1st Edition
The Numerical Method of Lines and Duality Principles Applied to Models in Physics and Engineering
The book includes theoretical and applied results of a generalization of the numerical method of lines. A Ginzburg-Landau type equation comprises the initial application, with detailed explanations about the establishment of the general line expressions. Approximate numerical procedures have been developed for a variety of equation types, including the related algorithms and software. The applications include the Ginzburg-Landau system in superconductivity, applications to the Navier-Stokes system in fluid mechanics and, among others, models in flight mechanics. In its second and final parts, the book develops duality principles and numerical results for other similar and related models.
The book is meant for applied mathematicians, physicists and engineers interested in numerical methods and concerning duality theory. It is expected the text will serve as a valuable auxiliary project tool for some important engineering and physics fields of research.
SECTION I: THE GENERALIZED METHOD OF LINES. The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation. An Approximate Proximal Numerical Procedure Concerning the Generalized Method of Lines. Approximate Numerical Procedures for the Navier-Stokes System through the Generalized Method of Lines. An Approximate Numerical Method for Ordinary Differential Equation Systems with Applications to a Flight Mechanics Model. SECTION II: CALCULUS OF VARIATIONS, CONVEX ANALYSIS AND RESTRICTED OPTIMIZATION. Basic Topics on the Calculus of Variations. More topics on the Calculus of Variations. Convex Analysis and Duality Theory. Constrained Variational Optimization. On Lagrange Multiplier Theorems for Non-Smooth Optimization for a Large Class of Variational Models in Banach Spaces. SECTION III: DUALITY PRINCIPLES AND RELATED NUMERICAL EXAMPLES THROUGH THE ENERALIZED METHOD OF LINES. A Convex Dual Formulation for a Large Class of Non-Convex Models in Variational Optimization. Duality Principles and Numerical Procedures for a Large Class of Non-Convex Models in the Calculus of Variations. Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations. A Note on the Korn’s Inequality in a n-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model. References.
Biography
Fabio Silva Botelho obtained a Ph.D. in Mathematics from Virginia Tech, USA in 2009. Prior to that he got an undergraduate (1992) and master degrees (1996) in Aeronautical Engineering both from the Technological Institute of Aeronautics, ITA, SP, Brazil.
From 2004 to 2015 he worked as an Assistant Professor at the Mathematics Department of Federal University of Pelotas in Brazil. Since April 2015, he has been working as an Adjunct Professor at the Mathematics Department of Federal University of Santa Catarina, in Florianopolis, SC, Brazil.
He is the author of three books - Functional Analysis and Applied Optimization in Banach Spaces (2014), Real Analysis and Applications (2018) published by Springer; and Functional Analysis, Calculus Variations and Numerical Methods for Models in Physics and Engineering (2020), published by CRC Press.
