Volterra Integral Equations

An Introduction to Theory and Applications

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Author: Hermann Brunner

Publisher: Cambridge University Press

ISBN: 1107098726

Category: Mathematics

Page: 410

View: 9244

This book offers a comprehensive introduction to the theory of linear and nonlinear Volterra integral equations. It includes applications and an extensive bibliography.

Collocation Methods for Volterra Integral and Related Functional Differential Equations

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Author: Hermann Brunner

Publisher: Cambridge University Press

ISBN: 9780521806152

Category: Mathematics

Page: 597

View: 7790

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.

Invitation to Ergodic Theory

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Author: César Ernesto Silva

Publisher: American Mathematical Soc.

ISBN: 0821844202

Category: Mathematics

Page: 262

View: 5895

This book is an introduction to basic concepts in ergodic theory such as recurrence, ergodicity, the ergodic theorem, mixing, and weak mixing. It does not assume knowledge of measure theory; all the results needed from measure theory are presented from scratch. In particular, the book includes a detailed construction of the Lebesgue measure on the real line and an introduction to measure spaces up to the Caratheodory extension theorem. It also develops the Lebesgue theory of integration, including the dominated convergence theorem and an introduction to the Lebesgue $Lp$spaces.

Simulating Hamiltonian Dynamics

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Author: Benedict Leimkuhler,Sebastian Reich

Publisher: Cambridge University Press

ISBN: 9780521772907

Category: Mathematics

Page: 379

View: 2859

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.

Introduction to Model Spaces and their Operators

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Author: Stephan Ramon Garcia,Javad Mashreghi,William T. Ross

Publisher: Cambridge University Press

ISBN: 1107108748

Category: Mathematics

Page: 335

View: 3426

A self-contained textbook which opens up this challenging field to newcomers and points to areas of future research.

The Analysis of Fractional Differential Equations

An Application-Oriented Exposition Using Differential Operators of Caputo Type

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Author: Kai Diethelm

Publisher: Springer

ISBN: 3642145744

Category: Mathematics

Page: 247

View: 7954

Fractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interaction between these two communities. In particular, typical mathematical works provide extensive findings on aspects with comparatively little significance in applications, and the engineering literature often lacks mathematical detail and precision. This book bridges the gap between the two communities. It concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough and mathematically rigorous study of their properties and of the corresponding differential equations. The text is a useful tool for mathematicians and researchers from the applied sciences alike. It can also be used as a basis for teaching graduate courses on fractional differential equations.

Radial Basis Functions

Theory and Implementations

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Author: Martin D. Buhmann

Publisher: Cambridge University Press

ISBN: 9781139435246

Category: Mathematics

Page: N.A

View: 5775

In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing, it is necessary to estimate parameters, usually multidimensional, by approximation and interpolation. Radial basis functions are a powerful tool which work well in very general circumstances and so are becoming of widespread use as the limitations of other methods, such as least squares, polynomial interpolation or wavelet-based, become apparent. The author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.

Difference Equations by Differential Equation Methods

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Author: Peter E. Hydon

Publisher: Cambridge University Press

ISBN: 1139991701

Category: Mathematics

Page: N.A

View: 7175

Most well-known solution techniques for differential equations exploit symmetry in some form. Systematic methods have been developed for finding and using symmetries, first integrals and conservation laws of a given differential equation. Here the author explains how to extend these powerful methods to difference equations, greatly increasing the range of solvable problems. Beginning with an introduction to elementary solution methods, the book gives readers a clear explanation of exact techniques for ordinary and partial difference equations. The informal presentation is suitable for anyone who is familiar with standard differential equation methods. No prior knowledge of difference equations or symmetry is assumed. The author uses worked examples to help readers grasp new concepts easily. There are 120 exercises of varying difficulty and suggestions for further reading. The book goes to the cutting edge of research; its many new ideas and methods make it a valuable reference for researchers in the field.

Multiscale Methods for Fredholm Integral Equations

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

Publisher: Cambridge University Press

ISBN: 1107103479

Category: Mathematics

Page: 552

View: 7273

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

Abel Integral Equations

Analysis and Applications

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Author: Rudolf Gorenflo,Sergio Vessella

Publisher: Springer

ISBN: 3540469494

Category: Mathematics

Page: 222

View: 6325

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.

Symmetry, Phase Modulation and Nonlinear Waves

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Author: Thomas J. Bridges

Publisher: Cambridge University Press

ISBN: 1107188849

Category: Mathematics

Page: 242

View: 2101

Nonlinear waves are pervasive in nature, but are often elusive when they are modelled and analysed. This book develops a natural approach to the problem based on phase modulation. It is both an elaboration of the use of phase modulation for the study of nonlinear waves and a compendium of background results in mathematics, such as Hamiltonian systems, symplectic geometry, conservation laws, Noether theory, Lagrangian field theory and analysis, all of which combine to generate the new theory of phase modulation. While the build-up of theory can be intensive, the resulting emergent partial differential equations are relatively simple. A key outcome of the theory is that the coefficients in the emergent modulation equations are universal and easy to calculate. This book gives several examples of the implications in the theory of fluid mechanics and points to a wide range of new applications.

