Numerical Approximation Methods

Semiclassical approximation addresses the important relationship between quantum and classical mechanics. There has been a very strong development in the mathematical theory, mainly thanks to methods of microlocal analysis. This book develops the basic methods, including the WKB-method, stationary phase and h-pseudodifferential operators. The applications include results on the tunnel effect, the asymptotics of eigenvalues in relation to classical trajectories and normal forms, plus slow perturbations of periodic Schrödinger operators appearing in solid state physics. No previous specialized knowledge in quantum mechanics or microlocal analysis is assumed, and only general facts about spectral theory in Hilbert space, distributions, Fourier transforms and some differential geometry belong to the prerequisites. This book is addressed to researchers and graduate students in mathematical analysis, as well as physicists who are interested in rigorous results. A fairly large fraction can be

This book aims to give a self-contained presentation of a number of results, which relate the volume of convex bodies in n-dimensional Euclidean space and the geometry of the corresponding finite-dimensional normed spaces. The methods employ classical ideas from the theory of convex sets, probability theory, approximation theory and the local theory of Banach spaces. The book is in two parts. The first presents self-contained proofs of the quotient of the subspace theorem, the inverse Santalo inequality and the inverse Brunn-Minkowski inequality. The second part gives a detailed exposition of the recently introduced classes of Banach spaces of weak cotype 2 or weak type 2, and the intersection of the classes (weak Hilbert space). The book is based on courses given in Paris and in Texas.

Systems of conservation laws arise naturally in physics and chemistry. To understand them and their consequences (shock waves, finite velocity wave propagation) properly in mathematical terms requires, however, knowledge of a broad range of topics. This book sets up the foundations of the modern theory of conservation laws, describing the physical models and mathematical methods, leading to the Glimm scheme. Building on this the author then takes the reader to the current state of knowledge in the subject. The maximum principle is considered from the viewpoint of numerical schemes and also in terms of viscous approximation. Small waves are studied using geometrical optics methods. Finally, the initial-boundary problem is considered in depth. Throughout, the presentation is reasonably self-contained, with large numbers of exercises and full discussion of all the ideas. This will make it ideal as a text for graduate courses in the area of partial differential equations.

The application of modern methods in numerical mathematics on problems in chemical engineering is essential for designing, analyzing and running chemical processes and even entire plants. Scientific C

There has in recent years been a remarkable growth of interest in the area of discrete integrable systems. Much progress has been made by applying symmetry groups to the study of differential equations, and connections have been made to other topics such as numerical methods, cellular automata and mathematical physics. This volume comprises state of the art articles from almost all the leading workers in this important and rapidly developing area, making it a necessary resource for all researchers interested in discrete integrable systems or related subjects.

Presenting innovative, promising and unconventional techniques, Post-modern Electromagnetics outlines the essential tools for the creation of numerical methods. Hafner offers a comparative analysis of

This book generalises the classical theory of orthogonal polynomials on the complex unit circle, or on the real line to orthogonal rational functions whose poles are among a prescribed set of complex numbers. The first part treats the case where these poles are all outside the unit disk or in the lower half plane. Classical topics such as recurrence relations, numerical quadrature, interpolation properties, Favard theorems, convergence, asymptotics, and moment problems are generalised and treated in detail. The same topics are discussed for the different situation where the poles are located on the unit circle or on the extended real line. In the last chapter, several applications are mentioned including linear prediction, Pisarenko modelling, lossless inverse scattering, and network synthesis. This theory has many applications in theoretical real and complex analysis, approximation theory, numerical analysis, system theory, and in electrical engineering.

An integral part of engineering design is the development of models that describe physical behavior or phenomena in mathematical terms. As engineering systems have become more complex, classic linear methods of modeling and analysis have proved inadequate, creating a need for nonlinear models to solve design problems. This 1999 text provides an introduction to mathematical modeling of linear and nonlinear systems, with an emphasis on the solution of nonlinear design problems. While encouraging the use of the computer as a tool for modeling and analysis, the aim is to discuss the basic concepts underlying computer techniques and to seek analytical solutions. Among topics covered are exact solution, numerical solution, graphical solution, and approximate solution methods; and the stability of nonlinear systems. Numerous examples show how to apply modeling methods to real engineering systems. The book also includes end-of-chapter problems and case studies of challenging design problems. I

Presenting sufficient material for a second two-semester sequence in the mechanics of materials without unduly complicated analytical or numerical methods, Cook (U. of Wisconsin-Madison) and Young ack

