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Books in Ordinary differential equations

  • Difference Equations and Applications

    • 1st Edition
    • Youssef N. Raffoul
    • English
    Difference Equations and Applications provides unique coverage of high-level topics in the application of difference equations and dynamical systems. The book begins with extensive coverage of the calculus of difference equations, including contemporary topics on l_p stability, exponential stability, and parameters that can be used to qualitatively study solutions to non-linear difference equations, including variations of parameters and equations with constant coefficients, before moving on to the Z-Transform and its various functions, scalings, and applications. It covers systems, Lyapunov functions, and stability, a subject rarely covered in competitor titles, before concluding with a comprehensive section on new variations of parameters.Exercises are provided after each section, ranging from an easy to medium level of difficulty. When finished, students are set up to conduct meaningful research in discrete dynamical systems. In summary, this book is a comprehensive resource that delves into the mathematical theory of difference equations while highlighting their practical applications in various dynamic systems. It is highly likely to be of interest to students, researchers, and professionals in fields where discrete modeling and analysis are essential.

  • Introductory Differential Equations

    • 6th Edition
    • Martha L. Abell + 1 more
    • English
    **2025 Textbook and Academic Authors Association (TAA) McGuffey Longevity Award Winner**Introductory Differential Equations, Sixth Edition provides the foundations to assist students in learning not only how to read and understand differential equations, but also how to read technical material in more advanced texts as they progress through their studies. The book's accessible explanations and many robust sample problems are appropriate for a first semester course in introductory ordinary differential equations (including Laplace transforms), for a second course in Fourier series and boundary value problems, and for students with no background on the subject.

  • Advanced Differential Equations

    • 1st Edition
    • Youssef N. Raffoul
    • English
    Advanced Differential Equations provides coverage of high-level topics in ordinary differential equations and dynamical systems. The book delivers difficult material in an accessible manner, utilizing easier, friendlier notations and multiple examples. Sections focus on standard topics such as existence and uniqueness for scalar and systems of differential equations, the dynamics of systems, including stability, with examples and an examination of the eigenvalues of an accompanying linear matrix, as well as coverage of existing literature. From the eigenvalues' approach, to coverage of the Lyapunov direct method, this book readily supports the study of stable and unstable manifolds and bifurcations. Additional sections cover the study of delay differential equations, extending from ordinary differential equations through the extension of Lyapunov functions to Lyapunov functionals. In this final section, the text explores fixed point theory, neutral differential equations, and neutral Volterra integro-differential equations.

  • A Modern Introduction to Differential Equations

    • 3rd Edition
    • Henry J. Ricardo
    • English
    A Modern Introduction to Differential Equations, Third Edition, provides an introduction to the basic concepts of differential equations. The book begins by introducing the basic concepts of differential equations, focusing on the analytical, graphical and numerical aspects of first-order equations, including slope fields and phase lines. The comprehensive resource then covers methods of solving second-order homogeneous and nonhomogeneous linear equations with constant coefficients, systems of linear differential equations, the Laplace transform and its applications to the solution of differential equations and systems of differential equations, and systems of nonlinear equations. Throughout the text, valuable pedagogical features support learning and teaching. Each chapter concludes with a summary of important concepts, and figures and tables are provided to help students visualize or summarize concepts. The book also includes examples and updated exercises drawn from biology, chemistry, and economics, as well as from traditional pure mathematics, physics, and engineering.

  • Foundations of Mathematical System Dynamics

    The Fundamental Theory of Causal Recursion and Its Application to Social Science and Economics
    • 1st Edition
    • Volume 2
    • George J. Klir
    • English
    This book is a foundational study of causality as conceived in the mathematical sciences. It is shown that modern mathematical dynamics involves a formulation of the fundamental concept of causality, and an exhaustive classification of causal systems. Among them are the 'self-steering' and 'self-regulating' systems, which together form the class of purposive systems, on whose specific properties the book then focuses. These properties are the mathematical-dynamic... foundations of the behavioural and social sciences. This is the definitive book on causality and purposive processes by the originator of the mathematical concept of self-steering.

  • Analytical Solution Methods for Boundary Value Problems

    • 1st Edition
    • A.S. Yakimov
    • English
    Analytical Solution Methods for Boundary Value Problems is an extensively revised, new English language edition of the original 2011 Russian language work, which provides deep analysis methods and exact solutions for mathematical physicists seeking to model germane linear and nonlinear boundary problems. Current analytical solutions of equations within mathematical physics fail completely to meet boundary conditions of the second and third kind, and are wholly obtained by the defunct theory of series. These solutions are also obtained for linear partial differential equations of the second order. They do not apply to solutions of partial differential equations of the first order and they are incapable of solving nonlinear boundary value problems. Analytical Solution Methods for Boundary Value Problems attempts to resolve this issue, using quasi-linearization methods, operational calculus and spatial variable splitting to identify the exact and approximate analytical solutions of three-dimensional non-linear partial differential equations of the first and second order. The work does so uniquely using all analytical formulas for solving equations of mathematical physics without using the theory of series. Within this work, pertinent solutions of linear and nonlinear boundary problems are stated. On the basis of quasi-linearization, operational calculation and splitting on spatial variables, the exact and approached analytical solutions of the equations are obtained in private derivatives of the first and second order. Conditions of unequivocal resolvability of a nonlinear boundary problem are found and the estimation of speed of convergence of iterative process is given. On an example of trial functions results of comparison of the analytical solution are given which have been obtained on suggested mathematical technology, with the exact solution of boundary problems and with the numerical solutions on well-known methods.

