Note: Cover -- Contents -- Chapter 1. The Stefan Problem and its Classical Formulation -- 1.1 Some Stefan and Stefan-like Problems -- 1.2 Free Boundary Problems with Free Boundaries of Codimension-two -- 1.3 The Classical Stefan Problem in One-dimension and the Neumann Solution -- 1.4 Classical Formulation of Multi-dimensional Stefan Problems -- Chapter 2. Thermodynamical and Metallurgical Aspects of Stefan Problems -- 2.1 Thermodynamical Aspects -- 2.2 Some Metallurgical Aspects of Stefan Problems -- 2.3 Morphological Instability of the Solid--Liquid Interface -- 2.4 Non-material Singular Surface: Generalized Stefan Condition -- Chapter 3. Extended Classical Formulations of n-phase Stefan Problems with n > -- 1 -- 3.1 One-phase Problems -- 3.2 Extended Classical Formulations of Two-phase Stefan Problems -- 3.3 Stefan problems with Implicit Free Boundary Conditions -- Chapter 4. Stefan Problem with Supercooling: Classical Formulation and Analysis -- 4.1 Introduction -- 4.2 A Phase-field Model for Solidification using Landau Ginzburg Free Energy Functional -- 4.3 Some Thermodynamically Consistent Phase-field and Phase Relaxation Models of Solidification -- 4.4 Solidification of Supercooled Liquid Without Curvature Effect and Kinetic Undercooling: Analysis of the Solution -- 4.5 Analysis of Supercooled Stefan Problems with the Modified Gibbs Thomson Relation -- Chapter 5. Superheating due to Volumetric Heat Sources: The Formulation and Analysis -- 5.1 The Classical Enthalpy Formulation of a One-dimensional Problem -- 5.2 The Weak Solution -- 5.3 Blow-up and Regularization -- Chapter 6. Steady-State and Degenerate Classical Stefan Problems -- 6.1 Some Steady-state Stefan Problems -- 6.2 Degenerate Stefan Problems -- Chapter 7. Elliptic and Parabolic Variational Inequalities -- 7.1 Introduction -- 7.2 The Elliptic Variational Inequality. , 7.3 The Parabolic Variational Inequality -- 7.4 Some Variational Inequality Formulations of Classical Stefan Problems -- Chapter 8. The Hyperbolic Stefan Problem -- 8.1 Introduction -- 8.2 Model I: Hyperbolic Stefan Problem with Temperature Continuity at the Interface -- 8.3 Model II: Formulation with Temperature Discontinuity at the Interface -- 8.4 Model III: Delay in the Response of Energy to Latent and Sensible Heats -- Chapter 9. Inverse Stefan Problems -- 9.1 Introduction -- 9.2 Well-posedness of the solution -- 9.3 Regularization -- 9.4 Determination of Unknown Parameters in Inverse Stefan Problems -- 9.5 Regularization of Inverse Heat Conduction Problems by Imposing Suitable Restrictions on the solution -- 9.6 Regularization of Inverse Stefan Problems Formulated as Equations in the form of Convolution Integrals -- 9.7 Inverse Stefan Problems Formulated as Defect Minimization Problems -- Chapter 10. Analysis of the Classical Solutions of Stefan Problems -- 10.1 One-dimensional One-phase Stefan Problems -- 10.2 One-dimensional Two-phase Stefan Problems -- 10.3 Analysis of the Classical Solutions of Multi-dimensional Stefan Problems -- Chapter 11. Regularity of the Weak Solutions of Some Stefan Problems -- 11.1 Regularity of the Weak solutions of One-dimensional Stefan Problems -- 11.2 Regularity of the Weak solutions of Multi-dimensional Stefan Problems -- Appendix A. Preliminaries -- Appendix B. Some Function Spaces and Norms -- Appendix C. Fixed Point Theorems and Maximum Principles -- Appendix D. Sobolev Spaces -- Bibliography -- Captions for Figures -- Subject Index.

