Download or read book Complete Second Order Linear Differential Equations in Hilbert Spaces written by Alexander Ya. Shklyar and published by Birkhäuser. This book was released on 2012-12-06 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: Incomplete second order linear differential equations in Banach spaces as well as first order equations have become a classical part of functional analysis. This monograph is an attempt to present a unified systematic theory of second order equations y" (t) + Ay' (t) + By (t) = 0 including well-posedness of the Cauchy problem as well as the Dirichlet and Neumann problems. Exhaustive yet clear answers to all posed questions are given. Special emphasis is placed on new surprising effects arising for complete second order equations which do not take place for first order and incomplete second order equations. For this purpose, some new results in the spectral theory of pairs of operators and the boundary behavior of integral transforms have been developed. The book serves as a self-contained introductory course and a reference book on this subject for undergraduate and post- graduate students and research mathematicians in analysis. Moreover, users will welcome having a comprehensive study of the equations at hand, and it gives insight into the theory of complete second order linear differential equations in a general context - a theory which is far from being fully understood.

Download or read book Second Order Partial Differential Equations in Hilbert Spaces written by Giuseppe Da Prato and published by Cambridge University Press. This book was released on 2002-07-25 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: Second order linear parabolic and elliptic equations arise frequently in mathematics and other disciplines. For example parabolic equations are to be found in statistical mechanics and solid state theory, their infinite dimensional counterparts are important in fluid mechanics, mathematical finance and population biology, whereas nonlinear parabolic equations arise in control theory. Here the authors present a state of the art treatment of the subject from a new perspective. The main tools used are probability measures in Hilbert and Banach spaces and stochastic evolution equations. There is then a discussion of how the results in the book can be applied to control theory. This area is developing very rapidly and there are numerous notes and references that point the reader to more specialised results not covered in the book. Coverage of some essential background material will help make the book self-contained and increase its appeal to those entering the subject.

Download or read book The Cauchy Problem for Higher Order Abstract Differential Equations written by Ti-Jun Xiao and published by Springer. This book was released on 2013-12-11 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main purpose of this book is to present the basic theory and some recent de velopments concerning the Cauchy problem for higher order abstract differential equations u(n)(t) + ~ AiU(i)(t) = 0, t ~ 0, { U(k)(O) = Uk, 0 ~ k ~ n-l. where AQ, Ab . . . , A - are linear operators in a topological vector space E. n 1 Many problems in nature can be modeled as (ACP ). For example, many n initial value or initial-boundary value problems for partial differential equations, stemmed from mechanics, physics, engineering, control theory, etc. , can be trans lated into this form by regarding the partial differential operators in the space variables as operators Ai (0 ~ i ~ n - 1) in some function space E and letting the boundary conditions (if any) be absorbed into the definition of the space E or of the domain of Ai (this idea of treating initial value or initial-boundary value problems was discovered independently by E. Hille and K. Yosida in the forties). The theory of (ACP ) is closely connected with many other branches of n mathematics. Therefore, the study of (ACPn) is important for both theoretical investigations and practical applications. Over the past half a century, (ACP ) has been studied extensively.

Download or read book Second Order Linear Differential Equations in Banach Spaces written by H.O. Fattorini and published by Elsevier. This book was released on 2011-08-18 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Operator Theory and Its Applications written by Alexander G. Ramm and published by American Mathematical Soc.. This book was released on 2000 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: Together with the papers on the abstract operator theory are many papers on the theory of differential operators, boundary value problems, inverse scattering and other inverse problems, and on applications to biology, chemistry, wave propagation, and many other areas."--BOOK JACKET.

Download or read book Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces written by Behzad Djafari Rouhani and published by CRC Press. This book was released on 2019-05-20 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the study of non-linear evolution and difference equations of first or second order governed by maximal monotone operator. This class of abstract evolution equations contains ordinary differential equations, as well as the unification of some important partial differential equations including heat equation, wave equation, Schrodinger equation, etc. The book contains a collection of the authors' work and applications in this field, as well as those of other authors.

