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The aim of this series is to provide a focus for publishing textbooks in applied mathematics at the advanced undergraduate and beginning graduate level. It is planned that the books will be devoted to covering certain mathematical techniques and theories and exploring their applications. The main audience for the series will be in departments of applied mathematics, engineering science or physics. It is important to recognize that for students, clear and careful exposition and a sound pedagogic presentation are frequently more relevant than surveying and synthesising recent developments. Authors will be encouraged to supply plenty of worked examples and exercises, of varying degrees of difficulty, for the reader. Books in the series should provide a solid understanding of how a given method can usefully be applied to help solve problems in physics and engineering.

  • Editorial Boards: M. J. Ablowitz, University of Colorado, Boulder, S. H. Davis, Northwestern University, Illinois , E. J. Hinch, University of Cambridge, A. Iserles, University of Cambridge, J. Ockendon, University of Oxford, P. J. Olver, University of Minnesota

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50 results in Cambridge Texts in Applied Mathematics

Page 1 of 3

Introduction to Complex Variables and Applications
  • Mark J. Ablowitz, Athanassios S. Fokas
  • Published online:
    18 August 2021
    Print publication:
    25 March 2021
  • The study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based on the authors' extensive teaching experience, it covers topics of keen interest to these students, including ordinary differential equations, as well as Fourier and Laplace transform methods for solving partial differential equations arising in physical applications. Many worked examples, applications, and exercises are included. With this foundation, students can progress beyond the standard course and explore a range of additional topics, including generalized Cauchy theorem, Painlevé equations, computational methods, and conformal mapping with circular arcs. Advanced topics are labeled with an asterisk and can be included in the syllabus or form the basis for challenging student projects.

The Fluid Dynamics of Cell Motility
  • Eric Lauga
  • Published online:
    09 September 2020
    Print publication:
    05 November 2020
  • Fluid dynamics plays a crucial role in many cellular processes, including the locomotion of cells such as bacteria and spermatozoa. These organisms possess flagella, slender organelles whose time periodic motion in a fluid environment gives rise to motility. Sitting at the intersection of applied mathematics, physics and biology, the fluid dynamics of cell motility is one of the most successful applications of mathematical tools to the understanding of the biological world. Based on courses taught over several years, it details the mathematical modelling necessary to understand cell motility in fluids, covering phenomena ranging from single-cell motion to instabilities in cell populations. Each chapter introduces mathematical models to rationalise experiments, uses physical intuition to interpret mathematical results, highlights the history of the field and discusses notable current research questions. All mathematical derivations are included for students new to the field, and end-of-chapter exercises help consolidate understanding and practise applying the concepts.

Think Before You Compute
  • A Prelude to Computational Fluid Dynamics
  • E. J. Hinch
  • Published online:
    28 April 2020
    Print publication:
    30 April 2020
  • Every fluid dynamicist will at some point need to use computation. Thinking about the physics, constraints and the requirements early on will be rewarded with benefits in time, effort, accuracy and expense. How these benefits can be realised is illustrated in this guide for would-be researchers and beginning graduate students to some of the standard methods and common pitfalls of computational fluid mechanics. Based on a lecture course that the author has developed over twenty years, the text is split into three parts. The quick introduction enables students to solve numerically a basic nonlinear problem by a simple method in just three hours. The follow-up part expands on all the key essentials, including discretisation (finite differences, finite elements and spectral methods), time-stepping and linear algebra. The final part is a selection of optional advanced topics, including hyperbolic equations, the representation of surfaces, the boundary integral method, the multigrid method, domain decomposition, the fast multipole method, particle methods and wavelets.

Stochastic Modelling of Reaction–Diffusion Processes
  • Radek Erban, S. Jonathan Chapman
  • Published online:
    04 November 2019
    Print publication:
    30 January 2020
  • This practical introduction to stochastic reaction-diffusion modelling is based on courses taught at the University of Oxford. The authors discuss the essence of mathematical methods which appear (under different names) in a number of interdisciplinary scientific fields bridging mathematics and computations with biology and chemistry. The book can be used both for self-study and as a supporting text for advanced undergraduate or beginning graduate-level courses in applied mathematics. New mathematical approaches are explained using simple examples of biological models, which range in size from simulations of small biomolecules to groups of animals. The book starts with stochastic modelling of chemical reactions, introducing stochastic simulation algorithms and mathematical methods for analysis of stochastic models. Different stochastic spatio-temporal models are then studied, including models of diffusion and stochastic reaction-diffusion modelling. The methods covered include molecular dynamics, Brownian dynamics, velocity jump processes and compartment-based (lattice-based) models.