Spectral Methods

Algorithms, Analysis and Applications

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Author: Jie Shen,Tao Tang,Li-Lian Wang

Publisher: Springer Science & Business Media

ISBN: 3540710418

Category: Mathematics

Page: 472

View: 8439

Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed presentation of basic spectral algorithms, as well as a systematical presentation of basic convergence theory and error analysis for spectral methods. Readers of this book will be exposed to a unified framework for designing and analyzing spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains. The book contains a large number of figures which are designed to illustrate various concepts stressed in the book. A set of basic matlab codes has been made available online to help the readers to develop their own spectral codes for their specific applications.

Linear Matrix Inequalities in System and Control Theory

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Author: Stephen Boyd,Laurent El Ghaoui,Eric Feron,Venkataramanan Balakrishnan

Publisher: SIAM

ISBN: 9781611970777

Category: Control theory

Page: 193

View: 5856

In this book the authors reduce a wide variety of problems arising in system and control theory to a handful of convex and quasiconvex optimization problems that involve linear matrix inequalities. These optimization problems can be solved using recently developed numerical algorithms that not only are polynomial-time but also work very well in practice; the reduction therefore can be considered a solution to the original problems. This book opens up an important new research area in which convex optimization is combined with system and control theory, resulting in the solution of a large number of previously unsolved problems.

The H-Function

Theory and Applications

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Author: A.M. Mathai,Ram Kishore Saxena,Hans J. Haubold

Publisher: Springer Science & Business Media

ISBN: 9781441909169

Category: Science

Page: 268

View: 9563

TheH-function or popularly known in the literature as Fox’sH-function has recently found applications in a large variety of problems connected with reaction, diffusion, reaction–diffusion, engineering and communication, fractional differ- tial and integral equations, many areas of theoretical physics, statistical distribution theory, etc. One of the standard books and most cited book on the topic is the 1978 book of Mathai and Saxena. Since then, the subject has grown a lot, mainly in the elds of applications. Due to popular demand, the authors were requested to - grade and bring out a revised edition of the 1978 book. It was decided to bring out a new book, mostly dealing with recent applications in statistical distributions, pa- way models, nonextensive statistical mechanics, astrophysics problems, fractional calculus, etc. and to make use of the expertise of Hans J. Haubold in astrophysics area also. It was decided to con ne the discussion toH-function of one scalar variable only. Matrix variable cases and many variable cases are not discussed in detail, but an insight into these areas is given. When going from one variable to many variables, there is nothing called a unique bivariate or multivariate analogue of a givenfunction. Whatever be the criteria used, there may be manydifferentfunctions quali ed to be bivariate or multivariate analogues of a given univariate function. Some of the bivariate and multivariateH-functions, currently in the literature, are also questioned by many authors.

Modeling with Itô Stochastic Differential Equations

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Author: E. Allen

Publisher: Springer Science & Business Media

ISBN: 1402059531

Category: Mathematics

Page: 230

View: 1367

This book explains a procedure for constructing realistic stochastic differential equation models for randomly varying systems in biology, chemistry, physics, engineering, and finance. Introductory chapters present the fundamental concepts of random variables, stochastic processes, stochastic integration, and stochastic differential equations. These concepts are explained in a Hilbert space setting which unifies and simplifies the presentation.

Probability Based High Temperature Engineering

Creep and Structural Fire Resistance

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Author: Leo Razdolsky

Publisher: Springer

ISBN: 3319419099

Category: Science

Page: 656

View: 9772

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.

Topology for Computing

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Author: Afra J. Zomorodian

Publisher: Cambridge University Press

ISBN: 9781139442633

Category: Computers

Page: N.A

View: 6138

The emerging field of computational topology utilizes theory from topology and the power of computing to solve problems in diverse fields. Recent applications include computer graphics, computer-aided design (CAD), and structural biology, all of which involve understanding the intrinsic shape of some real or abstract space. A primary goal of this book is to present basic concepts from topology and Morse theory to enable a non-specialist to grasp and participate in current research in computational topology. The author gives a self-contained presentation of the mathematical concepts from a computer scientist's point of view, combining point set topology, algebraic topology, group theory, differential manifolds, and Morse theory. He also presents some recent advances in the area, including topological persistence and hierarchical Morse complexes. Throughout, the focus is on computational challenges and on presenting algorithms and data structures when appropriate.

Introduction to the Modern Theory of Dynamical Systems

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Author: Anatole Katok,Boris Hasselblatt

Publisher: Cambridge University Press

ISBN: 9780521575577

Category: Mathematics

Page: 802

View: 6978

This book provides a self-contained comprehensive exposition of the theory of dynamical systems. The book begins with a discussion of several elementary but crucial examples. These are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods. The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. The third and fourth parts develop the theories of low-dimensional dynamical systems and hyperbolic dynamical systems in depth. The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate and up.