Books in Ordinary differential equations

Half-Linear Differential Equations

1st Edition
Volume 202
July 6, 2005
Ondrej Dosly + 1 more
English
eBook
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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
April 1, 2005
Ted A. Burton
English
eBook
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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
August 1, 2004
A. Canada + 2 more
English
Hardback
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eBook
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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.

Differential Equations, Dynamical Systems, and an Introduction to Chaos

2nd Edition
October 22, 2003
Morris W. Hirsch + 2 more
English
eBook
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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
June 26, 2003
R. Mennicken + 1 more
English
eBook
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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
June 20, 2001
C.T.H. Baker + 2 more
J.D. Pryce
English
eBook
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/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
August 3, 2000
A.D. Bruno
English
eBook
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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

1st Edition
Volume 15
April 1, 2000
M. Fortin + 1 more
English
eBook
9 7 8 0 0 8 0 8 7 5 3 6 1

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

1st Edition
Volume 92
April 1, 2000
I.W. Knowles + 1 more
English
eBook
9 7 8 0 0 8 0 8 7 2 0 3 2

This volume forms a record of the lectures given at this International Conference. Under the general heading of the equations of mathematical physics, contributions are included on a broad range of topics in the theory and applications of ordinary and partial differential equations, including both linear and non-linear equations. The topics cover a wide variety of methods (spectral, theoretical, variational, topological, semi-group), and a equally wide variety of equations including the Laplace equation, Navier-Stokes equations, Boltzmann's equation, reaction-diffusion equations, Schroedinger equations and certain non-linear wave equations. A number of papers are devoted to multi-particle scattering theory, and to inverse theory. In addition, many of the plenary lectures contain a significant amount of survey material on a wide variety of these topics.

Ordinary Differential Equations and Applications

1st Edition
June 1, 1999
W S Weiglhofer + 1 more
English
Paperback
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eBook
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This introductory text presents ordinary differential equations with a modern approach to mathematical modelling in a one semester module of 20–25 lectures.

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