  • Comparison and Oscillation Theory of Linear Differential Equations

    • 1st Edition
    • C. A. Swanson
    • Richard Bellman
    • English
    Mathematics in Science and Engineering, Volume 48: Comparison and Oscillation Theory of Linear Differential Equations deals primarily with the zeros of solutions of linear differential equations. This volume contains five chapters. Chapter 1 focuses on comparison theorems for second order equations, while Chapter 2 treats oscillation and nonoscillation theorems for second order equations. Separation, comparison, and oscillation theorems for fourth order equations are covered in Chapter 3. In Chapter 4, ordinary equations and systems of differential equations are reviewed. The last chapter discusses the result of the first analog of a Sturm-type comparison theorem for an elliptic partial differential equation. This publication is intended for college seniors or beginning graduate students who are well-acquainted with advanced calculus, complex analysis, linear algebra, and linear differential equations.

  • Boundary Value Problems for Systems of Differential, Difference and Fractional Equations

    Positive Solutions
    • 1st Edition
    • Johnny Henderson + 1 more
    • English
    Boundary Value Problems for Systems of Differential, Difference and Fractional Equations: Positive Solutions discusses the concept of a differential equation that brings together a set of additional constraints called the boundary conditions. As boundary value problems arise in several branches of math given the fact that any physical differential equation will have them, this book will provide a timely presentation on the topic. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.

  • Ordinary Differential Equations

    Introduction to the Theory of Ordinary Differential Equations in the Real Domain
    • 1st Edition
    • Volume 13
    • J. Kurzweil
    • English
    The author, Professor Kurzweil, is one of the world's top experts in the area of ordinary differential equations - a fact fully reflected in this book. Unlike many classical texts which concentrate primarily on methods of integration of differential equations, this book pursues a modern approach: the topic is discussed in full generality which, at the same time, permits us to gain a deep insight into the theory and to develop a fruitful intuition. The basic framework of the theory is expanded by considering further important topics like stability, dependence of a solution on a parameter, Carathéodory's theory and differential relations.The book is very well written, and the prerequisites needed are minimal - some basics of analysis and linear algebra. As such, it is accessible to a wide circle of readers, in particular to non-mathematicians.

  • Nonlinear Equations in Abstract Spaces

    Proceedings of an International Symposium on Nonlinear Equations in Abstract Spaces, Held at the University of Texas at Arlington, Arlington, Texas, June 8–10, 1977
    • 1st Edition
    • V. Lakshmikantham
    • English
    Many problems in partial differential equations which arise from physical models can be considered as ordinary differential equations in appropriate infinite dimensional spaces, for which elegant theories and powerful techniques have recently been developed. This book gives a detailed account of the current state of the theory of nonlinear differential equations in a Banach space, and discusses existence theory for differential equations with continuous and discontinuous right-hand sides. Of special importance is the first systematic presentation of the very important and complex theory of multivalued discontinuous differential equations.

  • Numerical Methods for Differential Systems

    Recent Developments in Algorithms, Software, and Applications
    • 1st Edition
    • L. Lapidus + 1 more
    • English
    Numerical Methods for Differential Systems: Recent Developments in Algorithms, Software, and Applications reviews developments in algorithms, software, and applications of numerical methods for differential systems. Topics covered include numerical algorithms for ordinary and partial differential equations (ODE/PDEs); theoretical approaches to the solution of nonlinear algebraic and boundary value problems via associated differential systems; integration algorithms for initial-value ODEs with particular emphasis on stiff systems; finite difference algorithms; and general- and special-purpose computer codes for ODE/PDEs. Comprised of 15 chapters, this book begins with an introduction to high-order A-stable averaging algorithms for stiff differential systems, followed by a discussion on second derivative multistep formulas based on g-splines; numerical integration of linearized stiff ODEs; and numerical solution of large systems of stiff ODEs in a modular simulation framework. Subsequent chapters focus on numerical methods for mass action kinetics; a systematized collection of codes for solving two-point boundary value problems; general software for PDEs; and the choice of algorithms in automated method of lines solution of PDEs. The final chapter is devoted to quality software for ODEs. This monograph should be of interest to mathematicians, chemists, and chemical engineers.

  • Differential Forms

    Theory and Practice
    • 2nd Edition
    • Steven H. Weintraub
    • English
    Differential forms are a powerful mathematical technique to help students, researchers, and engineers solve problems in geometry and analysis, and their applications. They both unify and simplify results in concrete settings, and allow them to be clearly and effectively generalized to more abstract settings. Differential Forms has gained high recognition in the mathematical and scientific community as a powerful computational tool in solving research problems and simplifying very abstract problems. Differential Forms, Second Edition, is a solid resource for students and professionals needing a general understanding of the mathematical theory and to be able to apply that theory into practice.