Note: Front Cover -- The Classical Stefan Problem: Basic Concepts, Modelling and Analysis with Quasi-Analytical Solutions and Methods -- Copyright -- Dedication -- Contents -- List of Symbols -- Preface to the New Edition -- Preface -- Acknowledgements -- Chapter 1: The Stefan Problem and Its Classical Formulation -- 1.1 Some Stefan and Stefan-Like Problems -- Linearization of the Above Problem -- 1.2 Free Boundary Problems With Free Boundariesof Codimension-Two -- 1.3 The Classical Stefan Problem in One-Dimension and the Neumann Solution -- 1.3.1 Melting Problem -- 1.3.2 Neumann Solution -- 1.4 Classical Formulation of Multidimensional Stefan Problems -- 1.4.1 Two-Phase Stefan Problem in Multiple Dimensions -- 1.4.2 Alternative Forms of the Stefan Condition -- 1.4.3 The Kirchhoff's Transformation -- 1.4.4 Boundary Conditions at the Fixed Boundary -- (A) Standard Boundary Conditions -- (AI) Type I Boundary Condition -- (AII) Boundary Condition of Radiative-Convective Type -- (B) Nonstandard Boundary Conditions -- (BI) Nonlocal Boundary Condition -- (BII) Boundary Condition of the Fifth Type -- (C) Boundary Conditions With Multivalued Functions -- 1.4.5 Conditions at the Free Boundary -- Implicit Free Boundary Condition -- 1.4.6 The Classical Solution -- 1.4.7 Conservation Laws and the Motion of the Melt -- Control Volume and Applications of Conservation Laws -- Conservation of Mass: Equation of Continuity -- Equation of Conservation of Linear Momentum -- The Equation of Conservation of Energy -- Chapter 2: Thermodynamical and Metallurgical Aspects of Stefan Problems -- 2.1 Thermodynamical Aspects -- 2.1.1 Microscopic and Macroscopic Models -- 2.1.2 Laws of Classical Thermodynamics -- First Law of Thermodynamics -- Second Law of Thermodynamics: Entropy -- 2.1.3 Some Thermodynamic Variables and Thermal Parameters. , 2.1.4 Equilibrium Temperature: Clapeyron's Equation -- 2.2 Some Metallurgical Aspects of Stefan Problems -- 2.2.1 Nucleation and Supercooling -- Degree of Supercooling -- 2.2.2 The Effect of Interface Curvature -- 2.2.3 Nucleation of Melting, Effect of Interface Kinetics and Glassy Solids -- 2.3 Morphological Instability of the Solid-Liquid Interface -- 2.4 Nonmaterial Singular Surface: Generalized Stefan Condition -- Conservation of Forces in R -- Conservation of Energy in the Singular Surface Γ -- Chapter 3: Extended Classical Formulations of n-Phase Stefan Problems With n ≥1 -- 3.1 One-Phase Problems -- 3.1.1 An Extended Formulation of One-Dimensional One-Phase Problem -- 3.1.2 Solidification of Supercooled Liquid -- 3.1.3 Multidimensional One-Phase Problems -- A Three-Dimensional Ablation Problem -- A Signorini-Type Boundary Condition -- 3.2 Extended Classical Formulations of Two-Phase Stefan Problems -- 3.2.1 An Extended Formulation of the One-Dimensional Two-Phase Problem -- 3.2.2 Multidimensional Stefan Problems of Classes II and III -- 3.2.3 Classical Stefan Problems With n-Phases, n> -- 2 -- Some One-Dimensional Problems With More Than Two Phases -- 3.2.4 Solidification With Transition Temperature Range -- 3.3 Stefan Problems With Implicit Free Boundary Conditions -- 3.3.1 Schatz Transformations and Implicit Free Boundary Conditions -- Conversion of a Stefan-Type Problem to a Stefan Problem -- 3.3.2 Unconstrained and Constrained Oxygen-Diffusion Problem -- Constrained and Unconstrained ODP -- ODP in a Radially Symmetric Domain -- Quasi-Static Two-Dimensional ODP and the Hele-Shaw Problem -- Chapter 4: Stefan Problem With Supercooling: Classical Formulation and Analysis -- 4.1 Introduction -- 4.2 A Phase-Field Model for Solidification Using Landau-Ginzburg Free Energy Functional. , 4.3 Some Thermodynamically Consistent Phase-Field and Phase Relaxation Models of Solidification -- Entropy Functional -- Some Thermodynamically Consistent Phase Relaxation Modelsfor Supercooling -- Superheating and Supercooling Effects -- 4.4 Solidification of Supercooled Liquid Without Curvature Effect and Kinetic Undercooling: Analysis of the Solution -- 4.4.1 One-Dimensional One-Phase Solidification of Supercooled Liquid (SSP) -- 4.4.2 Regularization of a Blow-Up in SSP by Looking at CODP -- 4.4.3 Analysis of Problems With Some Changes in the Initialand Boundary Conditions in SSP -- 4.5 Analysis of SSPs With the Modified Gibbs-Thomson Relation -- 4.5.1 Introduction -- 4.5.2 One-Dimensional One-Phase SSPs With the Modified Gibbs-Thomson Relation -- 4.5.3 One-Dimensional Two-Phase Stefan Problems Withthe Modified Gibbs-Thomson Relation -- 4.5.4 Multidimensional SSPs and Problems With the Modified Gibbs-Thomson Relation -- Concluding Remarks -- 4.5.5 Weak