Collocation Methods for Volterra Integral and Related Functional Differential Equations

Author: Hermann Brunner

Publisher: Cambridge University Press

ISBN: 9780521806152

Category: Mathematics

Page: 597

View: 2092

Collocation based on piecewise polynomial approximation represents a powerful class of methods for the numerical solution of initial-value problems for functional differential and integral equations arising in a wide spectrum of applications, including biological and physical phenomena. The present book introduces the reader to the general principles underlying these methods and then describes in detail their convergence properties when applied to ordinary differential equations, functional equations with (Volterra type) memory terms, delay equations, and differential-algebraic and integral-algebraic equations. Each chapter starts with a self-contained introduction to the relevant theory of the class of equations under consideration. Numerous exercises and examples are supplied, along with extensive historical and bibliographical notes utilising the vast annotated reference list of over 1300 items. In sum, Hermann Brunner has written a treatise that can serve as an introduction for students, a guide for users, and a comprehensive resource for experts.
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Numerical Mathematics and Advanced Applications 2009

Proceedings of ENUMATH 2009, the 8th European Conference on Numerical Mathematics and Advanced Applications, Uppsala, July 2009

Author: Gunilla Kreiss,Per Lötstedt,Axel Målqvist,Maya Neytcheva

Publisher: Springer Science & Business Media

ISBN: 9783642117954

Category: Mathematics

Page: 939

View: 7227

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Probability Based High Temperature Engineering

Creep and Structural Fire Resistance

Author: Leo Razdolsky

Publisher: Springer

ISBN: 3319419099

Category: Science

Page: 656

View: 5091

This volume on structural fire resistance is for aerospace, structural, and fire prevention engineers; architects, and educators. It bridges the gap between prescriptive- and performance-based methods and simplifies very complex and comprehensive computer analyses to the point that the structural fire resistance and high temperature creep deformations will have a simple, approximate analytical expression that can be used in structural analysis and design. The book emphasizes methods of the theory of engineering creep (stress-strain diagrams) and mathematical operations quite distinct from those of solid mechanics absent high-temperature creep deformations, in particular the classical theory of elasticity and structural engineering. Dr. Razdolsky’s previous books focused on methods of computing the ultimate structural design load to the different fire scenarios. The current work is devoted to the computing of the estimated ultimate resistance of the structure taking into account the effect of high temperature creep deformations. An essential resource for aerospace structural engineers who wish to improve their understanding of structure exposed to flare up temperatures and severe fires, the book also serves as a textbook for introductory courses in fire safety in civil or structural engineering programs, vital reading for the PhD students in aerospace fire protection and structural engineering, and a case study of a number of high-profile fires (the World Trade Center, Broadgate Phase 8, One Meridian Plaza; Mandarin Towers). Probability Based High Temperature Engineering: Creep and Structural Fire Resistance successfully bridges the information gap between aerospace, structural, and engineers; building inspectors, architects, and code officials.
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Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

Author: Josef Dick,Frances Y. Kuo,Henryk Woźniakowski

Publisher: Springer

ISBN: 3319724568

Category: Mathematics

Page: 1309

View: 2051

This book is a tribute to Professor Ian Hugh Sloan on the occasion of his 80th birthday. It consists of nearly 60 articles written by international leaders in a diverse range of areas in contemporary computational mathematics. These papers highlight the impact and many achievements of Professor Sloan in his distinguished academic career. The book also presents state of the art knowledge in many computational fields such as quasi-Monte Carlo and Monte Carlo methods for multivariate integration, multi-level methods, finite element methods, uncertainty quantification, spherical designs and integration on the sphere, approximation and interpolation of multivariate functions, oscillatory integrals, and in general in information-based complexity and tractability, as well as in a range of other topics. The book also tells the life story of the renowned mathematician, family man, colleague and friend, who has been an inspiration to many of us. The reader may especially enjoy the story from the perspective of his family, his wife, his daughter and son, as well as grandchildren, who share their views of Ian. The clear message of the book is that Ian H. Sloan has been a role model in science and life.
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The Mathematical Foundations of Mixing