This text takes a practical, step-by-step approach to algebraic curves and surface interpolation motivated by the understanding of the many practical applications in engineering analysis, approximation, and curve plotting problems. Because of its usefulness for computing, the algebraic approach is the main theme, but a brief discussion of the synthetic approach is also presented as a way of gaining additional insight before proceeding with the algebraic manipulation. The authors start with simple interpolation, including splines, and extend this in an intuitive fashion to the production of conic sections. They then introduce projective co-ordinates as tools for dealing with higher order curves and singular points. They present many applications and concrete examples, including parabolic interpolation, geometric approximation, and the numerical solution of trajectory problems. In the final chapter they apply the basic theory to the construction of finite element basis functions and surf

Numerical methods are indispensable tools in the analysis of complex fluid flows. This book focuses on computational techniques for high-speed gas flows, especially gas flows containing shocks and other steep gradients. The book decomposes complicated numerical methods into simple modular parts, showing how each part fits and how each method relates to or differs from others. The text begins with a review of gasdynamics and computational techniques. Next come basic principles of computational gasdynamics. The last two parts cover basic techniques and advanced techniques. Senior and graduate level students, especially in aerospace engineering, as well as researchers and practising engineers, will find a wealth of invaluable information on high-speed gas flows in this text.

Numerical methods are indispensable tools in the analysis of complex fluid flows. This book focuses on computational techniques for high-speed gas flows, especially gas flows containing shocks and other steep gradients. The book decomposes complicated numerical methods into simple modular parts, showing how each part fits and how each method relates to or differs from others. The text begins with a review of gasdynamics and computational techniques. Next come basic principles of computational gasdynamics. The last two parts cover basic techniques and advanced techniques. Senior and graduate level students, especially in aerospace engineering, as well as researchers and practising engineers, will find a wealth of invaluable information on high-speed gas flows in this text.

This book presents a comprehensive and coherent account of the theory of quantum fields on a lattice, an essential technique for the study of the strong and electroweak nuclear interactions. Quantum field theory describes basic physical phenomena over an extremely wide range of length or energy scales. Quantum fields exist in space and time, which can be approximated by a set of lattice points. This approximation allows the application of powerful analytical and numerical techniques, and has provided a powerful tool for the study of both the strong and the electroweak interaction. After introductory chapters on scalar fields, gauge fields and fermion fields, the book studies quarks and gluons in QCD and fermions and bosons in the electroweak theory. The last chapter is devoted to numerical simulation algorithms which have been used in recent large-scale numerical simulations. The book will be valuable for graduate students and researchers in theoretical physics, elementary particle phy

This book reviews one-dimensional reactions, dynamics, diffusion and adsorption. In studies of complex systems in biology, chemistry and physics, understanding can be gained by analytical and numerical analyses of simple models. This book presents review articles at an advanced research level, describing results for one-dimensional models of dynamical processes such as chemical reactions and catalysis, kinetic ising models, phase separation and cluster growth, monolayer and multilayer adsorption with added relaxation, surface and hard-core particle dynamics, diffusional transport, random systems and experimental results. It also covers experimental results for systems ranging from chemical reactions to adsorption and reactions on polymer chains, steps on crystalline surfaces, and DNA. All chapters are written by leading scientists in the field. They present a self-contained review of this subject that will guide a reader from basic concepts, ideas, methods and models to the forefront o

This comprehensive reference presents the latest techniques for numerical analysis of forming operations. This is the perfect tool for those who wish to investigate new analytical methods for forming.

Advances in computer technology have conveniently coincided with trends in numerical analysis toward increased complexity of computational algorithms based on finite difference methods. It is no longe

The first edition of this book was reviewed in 1982 as 'the most extensive treatment of Padé approximants actually available'. This second edition has been thoroughly updated, with a substantial chapter on multiseries approximants. Applications to statistical mechanics and critical phenomena are extensively covered, and there are extended sections devoted to circuit design, matrix Pade approximation, and computational methods. This succinct and straightforward treatment will appeal to scientists, engineers, and mathematicians alike.

This book, which is published in two volumes, studies heat transfer problems by modern numerical methods. Basic mathematical models of heat transfer are considered. The main approaches, to the analysi

This book, which is published in two volumes, studies heat transfer problems by modern numerical methods. Basic mathematical models of heat transfer are considered. The main approaches to the analysis

In exploring the pattern and methods of Aztec expansion, Ross Hassig focuses on political and economic factors. Because they lacked numerical superiority, faced logistical problems presented by the te

Acta Numerica has established itself as the prime forum for the presentation of definitive reviews of numerical analysis topics. The invited review papers, by leaders in their respective fields, allow researchers and graduate students alike quickly to grasp trends and developments. Highlights of the 1995 issue include articles on sequential quadratic programming, mesh adaption, free boundary problems and particle methods in continuum computations.

