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- Mathematics
- 2012-12-06 2012-12-06

## Complete Second Order Linear Differential Equations in Hilbert Spaces

**Author**: Alexander Ya. Shklyar**Publisher:** Birkhäuser**ISBN:** 3034891873**Category : **Mathematics**Languages : **en**Pages : **225

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**Book Description**

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.

## Complete Second Order Linear Differential Equations in Hilbert Spaces

**Author**: Alexander Ya. Shklyar**Publisher:** Birkhäuser**ISBN:** 3034891873**Category : **Mathematics**Languages : **en**Pages : **225

Get Book Here

**Book Description**

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.

## Second Order Partial Differential Equations in Hilbert Spaces

**Author**: Giuseppe Da Prato**Publisher:** Cambridge University Press**ISBN:** 1139433431**Category : **Mathematics**Languages : **en**Pages : **397

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**Book Description**

State of the art treatment of a subject which has applications in mathematical physics, biology and finance. Includes discussion of applications to control theory. There are numerous notes and references that point to further reading. Coverage of some essential background material helps to make the book self contained.

## The Cauchy Problem for Higher Order Abstract Differential Equations

**Author**: Ti-Jun Xiao**Publisher:** Springer**ISBN:** 3540494790**Category : **Mathematics**Languages : **en**Pages : **314

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**Book Description**

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.

## Second Order Linear Differential Equations in Banach Spaces

**Author**: H.O. Fattorini**Publisher:** Elsevier**ISBN:** 9780080872193**Category : **Mathematics**Languages : **en**Pages : **313

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**Book Description**

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.

## Operator Theory and Its Applications

**Author**: Alexander G. Ramm**Publisher:** American Mathematical Soc.**ISBN:** 0821819909**Category : **Mathematics**Languages : **en**Pages : **594

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**Book Description**

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.

## Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces

**Author**: Behzad Djafari Rouhani**Publisher:** CRC Press**ISBN:** 0429528884**Category : **Mathematics**Languages : **en**Pages : **131

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**Book Description**

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.

## Introduction to Partial Differential Equations and Hilbert Space Methods

**Author**: Karl E. Gustafson**Publisher:** Courier Corporation**ISBN:** 0486140873**Category : **Mathematics**Languages : **en**Pages : **500

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**Book Description**

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.

## Functional Analysis in China

**Author**: Bingren Li**Publisher:** Springer Science & Business Media**ISBN:** 9400901852**Category : **Mathematics**Languages : **en**Pages : **391

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**Book Description**

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.

## Tests for Correctness and Weak Correctness of the Couchy Problem for Complete Second Order Linear Differential Equations in Hilbert Spaces

**Author**: Aleksandr Jakovlevič Škljar**Publisher:** **ISBN:** **Category : ****Languages : **en**Pages : **33

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**Book Description**

## Hilbert Space, Boundary Value Problems, and Orthogonal Polynomials

**Author**: Allan M. Krall**Publisher:** Springer Science & Business Media**ISBN:** 9783764367015**Category : **Mathematics**Languages : **en**Pages : **374

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**Book Description**

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.