The Linked Twist Map as a Paradigm in Applications: Micro to Macro, Fluids to Solids

Author: Rob Sturman,Julio M. Ottino,Stephen Wiggins

Publisher: Cambridge University Press

ISBN: 1139459201

Category: Mathematics

Page: N.A

View: 478

Mixing processes occur in many technological and natural applications, with length and time scales ranging from the very small to the very large. The diversity of problems can give rise to a diversity of approaches. Are there concepts that are central to all of them? Are there tools that allow for prediction and quantification? The authors show how a variety of flows in very different settings possess the characteristic of streamline crossing. This notion can be placed on firm mathematical footing via Linked Twist Maps (LTMs), which is the central organizing principle of this book. The authors discuss the definition and construction of LTMs, provide examples of specific mixers that can be analyzed in the LTM framework and introduce a number of mathematical techniques which are then brought to bear on the problem of fluid mixing. In a final chapter, they present a number of open problems and new directions.
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Multiscale Methods for Fredholm Integral Equations

Author: Zhongying Chen,Charles A. Micchelli,Yuesheng Xu

Publisher: Cambridge University Press

ISBN: 1107103479

Category: Mathematics

Page: 552

View: 6593

Presents the state of the art in the study of fast multiscale methods for solving these equations based on wavelets.
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Volterra Integral Equations

An Introduction to Theory and Applications

Author: Hermann Brunner

Publisher: Cambridge University Press

ISBN: 1107098726

Category: Mathematics

Page: 410

View: 4730

This book offers a comprehensive introduction to the theory of linear and nonlinear Volterra integral equations. It includes applications and an extensive bibliography.
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Spectral Methods for Time-Dependent Problems

Author: Jan S. Hesthaven,Sigal Gottlieb,David Gottlieb

Publisher: Cambridge University Press

ISBN: 113945952X

Category: Mathematics

Page: N.A

View: 5236

Spectral methods are well-suited to solve problems modeled by time-dependent partial differential equations: they are fast, efficient and accurate and widely used by mathematicians and practitioners. This class-tested 2007 introduction, the first on the subject, is ideal for graduate courses, or self-study. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including discussions of stability, boundary conditions, filtering, and the extension from the linear to the nonlinear situation). Computational solution techniques for integration in time are dealt with by Runge-Kutta type methods. Several chapters are devoted to material not previously covered in book form, including stability theory for polynomial methods, techniques for problems with discontinuous solutions, round-off errors and the formulation of spectral methods on general grids. These will be especially helpful for practitioners.
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Simulating Hamiltonian Dynamics

Author: Benedict Leimkuhler,Sebastian Reich

Publisher: Cambridge University Press

ISBN: 9780521772907

Category: Mathematics

Page: 379

View: 7382

Geometric integrators are time-stepping methods, designed such that they exactly satisfy conservation laws, symmetries or symplectic properties of a system of differential equations. In this book the authors outline the principles of geometric integration and demonstrate how they can be applied to provide efficient numerical methods for simulating conservative models. Beginning from basic principles and continuing with discussions regarding the advantageous properties of such schemes, the book introduces methods for the N-body problem, systems with holonomic constraints, and rigid bodies. More advanced topics treated include high-order and variable stepsize methods, schemes for treating problems involving multiple time-scales, and applications to molecular dynamics and partial differential equations. The emphasis is on providing a unified theoretical framework as well as a practical guide for users. The inclusion of examples, background material and exercises enhance the usefulness of the book for self-instruction or as a text for a graduate course on the subject.
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Iterative Krylov Methods for Large Linear Systems

Author: H. A. van der Vorst

Publisher: Cambridge University Press

ISBN: 9780521818285

Category: Mathematics

Page: 221

View: 9975

Overview of iterative solutions methods for systems of linear equations. For graduate students and researchers.
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Computing Highly Oscillatory Integrals