  • Differential Equations, Dynamical Systems, and an Introduction to Chaos

    • 3rd Edition
    • Morris W. Hirsch + 2 more
    • English
    Hirsch, Devaney, and Smale’s classic Differential Equations, Dynamical Systems, and an Introduction to Chaos has been used by professors as the primary text for undergraduate and graduate level courses covering differential equations. It provides a theoretical approach to dynamical systems and chaos written for a diverse student population among the fields of mathematics, science, and engineering. Prominent experts provide everything students need to know about dynamical systems as students seek to develop sufficient mathematical skills to analyze the types of differential equations that arise in their area of study. The authors provide rigorous exercises and examples clearly and easily by slowly introducing linear systems of differential equations. Calculus is required as specialized advanced topics not usually found in elementary differential equations courses are included, such as exploring the world of discrete dynamical systems and describing chaotic systems.

  • Asymptotic Analysis of Singular Perturbations

    • 1st Edition
    • Volume 9
    • W. Eckhaus
    • English

  • The Inverse Scattering Transformation and The Theory of Solitons

    • 1st Edition
    • Volume 50
    • W. Eckhaus + 1 more
    • English

  • Second Order Linear Differential Equations in Banach Spaces

    • 1st Edition
    • Volume 108
    • H.O. Fattorini
    • English
    Second order linear differential equations in Banach spaces can be used for modelling such second order equations of mathematical physics as the wave equation, the Klein-Gordon equation, et al. In this way, a unified treatment can be given to subjects such as growth of solutions, singular perturbation of parabolic, hyperbolic and Schrödinger type initial value problems, and the like. The book covers in detail these subjects as well as the applications to each specific problem.

  • Spectral Transform and Solitons

    • 1st Edition
    • Volume 13
    • F. Calogero + 1 more
    • English

  • Introductory Differential Equations

    with Boundary Value Problems
    • 3rd Edition
    • Martha L. Abell + 1 more
    • English
    This text is for courses that are typically called (Introductory) Differential Equations, (Introductory) Partial Differential Equations, Applied Mathematics, Fourier Series and Boundary Value Problems. The text is appropriate for two semester courses: the first typically emphasizes ordinary differential equations and their applications while the second emphasizes special techniques (like Laplace transforms) and partial differential equations. The texts follows a "traditional" curriculum and takes the "traditional" (rather than "dynamical systems") approach. Introductory Differential Equations is a text that follows a traditional approach and is appropriate for a first course in ordinary differential equations (including Laplace transforms) and a second course in Fourier series and boundary value problems. Note that some schools might prefer to move the Laplace transform material to the second course, which is why we have placed the chapter on Laplace transforms in its location in the text. Ancillaries like Differential Equations with Mathematica and/or Differential Equations with Maple would be recommended and/or required ancillaries depending on the school, course, or instructor.

  • A Modern Introduction to Differential Equations

    • 2nd Edition
    • Henry J. Ricardo
    • English
    A Modern Introduction to Differential Equations, Second Edition, provides an introduction to the basic concepts of differential equations. The book begins by introducing the basic concepts of differential equations, focusing on the analytical, graphical, and numerical aspects of first-order equations, including slope fields and phase lines. The discussions then cover methods of solving second-order homogeneous and nonhomogeneous linear equations with constant coefficients; systems of linear differential equations; the Laplace transform and its applications to the solution of differential equations and systems of differential equations; and systems of nonlinear equations. Each chapter concludes with a summary of the important concepts in the chapter. Figures and tables are provided within sections to help students visualize or summarize concepts. The book also includes examples and exercises drawn from biology, chemistry, and economics, as well as from traditional pure mathematics, physics, and engineering. This book is designed for undergraduate students majoring in mathematics, the natural sciences, and engineering. However, students in economics, business, and the social sciences with the necessary background will also find the text useful.

  • Handbook of Differential Equations: Ordinary Differential Equations

    • 1st Edition
    • Volume 4
    • Flaviano Battelli + 1 more
    • English
    This handbook is the fourth volume in a series of volumes devoted to self-contained and up-to-date surveys in the theory of ordinary differential equations, with an additional effort to achieve readability for mathematicians and scientists from other related fields so that the chapters have been made accessible to a wider audience.

  • Stochastic Differential Equations and Applications

    • 2nd Edition
    • X Mao
    • English
    This advanced undergraduate and graduate text has now been revised and updated to cover the basic principles and applications of various types of stochastic systems, with much on theory and applications not previously available in book form. The text is also useful as a reference source for pure and applied mathematicians, statisticians and probabilists, engineers in control and communications, and information scientists, physicists and economists.

  • Stochastic Modelling in Process Technology

    • 1st Edition
    • Volume 211
    • Herold G. Dehling + 2 more
    • English
    There is an ever increasing need for modelling complex processes reliably. Computational modelling techniques, such as CFD and MD may be used as tools to study specific systems, but their emergence has not decreased the need for generic, analytical process models. Multiphase and multicomponent systems, and high-intensity processes displaying a highly complex behaviour are becoming omnipresent in the processing industry. This book discusses an elegant, but little-known technique for formulating process models in process technology: stochastic process modelling. The technique is based on computing the probability distribution for a single particle's position in the process vessel, and/or the particle's properties, as a function of time, rather than - as is traditionally done - basing the model on the formulation and solution of differential conservation equations. Using this technique can greatly simplify the formulation of a model, and even make modelling possible for processes so complex that the traditional method is impracticable. Stochastic modelling has sporadically been used in various branches of process technology under various names and guises. This book gives, as the first, an overview of this work, and shows how these techniques are similar in nature, and make use of the same basic mathematical tools and techniques. The book also demonstrates how stochastic modelling may be implemented by describing example cases, and shows how a stochastic model may be formulated for a case, which cannot be described by formulating and solving differential balance equations.

