Mathematical Analysis and Numerical Methods for Science and Technology: Volume 1 Physical Origins and Classical Methods

These 6 volumes -- the result of a 10 year collaboration between the authors, both distinguished international figures -- compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. The advent of high-speed computers has made it possible to calculate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way.

Table of Contents:
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.