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On The Numerical Solution Of Nonlinear Riemann Hilbert Problems
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Book Synopsis Riemann-Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions by : Thomas Trogdon
Download or read book Riemann-Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions written by Thomas Trogdon and published by SIAM. This book was released on 2015-12-22 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis On the Numerical Solution of Nonlinear Riemann Hilbert Problems by : Elias Wegert
Download or read book On the Numerical Solution of Nonlinear Riemann Hilbert Problems written by Elias Wegert and published by . This book was released on 1996 with total page 109 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Riemann-Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions by : Thomas D. Trogdon
Download or read book Riemann-Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions written by Thomas D. Trogdon and published by . This book was released on 2013 with total page 318 pages. Available in PDF, EPUB and Kindle. Book excerpt: The computation of special functions has important implications throughout engineering and the physical sciences. Nonlinear special functions include the solutions of integrable partial differential equations and the Painleve transcendents. Many problems in water wave theory, nonlinear optics and statistical mechanics are reduced to the study of a nonlinear special function in particular limits. The universal object that these functions share is a Riemann-Hilbert representation: the nonlinear special function can be recovered from the solution of a Riemann-Hilbert problem (RHP). A RHP consists of finding a piecewise-analytic function in the complex plane when the behavior of its discontinuities is specified. In this dissertation, the applied theory of Riemann-Hilbert problems, using both Holder and Lebesgue spaces, is reviewed. The numerical solution of RHPs is discussed. Furthermore, the uniform approximation theory for the numerical solution of RHPs is presented, proving that in certain cases the convergence of the numerical method is uniform with respect to a parameter. This theory shares close relation to the method of nonlinear steepest descent for RHPs. The inverse scattering transform for the Korteweg-de Vries and Nonlinear Schroedinger equation is made effective by solving the associated RHPs numerically. This technique is extended to solve the Painleve II equation numerically. Similar Riemann-Hilbert techniques are used to compute the so-called finite-genus solutions of the Korteweg-de Vries equation. This involves ideas from Riemann surface theory. Finally, the methodology is applied to compute orthogonal polynomials with exponential weights. This allows for the computation of statistical quantities stemming from random matrix ensembles.
Book Synopsis Numerical Method for the Riemann-Hilbert Problem of Nonlinear Elliptic Complex Equations of First Order by : Wen Guo-chun
Download or read book Numerical Method for the Riemann-Hilbert Problem of Nonlinear Elliptic Complex Equations of First Order written by Wen Guo-chun and published by . This book was released on 1991 with total page 10 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Nonlinear Boundary Value Problems for Holomorphic Functions and Singular Integral Equations by : Elias Wegert
Download or read book Nonlinear Boundary Value Problems for Holomorphic Functions and Singular Integral Equations written by Elias Wegert and published by Wiley-VCH. This book was released on 1992-05-08 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covers various topics in nonlinear boundary value problems for holomorphic functions, including existence and uniqueness results, questions concerning parameter dependence, regularity theorems, several procedures for numerically solving such problems, and applications to nonlinear singular integral equations. The emphasis is mainly on geometric aspects. Numerical methods are discussed. This text requires only an elementary knowledge of function theory. Includes a 13-page bibliography. Distributed in the US by VCH. Annotation copyright by Book News, Inc., Portland, OR
Book Synopsis The Riemann-Hilbert Problem by : D. V. Anosov
Download or read book The Riemann-Hilbert Problem written by D. V. Anosov and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Riemann-Hilbert problem (Hilbert's 21st problem) belongs to the theory of linear systems of ordinary differential equations in the complex domain. The problem concerns the existence of a Fuchsian system with prescribed singularities and monodromy. Hilbert was convinced that such a system always exists. However, this turned out to be a rare case of a wrong forecast made by him. In 1989 the second author (A. B.) discovered a counterexample, thus obtaining a negative solution to Hilbert's 21st problem in its original form.
