Advanced Engineering Analysis

The following sections are included:

• Preface

• Contents

The following sections are included:

• Introduction

• Euler's Equation for the Simplest Problem

• Properties of Extremals of the Simplest Functional

• Ritz's Method

• Natural Boundary Conditions

• Extensions to More General Functionals

• Functionals Depending on Functions in Many Variables

• A Functional with Integrand Depending on Partial Derivatives of Higher Order

• The First Variation

• Isoperimetric Problems

• General Form of the First Variation

• Movable Ends of Extremals

• Broken Extremals: Weierstrass–Erdmann Conditions and Related Problems

• Sufficient Conditions for Minimum

• Exercises

The following sections are included:

• Elementary Problems for Elastic Structures

• Some Extremal Principles of Mechanics

• Conservation Laws

• Conservation Laws and Noether's Theorem

• Functionals Depending on Higher Derivatives of y

• Noether's Theorem, General Case

• Generalizations

• Exercises

The following sections are included:

• A Variational Problem as a Problem of Optimal Control

• General Problem of Optimal Control

• Simplest Problem of Optimal Control

• Fundamental Solution of a Linear Ordinary Differential Equation

• The Simplest Problem, Continued

• Pontryagin's Maximum Principle for the Simplest Problem

• Some Mathematical Preliminaries

• General Terminal Control Problem

• Pontryagin's Maximum Principle for the Terminal Optimal Problem

• Generalization of the Terminal Control Problem

• Small Variations of Control Function for Terminal Control Problem

• A Discrete Version of Small Variations of Control Function for Generalized Terminal Control Problem

• Optimal Time Control Problems

• Final Remarks on Control Problems

• Exercises

The following sections are included:

• A Normed Space as a Metric Space

• Dimension of a Linear Space and Separability

• Cauchy Sequences and Banach Spaces

• The Completion Theorem

• L^p Spaces and the Lebesgue Integral

• Sobolev Spaces

• Compactness

• Inner Product Spaces, Hilbert Spaces

• Operators and Functionals

• Contraction Mapping Principle

• Some Approximation Theory

• Orthogonal Decomposition of a Hilbert Space and the Riesz Representation Theorem

• Basis, Gram–Schmidt Procedure, and Fourier Series in Hilbert Space

• Weak Convergence

• Adjoint and Self-Adjoint Operators

• Compact Operators

• Closed Operators

• On the Sobolev Imbedding Theorem

• Some Energy Spaces in Mechanics

• Introduction to Spectral Concepts

• The Fredholm Theory in Hilbert Spaces

• Exercises

The following sections are included:

• Some Mechanics Problems from the Standpoint of the Calculus of Variations; the Virtual Work Principle

• Generalized Solution of the Equilibrium Problem for a Clamped Rod with Springs

• Equilibrium Problem for a Clamped Membrane and its Generalized Solution

• Equilibrium of a Free Membrane

• Some Other Equilibrium Problems of Linear Mechanics

• The Ritz and Bubnov–Galerkin Methods

• The Hamilton–Ostrogradski Principle and Generalized Setup of Dynamical Problems in Classical Mechanics

• Generalized Setup of Dynamic Problem for Membrane

• Other Dynamic Problems of Linear Mechanics

• The Fourier Method

• An Eigenfrequency Boundary Value Problem Arising in Linear Mechanics

• The Spectral Theorem

• The Fourier Method, Continued

• Equilibrium of a von Kármán Plate

• A Unilateral Problem

• Exercises

The following sections are included:

• Appendix A: Hints for Selected Exercises

• Bibliography

• Index