  • Viability, Invariance and Applications

    • 1st Edition
    • Volume 207
    • Ovidiu Carja + 2 more
    • English
    The book is an almost self-contained presentation of the most important concepts and results in viability and invariance. The viability of a set K with respect to a given function (or multi-function) F, defined on it, describes the property that, for each initial data in K, the differential equation (or inclusion) driven by that function or multi-function) to have at least one solution. The invariance of a set K with respect to a function (or multi-function) F, defined on a larger set D, is that property which says that each solution of the differential equation (or inclusion) driven by F and issuing in K remains in K, at least for a short time.The book includes the most important necessary and sufficient conditions for viability starting with Nagumo’s Viability Theorem for ordinary differential equations with continuous right-hand sides and continuing with the corresponding extensions either to differential inclusions or to semilinear or even fully nonlinear evolution equations, systems and inclusions. In the latter (i.e. multi-valued) cases, the results (based on two completely new tangency concepts), all due to the authors, are original and extend significantly, in several directions, their well-known classical counterparts.

  • Handbook of Differential Equations: Ordinary Differential Equations

    • 1st Edition
    • Volume 3
    • A. Canada + 2 more
    • English
    This handbook is the third volume in a series of volumes devoted to self contained and up-to-date surveys in the tehory of ordinary differential equations, written by leading researchers in the area. All contributors have made an additional effort to achieve readability for mathematicians and scientists from other related fields so that the chapters have been made accessible to a wide audience. These ideas faithfully reflect the spirit of this multi-volume and hopefully it becomes a very useful tool for reseach, learing and teaching. This volumes consists of seven chapters covering a variety of problems in ordinary differential equations. Both pure mathematical research and real word applications are reflected by the contributions to this volume.

  • Two-Point Boundary Value Problems: Lower and Upper Solutions

    • 1st Edition
    • Volume 205
    • C. De Coster + 1 more
    • English
    This book introduces the method of lower and upper solutions for ordinary differential equations. This method is known to be both easy and powerful to solve second order boundary value problems. Besides an extensive introduction to the method, the first half of the book describes some recent and more involved results on this subject. These concern the combined use of the method with degree theory, with variational methods and positive operators. The second half of the book concerns applications. This part exemplifies the method and provides the reader with a fairly large introduction to the problematic of boundary value problems. Although the book concerns mainly ordinary differential equations, some attention is given to other settings such as partial differential equations or functional differential equations. A detailed history of the problem is described in the introduction.

  • Theory and Applications of Fractional Differential Equations

    • 1st Edition
    • Volume 204
    • A.A. Kilbas + 2 more
    • English
    This monograph provides the most recent and up-to-date developments on fractional differential and fractional integro-differential equations involving many different potentially useful operators of fractional calculus.The subject of fractional calculus and its applications (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering.Some of the areas of present-day applications of fractional models include Fluid Flow, Solute Transport or Dynamical Processes in Self-Similar and Porous Structures, Diffusive Transport akin to Diffusion, Material Viscoelastic Theory, Electromagnetic Theory, Dynamics of Earthquakes, Control Theory of Dynamical Systems, Optics and Signal Processing, Bio-Sciences, Economics, Geology, Astrophysics, Probability and Statistics, Chemical Physics, and so on.In the above-mentioned areas, there are phenomena with estrange kinetics which have a microscopic complex behaviour, and their macroscopic dynamics can not be characterized by classical derivative models.The fractional modelling is an emergent tool which use fractional differential equations including derivatives of fractional order, that is, we can speak about a derivative of order 1/3, or square root of 2, and so on. Some of such fractional models can have solutions which are non-differentiable but continuous functions, such as Weierstrass type functions. Such kinds of properties are, obviously, impossible for the ordinary models.What are the useful properties of these fractional operators which help in the modelling of so many anomalous processes? From the point of view of the authors and from known experimental results, most of the processes associated with complex systems have non-local dynamics involving long-memory in time, and the fractional integral and fractional derivative operators do have some of those characteristics.This book is written primarily for the graduate students and researchers in many different disciplines in the mathematical, physical, engineering and so many others sciences, who are interested not only in learning about the various mathematical tools and techniques used in the theory and widespread applications of fractional differential equations, but also in further investigations which emerge naturally from (or which are motivated substantially by) the physical situations modelled mathematically in the book.This monograph consists of a total of eight chapters and a very extensive bibliography. The main objective of it is to complement the contents of the other books dedicated to the study and the applications of fractional differential equations. The aim of the book is to present, in a systematic manner, results including the existence and uniqueness of solutions for the Cauchy type problems involving nonlinear ordinary fractional differential equations, explicit solutions of linear differential equations and of the corresponding initial-value problems through different methods, closed-form solutions of ordinary and partial differential equations, and a theory of the so-called sequential linear fractional differential equations including a generalization of the classical Frobenius method, and also to include an interesting set of applications of the developed theory.Key features:- It is mainly application oriented.- It contains a complete theory of Fractional Differential Equations.- It can be used as a postgraduate-level textbook in many different disciplines within science and engineering.- It contains an up-to-date bibliography.- It provides problems and directions for further investigations.- Fractional Modelling is an emergent tool with demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering.- It contains many examples.- and so on!

