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by Eduardo Souza de Cursi
Variational Methods for Engineers with Matlab
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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 (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
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