The Cauchy Problem

Author : Ti-Jun Xiao,Jin Liang
Publisher : Springer Science & Business Media
Page : 324 pages
File Size : 45,6 Mb
Release : 1998-11-18
Category : Mathematics
ISBN : 3540652388

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The Cauchy Problem for Higher Order Abstract Differential Equations by Ti-Jun Xiao,Jin Liang Pdf

This monograph is the first systematic exposition of the theory of the Cauchy problem for higher order abstract linear differential equations, which covers all the main aspects of the developed theory. The main results are complete with detailed proofs and established recently, containing the corresponding theorems for first and incomplete second order cases and therefore for operator semigroups and cosine functions. They will find applications in many fields. The special power of treating the higher order problems directly is demonstrated, as well as that of the vector-valued Laplace transforms in dealing with operator differential equations and operator families. The reader is expected to have a knowledge of complex and functional analysis.

Author : Irina V. Melnikova,Alexei Filinkov
Publisher : CRC Press
Page : 264 pages
File Size : 42,8 Mb
Release : 2001-03-27
Category : Mathematics
ISBN : 9781420035490

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Abstract Cauchy Problems by Irina V. Melnikova,Alexei Filinkov Pdf

Although the theory of well-posed Cauchy problems is reasonably understood, ill-posed problems-involved in a numerous mathematical models in physics, engineering, and finance- can be approached in a variety of ways. Historically, there have been three major strategies for dealing with such problems: semigroup, abstract distribution, and regularizat

Author : Sigeru Mizohata
Publisher : Academic Press
Page : 186 pages
File Size : 55,5 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483269061

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On the Cauchy Problem by Sigeru Mizohata Pdf

Notes and Reports in Mathematics in Science and Engineering, Volume 3: On the Cauchy Problem focuses on the processes, methodologies, and mathematical approaches to Cauchy problems. The publication first elaborates on evolution equations, Lax-Mizohata theorem, and Cauchy problems in Gevrey class. Discussions focus on fundamental proposition, proof of theorem 4, Gevrey property in t of solutions, basic facts on pseudo-differential, and proof of theorem 3. The book then takes a look at micro-local analysis in Gevrey class, including proof and consequences of theorem 1. The manuscript examines Schrödinger type equations, as well as general view-points on evolution equations. Numerical representations and analyses are provided in the explanation of these type of equations. The book is a valuable reference for mathematicians and researchers interested in the Cauchy problem.

Author : Robert T. Glassey
Publisher : SIAM
Page : 246 pages
File Size : 48,5 Mb
Release : 1996-01-01
Category : Science
ISBN : 9780898713671

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The Cauchy Problem in Kinetic Theory by Robert T. Glassey Pdf

Studies the basic equations of kinetic theory in all of space, and contains up-to-date, state-of-the-art treatments of initial-value problems for the major kinetic equations. This is the only existing book to treat Boltzmann-type problems and Vlasov-type problems together. Although describing very different phenomena, these equations share the same streaming term.

Author : Jacques Hadamard
Publisher : Courier Corporation
Page : 320 pages
File Size : 51,9 Mb
Release : 2014-08-25
Category : Mathematics
ISBN : 9780486781488

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Lectures on Cauchy's Problem in Linear Partial Differential Equations by Jacques Hadamard Pdf

Would well repay study by most theoretical physicists." — Physics Today "An overwhelming influence on subsequent work on the wave equation." — Science Progress "One of the classical treatises on hyperbolic equations." — Royal Naval Scientific Service Delivered at Columbia University and the Universities of Rome and Zürich, these lectures represent a pioneering investigation. Jacques Hadamard based his research on prior studies by Riemann, Kirchhoff, and Volterra. He extended and improved Volterra's work, applying its theories relating to spherical and cylindrical waves to all normal hyperbolic equations instead of only to one. Topics include the general properties of Cauchy's problem, the fundamental formula and the elementary solution, equations with an odd number of independent variables, and equations with an even number of independent variables and the method of descent.

Author : Ti-Jun Xiao,Jin Liang
Publisher : Springer
Page : 314 pages
File Size : 53,8 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9783540494799

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The Cauchy Problem for Higher Order Abstract Differential Equations by Ti-Jun Xiao,Jin Liang Pdf

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.

Author : Hector O. Fattorini
Publisher : Cambridge University Press
Page : 664 pages
File Size : 55,6 Mb
Release : 1983
Category : Mathematics
ISBN : 9780521302388

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The Cauchy Problem by Hector O. Fattorini Pdf

This volume deals with the Cauchy or initial value problem for linear differential equations. It treats in detail some of the applications of linear space methods to partial differential equations, especially the equations of mathematical physics such as the Maxwell, Schrödinger and Dirac equations. Background material presented in the first chapter makes the book accessible to mathematicians and physicists who are not specialists in this area as well as to graduate students.

Author : Hans Ringström
Publisher : European Mathematical Society
Page : 310 pages
File Size : 46,9 Mb
Release : 2009
Category : Mathematics
ISBN : 3037190531

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The Cauchy Problem in General Relativity by Hans Ringström Pdf

The general theory of relativity is a theory of manifolds equipped with Lorentz metrics and fields which describe the matter content. Einstein's equations equate the Einstein tensor (a curvature quantity associated with the Lorentz metric) with the stress energy tensor (an object constructed using the matter fields). In addition, there are equations describing the evolution of the matter. Using symmetry as a guiding principle, one is naturally led to the Schwarzschild and Friedmann-Lemaitre-Robertson-Walker solutions, modelling an isolated system and the entire universe respectively. In a different approach, formulating Einstein's equations as an initial value problem allows a closer study of their solutions. This book first provides a definition of the concept of initial data and a proof of the correspondence between initial data and development. It turns out that some initial data allow non-isometric maximal developments, complicating the uniqueness issue. The second half of the book is concerned with this and related problems, such as strong cosmic censorship. The book presents complete proofs of several classical results that play a central role in mathematical relativity but are not easily accessible to those without prior background in the subject. Prerequisites are a good knowledge of basic measure and integration theory as well as the fundamentals of Lorentz geometry. The necessary background from the theory of partial differential equations and Lorentz geometry is included.