  • Handbook of Differential Equations: Ordinary Differential Equations

    • 1st Edition
    • Volume 2
    • A. Canada + 2 more
    • English
    This handbook is the second volume in a series devoted to self contained and up-to-date surveys in the theory of ordinary differential equations, writtenby leading researchers in the area. All contributors have made an additional effort to achieve readability for mathematicians and scientists from other related fields, in order to make the chapters of the volume accessible to a wide audience.

  • Half-Linear Differential Equations

    • 1st Edition
    • Volume 202
    • Ondrej Dosly + 1 more
    • English
    The book presents a systematic and compact treatment of the qualitative theory of half-lineardifferent... equations. It contains the most updated and comprehensive material and represents the first attempt to present the results of the rapidly developing theory of half-linear differential equations in a unified form. The main topics covered by the book are oscillation and asymptotic theory and the theory of boundary value problems associated with half-linear equations, but the book also contains a treatment of related topics like PDE’s with p-Laplacian, half-linear difference equations and various more general nonlinear differential equations.

  • Volterra Integral and Differential Equations

    • 2nd Edition
    • Volume 202
    • Ted A. Burton
    • English
    Most mathematicians, engineers, and many other scientists are well-acquainted with theory and application of ordinary differential equations. This book seeks to present Volterra integral and functional differential equations in that same framwork, allowing the readers to parlay their knowledge of ordinary differential equations into theory and application of the more general problems. Thus, the presentation starts slowly with very familiar concepts and shows how these are generalized in a natural way to problems involving a memory. Liapunov's direct method is gently introduced and applied to many particular examples in ordinary differential equations, Volterra integro-differential equations, and functional differential equations. By Chapter 7 the momentum has built until we are looking at problems on the frontier. Chapter 7 is entirely new, dealing with fundamental problems of the resolvent, Floquet theory, and total stability. Chapter 8 presents a solid foundation for the theory of functional differential equations. Many recent results on stability and periodic solutions of functional differential equations are given and unsolved problems are stated.

  • Handbook of Differential Equations: Ordinary Differential Equations

    • 1st Edition
    • Volume 1
    • A. Canada + 2 more
    • English
    The book contains seven survey papers about ordinary differential equations.The common feature of all papers consists in the fact that nonlinear equations are focused on. This reflects the situation in modern mathematical modelling - nonlinear mathematical models are more realistic and describe the real world problems more accurately. The implications are that new methods and approaches have to be looked for, developed and adopted in order to understand and solve nonlinear ordinary differential equations.The purpose of this volume is to inform the mathematical community and also other scientists interested in and using the mathematical apparatus of ordinary differential equations, about some of these methods and possible applications.

  • Nonlinear Theory of Pseudodifferential Equations on a Half-line

    • 1st Edition
    • Volume 194
    • Nakao Hayashi + 1 more
    • English
    This book is the first attempt to develop systematically a general theory of the initial-boundary value problems for nonlinear evolution equations with pseudodifferential operators Ku on a half-line or on a segment. We study traditionally important problems, such as local and global existence of solutions and their properties, in particular much attention is drawn to the asymptotic behavior of solutions for large time. Up to now the theory of nonlinear initial-boundary value problems with a general pseudodifferential operator has not been well developed due to its difficulty. There are many open natural questions. Firstly how many boundary data should we pose on the initial-boundary value problems for its correct solvability? As far as we know there are few results in the case of nonlinear nonlocal equations. The methods developed in this book are applicable to a wide class of dispersive and dissipative nonlinear equations, both local and nonlocal.

  • Differential Equations, Dynamical Systems, and an Introduction to Chaos

    • 2nd Edition
    • Morris W. Hirsch + 2 more
    • English
    Differential Equations, Dynamical Systems, and an Introduction to Chaos, Second Edition, provides a rigorous yet accessible introduction to differential equations and dynamical systems. The original text by three of the world's leading mathematicians has become the standard textbook for graduate courses in this area. Thirty years in the making, this Second Edition brings students to the brink of contemporary research, starting from a background that includes only calculus and elementary linear algebra. The book explores the dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. It presents the simplification of many theorem hypotheses and includes bifurcation theory throughout. It contains many new figures and illustrations; a simplified treatment of linear algebra; detailed discussions of the chaotic behavior in the Lorenz attractor, the Shil'nikov systems, and the double scroll attractor; and increased coverage of discrete dynamical systems. This book will be particularly useful to advanced students and practitioners in higher mathematics.

