Some Inverse Problems for Hyperbolic Partial Differential Equations PDF Download
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Author: Zachary J. Bailey
Publisher:
ISBN: 9780438241442
Category :
Languages : en
Pages : 97
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Book Description
We consider four inverse problems for hyperbolic PDEs with two of them associated with one space dimension and two of them associated with three space dimensions. ☐ The first two problems are inverse problems associated to one space dimensional hyperbolic systems of PDEs with complex coefficients where the goal is the recovery of a single complex coefficient from either the reflection data or the transmission data. We show that the map sending the coefficient to the reflection/transmission data is injective and stable and we also characterize the range of this map for the transmission data case. ☐ The other two problems are associated with a single hyperbolic PDE with a zero order coefficient and the goal is the recovery of this coefficient from two different types of ``backscattering data'' - backscattering data coming from a fixed offset distribution of sources and receivers on the boundary or backscattering data coming from a single incoming spherical wave. For these problems we prove a stability result provided the difference of the two coefficients is horizontally or angularly controlled respectively. ☐ Our work adapts the techniques used by Eemeli Blåsten, Rakesh and Gunther Uhlmann to solve problems similar to theirs.
Author: Zachary J. Bailey
Publisher:
ISBN: 9780438241442
Category :
Languages : en
Pages : 97
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Book Description
We consider four inverse problems for hyperbolic PDEs with two of them associated with one space dimension and two of them associated with three space dimensions. ☐ The first two problems are inverse problems associated to one space dimensional hyperbolic systems of PDEs with complex coefficients where the goal is the recovery of a single complex coefficient from either the reflection data or the transmission data. We show that the map sending the coefficient to the reflection/transmission data is injective and stable and we also characterize the range of this map for the transmission data case. ☐ The other two problems are associated with a single hyperbolic PDE with a zero order coefficient and the goal is the recovery of this coefficient from two different types of ``backscattering data'' - backscattering data coming from a fixed offset distribution of sources and receivers on the boundary or backscattering data coming from a single incoming spherical wave. For these problems we prove a stability result provided the difference of the two coefficients is horizontally or angularly controlled respectively. ☐ Our work adapts the techniques used by Eemeli Blåsten, Rakesh and Gunther Uhlmann to solve problems similar to theirs.
Author: V. G. Romanov
Publisher: Springer Science & Business Media
ISBN: 364280781X
Category : Mathematics
Languages : en
Pages : 160
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Book Description
There are currently many practical situations in which one wishes to determine the coefficients in an ordinary or partial differential equation from known functionals of its solution. These are often called "inverse problems of mathematical physics" and may be contrasted with problems in which an equation is given and one looks for its solution under initial and boundary conditions. Although inverse problems are often ill-posed in the classical sense, their practical importance is such that they may be considered among the pressing problems of current mathematical re search. A. N. Tihonov showed [82], [83] that there is a broad class of inverse problems for which a particular non-classical definition of well-posed ness is appropriate. This new definition requires that a solution be unique in a class of solutions belonging to a given subset M of a function space. The existence of a solution in this set is assumed a priori for some set of data. The classical requirement of continuous dependence of the solution on the data is retained but it is interpreted differently. It is required that solutions depend continuously only on that data which does not take the solutions out of M.
Author: Victor Isakov
Publisher: Springer
ISBN: 3319516582
Category : Mathematics
Languages : en
Pages : 414
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Book Description
A comprehensive description of the current theoretical and numerical aspects of inverse problems in partial differential equations. Applications include recovery of inclusions from anomalies of their gravity fields, reconstruction of the interior of the human body from exterior electrical, ultrasonic, and magnetic measurement. By presenting the data in a readable and informative manner, the book introduces both scientific and engineering researchers as well as graduate students to the significant work done in this area in recent years, relating it to broader themes in mathematical analysis.
Author: Barbara Kaltenbacher
Publisher: American Mathematical Society
ISBN: 1470472775
Category : Mathematics
Languages : en
Pages : 522
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Book Description
As the title of the book indicates, this is primarily a book on partial differential equations (PDEs) with two definite slants: toward inverse problems and to the inclusion of fractional derivatives. The standard paradigm, or direct problem, is to take a PDE, including all coefficients and initial/boundary conditions, and to determine the solution. The inverse problem reverses this approach asking what information about coefficients of the model can be obtained from partial information on the solution. Answering this question requires knowledge of the underlying physical model, including the exact dependence on material parameters. The last feature of the approach taken by the authors is the inclusion of fractional derivatives. This is driven by direct physical applications: a fractional derivative model often allows greater adherence to physical observations than the traditional integer order case. The book also has an extensive historical section and the material that can be called "fractional calculus" and ordinary differential equations with fractional derivatives. This part is accessible to advanced undergraduates with basic knowledge on real and complex analysis. At the other end of the spectrum, lie nonlinear fractional PDEs that require a standard graduate level course on PDEs.
