Mathematical Analysis and Numerical Methods for Science and Technology

I. Physical Examples.- A. The Physical Models.- § 1. Classical Fluids and the Navier-Stokes System.- 1. Introduction: Mechanical Origin.- 2. Corresponding Mathematical Problem.- 3. Linearisation. Stokes' Equations.- 4. Case of a Perfect Fluid. Euler's Equations.- 5. Case of Stationary Flows. Examples of Linear Problems.- 6. Non-Stationary Flows Leading to the Equations of Viscous Diffusion.- 7. Conduction of Heat. Linear Example in the Mechanics of Fluids.- 8. Example of Acoustic Propagation.- 9. Example with Boundary Conditions on Oblique Derivatives.- Review.- §2. Linear Elasticity.- 1. Introduction: Elasticity; Hyperelasticity.- 2. Linear (not Necessarily Isotropic) Elasticity.- 3. Isotropic Linear Elasticity (or Classical Elasticity).- 4. Stationary Problems in Classical Elasticity.- 5. Dynamical Problems in Classical Elasticity.- 6. Problems of Thermal Diffusion. Classical Thermoelasticity.- Review.- §3. Linear Viscoelasticity.- 1. Introduction.- 2. Materials with Short Memory.- 3. Materials with Long Memory.- 4. Particular Case of Isotropic Media.- 5. Stationary Problems in Classical Viscoelasticity.- Review.- §4. Electromagnetism and Maxwell's Equations.- 1. Fundamental Equations of Electromagnetism.- 2. Macroscopic Equations: Electromagnetism in Continuous Media.- 3. Potentials. Gauge Transformation (Case of the Entire Space IR3x × IRt).- 4. Some Evolution Problems.- 5. Static Electromagnetism.- 6. Stationary Problems.- Review.- §5. Neutronics. Equations of Transport and Diffusion.- 1. Problems of the Transport of Neutrons.- 2. Problems of Neutron Diffusion.- 3. Stationary Problems.- Review.- §6. Quantum Physics.- 1. The Fundamental Principles of Modelling.- 2. Systems Consisting of One Particle.- 3. Systems of Several Particles.- Review.- Appendix. Concise Elements Concerning Some Mathematical Ideas Used in this §6.- Appendix "Mechanics". Elements Concerning the Problems of Mechanics.- §1. Indicial Calculus. Elementary Techniques of the Tensor Calculus.- 1. Orientation Tensor or Fundamental Alternating Tensor in IR3.- 2. Possibilities of Decompositions of a Second Order Tensor.- 3. Generalized Divergence Theorem.- 4. Ideas About Wrenches.- §2. Notation, Language and Conventions in Mechanics.- 1. Lagrangian and Eulerian Coordinates.- 2. Notions of Displacement and of Strain.- 3. Notions of Velocity and of Rate of Strain.- 4. Notions of Particle Derivative, of Acceleration and of Dilatation.- 5. Notions of Trajectory and of Stream Line.- §3. Ideas Concerning the Principle of Virtual Power.- 1. Introduction: Schematization of Forces.- 2. Preliminary Definitions.- 3. Fundamental Statements.- 4. Theory of the First Gradient.- 5. Application to the Formulation of Curvilinear Media.- 6. Application to the Formulation of the Theory of Thin Plates.- Linear and Non-Linear Problems in §1 to §6 of this Chapter IA.- B. First Examination of the Mathematical Models.- § 1. The Principal Types of Linear Partial Differential Equations Seen in Chapter IA.- 1. Equation of Diffusion Type.- 2. Equation of the Type of Wave Equations.- 3. Schrödinger Equation.- 4. The Equation Au = f in which A is a Linear Operator not Depending on the Time and f is Given (Stationary Equations).- §2. Global Constraints Imposed on the Solutions of a Problem: Inclusion in a Function Space; Boundary Conditions; Initial Conditions.- 1. Introduction. Function Spaces.- 2. Initial Conditions and Evolution Problems.- 3. Boundary Conditions.- 4. Transmission Conditions.- 5. Problems Involving Time-Derivatives of the Unknown Function u on the Boundary.- 6. Problems of Time Delay.- Review of Chapter IB.- II. The Laplace Operator Introduction.- §1. The Laplace Operator.- 1. Poisson's Equation.- 2. Examples in Mechanics and Electrostatics.- 3. Green's Formulae: The Classical Framework.- 4. The Laplacian in Polar Coordinates.- §2. Harmonic Functions.- 1. Definitions. Examples. Elementary Solutions.- 2. Gauss' Theorem. Formulae of the Mean. The Maximum Principle.- 3. Poisson's Integral Formula; Regularity of Harmonic Functions; Harnack's Inequality.- 4. Characterisation of Harmonic Functions. Elimination of Singularities.- 5. Kelvin's Transformation; Application to Harmonic Functions in an Unbounded Set; Conformai Transformation.- 6. Some Physical Interpretations (in Mechanics and Electrostatics).- §3. Newtonian Potentials.- 1. Generalities on the Newtonian Potentials of a Distribution with Compact Support.- 2. Study of Local Regularity of Solutions of Poisson's Equation.- 3. Regularity of Simple and Double Layer Potentials.- 4. Newtonian Potential of a Distribution Without Compact Support.- 5. Some Physical Interpretations (in Mechanics and Electrostatics).- §4. Classical Theory of Dirichlet's Problem.- 1. Generalities on Dirichlet's Problem P(?,?,) in the Case ? Bounded: Classical Solution, Examples, Outline of Perron's Method, Generalized Solutions, Regular Point of the Boundary, Barrier Function.- 2. Generalities on the Dirichlet Problem P(?,?, f) and the Green's Function of ?, a Bounded Open Set.- 3. Generalities on Dirichlefs Problem in an Unbounded Open Set.- 4. The Neumann Problem; Mixed Problem; Hopf's Maximum Principle; Examples.- 5. Solution by Simple and Double Layer Potentials: Fredholm's Integral Method.- 6. Sub-Harmonic Functions. Perron's Method.- §5. Capacities.- 1. Interior and Exterior Capacity Operators.- 2. Electrical Equilibrium; Coefficients of Capacitance.- 3. Capacity of a Part of an Open Set in IRn.- §6. Regularity.- 1. Regularity of the Solutions of Dirichlet and Neumann Problems.- 2. Analytic Regularity and Trace on the Boundary of a Harmonic Function.- 3. Dirichlet Problem with Given Measures or Discontinuous Functions. Herglotz's Theorem.- 4. Neumann Problem with Given Measures.- 5. Dependence of Solutions of Dirichlet Problems as a Function of the Open Set: Hadamard's Formula.- §7. Other Methods of Solution of the Dirichlet Problem.- 1. Case of a Convex Open Set: Neumann's Integral Method.- 2. Alternating Procedure of Schwarz.- 3. Method of Separation of Variables. Harmonic Polynomials. Spherical Harmonic Function.- 4. Dirichlet's Method.- 5. Symmetry Methods and Method of Images.- §8. Elliptic Equations of the Second Order.- 1. The Divergence Form, Green's Formula.- 2. Different Concepts of Solutions, Boundary Value Problems, Transmission Conditions.- 3. General Results on the Regularity of Elliptic Problems of the Second Order.- 4. Results on Existence and Uniqueness of Solutions of Strictly Elliptic Boundary Value Problems of the Second Order on a Bounded Open Set.- 5. Harnack's Inequality and the Maximum Principle.- 6. Green's Functions.- 7. Helmholtz's Equation.- Review of Chapter II.- Table of Notations.- of Volumes 2-6.