Download or read book Introduction to Partial Differential Equations and Hilbert Space Methods written by Karl E. Gustafson and published by Courier Corporation. This book was released on 2012-04-26 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt: Easy-to-use text examines principal method of solving partial differential equations, 1st-order systems, computation methods, and much more. Over 600 exercises, with answers for many. Ideal for a 1-semester or full-year course.

Download or read book Introduction To Second Order Partial Differential Equations An Classical And Variational Solutions written by Doina Cioranescu and published by World Scientific Publishing Company. This book was released on 2017-11-27 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book extensively introduces classical and variational partial differential equations (PDEs) to graduate and post-graduate students in Mathematics. The topics, even the most delicate, are presented in a detailed way. The book consists of two parts which focus on second order linear PDEs. Part I gives an overview of classical PDEs, that is, equations which admit strong solutions, verifying the equations pointwise. Classical solutions of the Laplace, heat, and wave equations are provided. Part II deals with variational PDEs, where weak (variational) solutions are considered. They are defined by variational formulations of the equations, based on Sobolev spaces. A comprehensive and detailed presentation of these spaces is given. Examples of variational elliptic, parabolic, and hyperbolic problems with different boundary conditions are discussed.

Download or read book Functional Analysis in China written by Bingren Li and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt: Functional Analysis has become one of the main branches in Chinese mathematics. Many outstanding contributions and results have been achieved over the past sixty years. This authoritative collection is complementary to Western studies in this field, and seeks to summarise and introduce the historical progress of the development of Functional Analysis in China from the 1940s to the present. A broad range of topics is covered, such as nonlinear functional analysis, linear operator theory, theory of operator algebras, applications including the solvability of some partial differential equations, and special spaces that contain Banach spaces and topological vector spaces. Some of these papers have made a significant impact on the mathematical community worldwide. Audience: This volume will be of interest to mathematicians, physicists and engineers at postgraduate level.

Download or read book Hilbert Space Boundary Value Problems and Orthogonal Polynomials written by Allan M. Krall and published by Springer Science & Business Media. This book was released on 2002 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph consists of three parts: - the abstract theory of Hilbert spaces, leading up to the spectral theory of unbounded self-adjoined operators; - the application to linear Hamiltonian systems, giving the details of the spectral resolution; - further applications such as to orthogonal polynomials and Sobolev differential operators. Written in textbook style this up-to-date volume is geared towards graduate and postgraduate students and researchers interested in boundary value problems of linear differential equations or in orthogonal polynomials.

Download or read book Tests for Correctness and Weak Correctness of the Couchy Problem for Complete Second Order Linear Differential Equations in Hilbert Spaces written by Aleksandr Jakovlevič Škljar and published by . This book was released on 1993 with total page 33 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Linear Second Order Elliptic Operators written by Julian Lopez-gomez and published by World Scientific Publishing Company. This book was released on 2013-04-24 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main goal of the book is to provide a comprehensive and self-contained proof of the, relatively recent, theorem of characterization of the strong maximum principle due to Molina-Meyer and the author, published in Diff. Int. Eqns. in 1994, which was later refined by Amann and the author in a paper published in J. of Diff. Eqns. in 1998. Besides this characterization has been shown to be a pivotal result for the development of the modern theory of spatially heterogeneous nonlinear elliptic and parabolic problems; it has allowed us to update the classical theory on the maximum and minimum principles by providing with some extremely sharp refinements of the classical results of Hopf and Protter-Weinberger. By a celebrated result of Berestycki, Nirenberg and Varadhan, Comm. Pure Appl. Maths. in 1994, the characterization theorem is partially true under no regularity constraints on the support domain for Dirichlet boundary conditions.Instead of encyclopedic generality, this book pays special attention to completeness, clarity and transparency of its exposition so that it can be taught even at an advanced undergraduate level. Adopting this perspective, it is a textbook; however, it is simultaneously a research monograph about the maximum principle, as it brings together for the first time in the form of a book, the most paradigmatic classical results together with a series of recent fundamental results scattered in a number of independent papers by the author of this book and his collaborators.Chapters 3, 4, and 5 can be delivered as a classical undergraduate, or graduate, course in Hilbert space techniques for linear second order elliptic operators, and Chaps. 1 and 2 complete the classical results on the minimum principle covered by the paradigmatic textbook of Protter and Weinberger by incorporating some recent classification theorems of supersolutions by Walter, 1989, and the author, 2003. Consequently, these five chapters can be taught at an undergraduate, or graduate, level. Chapters 6 and 7 study the celebrated theorem of Krein-Rutman and infer from it the characterizations of the strong maximum principle of Molina-Meyer and Amann, in collaboration with the author, which have been incorporated to a textbook by the first time here, as well as the results of Chaps. 8 and 9, polishing some recent joint work of Cano-Casanova with the author. Consequently, the second half of the book consists of a more specialized monograph on the maximum principle and the underlying principal eigenvalues.

