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Variational Methods for Engineers with Matlab

Book Description

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.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Introduction
    1. 1.1 Riemann integrals
    2. 1.2 Lebesgue integrals
    3. 1.3 Matlab® classes for a Riemann integral by trapezoidal integration
    4. 1.4 Matlab® classes for Lebesgue’s integral
    5. 1.5 Matlab® classes for evaluation of the integrals when/is defined by a subprogram
    6. 1.6 Matlab® classes for partitions including the evaluation of the integrals
    1. 2.1 Linear systems
    2. 2.2 Algebraic equations depending upon a parameter
    3. 2.3 Exercises
    1. 3.1 Vector spaces
    2. 3.2 Distance, norm and scalar product
    3. 3.3 Continuous maps
    4. 3.4 Sequences and convergence
    5. 3.5 Hilbert spaces and completeness
    6. 3.6 Open and closed sets
    7. 3.7 Orthogonal projection
    8. 3.8 Series and separable spaces
    9. 3.9 Duality
    10. 3.10 Generating a Hilbert basis
    11. 3.11 Exercises
    1. 4.1 The L<sup xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ibooks="http://vocabulary.itunes.apple.com/rdf/ibooks/vocabulary-extensions-1.0">2</sup> (&#8486;) space (Ω) space
    2. 4.2 Weak derivatives
    3. 4.3 Sobolev spaces
    4. 4.4 Variational equations involving elements of a functional space
    5. 4.5 Reducing multiple indexes to a single one
    6. 4.6 Existence and uniqueness of the solution of a variational equation
    7. 4.7 Linear variational equations in separable spaces
    8. 4.8 Parametric variational equations
    9. 4.9 A Matlab® class for variational equations
    10. 4.10 Exercises
    1. 5.1 A simple situation: the oscillator with one degree of freedom
    2. 5.2 Connection between differential equations and variational equations
    3. 5.3 Variational approximation of differential equations
    4. 5.4 Evolution partial differential equations
    5. 5.5 Exercises
    1. 6.1 A simple example
    2. 6.2 Functional definition of Dirac’s delta
    3. 6.3 Approximations of Dirac’s delta
    4. 6.4 Smoothed particle approximations of Dirac’s delta
    5. 6.5 Derivation using Dirac’s delta approximations
    6. 6.6 A Matlab® class for smoothed particle approximations
    7. 6.7 Green’s functions
    1. 7.1 Differentials
    2. 7.2 Gâteaux derivatives of functionals
    3. 7.3 Convex functionals
    4. 7.4 Standard methods for the determination of Gâteaux derivatives
    5. 7.5 Numerical evaluation and use of Gâteaux differentials
    6. 7.6 Minimum of the energy
    7. 7.7 Lagrange’s multipliers
    8. 7.8 Primal and dual problems
    9. 7.9 Matlab® determination of minimum energy solutions
    10. 7.10 First-order control problems
    11. 7.11 Second-order control problems
    12. 7.12 A variational approach for multiobjective optimization
    13. 7.13 Matlab® implementation of the variational approach for biobjective optimization
    14. 7.14 Exercises