors to the theory, of the inverse scattering transform and its application to problems of nonlinear dispersive waves that arise in fluid dynamics, plasma physics, nonlinear optics, particle physics, crystal lattice theory, nonlinear circuit theory and other areas. A soliton is a localised pulse-like nonlinear wave that possesses remarkable stability properties. Typically, problems that admit soliton solutions are in the form of evolution equations that describe how some variable or set of variables evolve in time from a given state. The equations may take a variety of forms, for example, PDEs, differential difference equations, partial difference equations, and integrodifferential equations, as well as coupled ODEs of finite order. What is surprising is that, although these problems are nonlinear, the general solution that evolves from almost arbitrary initial data may be obtained without approximation.

Generalized Analytic Functions

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Author: I. N. Vekua

Publisher: Elsevier

ISBN: 1483184676

Category: Mathematics

Page: 698

View: 6141

Generalized Analytic Functions is concerned with foundations of the general theory of generalized analytic functions and some applications to problems of differential geometry and theory of shells. Some classes of functions and operators are discussed, along with the reduction of a positive differential quadratic form to the canonical form. Boundary value problems and infinitesimal bendings of surfaces are also considered. Comprised of six chapters, this volume begins with a detailed treatment of various problems of the general theory of generalized analytic functions as as well as boundary value problems. The reader is introduced to some classes of functions and functional spaces, with emphasis on functions of two independent variables. Subsequent chapters focus on the problem of reducing a positive differential quadratic form to the canonical form; basic properties of solutions of elliptic systems of partial differential equations of the first order, in a two-dimensional domain; and some boundary value problems for an elliptic system of equations of the first order and for an elliptic equation of the second order, in a two-dimensional domain. The final part of the book deals with problems of the theory of surfaces and the membrane theory of shells. This book is intended for students of advanced courses of the mechanico-mathematical faculties, postgraduates, and research workers.

Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions

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Author: v Mityushev,S V Rogosin

Publisher: CRC Press

ISBN: 9781584880578

Category: Mathematics

Page: 296

View: 9811

Constructive methods developed in the framework of analytic functions effectively extend the use of mathematical constructions, both within different branches of mathematics and to other disciplines. This monograph presents some constructive methods-based primarily on original techniques-for boundary value problems, both linear and nonlinear. From among the many applications to which these methods can apply, the authors focus on interesting problems associated with composite materials with a finite number of inclusions. How far can one go in the solutions of problems in nonlinear mechanics and physics using the ideas of analytic functions? What is the difference between linear and nonlinear cases from the qualitative point of view? What kinds of additional techniques should one use in investigating nonlinear problems? Constructive Methods for Linear and Nonlinear Boundary Value Problems serves to answer these questions, and presents many results to Westerners for the first time. Among the most interesting of these is the complete solution of the Riemann-Hilbert problem for multiply connected domains. The results offered in Constructive Methods for Linear and Nonlinear Boundary Value Problems are prepared for direct application. A historical survey along with background material, and an in-depth presentation of practical methods make this a self-contained volume useful to experts in analytic function theory, to non-specialists, and even to non-mathematicians who can apply the methods to their research in mechanics and physics.

Fundamentals of Numerical Computation

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Author: Tobin A. Driscoll,Richard J. Braun

Publisher: SIAM

ISBN: 1611975077

Category: Science

Page: 553

View: 7231

Fundamentals of Numerical Computation is an advanced undergraduate-level introduction to the mathematics and use of algorithms for the fundamental problems of numerical computation: linear algebra, finding roots, approximating data and functions, and solving differential equations. The book is organized with simpler methods in the first half and more advanced methods in the second half, allowing use for either a single course or a sequence of two courses. The authors take readers from basic to advanced methods, illustrating them with over 200 self-contained MATLAB functions and examples designed for those with no prior MATLAB experience. Although the text provides many examples, exercises, and illustrations, the aim of the authors is not to provide a cookbook per se, but rather an exploration of the principles of cooking. The authors have developed an online resource that includes well-tested materials related to every chapter. Among these materials are lecture-related slides and videos, ideas for student projects, laboratory exercises, computational examples and scripts, and all the functions presented in the book. The book is intended for advanced undergraduates in math, applied math, engineering, or science disciplines, as well as for researchers and professionals looking for an introduction to a subject they missed or overlooked in their education.