  • Non-Self-Adjoint Boundary Eigenvalue Problems

    • 1st Edition
    • Volume 192
    • R. Mennicken + 1 more
    • English
    This monograph provides a comprehensive treatment of expansion theorems for regular systems of first order differential equations and n-th order ordinary differential equations.In 10 chapters and one appendix, it provides a comprehensive treatment from abstract foundations to applications in physics and engineering. The focus is on non-self-adjoint problems. Bounded operators are associated to these problems, and Chapter 1 provides an in depth investigation of eigenfunctions and associated functions for bounded Fredholm valued operators in Banach spaces. Since every n-th order differential equation is equivalentto a first order system, the main techniques are developed for systems. Asymptotic fundamentalsystems are derived for a large class of systems of differential equations. Together with boundaryconditions, which may depend polynomially on the eigenvalue parameter, this leads to the definition of Birkhoff and Stone regular eigenvalue problems. An effort is made to make the conditions relatively easy verifiable; this is illustrated with several applications in chapter 10.The contour integral method and estimates of the resolvent are used to prove expansion theorems.For Stone regular problems, not all functions are expandable, and again relatively easy verifiableconditions are given, in terms of auxiliary boundary conditions, for functions to be expandable.Chapter 10 deals exclusively with applications; in nine sections, various concrete problems such asthe Orr-Sommerfeld equation, control of multiple beams, and an example from meteorology are investigated.Key features:• Expansion Theorems for Ordinary Differential Equations• Discusses Applications to Problems from Physics and Engineering• Thorough Investigation of Asymptotic Fundamental Matrices and Systems• Provides a Comprehensive Treatment• Uses the Contour Integral Method• Represents the Problems as Bounded Operators• Investigates Canonical Systems of Eigen- and Associated Vectors for Operator Functions

  • Ordinary Differential Equations and Integral Equations

    • 1st Edition
    • Volume 6
    • C.T.H. Baker + 2 more
    • J.D. Pryce
    • English
    /homepage/sac/cam/na... Set now available at special set price !This volume contains contributions in the area of differential equations and integral equations. Many numerical methods have arisen in response to the need to solve "real-life" problems in applied mathematics, in particular problems that do not have a closed-form solution. Contributions on both initial-value problems and boundary-value problems in ordinary differential equations appear in this volume. Numerical methods for initial-value problems in ordinary differential equations fall naturally into two classes: those which use one starting value at each step (one-step methods) and those which are based on several values of the solution (multistep methods).John Butcher has supplied an expert's perspective of the development of numerical methods for ordinary differential equations in the 20th century. Rob Corless and Lawrence Shampine talk about established technology, namely software for initial-value problems using Runge-Kutta and Rosenbrock methods, with interpolants to fill in the solution between mesh-points, but the 'slant' is new - based on the question, "How should such software integrate into the current generation of Problem Solving Environments?"Natali... Borovykh and Marc Spijker study the problem of establishing upper bounds for the norm of the nth power of square matrices.The dynamical system viewpoint has been of great benefit to ODE theory and numerical methods. Related is the study of chaotic behaviour.Willy Govaerts discusses the numerical methods for the computation and continuation of equilibria and bifurcation points of equilibria of dynamical systems.Arieh Iserles and Antonella Zanna survey the construction of Runge-Kutta methods which preserve algebraic invariant functions.Valeria Antohe and Ian Gladwell present numerical experiments on solving a Hamiltonian system of Hénon and Heiles with a symplectic and a nonsymplectic method with a variety of precisions and initial conditions.Stiff differential equations first became recognized as special during the 1950s. In 1963 two seminal publications laid to the foundations for later development: Dahlquist's paper on A-stable multistep methods and Butcher's first paper on implicit Runge-Kutta methods.Ernst Hairer and Gerhard Wanner deliver a survey which retraces the discovery of the order stars as well as the principal achievements obtained by that theory.Guido Vanden Berghe, Hans De Meyer, Marnix Van Daele and Tanja Van Hecke construct exponentially fitted Runge-Kutta methods with s stages.Differential-... equations arise in control, in modelling of mechanical systems and in many other fields.Jeff Cash describes a fairly recent class of formulae for the numerical solution of initial-value problems for stiff and differential-algebra... systems.Shengtai Li and Linda Petzold describe methods and software for sensitivity analysis of solutions of DAE initial-value problems.Again in the area of differential-algebra... systems, Neil Biehn, John Betts, Stephen Campbell and William Huffman present current work on mesh adaptation for DAE two-point boundary-value problems.Contrasting approaches to the question of how good an approximation is as a solution of a given equation involve (i) attempting to estimate the actual error (i.e., the difference between the true and the approximate solutions) and (ii) attempting to estimate the defect - the amount by which the approximation fails to satisfy the given equation and any side-conditions.The paper by Wayne Enright on defect control relates to carefully analyzed techniques that have been proposed both for ordinary differential equations and for delay differential equations in which an attempt is made to control an estimate of the size of the defect.Many phenomena incorporate noise, and the numerical solution of stochastic differential equations has developed as a relatively new item of study in the area.Keven Burrage, Pamela Burrage and Taketomo Mitsui review the way numerical methods for solving stochastic differential equations (SDE's) are constructed.One of the more recent areas to attract scrutiny has been the area of differential equations with after-effect (retarded, delay, or neutral delay differential equations) and in this volume we include a number of papers on evolutionary problems in this area.The paper of Genna Bocharov and Fathalla Rihan conveys the importance in mathematical biology of models using retarded differential equations.The contribution by Christopher Baker is intended to convey much of the background necessary for the application of numerical methods and includes some original results on stability and on the solution of approximating equations.Alfredo Bellen, Nicola Guglielmi and Marino Zennaro contribute to the analysis of stability of numerical solutions of nonlinear neutral differential equations.Koen Engelborghs, Tatyana Luzyanina, Dirk Roose, Neville Ford and Volker Wulf consider the numerics of bifurcation in delay differential equations.Evelyn Buckwar contributes a paper indicating the construction and analysis of a numerical strategy for stochastic delay differential equations (SDDEs).This volume contains contributions on both Volterra and Fredholm-type integral equations.Christophe... Baker responded to a late challenge to craft a review of the theory of the basic numerics of Volterra integral and integro-differential equations.Simon Shaw and John Whiteman discuss Galerkin methods for a type of Volterra integral equation that arises in modelling viscoelasticity.A subclass of boundary-value problems for ordinary differential equation comprises eigenvalue problems such as Sturm-Liouville problems (SLP) and Schrödinger equations.Liviu Ixaru describes the advances made over the last three decades in the field of piecewise perturbation methods for the numerical solution of Sturm-Liouville problems in general and systems of Schrödinger equations in particular.Alan Andrew surveys the asymptotic correction method for regular Sturm-Liouville problems.Leon Greenberg and Marco Marletta survey methods for higher-order Sturm-Liouville problems.R. Moore in the 1960s first showed the feasibility of validated solutions of differential equations, that is, of computing guaranteed enclosures of solutions.Boundary integral equations. Numerical solution of integral equations associated with boundary-value problems has experienced continuing interest.Peter Junghanns and Bernd Silbermann present a selection of modern results concerning the numerical analysis of one-dimensional Cauchy singular integral equations, in particular the stability of operator sequences associated with different projection methods.Johannes Elschner and Ivan Graham summarize the most important results achieved in the last years about the numerical solution of one-dimensional integral equations of Mellin type of means of projection methods and, in particular, by collocation methods.A survey of results on quadrature methods for solving boundary integral equations is presented by Andreas Rathsfeld.Wolfgang Hackbusch and Boris Khoromski present a novel approach for a very efficient treatment of integral operators.Ernst Stephan examines multilevel methods for the h-, p- and hp- versions of the boundary element method, including pre-conditioning techniques.George Hsiao, Olaf Steinbach and Wolfgang Wendland analyze various boundary element methods employed in local discretization schemes.