Self-Exciting Fluid Dynamos
  • Keith Moffatt, Emmanuel Dormy
  • Published online:
    13 May 2019
    Print publication:
    25 April 2019
  • Exploring the origins and evolution of magnetic fields in planets, stars and galaxies, this book gives a basic introduction to magnetohydrodynamics and surveys the observational data, with particular focus on geomagnetism and solar magnetism. Pioneering laboratory experiments that seek to replicate particular aspects of fluid dynamo action are also described. The authors provide a complete treatment of laminar dynamo theory, and of the mean-field electrodynamics that incorporates the effects of random waves and turbulence. Both dynamo theory and its counterpart, the theory of magnetic relaxation, are covered. Topological constraints associated with conservation of magnetic helicity are thoroughly explored and major challenges are addressed in areas such as fast-dynamo theory, accretion-disc dynamo theory and the theory of magnetostrophic turbulence. The book is aimed at graduate-level students in mathematics, physics, Earth sciences and astrophysics, and will be a valuable resource for researchers at all levels.

Geometric and Topological Inference
  • Jean-Daniel Boissonnat, Frédéric Chazal, Mariette Yvinec
  • Published online:
    14 September 2018
    Print publication:
    27 September 2018
  • Geometric and topological inference deals with the retrieval of information about a geometric object using only a finite set of possibly noisy sample points. It has connections to manifold learning and provides the mathematical and algorithmic foundations of the rapidly evolving field of topological data analysis. Building on a rigorous treatment of simplicial complexes and distance functions, this self-contained book covers key aspects of the field, from data representation and combinatorial questions to manifold reconstruction and persistent homology. It can serve as a textbook for graduate students or researchers in mathematics, computer science and engineering interested in a geometric approach to data science.

Microhydrodynamics, Brownian Motion, and Complex Fluids
  • Michael D. Graham
  • Published online:
    27 August 2018
    Print publication:
    13 September 2018
  • This is an introduction to the dynamics of fluids at small scales, the physical and mathematical underpinnings of Brownian motion, and the application of these subjects to the dynamics and flow of complex fluids such as colloidal suspensions and polymer solutions. It brings together continuum mechanics, statistical mechanics, polymer and colloid science, and various branches of applied mathematics, in a self-contained and integrated treatment that provides a foundation for understanding complex fluids, with a strong emphasis on fluid dynamics. Students and researchers will find that this book is extensively cross-referenced to illustrate connections between different aspects of the field. Its focus on fundamental principles and theoretical approaches provides the necessary groundwork for research in the dynamics of flowing complex fluids.

Numerical Linear Algebra
  • An Introduction
  • Holger Wendland
  • Published online:
    20 November 2017
    Print publication:
    16 November 2017
  • This self-contained introduction to numerical linear algebra provides a comprehensive, yet concise, overview of the subject. It includes standard material such as direct methods for solving linear systems and least-squares problems, error, stability and conditioning, basic iterative methods and the calculation of eigenvalues. Later chapters cover more advanced material, such as Krylov subspace methods, multigrid methods, domain decomposition methods, multipole expansions, hierarchical matrices and compressed sensing. The book provides rigorous mathematical proofs throughout, and gives algorithms in general-purpose language-independent form. Requiring only a solid knowledge in linear algebra and basic analysis, this book will be useful for applied mathematicians, engineers, computer scientists, and all those interested in efficiently solving linear problems.

Introduction to Magnetohydrodynamics
  • 2nd edition
  • P. A. Davidson
  • Published online:
    09 February 2017
    Print publication:
    22 December 2016
  • 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.

Discrete Systems and Integrability
  • J. Hietarinta, N. Joshi, F. W. Nijhoff
  • Published online:
    05 September 2016
    Print publication:
    19 August 2016
  • 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 Bäcklund transformations; difference equations and special functions; multidimensional consistency of integrable lattice equations; associated linear problems (Lax pairs); connections with Padé approximants and convergence algorithms; singularities and geometry; Hirota's bilinear formalism for lattices; intriguing properties of discrete Painlevé 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.