Author : Florent J. Bureau
Publisher : Unknown
Page : 138 pages
File Size : 45,6 Mb
Release : 1961
Category : Cauchy problem
ISBN : UVA:X002234758

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The Cauchy Problem for Partial Differential Equations of the Second Order and the Method of Ascent by Florent J. Bureau Pdf

A method of ascent is used to solve the Cauchy problem for linear partial differential equations of the second order in p space variables with constant coefficients i.e., the pure wave equation, the damped wave equation, and the heat equation. This method consists of inferring the solution of the problem referred to from the well known solution of the same problem for one space variable. The commutability of repeated pf integral, the solution deduced by the method of singularities for the Cauchy problem for the damped wave equation, and the solution of singular integral equations of the Volterra type are also considered. (Author).

Author : Claude Zuily
Publisher : Birkhauser
Page : 200 pages
File Size : 55,6 Mb
Release : 1983
Category : Cauchy problem
ISBN : UCAL:B4406796

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Uniqueness and Non-uniqueness in the Cauchy Problem by Claude Zuily Pdf

Author : Nikolaĭ Nikolaevich Tarkhanov
Publisher : De Gruyter Akademie Forschung
Page : 488 pages
File Size : 46,5 Mb
Release : 1995
Category : Mathematics
ISBN : UOM:39015037696294

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The Cauchy Problem for Solutions of Elliptic Equations by Nikolaĭ Nikolaevich Tarkhanov Pdf

The book is an attempt to bring together various topics in partial differential equations related to the Cauchy problem for solutions of an elliptic equation. Ever since Hadamard, the Cauchy problem for solutions of elliptic equations has been known to be ill-posed.

Author : Anonim
Publisher : Academic Press
Page : 343 pages
File Size : 42,9 Mb
Release : 1977-01-13
Category : Mathematics
ISBN : 9780080956367

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Singular and Degenerate Cauchy Problems by Anonim Pdf

In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation; methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory. As a result, the book represents a blend of new methods in general computational analysis, and specific, but also generic, techniques for study of systems theory ant its particular branches, such as optimal filtering and information compression. - Best operator approximation, - Non-Lagrange interpolation, - Generic Karhunen-Loeve transform - Generalised low-rank matrix approximation - Optimal data compression - Optimal nonlinear filtering

Author : Wolfgang Arendt,Charles J.K. Batty,Frank Neubrander
Publisher : Springer Science & Business Media
Page : 526 pages
File Size : 45,5 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9783034850759

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Vector-valued Laplace Transforms and Cauchy Problems by Wolfgang Arendt,Charles J.K. Batty,Frank Neubrander Pdf

Linear evolution equations in Banach spaces have seen important developments in the last two decades. This is due to the many different applications in the theory of partial differential equations, probability theory, mathematical physics, and other areas, and also to the development of new techniques. One important technique is given by the Laplace transform. It played an important role in the early development of semigroup theory, as can be seen in the pioneering monograph by Rille and Phillips [HP57]. But many new results and concepts have come from Laplace transform techniques in the last 15 years. In contrast to the classical theory, one particular feature of this method is that functions with values in a Banach space have to be considered. The aim of this book is to present the theory of linear evolution equations in a systematic way by using the methods of vector-valued Laplace transforms. It is simple to describe the basic idea relating these two subjects. Let A be a closed linear operator on a Banach space X. The Cauchy problern defined by A is the initial value problern (t 2 0), (CP) {u'(t) = Au(t) u(O) = x, where x E X is a given initial value. If u is an exponentially bounded, continuous function, then we may consider the Laplace transform 00 u(>. ) = 1 e-). . tu(t) dt of u for large real>. .

Author : V.I. Shalashilin,E. B. Kuznetsov
Publisher : Springer Science & Business Media
Page : 246 pages
File Size : 51,5 Mb
Release : 2003-09-30
Category : Mathematics
ISBN : 1402015429

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Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics by V.I. Shalashilin,E. B. Kuznetsov Pdf

The optimal continuation parameter provides the best conditions in a linearized system of equations at any moment of the continuation process. This is one of the first books in which the best parametrization is regarded systematically for a wide class of problems. It is of interest to scientists, specialists, and postgraduate students of applied and numerical mathematics and mechanics.

Author : Irina V. Melnikova
Publisher : CRC Press
Page : 286 pages
File Size : 55,5 Mb
Release : 2016-04-27
Category : Mathematics
ISBN : 9781498785853

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Stochastic Cauchy Problems in Infinite Dimensions by Irina V. Melnikova Pdf

Stochastic Cauchy Problems in Infinite Dimensions: Generalized and Regularized Solutions presents stochastic differential equations for random processes with values in Hilbert spaces. Accessible to non-specialists, the book explores how modern semi-group and distribution methods relate to the methods of infinite-dimensional stochastic analysis. It also shows how the idea of regularization in a broad sense pervades all these methods and is useful for numerical realization and applications of the theory. The book presents generalized solutions to the Cauchy problem in its initial form with white noise processes in spaces of distributions. It also covers the "classical" approach to stochastic problems involving the solution of corresponding integral equations. The first part of the text gives a self-contained introduction to modern semi-group and abstract distribution methods for solving the homogeneous (deterministic) Cauchy problem. In the second part, the author solves stochastic problems using semi-group and distribution methods as well as the methods of infinite-dimensional stochastic analysis.