The Numerical Method of Lines and Duality Principles Applied to Models in Physics and Engineering

Cover
Title Page
Copyright Page
Preface
Acknowledgments
Table of Contents
Section I: The Generalized Method of Lines
	1. The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation
		1.1 Introduction
		1.2 On the numerical procedures for Ginzburg-Landau type ODEs
		1.3 Numerical results for related P.D.E.s
			1.3.1 A related P.D.E on a special class of domains
			1.3.2 About the matrix version of G.M.O.L.
			1.3.3 Numerical results for the concerning partial differential equation
		1.4 A numerical example concerning a Ginzburg-Landau type equation
			1.4.1 About the concerning improvement for the generalized method of lines
			1.4.2 Software in Mathematica for solving such an equation
		1.5 The generalized method of lines for a more general domain
		1.6 Conclusion
	2. An Approximate Proximal Numerical Procedure Concerning the Generalized Method of Lines
		2.1 Introduction
		2.2 The numerical method
		2.3 A numerical example
		2.4 A general proximal explicit approach
		2.5 Conclusion
	3. Approximate Numerical Procedures for the Navier-Stokes System through the Generalized Method of Lines
		3.1 Introduction
		3.2 Details about an equivalent elliptic system
		3.3 An approximate proximal approach
		3.4 A software in MATHEMATICA related to the previous algorithm
		3.5 The software and numerical results for a more specific example
		3.6 Numerical results through the original conception of the generalized method of lines for the Navier-Stokes system
		3.7 Conclusion
	4. An Approximate Numerical Method for Ordinary Differential Equation Systems with Applications to a Flight Mechanics Model
		4.1 Introduction
		4.2 Applications to a flight mechanics model
		4.3 Acknowledgements
Section II: Calculus of Variations, Convex Analysis and Restricted Optimization
	5. Basic Topics on the Calculus of Variations
		5.1 Banach spaces
		5.2 The Gâteaux variation
		5.3 Minimization of convex functionals
		5.4 Sufficient conditions of optimality for the convex case
		5.5 Natural conditions, problems with free extremals
		5.6 The du Bois-Reymond lemma
		5.7 Calculus of variations, the case of scalar functions on Rn
		5.8 The second Gâteaux variation
		5.9 First order necessary conditions for a local minimum
		5.10 Continuous functionals
		5.11 The Gâteaux variation, the formal proof of its formula
	6. More Topics on the Calculus of Variations
		6.1 Introductory remarks
		6.2 The Gâteaux variation, a more general case
		6.3 Fréchet differentiability
		6.4 The Legendre-Hadamard condition
		6.5 The Weierstrass condition for n = 1
		6.6 The Weierstrass condition, the general case
		6.7 The Weierstrass-Erdmann conditions
		6.8 Natural boundary conditions
	7. Convex Analysis and Duality Theory
		7.1 Convex sets and functions
		7.2 Weak lower semi-continuity
		7.3 Polar functionals and related topics on convex analysis
		7.4 The Legendre transform and the Legendre functional
		7.5 Duality in convex optimization
		7.6 The min-max theorem
		7.7 Relaxation for the scalar case
		7.8 Duality suitable for the vectorial case
			7.8.1 The Ekeland variational principle
		7.9 Some examples of duality theory in convex and non-convex analysis
	8. Constrained Variational Optimization
		8.1 Basic concepts
		8.2 Duality
		8.3 The Lagrange multiplier theorem
		8.4 Some examples concerning inequality constraints
		8.5 The Lagrange multiplier theorem for equality and inequality constraints
		8.6 Second order necessary conditions
		8.7 On the Banach fixed point theorem
		8.8 Sensitivity analysis
			8.8.1 Introduction
		8.9 The implicit function theorem
			8.9.1 The main results about Gâteaux differentiability
	9. On Lagrange Multiplier Theorems for Non-Smooth Optimization for a Large Class of Variational Models in Banach Spaces
		9.1 Introduction
		9.2 The Lagrange multiplier theorem for equality constraints and non-smooth optimization
		9.3 The Lagrange multiplier theorem for equality and inequality constraints for non-smooth optimization
		9.4 Conclusion
Section III: Duality Principles and Related Numerical Examples Through the Generalized Method of Lines
	10. A Convex Dual Formulation for a Large Class of Non-Convex Models in Variational Optimization
		10.1 Introduction
		10.2 The main duality principle, a convex dual variational formulation
	11. Duality Principles and Numerical Procedures for a Large Class of Non-Convex Models in the Calculus of Variations
		11.1 Introduction
		11.2 A general duality principle non-convex optimization
		11.3 Another duality principle for a simpler related model in phase transition with a respective numerical example
		11.4 A convex dual variational formulation for a third similar model
			11.4.1 The algorithm through which we have obtained the numerical results
		11.5 An improvement of the convexity conditions for a non-convex related model through an approximate primal formulation
		11.6 An exact convex dual variational formulation for a non-convex primal one
		11.7 Another primal dual formulation for a related model
		11.8 A third primal dual formulation for a related model
		11.9 A fourth primal dual formulation for a related model
		11.10 One more primal dual formulation for a related model
		11.11 Another primal dual formulation for a related model
	12. Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations
		12.1 Introduction
		12.2 The main duality principle, a convex dual formulation and the concerning proximal primal functional
		12.3 A primal dual variational formulation
		12.4 One more duality principle and a concerning primal dual variational formulation
			12.4.1 Introduction
			12.4.2 The main duality principle and a related primal dual variational formulation
		12.5 One more dual variational formulation
		12.6 Another dual variational formulation
		12.7 A related numerical computation through the generalized method of lines
			12.7.1 About a concerning improvement for the generalized method of lines
			12.7.2 Software in Mathematica for solving such an equation
			12.7.3 Some plots concerning the numerical results
		12.8 Conclusion
	13. A Note on the Korn’s Inequality in a N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model
		13.1 Introduction
		13.2 The main results, the Korn inequalities
		13.3 An existence result for a non-linear model of plates
		13.4 On the existence of a global minimizer
		13.5 Conclusion
References
Index