Book Synopsis Painleve Transcendents by : A. S. Fokas
Download or read book Painleve Transcendents written by A. S. Fokas and published by American Mathematical Soc.. This book was released on 2006 with total page 570 pages. Available in PDF, EPUB and Kindle. Book excerpt: At the turn of the twentieth century, the French mathematician Paul Painleve and his students classified second order nonlinear ordinary differential equations with the property that the location of possible branch points and essential singularities of their solutions does not depend on initial conditions. It turned out that there are only six such equations (up to natural equivalence), which later became known as Painleve I-VI. Although these equations were initially obtainedanswering a strictly mathematical question, they appeared later in an astonishing (and growing) range of applications, including, e.g., statistical physics, fluid mechanics, random matrices, and orthogonal polynomials. Actually, it is now becoming clear that the Painleve transcendents (i.e., the solutionsof the Painleve equations) play the same role in nonlinear mathematical physics that the classical special functions, such as Airy and Bessel functions, play in linear physics. The explicit formulas relating the asymptotic behaviour of the classical special functions at different critical points, play a crucial role in the applications of these functions. It is shown in this book, that even though the six Painleve equations are nonlinear, it is still possible, using a new technique called theRiemann-Hilbert formalism, to obtain analogous explicit formulas for the Painleve transcendents. This striking fact, apparently unknown to Painleve and his contemporaries, is the key ingredient for the remarkable applicability of these ``nonlinear special functions''. The book describes in detail theRiemann-Hilbert method and emphasizes its close connection to classical monodromy theory of linear equations as well as to modern theory of integrable systems. In addition, the book contains an ample collection of material concerning the asymptotics of the Painleve functions and their various applications, which makes it a good reference source for everyone working in the theory and applications of Painleve equations and related areas.
Book Synopsis Painlevé Transcendents by : Athanassios S. Fokas
Download or read book Painlevé Transcendents written by Athanassios S. Fokas and published by American Mathematical Society. This book was released on 2023-11-20 with total page 570 pages. Available in PDF, EPUB and Kindle. Book excerpt: At the turn of the twentieth century, the French mathematician Paul Painlevé and his students classified second order nonlinear ordinary differential equations with the property that the location of possible branch points and essential singularities of their solutions does not depend on initial conditions. It turned out that there are only six such equations (up to natural equivalence), which later became known as Painlevé I–VI. Although these equations were initially obtained answering a strictly mathematical question, they appeared later in an astonishing (and growing) range of applications, including, e.g., statistical physics, fluid mechanics, random matrices, and orthogonal polynomials. Actually, it is now becoming clear that the Painlevé transcendents (i.e., the solutions of the Painlevé equations) play the same role in nonlinear mathematical physics that the classical special functions, such as Airy and Bessel functions, play in linear physics. The explicit formulas relating the asymptotic behaviour of the classical special functions at different critical points play a crucial role in the applications of these functions. It is shown in this book that even though the six Painlevé equations are nonlinear, it is still possible, using a new technique called the Riemann-Hilbert formalism, to obtain analogous explicit formulas for the Painlevé transcendents. This striking fact, apparently unknown to Painlevé and his contemporaries, is the key ingredient for the remarkable applicability of these “nonlinear special functions”. The book describes in detail the Riemann-Hilbert method and emphasizes its close connection to classical monodromy theory of linear equations as well as to modern theory of integrable systems. In addition, the book contains an ample collection of material concerning the asymptotics of the Painlevé functions and their various applications, which makes it a good reference source for everyone working in the theory and applications of Painlevé equations and related areas.
Book Synopsis Approximate Methods and Numerical Analysis for Elliptic Complex Equation by : Guo Chun Wen
Download or read book Approximate Methods and Numerical Analysis for Elliptic Complex Equation written by Guo Chun Wen and published by CRC Press. This book was released on 1999-06-11 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical methods for elliptic partial differential equations have been the subject of many books in recent years, but few have treated the subject of complex equations. In this important new book, the author introduces the theory of, and approximate methods for, nonlinear elliptic complex equations in multiple connected domains. Constructive methods are systematically applied to proper boundary value problems which include very general boundary conditions. Approximate and numerical methods, such as the Newton imbedding method, the continuity method, the finite element method, the difference method and the boundary integral method, as well as their applications, are discussed in detail. The book will be of interest to all scientists studying the theory or applications of complex analysis.
Book Synopsis Symmetries and Integrability of Difference Equations by : Decio Levi
Download or read book Symmetries and Integrability of Difference Equations written by Decio Levi and published by Springer. This book was released on 2017-06-30 with total page 441 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The Riemann-Hilbert Problem by : D. V. Anosov
Download or read book The Riemann-Hilbert Problem written by D. V. Anosov and published by Vieweg+Teubner Verlag. This book was released on 1994-01-01 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Riemann-Hilbert problem (Hilbert's 21st problem) belongs to the theory of linear systems of ordinary differential equations in the complex domain. The problem concerns the existence of a Fuchsian system with prescribed singularities and monodromy. Hilbert was convinced that such a system always exists. However, this turned out to be a rare case of a wrong forecast made by him. In 1989 the second author (A. B.) discovered a counterexample, thus obtaining a negative solution to Hilbert's 21st problem in its original form.