Download or read book Differential Equations Fourier Series and Hilbert Spaces written by Raffaele Chiappinelli and published by Walter de Gruyter GmbH & Co KG. This book was released on 2023-09-18 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended to be used as a rather informal, and surely not complete, textbook on the subjects indicated in the title. It collects my Lecture Notes held during three academic years at the University of Siena for a one semester course on "Basic Mathematical Physics", and is organized as a short presentation of few important points on the arguments indicated in the title. It aims at completing the students' basic knowledge on Ordinary Differential Equations (ODE) - dealing in particular with those of higher order - and at providing an elementary presentation of the Partial Differential Equations (PDE) of Mathematical Physics, by means of the classical methods of separation of variables and Fourier series. For a reasonable and consistent discussion of the latter argument, some elementary results on Hilbert spaces and series expansion in othonormal vectors are treated with some detail in Chapter 2. Prerequisites for a satisfactory reading of the present Notes are not only a course of Calculus for functions of one or several variables, but also a course in Mathematical Analysis where - among others - some basic knowledge of the topology of normed spaces is supposed to be included. For the reader's convenience some notions in this context are explicitly recalled here and there, and in particular as an Appendix in Section 1.4. An excellent reference for this general background material is W. Rudin's classic Principles of Mathematical Analysis. On the other hand, a complete discussion of the results on ODE and PDE that are here just sketched are to be found in other books, specifically and more deeply devoted to these subjects, some of which are listed in the Bibliography. In conclusion and in brief, my hope is that the present Notes can serve as a second quick reading on the theme of ODE, and as a first introductory reading on Fourier series, Hilbert spaces, and PDE

Download or read book Two Point Boundary Value Problems Lower and Upper Solutions written by C. De Coster and published by Elsevier. This book was released on 2006-03-21 with total page 502 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.· Presents the fundamental features of the method· Construction of lower and upper solutions in problems· Working applications and illustrated theorems by examples· Description of the history of the method and Bibliographical notes

Download or read book Second Order Partial Differential Equations in Hilbert Spaces London Mathematical Society Lecture Note Series written by Giuseppe Da Prato and published by . This book was released on 2002 with total page 397 pages. Available in PDF, EPUB and Kindle. Book excerpt: Second order linear parabolic and elliptic equations arise frequently in mathematics and other disciplines. For example parabolic equations are to be found in statistical mechanics and solid state theory, their infinite dimensional counterparts are important in fluid mechanics, mathematical finance and population biology, whereas nonlinear parabolic equations arise in control theory. Here the authors present a state of the art treatment of the subject from a new perspective. The main tools used are probability measures in Hilbert and Banach spaces and stochastic evolution equations. There is t.

Download or read book Tests for Correctness and Weak Correctness of the Cauchy Problem for Complete Second Order Linear Differential Equations in Hilbert Spaces written by A. Ya Shklyar and published by . This book was released on 1993 with total page 33 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Playing Around Resonance written by Alessandro Fonda and published by Birkhäuser. This book was released on 2016-11-11 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an up-to-date description of the methods needed to face the existence of solutions to some nonlinear boundary value problems. All important and interesting aspects of the theory of periodic solutions of ordinary differential equations related to the physical and mathematical question of resonance are treated. The author has chosen as a model example the periodic problem for a second order scalar differential equation. In a paedagogical style the author takes the reader step by step from the basics to the most advanced existence results in the field.