The Selected Works of Roderick S C Wong

(In 3 Volumes)

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Author: Dan Dai,Hui-Hui Dai,Tong Yang,Ding-Xuan Zhou

Publisher: World Scientific

ISBN: 9814656062

Category: Mathematics

Page: 1540

View: 7622

This collection, in three volumes, presents the scientific achievements of Roderick S C Wong, spanning 45 years of his career. It provides a comprehensive overview of the author's work which includes significant discoveries and pioneering contributions, such as his deep analysis on asymptotic approximations of integrals and uniform asymptotic expansions of orthogonal polynomials and special functions; his important contributions to perturbation methods for ordinary differential equations and difference equations; and his advocation of the Riemann–Hilbert approach for global asymptotics of orthogonal polynomials. The book is an essential source of reference for mathematicians, statisticians, engineers, and physicists. It is also a suitable reading for graduate students and interested senior year undergraduate students. Contents:Volume 1:The Asymptotic Behaviour of μ(z, β,α)A Generalization of Watson's LemmaLinear Equations in Infinite MatricesAsymptotic Solutions of Linear Volterra Integral Equations with Singular KernelsOn Infinite Systems of Linear Differential EquationsError Bounds for Asymptotic Expansions of HankelExplicit Error Terms for Asymptotic Expansions of StieltjesExplicit Error Terms for Asymptotic Expansions of MellinAsymptotic Expansion of Multiple Fourier TransformsExact Remainders for Asymptotic Expansions of FractionalAsymptotic Expansion of the Hilbert TransformError Bounds for Asymptotic Expansions of IntegralsDistributional Derivation of an Asymptotic ExpansionOn a Method of Asymptotic Evaluation of Multiple IntegralsAsymptotic Expansion of the Lebesgue Constants Associated with Polynomial InterpolationQuadrature Formulas for Oscillatory Integral TransformsGeneralized Mellin Convolutions and Their Asymptotic Expansions,A Uniform Asymptotic Expansion of the Jacobi Polynomials with Error BoundsAsymptotic Expansion of a Multiple IntegralAsymptotic Expansion of a Double Integral with a Curve of Stationary PointsSzegö's Conjecture on Lebesgue Constants for Legendre SeriesUniform Asymptotic Expansions of Laguerre PolynomialsTransformation to Canonical Form for Uniform Asymptotic ExpansionsMultidimensional Stationary Phase Approximation: Boundary Stationary PointTwo-Dimensional Stationary Phase Approximation: Stationary Point at a CornerAsymptotic Expansions for Second-Order Linear Difference EquationsAsymptotic Expansions for Second-Order Linear Difference Equations, IIAsymptotic Behaviour of the Fundamental Solution to ∂u/∂t = –(–Δ)muA Bernstein-Type Inequality for the Jacobi PolynomialError Bounds for Asymptotic Expansions of Laplace ConvolutionsVolume 2:Asymptotic Behavior of the Pollaczek Polynomials and Their ZerosJustification of the Stationary Phase Approximation in Time-Domain AsymptoticsAsymptotic Expansions of the Generalized Bessel PolynomialsUniform Asymptotic Expansions for Meixner Polynomials"Best Possible" Upper and Lower Bounds for the Zeros of the Bessel Function Jν(x)Justification of a Perturbation Approximation of the Klein–Gordon EquationSmoothing of Stokes's Discontinuity for the Generalized Bessel Function. IIUniform Asymptotic Expansions of a Double Integral: Coalescence of Two Stationary PointsUniform Asymptotic Formula for Orthogonal Polynomials with Exponential WeightOn the Asymptotics of the Meixner–Pollaczek Polynomials and Their ZerosGevrey Asymptotics and Stieltjes Transforms of Algebraically Decaying FunctionsExponential Asymptotics of the Mittag–Leffler FunctionOn the Ackerberg–O'Malley ResonanceAsymptotic Expansions for Second-Order Linear Difference Equations with a Turning PointOn a Two-Point Boundary-Value Problem with Spurious SolutionsShooting Method for Nonlinear Singularly Perturbed Boundary-Value ProblemsVolume 3:Asymptotic Expansion of the Krawtchouk Polynomials and Their ZerosOn a Uniform Treatment of Darboux's MethodLinear Difference Equations with Transition PointsUniform Asymptotics for Jacobi Polynomials with Varying Large Negative Parameters — A Riemann–Hilbert ApproachUniform Asymptotics of the Stieltjes–Wigert Polynomials via the Riemann–Hilbert ApproachA Singularly Perturbed Boundary-Value Problem Arising in Phase TransitionsOn the Number of Solutions to Carrier's ProblemAsymptotic Expansions for Riemann–Hilbert ProblemsOn the Connection Formulas of the Third Painlevé TranscendentHyperasymptotic Expansions of the Modified Bessel Function of the Third Kind of Purely Imaginary OrderGlobal Asymptotics for Polynomials Orthogonal with Exponential Quartic WeightThe Riemann–Hilbert Approach to Global Asymptotics of Discrete Orthogonal Polynomials with Infinite NodesGlobal Asymptotics of the Meixner PolynomialsAsymptotics of Orthogonal Polynomials via Recurrence RelationsUniform Asymptotic Expansions for the Discrete Chebyshev PolynomialsGlobal Asymptotics of the Hahn PolynomialsGlobal Asymptotics of Stieltjes–Wigert Polynomials Readership: Undergraduates, gradudates and researchers in the areas of asymptotic approximations of integrals, singular perturbation theory, difference equations and Riemann–Hilbert approach. Key Features:This book provides a broader viewpoint of asymptoticsIt contains about half of the papers that Roderick Wong has written on asymptoticsIt demonstrates how analysis is used to make some formal results mathematically rigorousThis collection presents the scientific achievements of the authorKeywords:Asymptotic Analysis;Perturbation Method;Special Functions;Orthogonal Polynomials;Integral Transforms;Integral Equations;Ordinary Differential Equations;Difference Equations;Riemann–Hilbert Problem

