Mathematical Analysis and Numerical Methods for Science and Technology

XIV. Evolution Problems: Cauchy Problems in IRn.- §1. The Ordinary Cauchy Problems in Finite Dimensional Spaces.- 1. Linear Systems with Constant Coefficients.- 2. Linear Systems with Non Constant Coefficients.- §2. Diffusion Equations.- 1. Setting of Problem.- 2. The Method of the Fourier Transform.- 3. The Elementary Solution of the Heat Equation.- 4. Mathematical Properties of the Elementary Solution and the Semigroup Associated with the Heat Operator.- §3. Wave Equations.- 1. Model Problem: The Wave Equation in ?n.- 2. The Euler-Poisson-Darboux Equation.- 3. An Application of §2 and 3: Viscoelasticity.- §4. The Cauchy Problem for the Schrödinger Equation, Introduction.- 1. Model Problem 1. The Case of Zero Potential.- 2. Model Problem 2. The Case of a Harmonic Oscillator.- §5. The Cauchy Problem for Evolution Equations Related to Convolution Products.- 1. Setting of Problem.- 2. The Method of the Fourier Transform.- 3. The Dirac Equation for a Free Particle.- §6. An Abstract Cauchy Problem. Ovsyannikov's Theorem.- Review of Chapter XIV.- XV. Evolution Problems: The Method of Diagonalisation.- §1. The Fourier Method or the Method of Diagonalisation.- 1. The Case of the Space ?1(n = 1).- 2. The Case of Space Dimension n = 2.- 3. The Case of Arbitrary Dimension n.- Review.- §2. Variations. The Method of Diagonalisation for an Operator Having Continuous Spectrum.- 1. Review of Self-Adjoint Operators in Hilbert Spaces.- 2. General Formulation of the Problem.- 3. A Simple Example of the Problem with Continuous Spectrum.- §3. Examples of Application: The Diffusion Equation.- 1. Example of Application 1: The Monokinetic Diffusion Equation for Neutrons.- 2. Example of Application 2: The Age Equation in Problems of Slowing Down of Neutrons.- 3. Example of Application 3: Heat Conduction.- §4. The Wave Equation: Mathematical Examples and Examples of Application.- 1. The Case of Dimension n = 1.- 2. The Case of Arbitrary Dimension n.- 3. Examples of Applications for n = 1.- 4. Examples of Applications for n = 2. Vibrating Membranes.- 5. Application to Elasticity; the Dynamics of Thin Homogeneous Beams.- §5. The Schrödinger Equation.- 1. The Cauchy Problem for the Schrödinger Equation in a Domain ? = ]0, 1[? ?.- 2. A Harmonic Oscillator.- Review.- §6. Application with an Operator Having a Continuous Spectrum: Example.- Review of Chapter XV.- Appendix. Return to the Problem of Vibrating Strings.- XVI. Evolution Problems: The Method of the Laplace Transform.- §1. Laplace Transform of Distributions.- 1. Study of the Set If and Definition of the Laplace Transform.- 2. Properties of the Laplace Transform.- 3. Characterisation of Laplace Transforms of Distributions of L+ (?).- §2. Laplace Transform of Vector-valued Distributions.- 1. Distributions with Vector-valued Values.- 2. Fourier and Laplace Transforms of Vector-valued Distributions.- §3. Applications to First Order Evolution Problems.- 1. 'Vector-valued Distribution' Solutions of an Evolution Equation of First Order in t.- 2. The Method of Transposition.- 3. Application to First Order Evolution Equations. The Hilbert Space Case. L2 Solutions in Hilbert Space.- 4. The Case where A is Defined by a Sesquilinear Form a(u, v).- §4. Evolution Problems of Second Order in t.- 1. Direct Method.- 2. Use of Symbolic Calculus.- Review.- §5. Applications.- 1. Hydrodynamical Problems.- 2. A Problem of the Kinetics of Neutron Diffusion.- 3. Problems of Diffusion of an Electromagnetic Wave.- 4. Problems of Wave Propagation.- 5. Viscoelastic Problems.- 6. A Problem Related to the Schrödinger Equation.- 7. A Problem Related to Causality, Analyticity and Dispersion Relations.- 8. Remark 10.- Review of Chapter XVI.- XVII. Evolution Problems: The Method of Semigroups.- A. Study of Semigroups.- §1. Definitions and Properties of Semigroups Acting in a Banach Space.