Forward and Inverse Problems for Hyperbolic, Elliptic and Mixed Type Equations
Forward and Inverse Problems for Hyperbolic, Elliptic and Mixed Type Equations

Author: Alexander G. Megrabov

Publisher: Walter de Gruyter

Published: 2012-05-24

Total Pages: 244

ISBN-13: 3110944987

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Inverse problems are an important and rapidly developing direction in mathematics, mathematical physics, differential equations, and various applied technologies (geophysics, optic, tomography, remote sensing, radar-location, etc.). In this monograph direct and inverse problems for partial differential equations are considered. The type of equations focused are hyperbolic, elliptic, and mixed (elliptic-hyperbolic). The direct problems arise as generalizations of problems of scattering plane elastic or acoustic waves from inhomogeneous layer (or from half-space). The inverse problems are those of determination of medium parameters by giving the forms of incident and reflected waves or the vibrations of certain points of the medium. The method of research of all inverse problems is spectral-analytical, consisting in reducing the considered inverse problems to the known inverse problems for the Sturm-Liouville equation or the string equation. Besides the book considers discrete inverse problems. In these problems an arbitrary set of point sources (emissive sources, oscillators, point masses) is determined.

Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems
Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems

Author: Sergey I. Kabanikhin

Publisher: Walter de Gruyter

Published: 2013-04-09

Total Pages: 188

ISBN-13: 3110960710

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The authors consider dynamic types of inverse problems in which the additional information is given by the trace of the direct problem on a (usually time-like) surface of the domain. They discuss theoretical and numerical background of the finite-difference scheme inversion, the linearization method, the method of Gel'fand-Levitan-Krein, the boundary control method, and the projection method and prove theorems of convergence, conditional stability, and other properties of the mentioned methods.

Linear Sobolev Type Equations and Degenerate Semigroups of Operators
Linear Sobolev Type Equations and Degenerate Semigroups of Operators

Author: Georgy A. Sviridyuk

Publisher: Walter de Gruyter

Published: 2012-06-04

Total Pages: 224

ISBN-13: 3110915502

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Focusing on the mathematics, and providing only a minimum of explicatory comment, this volume contains six chapters covering auxiliary material, relatively p-radial operators, relatively p-sectorial operators, relatively σ-bounded operators, Cauchy problems for inhomogenous Sobolev-type equations, bounded solutions to Sobolev-type equations, and optimal control.

Inverse Problems of Mathematical Physics
Inverse Problems of Mathematical Physics

Author: Mikhail M. Lavrent'ev

Publisher: Walter de Gruyter

Published: 2012-05-07

Total Pages: 288

ISBN-13: 3110915529

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This monograph deals with the theory of inverse problems of mathematical physics and applications of such problems. Besides it considers applications and numerical methods of solving the problems under study. Descriptions of particular numerical experiments are also included.

Carleman Estimates for Coefficient Inverse Problems and Numerical Applications
Carleman Estimates for Coefficient Inverse Problems and Numerical Applications

Author: Michael V. Klibanov

Publisher: Walter de Gruyter

Published: 2012-04-17

Total Pages: 292

ISBN-13: 3110915545

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In this monograph, the main subject of the author's considerations is coefficient inverse problems. Arising in many areas of natural sciences and technology, such problems consist of determining the variable coefficients of a certain differential operator defined in a domain from boundary measurements of a solution or its functionals. Although the authors pay strong attention to the rigorous justification of known results, they place the primary emphasis on new concepts and developments.

Well-posed, Ill-posed, and Intermediate Problems with Applications
Well-posed, Ill-posed, and Intermediate Problems with Applications

Author: Petrov Yuri P.