Orthogonal Polynomials and Special Functions

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Author: Richard Askey

Publisher: SIAM

ISBN: 0898710189

Category: Mathematics

Page: 110

View: 5352

This volume presents the idea that one studies orthogonal polynomials and special functions to use them to solve problems.

Functions of Matrices

Theory and Computation

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Author: Nicholas J. Higham

Publisher: SIAM

ISBN: 0898717779

Category: Factorization (Mathematics)

Page: 425

View: 8209

A thorough and elegant treatment of the theory of matrix functions and numerical methods for computing them, including an overview of applications, new and unpublished research results, and improved algorithms. Key features include a detailed treatment of the matrix sign function and matrix roots; a development of the theory of conditioning and properties of the Fre;chet derivative; Schur decomposition; block Parlett recurrence; a thorough analysis of the accuracy, stability, and computational cost of numerical methods; general results on convergence and stability of matrix iterations; and a chapter devoted to the f(A)b problem. Ideal for advanced courses and for self-study, its broad content, references and appendix also make this book a convenient general reference. Contains an extensive collection of problems with solutions and MATLAB implementations of key algorithms.

Scattering, Two-Volume Set

Scattering and Inverse Scattering in Pure and Applied Science

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Author: E R Pike,Pierre Celestin Sabatier

Publisher: Academic Press

ISBN: 0126137609

Category:

Page: 1831

View: 5055

Scattering is the collision of two objects that results in a change of trajectory and energy. For example, in particle physics, such as electrons, photons, or neutrons are "scattered off" of a target specimen, resulting in a different energy and direction. In the field of electromagnetism, scattering is the random diffusion of electromagnetic radiation from air masses is an aid in the long-range sending of radio signals over geographic obstacles such as mountains. This type of scattering, applied to the field of acoustics, is the spreading of sound in many directions due to irregularities in the transmission medium. Volume I of Scattering will be devoted to basic theoretical ideas, approximation methods, numerical techniques and mathematical modeling. Volume II will be concerned with basic experimental techniques, technological practices, and comparisons with relevant theoretical work including seismology, medical applications, meteorological phenomena and astronomy. This reference will be used by researchers and graduate students in physics, applied physics, biophysics, chemical physics, medical physics, acoustics, geosciences, optics, mathematics, and engineering. This is the first encyclopedic-range work on the topic of scattering theory in quantum mechanics, elastodynamics, acoustics, and electromagnetics. It serves as a comprehensive interdisciplinary presentation of scattering and inverse scattering theory and applications in a wide range of scientific fields, with an emphasis, and details, up-to-date developments. Scattering also places an emphasis on the problems that are still in active current research. The first interdisciplinary reference source on scattering to gather all world expertise in this technique Covers the major aspects of scattering in a common language, helping to widening the knowledge of researchers across disciplines The list of editors, associate editors and contributors reads like an international Who's Who in the interdisciplinary field of scattering

Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrodinger Equation (AM-154)

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Author: Spyridon Kamvissis,Kenneth D.T-R McLaughlin,Peter D. Miller

Publisher: Princeton University Press

ISBN: 1400837189

Category: Mathematics

Page: 312

View: 4728

This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrödinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe. To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Hölder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.

Generalized Riemann Problems in Computational Fluid Dynamics

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Author: Matania Ben-Artzi,Joseph Falcovitz

Publisher: Cambridge University Press

ISBN: 9780521772969

Category: Mathematics

Page: 349

View: 8000

This 2003 monograph presents the GRP algorithm and is accessible to researchers and graduate students alike.

Orthogonal Polynomials and Special Functions

Computation and Applications

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Author: Francisco Marcellàn,Walter Van Assche

Publisher: Springer

ISBN: 3540367160

Category: Mathematics

Page: 422

View: 1075

Special functions and orthogonal polynomials in particular have been around for centuries. Can you imagine mathematics without trigonometric functions, the exponential function or polynomials? The present set of lecture notes contains seven chapters about the current state of orthogonal polynomials and special functions and gives a view on open problems and future directions.

Discrete and Continuous Nonlinear Schrödinger Systems

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Author: M. J. Ablowitz,B. Prinari,A. D. Trubatch

Publisher: Cambridge University Press

ISBN: 9780521534376

Category: Mathematics

Page: 257

View: 1412

This book presents a detailed mathematical analysis of scattering theory, obtains soliton solutions, and analyzes soliton interactions, both scalar and vector.

A Unified Approach to Boundary Value Problems

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Author: Athanassios S. Fokas

Publisher: SIAM

ISBN: 0898716519

Category: Mathematics

Page: 336

View: 7883

A novel approach to analysing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, based on ideas of the inverse scattering transform that the author introduced in 1997. This method is unique in also yielding novel integral representations for linear PDEs. Several new developments are addressed in the book, including a new transform method for linear evolution equations on the half-line and on the finite interval; analytical inversion of certain integrals such as the attenuated Radon transform and the Dirichlet-to-Neumann map for a moving boundary; integral representations for linear boundary value problems; analytical and numerical methods for elliptic PDEs in a convex polygon; and integrable nonlinear PDEs. An epilogue provides a list of problems on which the author's new approach has been used, offers open problems, and gives a glimpse into how the method might be applied to problems in three dimensions.

Discrete Differential Geometry

Integrable Structure

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Author: Alexander I. Bobenko,Yuri B. Suris

Publisher: American Mathematical Soc.

ISBN: 0821847007

Category: Mathematics

Page: 404

View: 9690

An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of Integrable systems. One of the main goals of this book Is to reveal this integrable structure of discrete differential geometry. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results. The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question "How do we discretize differential geometry?" arising in their specific field.

P-adic Analysis and Mathematical Physics

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Author: Vasili? Sergeevich Vladimirov,I. V. Volovich,E. I. Zelenov

Publisher: World Scientific

ISBN: 9789810208806

Category: Science

Page: 319

View: 1385

p-adic numbers play a very important role in modern number theory, algebraic geometry and representation theory. Lately p-adic numbers have attracted a great deal of attention in modern theoretical physics as a promising new approach for describing the non-Archimedean geometry of space-time at small distances.This is the first book to deal with applications of p-adic numbers in theoretical and mathematical physics. It gives an elementary and thoroughly written introduction to p-adic numbers and p-adic analysis with great numbers of examples as well as applications of p-adic numbers in classical mechanics, dynamical systems, quantum mechanics, statistical physics, quantum field theory and string theory.