Mathematical Analysis and Numerical Methods for Science and Technology: Volume 5 Evolution Problems I (Paperback, 2000)

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 Schrodinger 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 Schrodinger Equation.- 1. The Cauchy Problem for the Schrodinger 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 Schrodinger 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 Generato