Author: E. A. Coddington
Publisher:
ISBN: 9783662212622
Category :
Languages : en
Pages : 240

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Book Description

Author: E. A. Coddington
Publisher: Springer
ISBN: 354038670X
Category : Mathematics
Languages : en
Pages : 230

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Book Description
It is well known that two hermitian n x n matrices K, H, where H is positive definite, H> 0, can be simultaneously diagonalized. The key to the proof is to consider C[superscript]n, where C is the complex number field, as a Hilbert space [Fraktur capital]H [subscript]H with the inner product given by (f, g) = g*Hf, where f, g [lowercase Greek]Epsilon C[superscript]n, considered as a space of column vectors. Then the operator A = H−1K is selfadjoint in [Fraktur capital]H [subscript]H, and the spectral theorem readily yields the result. Of course such A, when K is not hermitian, can also be investigated in [Fraktur capital]H [subscript]H. We consider a similar problem where K, H are replaced by a pair of ordinary differential expressions L and M, where M> 0 in some sense. Two difficulties arise: (1) there are many natural choices for a selfadjoint H> 0 generated by M, and hence many choices for [Fraktur capital]H [subscript]H, and (2), once a choice for H has been made, there are many choices for the analogue of A. In our work we consider all possible choices for H> 0 and the analogue of A.

Author: Alan Coddington
Publisher:
ISBN: 9780387107066
Category :
Languages : en
Pages :

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Author: Earl A. Coddington
Publisher:
ISBN:
Category : Boundary value problems
Languages : en
Pages : 38

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Book Description

Author: Johnny Henderson
Publisher: Academic Press
ISBN: 0128036796
Category : Mathematics
Languages : en
Pages : 323

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Book Description
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. Explains the systems of second order and higher orders differential equations with integral and multi-point boundary conditions Discusses second order difference equations with multi-point boundary conditions Introduces Riemann-Liouville fractional differential equations with uncoupled and coupled integral boundary conditions

Author: Uri M. Ascher
Publisher: SIAM
ISBN: 9781611971231
Category : Mathematics
Languages : en
Pages : 620

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Book Description
This book is the most comprehensive, up-to-date account of the popular numerical methods for solving boundary value problems in ordinary differential equations. It aims at a thorough understanding of the field by giving an in-depth analysis of the numerical methods by using decoupling principles. Numerous exercises and real-world examples are used throughout to demonstrate the methods and the theory. Although first published in 1988, this republication remains the most comprehensive theoretical coverage of the subject matter, not available elsewhere in one volume. Many problems, arising in a wide variety of application areas, give rise to mathematical models which form boundary value problems for ordinary differential equations. These problems rarely have a closed form solution, and computer simulation is typically used to obtain their approximate solution. This book discusses methods to carry out such computer simulations in a robust, efficient, and reliable manner.

Author: Jussi Behrndt
Publisher: Springer Nature
ISBN: 3030367142
Category : Mathematics
Languages : en
Pages : 772

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Book Description
This open access book presents a comprehensive survey of modern operator techniques for boundary value problems and spectral theory, employing abstract boundary mappings and Weyl functions. It includes self-contained treatments of the extension theory of symmetric operators and relations, spectral characterizations of selfadjoint operators in terms of the analytic properties of Weyl functions, form methods for semibounded operators, and functional analytic models for reproducing kernel Hilbert spaces. Further, it illustrates these abstract methods for various applications, including Sturm-Liouville operators, canonical systems of differential equations, and multidimensional Schrödinger operators, where the abstract Weyl function appears as either the classical Titchmarsh-Weyl coefficient or the Dirichlet-to-Neumann map. The book is a valuable reference text for researchers in the areas of differential equations, functional analysis, mathematical physics, and system theory. Moreover, thanks to its detailed exposition of the theory, it is also accessible and useful for advanced students and researchers in other branches of natural sciences and engineering.

Author: A.K. Aziz
Publisher: Academic Press
ISBN: 1483267997
Category : Mathematics
Languages : en
Pages : 380

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Book Description
Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations covers the proceedings of the 1974 Symposium by the same title, held at the University of Maryland, Baltimore Country Campus. This symposium aims to bring together a number of numerical analysis involved in research in both theoretical and practical aspects of this field. This text is organized into three parts encompassing 15 chapters. Part I reviews the initial and boundary value problems. Part II explores a large number of important results of both theoretical and practical nature of the field, including discussions of the smooth and local interpolant with small K-th derivative, the occurrence and solution of boundary value reaction systems, the posteriori error estimates, and boundary problem solvers for first order systems based on deferred corrections. Part III highlights the practical applications of the boundary value problems, specifically a high-order finite-difference method for the solution of two-point boundary-value problems on a uniform mesh. This book will prove useful to mathematicians, engineers, and physicists.

Author: F. Zanolin
Publisher: Springer
ISBN: 3709126800
Category : Technology & Engineering
Languages : en
Pages : 214

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Book Description
The area covered by this volume represents a broad choice of some interesting research topics in the field of dynamical systems and applications of nonlinear analysis to ordinary and partial differential equations. The contributed papers, written by well known specialists, make this volume a useful tool both for the experts (who can find recent and new results) and for those who are interested in starting a research work in one of these topics (who can find some updated and carefully presented papers on the state of the art of the corresponding subject).

Author: Mark A. Pinsky
Publisher: American Mathematical Soc.
ISBN: 0821868896
Category : Mathematics
Languages : en
Pages : 545

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Book Description
Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems--rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate throughout the text. The notions of steady-state solution of closely related stationary solutions are developed for the heat equation; applications to the study of heat flow in the earth are presented. The problem of the vibrating string is studied in detail both in the Fourier transform setting and from the viewpoint of the explicit representation (d'Alembert formula). Additional chapters include the numerical analysis of solutions and the method of Green's functions for solutions of partial differential equations. The exposition also includes asymptotic methods (Laplace transform and stationary phase). With more than 200 working examples and 700 exercises (more than 450 with answers), the book is suitable for an undergraduate course in partial differential equations.