Author: Alfredo Deaño,Daan Huybrechs,Arieh Iserles

Publisher: SIAM

ISBN: 1611975123

Category: Mathematics

Page: 180

View: 4158

Highly oscillatory phenomena range across numerous areas in science and engineering and their computation represents a difficult challenge. A case in point is integrals of rapidly oscillating functions in one or more variables. The quadrature of such integrals has been historically considered very demanding. Research in the past 15 years (in which the authors played a major role) resulted in a range of very effective and affordable algorithms for highly oscillatory quadrature. This is the only monograph bringing together the new body of ideas in this area in its entirety. The starting point is that approximations need to be analyzed using asymptotic methods rather than by more standard polynomial expansions. As often happens in computational mathematics, once a phenomenon is understood from a mathematical standpoint, effective algorithms follow. As reviewed in this monograph, we now have at our disposal a number of very effective quadrature methods for highly oscillatory integrals--Filon-type and Levin-type methods, methods based on steepest descent, and complex-valued Gaussian quadrature. Their understanding calls for a fairly varied mathematical toolbox--from classical numerical analysis, approximation theory, and theory of orthogonal polynomials all the way to asymptotic analysis--yet this understanding is the cornerstone of efficient algorithms.
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Numerical Solution of Ordinary Differential Equations

Author: Kendall Atkinson,Weimin Han,David E. Stewart

Publisher: John Wiley & Sons

ISBN: 1118164520

Category: Mathematics

Page: 272

View: 7633

A concise introduction to numerical methodsand the mathematical framework neededto understand their performance Numerical Solution of Ordinary Differential Equations presents a complete and easy-to-follow introduction to classical topics in the numerical solution of ordinary differential equations. The book's approach not only explains the presented mathematics, but also helps readers understand how these numerical methods are used to solve real-world problems. Unifying perspectives are provided throughout the text, bringing together and categorizing different types of problems in order to help readers comprehend the applications of ordinary differential equations. In addition, the authors' collective academic experience ensures a coherent and accessible discussion of key topics, including: Euler's method Taylor and Runge-Kutta methods General error analysis for multi-step methods Stiff differential equations Differential algebraic equations Two-point boundary value problems Volterra integral equations Each chapter features problem sets that enable readers to test and build their knowledge of the presented methods, and a related Web site features MATLAB® programs that facilitate the exploration of numerical methods in greater depth. Detailed references outline additional literature on both analytical and numerical aspects of ordinary differential equations for further exploration of individual topics. Numerical Solution of Ordinary Differential Equations is an excellent textbook for courses on the numerical solution of differential equations at the upper-undergraduate and beginning graduate levels. It also serves as a valuable reference for researchers in the fields of mathematics and engineering.
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BIT.

Numerical mathematics

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Computers

Page: N.A

View: 3898

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Abel Integral Equations

Analysis and Applications

Author: Rudolf Gorenflo,Sergio Vessella

Publisher: Springer

ISBN: 3540469494

Category: Mathematics

Page: 222

View: 5028

In many fields of application of mathematics, progress is crucially dependent on the good flow of information between (i) theoretical mathematicians looking for applications, (ii) mathematicians working in applications in need of theory, and (iii) scientists and engineers applying mathematical models and methods. The intention of this book is to stimulate this flow of information. In the first three chapters (accessible to third year students of mathematics and physics and to mathematically interested engineers) applications of Abel integral equations are surveyed broadly including determination of potentials, stereology, seismic travel times, spectroscopy, optical fibres. In subsequent chapters (requiring some background in functional analysis) mapping properties of Abel integral operators and their relation to other integral transforms in various function spaces are investi- gated, questions of existence and uniqueness of solutions of linear and nonlinear Abel integral equations are treated, and for equations of the first kind problems of ill-posedness are discussed. Finally, some numerical methods are described. In the theoretical parts, emphasis is put on the aspects relevant to applications.
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