Singularities: Formation, Structure, and Propagation
  • J. Eggers, M. A. Fontelos
  • Published online:
    05 September 2015
    Print publication:
    10 September 2015
  • Many key phenomena in physics and engineering are described as singularities in the solutions to the differential equations describing them. Examples covered thoroughly in this book include the formation of drops and bubbles, the propagation of a crack and the formation of a shock in a gas. Aimed at a broad audience, this book provides the mathematical tools for understanding singularities and explains the many common features in their mathematical structure. Part I introduces the main concepts and techniques, using the most elementary mathematics possible so that it can be followed by readers with only a general background in differential equations. Parts II and III require more specialised methods of partial differential equations, complex analysis and asymptotic techniques. The book may be used for advanced fluid mechanics courses and as a complement to a general course on applied partial differential equations.

An Introduction to Polynomial and Semi-Algebraic Optimization
  • Jean Bernard Lasserre
  • Published online:
    05 February 2015
    Print publication:
    19 February 2015
  • This is the first comprehensive introduction to the powerful moment approach for solving global optimization problems (and some related problems) described by polynomials (and even semi-algebraic functions). In particular, the author explains how to use relatively recent results from real algebraic geometry to provide a systematic numerical scheme for computing the optimal value and global minimizers. Indeed, among other things, powerful positivity certificates from real algebraic geometry allow one to define an appropriate hierarchy of semidefinite (SOS) relaxations or LP relaxations whose optimal values converge to the global minimum. Several extensions to related optimization problems are also described. Graduate students, engineers and researchers entering the field can use this book to understand, experiment with and master this new approach through the simple worked examples provided.

An Introduction to Computational Stochastic PDEs
  • Gabriel J. Lord, Catherine E. Powell, Tony Shardlow
  • Published online:
    05 July 2014
    Print publication:
    11 August 2014
  • This book gives a comprehensive introduction to numerical methods and analysis of stochastic processes, random fields and stochastic differential equations, and offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis. Coverage includes traditional stochastic ODEs with white noise forcing, strong and weak approximation, and the multi-level Monte Carlo method. Later chapters apply the theory of random fields to the numerical solution of elliptic PDEs with correlated random data, discuss the Monte Carlo method, and introduce stochastic Galerkin finite-element methods. Finally, stochastic parabolic PDEs are developed. Assuming little previous exposure to probability and statistics, theory is developed in tandem with state-of-the-art computational methods through worked examples, exercises, theorems and proofs. The set of MATLAB® codes included (and downloadable) allows readers to perform computations themselves and solve the test problems discussed. Practical examples are drawn from finance, mathematical biology, neuroscience, fluid flow modelling and materials science.

Flow, Deformation and Fracture
  • Lectures on Fluid Mechanics and the Mechanics of Deformable Solids for Mathematicians and Physicists
  • Grigory Isaakovich Barenblatt
  • Published online:
    05 June 2014
    Print publication:
    05 June 2014
  • Over forty years of teaching experience are distilled into this text. The guiding principle is the wide use of the concept of intermediate asymptotics, which enables the natural introduction of the modeling of real bodies by continua. Beginning with a detailed explanation of the continuum approximation for the mathematical modeling of the motion and equilibrium of real bodies, the author continues with a general survey of the necessary methods and tools for analyzing models. Next, specific idealized approximations are presented, including ideal incompressible fluids, elastic bodies and Newtonian viscous fluids. The author not only presents general concepts but also devotes chapters to examining significant problems, including turbulence, wave-propagation, defects and cracks, fatigue and fracture. Each of these applications reveals essential information about the particular approximation. The author's tried and tested approach reveals insights that will be valued by every teacher and student of mechanics.

The Mathematics of Signal Processing
  • Steven B. Damelin, Willard Miller, Jr
  • Published online:
    05 June 2012
    Print publication:
    15 December 2011
  • Arising from courses taught by the authors, this largely self-contained treatment is ideal for mathematicians who are interested in applications or for students from applied fields who want to understand the mathematics behind their subject. Early chapters cover Fourier analysis, functional analysis, probability and linear algebra, all of which have been chosen to prepare the reader for the applications to come. The book includes rigorous proofs of core results in compressive sensing and wavelet convergence. Fundamental is the treatment of the linear system y=Φx in both finite and infinite dimensions. There are three possibilities: the system is determined, overdetermined or underdetermined, each with different aspects. The authors assume only basic familiarity with advanced calculus, linear algebra and matrix theory and modest familiarity with signal processing, so the book is accessible to students from the advanced undergraduate level. Many exercises are also included.