Publisher: Walter de Gruyter

Published: 2011-12-22

Total Pages: 245

ISBN-13: 3110195305

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This book deals with one of the key problems in applied mathematics, namely the investigation into and providing for solution stability in solving equations with due allowance for inaccuracies in set initial data, parameters and coefficients of a mathematical model for an object under study, instrumental function, initial conditions, etc., and also with allowance for miscalculations, including roundoff errors. Until recently, all problems in mathematics, physics and engineering were divided into two classes: well-posed problems and ill-posed problems. The authors introduce a third class of problems: intermediate ones, which are problems that change their property of being well- or ill-posed on equivalent transformations of governing equations, and also problems that display the property of being either well- or ill-posed depending on the type of the functional space used. The book is divided into two parts: Part one deals with general properties of all three classes of mathematical, physical and engineering problems with approaches to solve them; Part two deals with several stable models for solving inverse ill-posed problems, illustrated with numerical examples.

Operator Theory and Ill-Posed Problems
Operator Theory and Ill-Posed Problems

Author: Mikhail M. Lavrent'ev

Publisher: Walter de Gruyter

Published: 2011-12-22

Total Pages: 697

ISBN-13: 3110960729

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This book consists of three major parts. The first two parts deal with general mathematical concepts and certain areas of operator theory. The third part is devoted to ill-posed problems. It can be read independently of the first two parts and presents a good example of applying the methods of calculus and functional analysis. The first part "Basic Concepts" briefly introduces the language of set theory and concepts of abstract, linear and multilinear algebra. Also introduced are the language of topology and fundamental concepts of calculus: the limit, the differential, and the integral. A special section is devoted to analysis on manifolds. The second part "Operators" describes the most important function spaces and operator classes for both linear and nonlinear operators. Different kinds of generalized functions and their transformations are considered. Elements of the theory of linear operators are presented. Spectral theory is given a special focus. The third part "Ill-Posed Problems" is devoted to problems of mathematical physics, integral and operator equations, evolution equations and problems of integral geometry. It also deals with problems of analytic continuation. Detailed coverage of the subjects and numerous examples and exercises make it possible to use the book as a textbook on some areas of calculus and functional analysis. It can also be used as a reference textbook because of the extensive scope and detailed references with comments.

Operator Theory and Differential Equations
Operator Theory and Differential Equations

Author: Anatoly G. Kusraev

Publisher: Springer Nature

Published: 2021-01-13

Total Pages: 337

ISBN-13: 3030497631

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This volume features selected papers from The Fifteenth International Conference on Order Analysis and Related Problems of Mathematical Modeling, which was held in Vladikavkaz, Russia, on 15 - 20th July 2019. Intended for mathematicians specializing in operator theory, functional spaces, differential equations or mathematical modeling, the book provides a state-of-the-art account of various fascinating areas of operator theory, ranging from various classes of operators (positive operators, convolution operators, backward shift operators, singular and fractional integral operators, partial differential operators) to important applications in differential equations, inverse problems, approximation theory, metric theory of surfaces, the Hubbard model, social stratification models, and viscid incompressible fluids.

Ill-Posed Boundary-Value Problems
Ill-Posed Boundary-Value Problems

Author: Serikkali E. Temirbolat

Publisher: Walter de Gruyter

Published: 2012-06-04

Total Pages: 152

ISBN-13: 3110915510

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This monograph extends well-known facts to new classes of problems and works out novel approaches to the solution of these problems. It is devoted to the questions of ill-posed boundary-value problems for systems of various types of the first-order differential equations with constant coefficients and the methods for their solution.

Characterisation of Bio-Particles from Light Scattering
Characterisation of Bio-Particles from Light Scattering

Author: Valeri P. Maltsev

Publisher: Walter de Gruyter

Published: 2013-03-01

Total Pages: 144

ISBN-13: 3110915553

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The primary aim of this monograph is to provide a systematic state-of-the-art summary of the light scattering of bioparticles, including a brief consideration of analytical and numerical methods for computing electromagnetic scattering by single particles, a detailed discussion of the instrumental approach used in measurement of light scattering, an analysis of the methods used in solution of the inverse light scattering problem, and an introduction of the results dealing with practical analysis of biosamples. Considering the widespread need for this information in optics, remote sensing, engineering, medicine, and biology, the book is useful to many graduate students, scientists, and engineers working on various aspects of electromagnetic scattering and its applications.