Book Synopsis Recent Developments in Integrable Systems and Riemann-Hilbert Problems by : Kenneth T-R McLaughlin
Download or read book Recent Developments in Integrable Systems and Riemann-Hilbert Problems written by Kenneth T-R McLaughlin and published by American Mathematical Soc.. This book was released on 2003 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is a collection of papers presented at a special session on integrable systems and Riemann-Hilbert problems. The goal of the meeting was to foster new research by bringing together experts from different areas. Their contributions to the volume provide a useful portrait of the breadth and depth of integrable systems. Topics covered include discrete Painleve equations, integrable nonlinear partial differential equations, random matrix theory, Bose-Einstein condensation, spectral and inverse spectral theory, and last passage percolation models. In most of these articles, the Riemann-Hilbert problem approach plays a central role, which is powerful both analytically and algebraically. The book is intended for graduate students and researchers interested in integrable systems and its applications.
Book Synopsis Riemann Problems and Jupyter Solutions by : David I. Ketcheson
Download or read book Riemann Problems and Jupyter Solutions written by David I. Ketcheson and published by SIAM. This book was released on 2020-06-26 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book addresses an important class of mathematical problems (the Riemann problem) for first-order hyperbolic partial differential equations (PDEs), which arise when modeling wave propagation in applications such as fluid dynamics, traffic flow, acoustics, and elasticity. The solution of the Riemann problem captures essential information about these models and is the key ingredient in modern numerical methods for their solution. This book covers the fundamental ideas related to classical Riemann solutions, including their special structure and the types of waves that arise, as well as the ideas behind fast approximate solvers for the Riemann problem. The emphasis is on the general ideas, but each chapter delves into a particular application. Riemann Problems and Jupyter Solutions is available in electronic form as a collection of Jupyter notebooks that contain executable computer code and interactive figures and animations, allowing readers to grasp how the concepts presented are affected by important parameters and to experiment by varying those parameters themselves. The only interactive book focused entirely on the Riemann problem, it develops each concept in the context of a specific physical application, helping readers apply physical intuition in learning mathematical concepts. Graduate students and researchers working in the analysis and/or numerical solution of hyperbolic PDEs will find this book of interest. This includes mathematicians, as well as scientists and engineers, working on wave propagation problems. Educators interested in developing instructional materials using Jupyter notebooks will also find this book useful. The book is appropriate for courses in Numerical Methods for Hyperbolic PDEs and Analysis of Hyperbolic PDEs, and it can be a great supplement for courses in computational fluid dynamics, acoustics, and gas dynamics.
Book Synopsis On the nonlinear Riemann Hilbert problem by : Franc Forstnerič
Download or read book On the nonlinear Riemann Hilbert problem written by Franc Forstnerič and published by . This book was released on 1987 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Ernst Equation and Riemann Surfaces by : Christian Klein
Download or read book Ernst Equation and Riemann Surfaces written by Christian Klein and published by Springer Science & Business Media. This book was released on 2005-11-18 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: Exact solutions to Einstein’s equations have been useful for the understanding of general relativity in many respects. They have led to such physical concepts as black holes and event horizons, and helped to visualize interesting features of the theory. This volume studies the solutions to the Ernst equation associated to Riemann surfaces in detail. In addition, the book discusses the physical and mathematical aspects of this class analytically as well as numerically.
Book Synopsis Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics by : Dario A. Bini
Download or read book Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics written by Dario A. Bini and published by Birkhäuser. This book was released on 2017-03-21 with total page 757 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a collection of expository and research papers on various topics in matrix and operator theory, contributed by several experts on the occasion of Albrecht Böttcher’s 60th birthday. Albrecht Böttcher himself has made substantial contributions to the subject in the past. The book also includes a biographical essay, a complete bibliography of Albrecht Böttcher’s work and brief informal notes on personal encounters with him. The book is of interest to graduate and advanced undergraduate students majoring in mathematics, researchers in matrix and operator theory as well as engineers and applied mathematicians.
Book Synopsis Nonlinear Riemann-Hilbert Problems by : Gunter Semmler
Download or read book Nonlinear Riemann-Hilbert Problems written by Gunter Semmler and published by . This book was released on 2004 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
