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Inverse heat transfer problems. Transl. from the Russian. (English) Zbl 0979.80003
Berlin: Springer. xii, 348 p. (1994).
From the preface: This research monograph presents a systematic treatment of the theory of the propagation of transient electromagnetic fields (such as optical pulses) through dielectric media which exhibit both dispersion and absorption. The work divides naturally into two parts. Part I presents a summary of the fundamental theory of the radiation and propagation of rather general electromagnetic waves in causal, linear media which are homogeneous and isotropic but which otherwise have rather general dispersive and absorbing properties. In Part II, we specialize to the propagation of a plane, transient electromagnetic field in a homogeneous dielectric. Although we have made some contributions to the fundamental theory given in Part I, most of the results of our own research appear in Part II.
The purpose of the theory presented in Part II is to predict and to explain in explicit detail the dynamics of the field after it has propagated far enough through the medium to be in the mature-dispersion regime. It is the subject of a classic theory, based on the research conducted by A. Sommerfeld and L. Brillouin in the early 1900’s and described in modern textbooks on advanced electrodynamics. Over the past seventeen years, we have improved the classic theory in several ways. In particular, we have increased the accuracy of many of the approximations, we have applied modern mathematical methods to clarify some of the confusion present in the older theory, we have extended the theory to include the field dynamics in space-time regions that were excluded by the classic theory, we have applied numerical techniques with computers to guide our analysis and to check the accuracy of our analytic results, we have studied the dynamics of a number of special pulses, and we have developed a physical model which provides a simplified quantitative algorithm for obtaining all of the dynamics of the field and explains all of those dynamics in physical terms. All of these advances are included in this book.
Chapter headings are as follows:
Chapter 1. Statements and use of inverse problems in studying heat transfer processes and designing engineering units (Introduction to the problem, Simulation of heat transfer processes, Inverse heat transfer problems (IHTP), Practical Applications and the role of inverse problems in thermal investigations, The contents and structure of the book, Summary)
Chapter 2. Analysis of statements and solution methods for inverse heat transfer problems (Inverse problems formulation and stability of their solution, Existence of inverse problem solutions, Uniqueness of solution of inverse heat conduction problems, Degree of instability of a boundary inverse heat conduction problem, Conditionally-well-posed statement of inverse problems, Regularization principles of ill-posed inverse problem solutions, Summary)
Chapter 3. Analytical forms of boundary inverse heat conduction problems (Determination of transient boundary conditions in a one-dimensional case, Recovery of boundary conditions with a differential method of measurement, Analytical forms of multidimensional inverse problems, Statement of a two-dimensional inverse problem, Fictitious boundary method of solving inverse boundary problems, Summary)
Chapter 4. Direct algebraic method of determining transient heat loads (The recurrent algorithm construction, The boundary condition recoverability, Step regularization principle and limits of method applicability, The solution of an inverse heat conduction problem using some other methods of approximation and with disturbed data, Algorithmic presentation of a two-dimensional inverse heat conducting problem, Summary)
Chapter 5. Solution of boundary inverse heat conduction problems by direct numerical methods (Construction of difference algorithms, Stability criterion of the difference method for solving a boundary inverse problem, Investigation into the stability of numerical solution for inverse problems, An implicit scheme for inverse problem numerical solution, Artificial hyperbolization of the heat conduction equation in solving a boundary inverse problem, Summary)
Chapter 6. The extremal formulations and methods of solving inverse heat conduction problems (A boundary inverse problem in the extremal statement, The iterative regularization principle, Parametric optimization in solving inverse problems, Gradient methods of parametric optimization, Functional optimization in inverse problems, The selection of approximate solution and the general appraisement of gradient methods, Iterative algorithms for solving a linear inverse problem, Experimental investigation of algorithms, Numerical determination of heat loads under varying thermophysical properties of the body, Solution of a nonlinear inverse problem in statement II, The iteration technique of determining non-stationary heat loads in the two-dimensional case, Summary)
Chapter 7. Regularization of variational forms of inverse heat conduction problems (The regularized form of inverse problems, The construction of a regularizing operator, Regularization of the inverse problem finite-dimensional form, The admissible degree of smoothing and approximation sampling procedures, The reconstruction accuracy analysis of boundary heat conditions, By-Interval regularization of a nonlinear inverse problem, Regularized continuation of the solution of the nonlinear heat conduction equation, The regularization of a two-dimensional inverse problem, Summary)
Chapter 8. Iterative Regularization of inverse problems (On the rigorous basis of the iterative regularization, General formulation and integral forms of linear inverse heat conduction problems. Gradient of the residual functional, The general formulation of nonlinear IHCP. The problem of an increment of temperature field, Adjoint problems and gradient of a functional, Gradient algorithms with regard to a prior information, Examples of the construction of the algorithms for the solution of inverse problems, Computational experiments, Summary)
Conclusion, Bibliography, Additional bibliography, Subject index.

80A22 Stefan problems, phase changes, etc.
80-02 Research exposition (monographs, survey articles) pertaining to classical thermodynamics