A Physical Introduction to Suspension Dynamics
  • Élisabeth Guazzelli, Jeffrey F. Morris
  • Illustrated by Sylvie Pic
  • Published online:
    05 June 2012
    Print publication:
    24 November 2011
  • Understanding the behaviour of particles suspended in a fluid has many important applications across a range of fields, including engineering and geophysics. Comprising two main parts, this book begins with the well-developed theory of particles in viscous fluids, i.e. microhydrodynamics, particularly for single- and pair-body dynamics. Part II considers many-body dynamics, covering shear flows and sedimentation, bulk flow properties and collective phenomena. An interlude between the two parts provides the basic statistical techniques needed to employ the results of the first (microscopic) in the second (macroscopic). The authors introduce theoretical, mathematical concepts through concrete examples, making the material accessible to non-mathematicians. They also include some of the many open questions in the field to encourage further study. Consequently, this is an ideal introduction for students and researchers from other disciplines who are approaching suspension dynamics for the first time.

Nonlinear Dispersive Waves
  • Asymptotic Analysis and Solitons
  • Mark J. Ablowitz
  • Published online:
    05 June 2012
    Print publication:
    08 September 2011
  • The field of nonlinear dispersive waves has developed enormously since the work of Stokes, Boussinesq and Korteweg–de Vries (KdV) in the nineteenth century. In the 1960s, researchers developed effective asymptotic methods for deriving nonlinear wave equations, such as the KdV equation, governing a broad class of physical phenomena that admit special solutions including those commonly known as solitons. This book describes the underlying approximation techniques and methods for finding solutions to these and other equations. The concepts and methods covered include wave dispersion, asymptotic analysis, perturbation theory, the method of multiple scales, deep and shallow water waves, nonlinear optics including fiber optic communications, mode-locked lasers and dispersion-managed wave phenomena. Most chapters feature exercise sets, making the book suitable for advanced courses or for self-directed learning. Graduate students and researchers will find this an excellent entry to a thriving area at the intersection of applied mathematics, engineering and physical science.

Hydrodynamic Instabilities
  • François Charru
  • Translated by Patricia de Forcrand-Millard
  • Published online:
    05 August 2011
    Print publication:
    30 June 2011
  • The instability of fluid flows is a key topic in classical fluid mechanics because it has huge repercussions for applied disciplines such as chemical engineering, hydraulics, aeronautics, and geophysics. This modern introduction is written for any student, researcher, or practitioner working in the area, for whom an understanding of hydrodynamic instabilities is essential. Based on a decade's experience of teaching postgraduate students in fluid dynamics, this book brings the subject to life by emphasizing the physical mechanisms involved. The theory of dynamical systems provides the basic structure of the exposition, together with asymptotic methods. Wherever possible, Charru discusses the phenomena in terms of characteristic scales and dimensional analysis. The book includes numerous experimental studies, with references to videos and multimedia material, as well as over 150 exercises which introduce the reader to new problems.

Iterative Methods in Combinatorial Optimization
  • Lap Chi Lau, R. Ravi, Mohit Singh
  • Published online:
    05 June 2012
    Print publication:
    18 April 2011
  • With the advent of approximation algorithms for NP-hard combinatorial optimization problems, several techniques from exact optimization such as the primal-dual method have proven their staying power and versatility. This book describes a simple and powerful method that is iterative in essence and similarly useful in a variety of settings for exact and approximate optimization. The authors highlight the commonality and uses of this method to prove a variety of classical polyhedral results on matchings, trees, matroids and flows. The presentation style is elementary enough to be accessible to anyone with exposure to basic linear algebra and graph theory, making the book suitable for introductory courses in combinatorial optimization at the upper undergraduate and beginning graduate levels. Discussions of advanced applications illustrate their potential for future application in research in approximation algorithms.

Applied Solid Mechanics
  • Peter Howell, Gregory Kozyreff, John Ockendon
  • Published online:
    02 February 2010
    Print publication:
    11 December 2008
  • The world around us, natural or man-made, is built and held together by solid materials. Understanding their behaviour is the task of solid mechanics, which is in turn applied to many areas, from earthquake mechanics to industry, construction to biomechanics. The variety of materials (metals, rocks, glasses, sand, flesh and bone) and their properties (porosity, viscosity, elasticity, plasticity) is reflected by the concepts and techniques needed to understand them: a rich mixture of mathematics, physics and experiment. These are all combined in this unique book, based on years of experience in research and teaching. Starting from the simplest situations, models of increasing sophistication are derived and applied. The emphasis is on problem-solving and building intuition, rather than a technical presentation of theory. The text is complemented by over 100 carefully-chosen exercises, making this an ideal companion for students taking advanced courses, or those undertaking research in this or related disciplines.

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