Formulation With Supercooling and Superheating Effects -- Chapter 5: Superheating due to Volumetric Heat Sources -- 5.1 The Classical Enthalpy Formulation (CEF) -- 5.2 The Weak Solution (WS) -- 5.2.1 The WS and Its Relation to Classical Solution -- 5.2.2 Structure of the Mushy Region in the Presence of Heat Sources -- 5.3 Blow-Up and Regularization -- Chapter 6: Steady-State and Degenerate Classical Stefan Problems -- 6.1 Some Steady-State Stefan Problems -- 6.2 Degenerate Stefan Problems -- Quasi Steady-State Stefan Problems -- Degenerate Parabolic-Elliptic Problems -- 6.2.1 A Quasi Steady-State Problem (QSSP) and Its Relationto the HSP -- Chapter 7: Elliptic and Parabolic Variational Inequalities -- 7.1 Introduction -- 7.2 The Elliptic Variational Inequality -- 7.2.1 Definition and the Basic Function Spaces -- 7.2.2 Minimization of a Functional -- 7.2.3 The Complementarity Problem. , 7.2.4 Some Existence and Uniqueness Results Concerning Elliptic Inequalities -- Lions-Stampacchia Theorem -- Variational Equation -- 7.2.5 Equivalence of Different Inequality Formulations of an Obstacle Problem of the String -- Equivalence of Formulations (I), (II) and (III) -- 7.3 The Parabolic Variational Inequality -- 7.3.1 Formulation in Appropriate Spaces -- 7.4 Some Variational Inequality Formulations of Classical Stefan Problems -- 7.4.1 One-Phase Stefan Problems -- Duvait's Transformation -- One-Phase Continuous Casting Model and Its Variational Inequality Formulation -- Oxygen-Diffusion Problem -- 7.4.2 A Stefan Problem With a Quasi-Variational Inequality Formulation -- 7.4.3 The Variational Inequality Formulation of a Two-Phase Stefan Problem -- Chapter 8: The Hyperbolic Stefan Problem -- 8.1 Introduction -- 8.1.1 Relaxation Time and Relaxation Models -- 8.2 Model I: Hyperbolic Stefan Problem With Temperature Continuity at the Interface -- 8.2.1 The Mathematical Formulation -- A One-Dimensional Two-Phase Hyperbolic Stefan Problem -- Differential Equations -- Initial Conditions -- Boundary Conditions at the Fixed Boundaries -- Boundary Conditions at the Free Boundary -- 8.2.2 Some Existence, Uniqueness and Well-Posedness Results -- A One-Dimensional One-Phase Hyperbolic Stefan Problem -- Global Solution for the One-Phase Problem -- A Two-Phase Problem -- 8.3 Model II: Formulation With Temperature Discontinuity at the Interface -- 8.3.1 The Mathematical Formulation -- Admissibility Conditions -- 8.3.2 The Existence and Uniqueness of the Solution and Its Convergence as τ → 0 -- Dirichlet Problem -- Neumann Problem -- 8.4 Model III: Delay in the Response of Energy to Latent and Sensible Heats -- 8.4.1 The Classical and the Weak Formulations -- Derivation of Energy Conservation Equation for the Two-Phase Problem. , Chapter 9: Inverse Stefan Problems -- 9.1 Introduction -- 9.2 Well-Posedness of the Solution -- Nonexistence of the Solution -- Nonuniqueness of the Solution -- Continuous Dependence of the Solution on the Input Data -- 9.2.1 Approximate Solutions -- 9.3 Regularization -- 9.3.1 The Regularizing Operator and Generalized Discrepancy Principle -- 9.3.2 The Generalized Inverse -- 9.3.3 Regularization Methods -- 9.3.4 Rate of Convergence of a Regularization Method -- 9.4 Determination of Unknown Parameters in Inverse Stefan Problems -- 9.4.1 Unknown Parameters in the One-Phase Stefan Problems -- 9.4.2 Determination of Unknown Parameters in the Two-Phase Stefan Problems -- 9.5 Regularization of Inverse Heat conduction Problems by Imposing Suitable Restrictionson the Solution -- 9.6 Regularization of Inverse Stefan Problems Formulated as Equations in the Form of Convolution Integrals -- 9.7 Inverse Stefan Problems Formulated as Defect Minimization Problems -- Chapter 10: Analysis of the Classical Solutions of Stefan Problems -- 10.1 One-Dimensional One-Phase Stefan Problems -- 10.1.1 Analysis Using Integral Equation Formulations -- 10.1.2 Infinite Differentiability and Analyticity of the Free Boundary -- 10.1.3 Unilateral Boundary Conditions on the Fixed Boundary: Analysis Using Finite-Difference Schemes -- 10.1.4 Cauchy-Type Free Boundary Problems -- 10.1.5 Existence of Self-Similar Solutions of Some Stefan Problems -- 10.1.6 The Effect of Density Change -- 10.2 One-Dimensional Two-Phase Stefan Problems -- 10.2.1 Existence, Uniqueness and Stability Results -- 10.2.2 Differentiability and Analyticity of the Free Boundaryin the One-Dimensional Two-Phase Stefan Problems -- 10.2.3 One-Dimensional n-Phase Stefan Problems With n > -- 2 -- 10.3 Analysis of the Classical Solutionsof Multidimensional Stefan Problems. , 10.3.1 Existence and Uniqueness Results Valid for a Short Time.