- 1. Definition of a Semigroup of Class &0 (Resp. of a Group).- 2. Basic Properties of Semigroups of Class &0.- §2. The Infinitesimal Generator of a Semigroup.- 1. Examples.- 2. The Infinitesimal Generator of a Semigroup of Class &0.- §3. The Hille-Yosida Theorem.- 1. A Necessary Condition.- 2. The Hille-Yosida Theorem.- 3. Examples of Application of the Hille-Yosida Theorem.- §4. The Case of Groups of Class &0 and Stone's Theorem.- 1. The Characterisation of the Infinitesimal Generator of a Group of Class &0.- 2. Unitary Groups of Class &0. Stone's Theorem.- 3. Applications of Stone's Theorem.- 4. Conservative Operators and Isometric Semigroups in Hilbert Space.- Review.- §5. Differentiable Semigroups.- §6. Holomorphic Semigroups.- §7. Compact Semigroups.- 1. Definition and Principal Properties.- 2. Characterisation of Compact Semigroups.- 3. Examples of Compact Semigroups.- B. Cauchy Problems and Semigroups.- §1. Cauchy Problems.- §2. Asymptotic Behaviour of Solutions as t ? + ?. Conservation and Dissipation in Evolution Equations.- §3. Semigroups and Diffusion Problems.- §4. Groups and Evolution Equations.- 1. Wave Problems.- 2. Schrödinger Type Problems.- 3. Weak Asymptotic Behaviour, for t ? ± ?, of Solutions of Wave Type of Schrödinger Type Problems.- 4. The Cauchy Problem for Maxwell's Equations in an Open Set ? ? ?3.- §5. Evolution Operators in Quantum Physics. The Liouville-von Neumann Equation.- 1. Existence and Uniqueness of the Solution of the Cauchy Problem for the Liouville-von Neumann Equation in the Space of Trace Operators.- 2. The Evolution Equation of (Bounded) Observables in the Heisenberg Representation.- 3. Spectrum and Resolvent of the Operator h.- §6. Trotter's Approximation Theorem.- 1. Convergence of Semigroups.- 2. General Representation Theorem.- Summary of Chapter XVII.- XVIII. Evolution Problems: Variational Methods.- Orientation.- §1. Some Elements of Functional Analysis.- 1. Review of Vector-valued Distributions.- 2. The Space W(a, b; V, V').- 3. The Spaces W(a, b; X, Y).- 4. Extension to Banach Space Framework.- 5. An Intermediate Derivatives Theorem.- 6. Bidual. Reflexivity. Weak Convergence and Weak * Convergence.- §2. Galerkin Approximation of a Hilbert Space.- 1. Definition.- 2. Examples.- 3. The Outline of a Galerkin Method.- §3. Evolution Problems of First Order in t.- 1. Formulation of Problem (P).- 2. Uniqueness of the Solution of Problem (P).- 3. Existence of a Solution of Problem (P).- 4. Continuity with Respect to the Data.- 5. Appendix: Various Extensions - Liftings.- §4. Problems of First Order in t (Examples).- 1. Mathematical Example 1. Dirichlet Boundary Conditions.- 2. Mathematical Example 2. Neumann Boundary Conditions.- 3. Mathematical Example 3. Mixed Dirichlet-Neumann Boundary Conditions.- 4. Mathematical Example 4. Bilinear Form Depending on Time t.- 5. Evolution, Positivity and 'Maximum' of Solutions of Diffusion Equations in Lp(?), 1 ? p ? ?.- 6. Mathematical Example 5. A Problem of Oblique Derivatives.- 7. Example of Application. The Neutron Diffusion Equation.- 8. A Stability Result.- §5. Evolution Problems of Second Order in t.- 1. General Formulation of Problem (P1).- 2. Uniqueness in Problem (P1).- 3. Existence of a Solution of Problem (P1).- 4. Continuity with Respect to the Data.- 5. Formulation of Problem (P2).- §6. Problems of Second Order in t. Examples.- 1. Mathematical Example 1.- 2. Mathematical Example 2.- 3. Mathematical Example 3.- 4. Mathematical Example 4.- 5. Application Examples.- §7. Other Types of Equation.- 1. Schrödinger Type Equations.- 2. Evolution Equations with Delay.- 3. Some Integro-Differential Equations.- 4. Optimal Control and Problems where the Unknowns are Operators.- 5. The Problem of Coupled Parabolic-Hyperbolic Transmission.- 6. The Method of 'Extension with Respect to a Parameter'.- Review of Chapter XVIII.- Table of Notations.- of Volumes 1-4, 6.