Finance is one of the fastest growing areas in the modern banking and corporate world. This, together with the sophistication of modern financial products, provides a rapidly growing impetus for new mathematical models and modern mathematical methods; the area is an expanding source for novel and relevant 'real-world' mathematics. In this book the authors describe the modelling of financial derivative products from an applied mathematician's viewpoint, from modelling through analysis to elementary computation. A unified approach to modelling derivative products as partial differential equations is presented, using numerical solutions where appropriate. Some mathematics is assumed, but clear explanations are provided for material beyond elementary calculus, probability, and algebra. Over 140 exercises are included. This volume will become the standard introduction to this exciting new field for advanced undergraduate students.

This is a graduate text surveying both the theoretical and experimental aspects of chaotic behaviour. Over the course of the past two decades it has been discovered that relatively simple, deterministic, nonlinear mathematical models that describe dynamic phenomena in various physical, chemical, biological and other systems yield solutions which are aperiodic and depend very sensitively on the initial conditions. This phenomenon is known as deterministic chaos. The authors present chaos as a model of many seemingly random processes in nature. Basic notions from the theory of dynamical systems and bifurcation theory, together with the properties of chaotic solutions, are then described and are illustrated by examples. A review of the numerical methods used both in studies of mathematical models and in the interpretation of experimental data is also provided.

This book is a tool for the theoretical and numerical investigation of nonlinear dynamical systems modelled by means of ordinary differential and difference equations. The work is divided into two parts: a book, comprising a theoretical overview of the subject matter and a number of applications; and an integrated software program. Although the emphasis is laid on dynamical systems arising from economic motivation, and the applications are derived from these systems, both the text and the program will also be of use to researchers in other fields of study. The book first discusses the fundamental concepts and methods of chaos theory, and then applies these theoretical results and the facilities provided by the companion software program to models suggested by economic problems.

As an introduction to the subject for beginners or reference for experienced modelers, this volume provides a unified treatment of the models, governing equations and numerical methods, and a review o

Acta Numerica has established itself as the prime forum for the presentation of definitive reviews of numerical analysis topics. The invited papers by leaders in their respective fields allow researchers and graduate students alike quickly to grasp the then current trends and developments of 1994. Highlights of the year's issue are articles on domain decomposition, mesh adaption, pseudospectral methods and neural networks.

This volume is concerned with the understanding and modeling of combustion in high speed flows. The exploration of a dual approach combining asymptotic and numerical methods for the analysis of the

This book introduces the theory of the structural loading on ships and offshore structures caused by wind, waves and currents, and goes on to describe the applications of this theory in terms of real structures. The main topics described are linear-wave induced motions, loads on floating structures, numerical methods for ascertaining wave induced motions and loads, viscous wave loads and damping, station-keeping and water impact and entry. The applications of the theoretical principles are introduced with extensive use of exercises and examples. Applications covered include conventional ships, barges, high speed marine vehicles, semisubmersibles, tension leg platforms, moored or dynamic positioned ships, risers, buoys, fishing nets, jacket structures and gravity platforms.

This book presents statistical methods for analysis of the duration of events. The primary focus is on models for single-spell data, events in which individual agents are observed for a single duration. Some attention is also given to multiple-spell data. The first part of the book covers model specification, including both structural and reduced form models and models with and without neglected heterogeneity. The book next deals with likelihood based inference about such models, with sections on full and semiparametric specification. A final section treats graphical and numerical methods of specification testing. This is the first published exposition of current econometric methods for the study of duration data.

The interacting boson-fermion model has become in recent years the standard model for the description of atomic nuclei with an odd number of protons and/or neutrons. This book describes the mathematical framework on which the interacting boson-fermion model is built and presents applications to a variety of situations encountered in nuclei. The book addresses both the analytical and the numerical aspects of the problem. The analytical aspect requires the introduction of rather complex group theoretic methods, including the use of graded (or super) Lie algebras. The first (and so far only) example of supersymmetry occurring in nature is also discussed. The book is the first comprehensive treatment of the subject and will appeal to both theoretical and experimental physicists. The large number of explicit formulas for level energies, electromagnetic transition rates and intensities of transfer reactions presented in the book provide a simple but detailed way to analyse experimental data.

This introductory text presents the use of statistical methods as an integral part of biological investigation, yet one whose superficial complexities have deterred many biologists from using them. The author argues that the difficulties, such as they are, do not lie in mathematical manipulation, but in grasping a few simple, but unfamiliar concepts. He emphasizes the need for precisely defining problems and for careful selection of the most appropriate methods - a wide range of which are described and illustrated. Each chapter ends with a set of problems which are intended to help the student gain practical experience. No previous knowledge is assumed, and the student is encouraged to develop a competent and critical approach to analysing numerical data. In this second edition, the scope of the book has been extended, problems have been solved in a more satisfactory way, and a greater number of illustrative examples have been added.

