7 edition of Introduction to the theory and applications of functional differential equations found in the catalog.
Includes bibliographical references (p. 601-641) and index.
|Statement||by V. Kolmanovskii and A. Myshkis.|
|Series||Mathematics and its applications ;, v. 463, Mathematics and its applications (Kluwer Academic Publishers) ;, v. 463.|
|Contributions||Myshkis, A. D.|
|LC Classifications||QA372 .K774 1999|
|The Physical Object|
|Pagination||xvi, 648 p. :|
|Number of Pages||648|
|LC Control Number||98049064|
Ordinary differential equations[ edit ] Main articles: Ordinary differential equation and Linear differential equation An ordinary differential equation ODE is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. These results are applied to the Navier-Stokes equations, to a stability problem in fluid mechanics, to showing how a classical solution can be obtained, and to the Chafee-Infante problem as an example of a gradient system. Topics include first order equations and the method of characteristics, second order linear equations, wave and heat equations, Laplace and Poisson equations, and separation of variables. Example[ edit ] In classical mechanicsthe motion of a body is described by its position and velocity as the time value varies. These are the only real prerequisites for the whole book.
Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. For completeness, clarity, and rigor, Katok and Hasselblatt is without equal. Knot theorists will appreciate the discussion on kneading theory. The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. Perko treat this connection more thoroughly.
An equation containing only first derivatives is a first-order differential equation, an equation containing the second derivative is a second-order differential equation, and so on. By blending programming, mathematics and scientific applications, the book lays a solid foundation for practicing computational science. See our disclaimer An accessible yet rigorous introduction to partial differential equations This textbook provides beginning graduate students and advanced undergraduates with an accessible introduction to the rich subject of partial differential equations PDEs. Sturm-Liouville operators are nice examples of self-adjoint operators with compact resolvent and are reused in Chapter 4 as generators of strongly continuous semigroups.
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It presents a rigorous and clear explanation of the more elementary theoretical aspects of PDEs, while also drawing connections to deeper analysis and applications. In preparation for studies of nonlinear evolution equations, the invariant subspaces associated with the semigroups and the inhomogeneous problem are also examined.
The very important Smale horseshoe map and the construction of Markov partitions are discussed. Some elementary examples of dynamical systems are given in the first chapter, including definitions of the more important concepts such as topological transitivity and gradient flows.
Topics include first order equations and the method of characteristics, second order linear equations, wave and heat equations, Laplace and Poisson equations, and separation of variables. Provides an accessible yet rigorous introduction to partial differential equations Draws connections to advanced topics in analysis Covers applications to continuum mechanics An electronic solutions manual is available only to professors An online illustration package is available to professors "synopsis" may belong to another edition of this title.
See our disclaimer An accessible yet rigorous introduction to partial differential equations This textbook provides beginning graduate students and advanced undergraduates with an accessible introduction to the rich subject of partial differential equations PDEs.
Rachel Levy is associate professor of mathematics at Harvey Mudd College. With regard to the writing style, all formal statements, like Theorems, contain all assumptions except for those declared at the beginning of the section in which the statement appears.
Partial differential equations[ edit ] Main article: Partial differential equation A partial differential equation PDE is a differential equation that contains unknown multivariable functions and their partial derivatives. The methods include study of stable and unstable manifolds, bifurcations, index and degree, and construction of orbits as minima and minimaxes of action functionals.
There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. This is achieved by exploring hyperbolicity, transversality, global topological invariants, and variational methods.
About this title This textbook provides beginning graduate students and advanced undergraduates with an accessible introduction to the rich subject of partial differential equations PDEs. High school mathematics is a required background and it is advantageous to study classical and numerical one-variable calculus in parallel with reading this book.
The exposition is example and problem-oriented, where the applications are taken from mathematics, numerical calculus, statistics, physics, biology and finance. After an introductory chapter defining distributions and the operations that apply to them, Chapter 2 considers the calculus of distributions, especially limits, differentiation, integrations, and the interchange of limiting processes.
Chow and D. Most ODEs that are encountered in physics are linear, and, therefore, most special functions may be defined as solutions of linear differential equations see Holonomic function.
Any feedback would be very appreciated. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. Features A detailed and rigorous development of the theory of Sobolev spaces on Lipschitz domains, including the trace operator and the normal component of vector fields An integration of functional analysis concepts involving Hilbert spaces and the problems which can be solved with these concepts, rather than separating the two Introduction to the analytical tools needed for physical problems of interest like time-harmonic waves, Stokes and Darcy flow, surface differential equations, Maxwell cavity problems, etc.
The Cauchy Theorem is also used in a couple of places, with the most difficult version used in 4. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory cf.
Some deeper properties of distributions, such as their local character as derivatives of continuous functions, are given in Chapter 3. Superior illustrations accompany important concepts, and the anecdotes and examples throughout the book will keep students interested.
The main theme of the second part is the interplay between local analysis near individual e. Based on a graduate course given at the State University of New York at Stony Brook, this book has two objectives: to provide a comparatively elementary introduction to distribution theory and to describe the generalized Fourier and Laplace transformations and their applications to integrodifferential equations, difference equations, and passive systems.
The shadowing results have been misused in the literature, with many false statements about their applicability. It introduces and rigorously develops the central concepts and methods in dynamical systems in a hands-on and highly insightful fashion.
Chapter 12 is completely new and contains a guide to active topics of re- search. He is a fellow of the American Mathematical Society.
Other books e. The book also covers fundamental solutions, Green's functions and distributions, beginning functional analysis applied to elliptic PDEs, traveling wave solutions of selected parabolic PDEs, and scalar conservation laws and systems of hyperbolic PDEs.Aug 01, · This invaluable book is devoted to a rapidly developing area on the research of the qualitative theory of fractional differential equations.
It is self-contained and unified in presentation, and provides readers the necessary background material required to go further into the subject and explore.
The present book builds upon an earlier work of J. Hale, "Theory of Func tional Differential Equations" published in We have tried to maintain the spirit of that book and have retained approximately one-third of the material intact. One major change was a complete new presentation of lin ear systems (Chapters 6~9) for retarded and neutral functional differential equations.5/5(1).
Dec 06, · This textbook is addressed to graduate students in mathematics or other disciplines who wish to understand the essential concepts of functional analysis and their applications to partial differential equations. The book is intentionally concise, presenting all the fundamental concepts and results but omitting the more specialized topics.
Distribution theory, a relatively recent mathematical approach to classical Fourier analysis, not only opened up new areas of research but also helped promote the development of such mathematical disciplines as ordinary and partial differential equations, operational calculus, transformation theory, and functional analysis.
This book is an introduction to partial differential equations (PDEs) and the relevant functional analysis tools which they require. It is based on a course which has been taught at Michigan State University for a. Apr 05, · KENNETH L.
COOKE, in International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, 1 Introduction. Though differential-difference equations were encountered by such early analysts as Euler , and Poisson , a systematic development of the theory of such equations was not begun until E.
Schmidt published an important paper  about fifty years ago.