  • Power Geometry in Algebraic and Differential Equations

    • 1st Edition
    • Volume 57
    • A.D. Bruno
    • English
    The geometry of power exponents includes the Newton polyhedron, normal cones of its faces, power and logarithmic transformations. On the basis of the geometry universal algorithms for simplifications of systems of nonlinear equations (algebraic, ordinary differential and partial differential) were developed.The algorithms form a new calculus which allows to make local and asymptotical analysis of solutions to those systems.The efficiency of the calculus is demonstrated with regard to several complicated problems from Robotics, Celestial Mechanics, Hydrodynamics and Thermodynamics. The calculus also gives classical results obtained earlier intuitively and is an alternative to Algebraic Geometry, Differential Algebra, Lie group Analysis and Nonstandard Analysis.

  • Augmented Lagrangian Methods

    Applications to the Numerical Solution of Boundary-Value Problems
    • 1st Edition
    • Volume 15
    • M. Fortin + 1 more
    • English
    The purpose of this volume is to present the principles of the Augmented Lagrangian Method, together with numerous applications of this method to the numerical solution of boundary-value problems for partial differential equations or inequalities arising in Mathematical Physics, in the Mechanics of Continuous Media and in the Engineering Sciences.

  • Differential Equations with Maple V

    • 2nd Edition
    • Martha L. Abell + 1 more
    • English
    This book is an indespensable tool for anyone using Maple V in computing ordinary and partial differential equations.

  • Ordinary Differential Equations and Applications

    Mathematical Methods for Applied Mathematicians, Physicists, Engineers and Bioscientists
    • 1st Edition
    • W S Weiglhofer + 1 more
    • English
    This introductory text presents ordinary differential equations with a modern approach to mathematical modelling in a one semester module of 20–25 lectures.

  • Handbook of Differential Equations (CD-ROM Version 1 only)

    • 3rd Edition
    • Daniel Zwillinger
    • English
    This book and CD-ROM compile the most widely applicable methods for solving and approximating differential equations. The CD-ROM provides convenient access to these methods through electronic search capabilities, andtogether the book and CD-ROM contain numerous examples showing the methods use. Topics include ordinary differential equations, symplectic integration of differential equations, and the use of wavelets when numerically solving differential equations.

  • Handbook of Differential Equations

    • 3rd Edition
    • Daniel Zwillinger
    • English
    Handbook of Differential Equations, Third Edition compiles the most widely applicable methods for solving and approximating differential equations, also providing numerous examples that show methods being used. Topics include ordinary differential equations, symplectic integration of differential equations, and the use of wavelets when numerically solving differential equations.

