E-Book Overview
This book is issued from a 30 years’ experience on the presentation of variational methods to successive generations of students and researchers in Engineering. It gives a comprehensive, pedagogical and engineer-oriented presentation of the foundations of variational methods and of their use in numerical problems of Engineering. Particular applications to linear and nonlinear systems of equations, differential equations, optimization and control are presented. MATLAB programs illustrate the implementation and make the book suitable as a textbook and for self-study.
The evolution of knowledge, of the engineering studies and of the society in general has led to a change of focus from students and researchers. New generations of students and researchers do not have the same relations to mathematics as the previous ones. In the particular case of variational methods, the presentations used in the past are not adapted to the previous knowledge, the language and the centers of interest of the new generations. Since these methods remain a core knowledge – thus essential - in many fields (Physics, Engineering, Applied Mathematics, Economics, Image analysis …), a new presentation is necessary in order to address variational methods to the actual context.
E-Book Content
Table of Contents Cover Title Copyright Introduction Chapter 1. Integrals 1.1 Riemann integrals 1.2 Lebesgue integrals 1.3 Matlab® classes for a Riemann integral by trapezoidal integration 1.4 Matlab® classes for Lebesgue’s integral 1.5 Matlab® classes for evaluation of the integrals when/is defined by a subprogram 1.6 Matlab® classes for partitions including the evaluation of the integrals Chapter 2. Variational Methods for Algebraic Equations 2.1 Linear systems 2.2 Algebraic equations depending upon a parameter 2.3 Exercises Chapter 3. Hilbert Spaces for Engineers 3.1 Vector spaces 3.2 Distance, norm and scalar product 3.3 Continuous maps 3.4 Sequences and convergence 3.5 Hilbert spaces and completeness 3.6 Open and closed sets 3.7 Orthogonal projection 3.8 Series and separable spaces 3.9 Duality 3.10 Generating a Hilbert basis 3.11 Exercises Chapter 4. Functional Spaces for Engineers 4.1 The L2 (Ω) space 4.2 Weak derivatives 4.3 Sobolev spaces
4.4 Variational equations involving elements of a functional space 4.5 Reducing multiple indexes to a single one 4.6 Existence and uniqueness of the solution of a variational equation 4.7 Linear variational equations in separable spaces 4.8 Parametric variational equations 4.9 A Matlab® class for variational equations 4.10 Exercises Chapter 5. Variational Methods for Differential Equations 5.1 A simple situation: the oscillator with one degree of freedom 5.2 Connection between differential equations and variational equations 5.3 Variational approximation of differential equations 5.4 Evolution partial differential equations 5.5 Exercises Chapter 6. Dirac’s Delta 6.1 A simple example 6.2 Functional definition of Dirac’s delta 6.3 Approximations of Dirac’s delta 6.4 Smoothed particle approximations of Dirac’s delta 6.5 Derivation using Dirac’s delta approximations 6.6 A Matlab® class for smoothed particle approximations 6.7 Green’s functions Chapter 7. Functionals and Calculus of Variations 7.1 Differentials 7.2 Gâteaux derivatives of functionals 7.3 Convex functionals 7.4 Standard methods for the determination of Gâteaux derivatives 7.5 Numerical evaluation and use of Gâteaux differentials 7.6 Minimum of the energy 7.7 Lagrange’s multipliers 7.8 Primal and dual problems 7.9 Matlab® determination of minimum energy solutions 7.10 First-order control problems 7.11 Second-order control problems 7.12 A variational approach for multiobjective optimization
7.13 Matlab® implementation of the variational approach for biobjective optimization 7.14 Exercises Bibliography Index
List of Illustrations Chap