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Complex variables provide powerful methods for attacking problems that can be very difficult to solve in any other way, and it is the aim of this book to provide a thorough grounding in these methods and their application. Part I of this text provides an introduction to the subject, including analytic functions, integration, series, and residue calculus and also includes transform methods, ODEs in the complex plane, and numerical methods. Part II contains conformal mappings, asymptotic expansions, and the study of Riemann-Hilbert problems. The authors provide an extensive array of applications, illustrative examples and homework exercises. This 2003 edition was improved throughout and is ideal for use in undergraduate and introductory graduate level courses in complex variables.

A wide range of mathematical tools and ideas are drawn together in the study of nonlinear equations, and the results applied to diverse and countless problems in all the natural and social sciences.

A textbook presenting the theory and underlying techniques of perturbation methods in a manner suitable for senior undergraduates from a broad range of disciplines.

This book follows the macroscopic, phenomenological approach which proposes equations abstracted from generally accepted experimental facts, studies the adequacy of the consequences drawn from these equations to those facts and then provides useful tools for designers and engineers.

The book aims to tackle the solution of integral equations using a blend of abstract 'structural' results and more direct, down-to-earth mathematics.

This introduction to the mathematics of waves is for undergraduates in mathematics, physics or engineering. Further material on linear and nonlinear waves is also included for the benefit of graduates. The context and underlying physics is clearly explained; worked examples and exercises are supplied throughout, with solutions available to teachers.

Wave propagation and scattering are often very complex processes. One way to begin to understand them is to study wave propagation in the linear approximation. This is a book describing such propagation using, as a context, the equations of elasticity. Linear Elastic Waves is an advanced level textbook directed at applied mathematicians, seismologists, and engineers.

This is the first book to present a unified treatment of the mixing of fluids from a kinematical viewpoint.

By providing an introduction to nonlinear differential equations, Dr Glendinning aims to equip the student with the mathematical know-how needed to appreciate stability theory and bifurcations. His approach is readable and covers material both old and new to undergraduate courses. Included are treatments of the Poincare-Bendixson theorem, the Hopf bifurcation and chaotic systems. The unique treatment that is found in this book will prove to be an essential guide to stability and chaos.

A Prelude to Computational Fluid Dynamics. Mathematics, Fluid dynamics and solid mechanics, Thermal-fluids engineering

Many natural phenomena are described as singularities, for example, the formation of drops and bubbles, or the motion of cracks. Aimed at a broad audience of students and researchers in mathematics, physics and engineering, this book provides mathematical tools for understanding all aspects of singularities.

This first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant. While treating the material at an elementary level, the book also highlights many recent developments. Topics include: Darboux and Backlund transformations; difference equations and special functions; multidimensional consistency of integrable lattice equations; associated linear problems (Lax pairs); connections with Pade approximants and convergence algorithms; singularities and geometry; Hirota's bilinear formalism for lattices; intriguing properties of discrete Painleve equations; and the novel theory of Lagrangian multiforms. The book builds the material in an organic way, emphasizing interconnections between the various approaches, while the exposition is mostly done through explicit computations on key examples. Written by respected experts in the field, the numerous exercises and the thorough list of references will benefit upper-level undergraduate, and beginning graduate students as well as researchers from other disciplines.

This 2003 book describes and teaches the art of discovering scaling laws, starting from dimensional analysis and physical similarity. It demonstrates the concepts of intermediate asymptotics and the renormalisation group as natural consequences of self-similarity and shows how and when these notions and tools can be applied, and when they cannot.

This practical introduction covers mathematical methods for the analysis of stochastic models and their biological applications. Based on courses taught at the University of Oxford, the book can be used for self-study or as a supporting text for advanced undergraduate or beginning graduate-level courses in applied mathematics.

A pedagogical review of the mathematical modelling in fluid dynamics necessary to understand the motility of most microorganisms on Earth.

This book, first published in 2002, contains an introduction to hyperbolic partial differential equations and a powerful class of numerical methods for approximating their solution, including both linear problems and nonlinear conservation laws. This provides an excellent learning environment for understanding wave propagation phenomena and finite volume methods.

The theory of water waves has been a source of intriguing mathematical problems for at least 150 years. This text considers the classical problems in linear and non-linear water-wave theory, as well as more modern aspects - problems that give rise to soliton-type equations. Lastly it examines the effects of viscosity.

This text is for a one-semester introductory graduate course for students of operations research, mathematics, and computer science covers linear and integer programming, polytopes, matroids and matroid optimization, shortest paths, and network flows. The author focuses on the key mathematical ideas that lead to useful models and algorithms.

Used either by upper-undergraduate students, or as extra reading for any applied mathematician, this book illustrates how the reader's knowledge can be used to describe the world around them. Topics include distributions, asymptotic methods and the basics of modelling. Applications range from piano tuning to egg incubation and traffic flow.

This book begins from a non-traditional exposition of dimensional analysis, physical similarity theory and general theory of scaling phenomena, using classical examples to demonstrate that the onset of scaling is not until the influence of initial and/or boundary conditions has disappeared but when the system is still far from equilibrium.

Instability of flows and their transition to turbulence are widespread phenomena in engineering and nature, and are also important in many applied sciences. This is a textbook to introduce these phenomena at a level suitable for a graduate course, by modelling them mathematically, and describing numerical simulations and laboratory experiments.

Magnetohydrodynamics (MHD) plays a crucial role in astrophysics, planetary magnetism, engineering and controlled nuclear fusion. This comprehensive textbook emphasizes physical ideas, rather than mathematical detail, making it accessible to a broad audience. Starting from elementary chapters on fluid mechanics and electromagnetism, it takes the reader all the way through to the latest ideas in more advanced topics, including planetary dynamos, stellar magnetism, fusion plasmas and engineering applications. With the new edition, readers will benefit from additional material on MHD instabilities, planetary dynamos and applications in astrophysics, as well as a whole new chapter on fusion plasma MHD. The development of the material from first principles and its pedagogical style makes this an ideal companion for both undergraduate students and postgraduate students in physics, applied mathematics and engineering. Elementary knowledge of vector calculus is the only prerequisite.

The book is an introduction to theory of sound generation by fluid flow, specially written for a one semester course at advanced undergraduate or graduate level. Problems are provided at the end of each chapter, many of which can be used for extended student projects. A whole chapter is devoted to worked examples.

The aim of this book is to present the concepts, methods and applications of kinetic theory to rarefied gas dynamics. Each section is accompanied by problems which are intended to demonstrate the use of the material in the text. For graduate courses in aerospace engineering or applied mathematics.

This book explores the profound connections between a ubiquitous class of physically important waves known as solitons and the theory of transformations of a privileged class of surfaces. Punctuated with exercises, it is suitable for use in higher undergraduate or graduate level courses directed at applied mathematicians or mathematical physics.

The authors consider models of different 'acoustic' media as well as equations and behaviour of finite-amplitude waves. This book will be of interest not only to specialists in acoustics, but also to a wide audience of mathematicians, physicists, and engineers working on nonlinear waves in various physical systems.