  • Ordinary Differential Equations

    • 1st Edition
    • William Cox
    • English
    Building on introductory calculus courses, this text provides a sound foundation in the underlying principles of ordinary differential equations. Important concepts, including uniqueness and existence theorems, are worked through in detail and the student is encouraged to develop much of the routine material themselves, thus helping to ensure a solid understanding of the fundamentals required.The wide use of exercises, problems and self-assessment questions helps to promote a deeper understanding of the material and it is developed in such a way that it lays the groundwork for further study of partial differential equations.

  • Calculus and Ordinary Differential Equations

    • 1st Edition
    • David Pearson
    • English
    Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving to gain the confidence of students.

  • Delay Differential Equations

    With Applications in Population Dynamics
    • 1st Edition
    • Volume 191
    • Yang Kuang
    • English
    Delay Differential Equations emphasizes the global analysis of full nonlinear equations or systems. The book treats both autonomous and nonautonomous systems with various delays. Key topics addressed are the possible delay influence on the dynamics of the system, such as stability switching as time delay increases, the long time coexistence of populations, and the oscillatory aspects of the dynamics. The book also includes coverage of the interplay of spatial diffusion and time delays in some diffusive delay population models. The treatment presented in this monograph will be of great value in the study of various classes of DDEs and their multidisciplinary applications.

  • Symmetries and Laplacians

    Introduction to Harmonic Analysis, Group Representations and Applications
    • 1st Edition
    • Volume 174
    • D. Gurarie
    • English
    Designed as an introduction to harmonic analysis and group representations,this book covers a wide range of topics rather than delving deeply into anyparticular one. In the words of H. Weyl ...it is primarily meant forthe humble, who want to learn as new the things set forth therein, rather thanfor the proud and learned who are already familiar with the subject and merelylook for quick and exact information....The main objective is tointroduce the reader to concepts, ideas, results and techniques that evolvearound symmetry-groups, representations and Laplacians. Morespecifically, the main interest concerns geometrical objects and structures{X}, discrete or continuous, that possess sufficiently large symmetrygroup G, such as regular graphs (Platonic solids), lattices, andsymmetric Riemannian manifolds. All such objects have a natural Laplacian&Dgr;, a linear operator on functions over X, invariant underthe group action. There are many problems associated with Laplacians onX, such as continuous or discrete-time evolutions, on X,random walks, diffusion processes, and wave-propagation. This book containssufficient material for a 1 or 2-semester course.

  • Singular Perturbations I

    Spaces and Singular Perturbations on Manifolds Without Boundary
    • 1st Edition
    • Volume 23
    • L.S. Frank
    • English
    Singular perturbations, one of the central topics in asymptotic analysis, also play a special role in describing physical phenomena such as the propagation of waves in media in the presence of small energy dissipations or dispersions, the appearance of boundary or interior layers in fluid and gas dynamics, as well as in elasticity theory, semi-classical asymptotic approximations in quantum mechanics etc. Elliptic and coercive singular perturbations are of special interest for the asymptotic solution of problems which are characterized by boundary layer phenomena, e.g. the theory of thin buckling plates, elastic rods and beams. This first volume deals with linear singular perturbations (on smooth manifolds without boundary) considered as equicontinuous linear mappings between corresponding families of Sobolev-Slobodetski'... type spaces of vectorial order.

  • Scattering Theory for Hyperbolic Operators

    • 1st Edition
    • Volume 21
    • V. Petkov
    • English
    Scattering Theory for dissipative and time-dependent systems has been intensively studied in the last fifteen years. The results in this field, based on various tools and techniques, may be found in many published papers.This monograph presents an approach which can be applied to spaces of both even and odd dimension. The ideas on which the approach is based are connected with the RAGE type theorem, with Enss' decomposition of the phase space and with a time-dependent proof of the existence of the operator W which exploits the decay of the local energy of the perturbed and free systems. Some inverse scattering problems for time-dependent potentials, and moving obstacles with an arbitrary geometry, are also treated in the book.

  • Difference Schemes

    An Introduction to the Underlying Theory
    • 1st Edition
    • Volume 19
    • S.K. Godunov + 1 more
    • English
    Much applied and theoretical research in natural sciences leads to boundary-value problems stated in terms of differential equations. When solving these problems with computers, the differential problems are replaced approximately by difference schemes.This book is an introduction to the theory of difference schemes, and was written as a textbook for university mathematics and physics departments and for technical universities. Some sections of the book will be of interest to computations specialists.While stressing a mathematically rigorous treatment of model problems, the book also demonstrates the relation between theory and computer experiments, using difference schemes created for practical computations.

  • Singularities & Dynamical Systems

    • 1st Edition
    • Volume 103
    • S.N. Pnevmatikos
    • English
    This volume is an account of the lectures delivered at the international Conference ``Singularities and Dynamical Systems-83''. The main purpose of the Conference was to create conditions of scientific contact between mathematicians and physicists who have singularities and dynamical systems as common interests.

  • Coupled Nonlinear Oscillators

    • 1st Edition
    • Volume .
    • J. Chandra + 1 more
    • English

  • Stochastic Differential Equations and Diffusion Processes

    • 1st Edition
    • Volume 24
    • S. Watanabe + 1 more
    • English