Inverse Stefan problems

Description

Basic Designations -- 1 Statements of Quasilinear Inverse Stefan Problems -- 1.1 Classification of ill-posed inverse Stefan problems and their applications -- 1.2 One-phase boundary inverse Stefan problems with given phase boundaries -- 1.3 One-phase boundary inverse Stefan problems with unknown phase boundaries -- 1.4 Boundary inverse Stefan problems in the two-phase case -- 1.5 Coefficient inverse Stefan problems -- 2 The Regularization Variational Method for Solving Inverse Stefan Problems -- 2.1 Construction of approximate solutions on the basis of the quasi-solution method -- 2.2 Stability of approximate solutions -- 2.3 Differentiability of functionals in the variational formulations of inverse Stefan problems -- 3 Algorithms for the Numerical Solution of Inverse Stefan Problems -- 3.1 Principles of construction of algorithms -- 3.2 Numerical solution of the one-phase inverse Stefan problems with given phase boundaries. Determination of boundary regimes for continuous casting -- 3.3 Descriptive regularization algorithms for boundary inverse Stefan problems with unknown phase boundaries -- 3.4 Numerical solution of coefficient inverse Stefan problems. Determination of the intensity of laser sources -- 4 Properties of Operator Representations of Inverse Stefan Problems -- 4.1 On classical solvability of quasilinear moving boundary problems -- 4.2 A priori estimates of Hölder norms for the differential-difference analogs of linear parabolic equations -- 4.3 Unique solvability in the Hölder spaces of the differential-difference analogs of the quasilinear boundary-value problems -- 4.4 Global existence, uniqueness and stability in the Hölder spaces for the direct quasilinear Stefan problems.;In this monograph the theory and methods of solving inverse Stefan problems for quasilinear parabolic equations in regions with free boundaries are developed. The study of this new class of ill-posed problems is motivated by the needs of the mod­ eling and control of nonlinear processes with phase transitions in thermophysics and mechanics of continuous media. Inverse Stefan problems are important for the perfection of technologies both in high temperature processes (e.g., metallurgy, the aircraft industry, astronautics and power engineering) and in hydrology, exploitation of oil-gas fields, etc. The proposed book will complete a gap in these subjects in the preceding re­ searches of ill-posed problems. It contains the new theoretical and applied studies of a wide class of inverse Stefan problems. The statements of such problems on the determination of boundary functions and coefficients of the equation are considered for different types of additional information about their solution. The variational method of obtaining stable approximate solutions is proposed and established. It is implemented by an efficient computational scheme of descriptive regularization. This algorithm utilizes a priori knowledge of the qualitative structure of the sought solution and ensures a substantial saving in computational costs. It is tested on model and applied problems in nonlinear thermophysics. In particular, the results of calculations for important applications in continuous casting of ingots and in the melting of a plate with the help of laser technology are presented.