Author: Yurii Ya. Belov
Publisher: Walter de Gruyter
ISBN: 3110944634
Category : Mathematics
Languages : en
Pages : 220
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Book Description
This monograph is devoted to identification problems of coefficients in equations of mathematical physics. It invesitgates the existence and uniqueness of the solutions for identification coefficient problems in parabolic and hyperbolic equations and equation systems of composite type. The problems are studied with the Cauchy data and equations in which the Fourier transform with respect to the chosen variable is supposed to occur. Differential properties of the solutions for the original direct problems and their behavior under great values of time are studied on the basis of solution properties for direct problems. The identification problems with one or two unknown coefficients are also investigated. For initial boundary value conditions linear and nonlinear parabolic equations are studied.
Author: David L. Colton
Publisher: SIAM
ISBN: 9780898712520
Category : Mathematics
Languages : en
Pages : 234
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Book Description
Author: V. G Romanov
Publisher:
ISBN: 9783642807824
Category :
Languages : en
Pages : 164
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Book Description
Author: Alemdar Hasanov Hasanoğlu
Publisher: Springer
ISBN: 331962797X
Category : Mathematics
Languages : en
Pages : 264
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Book Description
This book presents a systematic exposition of the main ideas and methods in treating inverse problems for PDEs arising in basic mathematical models, though it makes no claim to being exhaustive. Mathematical models of most physical phenomena are governed by initial and boundary value problems for PDEs, and inverse problems governed by these equations arise naturally in nearly all branches of science and engineering. The book’s content, especially in the Introduction and Part I, is self-contained and is intended to also be accessible for beginning graduate students, whose mathematical background includes only basic courses in advanced calculus, PDEs and functional analysis. Further, the book can be used as the backbone for a lecture course on inverse and ill-posed problems for partial differential equations. In turn, the second part of the book consists of six nearly-independent chapters. The choice of these chapters was motivated by the fact that the inverse coefficient and source problems considered here are based on the basic and commonly used mathematical models governed by PDEs. These chapters describe not only these inverse problems, but also main inversion methods and techniques. Since the most distinctive features of any inverse problems related to PDEs are hidden in the properties of the corresponding solutions to direct problems, special attention is paid to the investigation of these properties.
Author: Yu. Ya Belov
Publisher: V.S.P. International Science
ISBN: 9789067643580
Category : Mathematics
Languages : en
Pages : 211
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Book Description
This monograph is devoted to identification problems of coefficients in equations of mathematical physics. It invesitgates the existence and uniqueness of the solutions for identification coefficient problems in parabolic and hyperbolic equations and equation systems of composite type. The problems are studied with the Cauchy data and equations in which the Fourier transform with respect to the chosen variable is supposed to occur. Differential properties of the solutions for the original direct problems and their behavior under great values of time are studied on the basis of solution properties for direct problems. The identification problems with one or two unknown coefficients are also investigated. For initial boundary value conditions linear and nonlinear parabolic equations are studied.
Author: Mourad Bellassoued
Publisher: Springer
ISBN: 4431566007
Category : Mathematics
Languages : en
Pages : 267
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Book Description
This book is a self-contained account of the method based on Carleman estimates for inverse problems of determining spatially varying functions of differential equations of the hyperbolic type by non-overdetermining data of solutions. The formulation is different from that of Dirichlet-to-Neumann maps and can often prove the global uniqueness and Lipschitz stability even with a single measurement. These types of inverse problems include coefficient inverse problems of determining physical parameters in inhomogeneous media that appear in many applications related to electromagnetism, elasticity, and related phenomena. Although the methodology was created in 1981 by Bukhgeim and Klibanov, its comprehensive development has been accomplished only recently. In spite of the wide applicability of the method, there are few monographs focusing on combined accounts of Carleman estimates and applications to inverse problems. The aim in this book is to fill that gap. The basic tool is Carleman estimates, the theory of which has been established within a very general framework, so that the method using Carleman estimates for inverse problems is misunderstood as being very difficult. The main purpose of the book is to provide an accessible approach to the methodology. To accomplish that goal, the authors include a direct derivation of Carleman estimates, the derivation being based essentially on elementary calculus working flexibly for various equations. Because the inverse problem depends heavily on respective equations, too general and abstract an approach may not be balanced. Thus a direct and concrete means was chosen not only because it is friendly to readers but also is much more relevant. By practical necessity, there is surely a wide range of inverse problems and the method delineated here can solve them. The intention is for readers to learn that method and then apply it to solving new inverse problems.