Four independent chapters present mechanical models, analytical methods, and numerical techniques for determining equilibria and periodic orbits for a self-excited, stick-slip oscillator; two coupled

Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory. Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory, and Random Systems. This two-volume work provides a comprehensive and timely survey of the application of the methods of quantum field theory to statistical physics, a very active and fruitful area of modern research. The first volume provides a pedagogical introduction to the subject, discussing Brownian motion, its anticommutative counterpart in the guise of Onsager's solution to the two-dimensional Ising model, the mean field or Landau approximation, scaling ideas exemplified by the Kosterlitz-Thouless theory for the XY transition, the continuous renormalization group applied to the standard phi-to the fourth theory (the simplest typical case) and lattice gauge theory as a pathway to the understanding of quark confinement in quantum chromodynamics. The second volume covers more diverse topics, including strong coupling expansions and the

Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory. Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory, and Random Systems. This two-volume work provides a comprehensive and timely survey of the application of the methods of quantum field theory to statistical physics, a very active and fruitful area of modern research. The first volume provides a pedagogical introduction to the subject, discussing Brownian motion, its anticommutative counterpart in the guise of Onsager's solution to the two-dimensional Ising model, the mean field or Landau approximation, scaling ideas exemplified by the Kosterlitz-Thouless theory for the XY transition, the continuous renormalization group applied to the standard phi-to the fourth theory (the simplest typical case) and lattice gauge theory as a pathway to the understanding of quark confinement in quantum chromodynamics. The second volume covers more diverse topics, including strong coupling expansions and the

An introductory undergraduate text demonstrating the use of geostatistical methods in the analysis of earth science data. The fundamental concepts are developed through the use of numerical examples d

First published in 1961 this monograph deals with the analysis of unsteady lift distributions of thin oscillating wings at transonic speeds. Such distributions are needed for the prediction of flutter, which tends to occur more frequently at speeds near that of sound than in any other speed regime. This book emphasises analytical methods that treat the linearised problem for simple wing planforms such as rectangular and delta wings. There is a thorough discussion of the limitations of the validity of linearised theory as well as the proper boundary conditions to be applied for the case of flows with shocks. Although the standard approach in engineering practice is to sense transonic flow calculations with the aid of direct numerical simulations on a high-speed computer, the treatment in this classic monograph is still of value in the physical insight it provides and in the reference 'exact' solutions it supplies for simplified cases that may serve as validation cases for complicated co

This textbook, covering the basic mathematics taught to first-year students of science and engineering, reflects the growing awareness that ancillary mathematics should not be taught in isolation from its applications. Topics covered include calculus, ordinary and partial differential equations and statistics. Each chapter starts with two or three examples setting the new techniques to be studied in the context of the scientific world; the mathematics is then presented, along with worked examples. Numerical methods are integrated with analytical techniques where appropriate. The resulting textbook provides the teacher with a rich and varied source of applications for classroom use and students with a textbook for self-learning, giving insight into the significance and role of mathematics in science and engineering.

The second part of this two-volume set contains advanced aspects of the quantitative theory of the dynamics of neurons. It begins with an introduction to the effects of reversal potentials on response to synaptic input. It then develops the theory of action potential generation based on the seminal Hodgkin-Huxley equations and gives methods for their solution in the space-clamped and non-space-clamped cases. The remainder of the book discusses stochastic models of neural activity and ends with a statistical analysis of neuronal data with emphasis on spike trains. The mathematics is more complex in this volume than in the first volume and involves numerical methods of solution of partial differential equations and the statistical analysis of point processes.

Here is an introductory text on metric spaces that is the first to be written for students who are as interested in the applications as in the theory. Knowledge of metric spaces is fundamental to understanding numerical methods (for example for solving differential equations) as well as analysis, yet most books at this level emphasise just the abstraction and theory. Dr Bryant uses applications to provide motivation and to sustain the development and discusses numerical procedures where appropriate. The reader is expected to have had some exposure to elementary analysis, but the author provides examples throughout to refresh the student's memory and to test and extend understanding. In short, this is an introductory textbook that will appeal to students of mathematics and engineering and will give them the required background for more advanced courses in both analysis and numerical analysis.

This is a version of Gevrey's classical treatise on the heat equations. Included in this volume are discussions of initial and/or boundary value problems, numerical methods, free boundary problems and parameter determination problems. The material is presented as a monograph and/or information source book. After the first six chapters of standard classical material, each chapter is written as a self-contained unit except for an occasional reference to elementary definitions, theorems and lemmas in previous chapters.