Functional Equations in Applied Sciences (Volume 199) (Mathematics in Science and Engineering, Volume 199)

✍ Scribed by Enrique Castillo, Andres Iglesias, Reyes Ruiz-Cobo

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Leaves: 407
• Edition: 1
• Category: Library

✦ Synopsis

The book provides the reader with the different types of functional equations that s/he can find in practice, showing, step by step, how they can be solved.
A general methodology for solving functional equations is provided in Chapter 2. The different types of functional equations are described and solved in Chapters 3 to 8. Many examples, coming from different fields, as geometry, science, engineering, economics, probability, statistics, etc, help the reader to change his/her mind in order to state problems as functional equations as an alternative to differential equations, and to state new problems in terms of functional equations or systems.
An interesting feature of the book is that it deals with functional networks, a powerful generalization of neural networks that allows solving many practical problems. The second part of the book, Chapters 9 to 13, is devoted to the applications of this important paradigm.
The book contains many examples and end of chapter exercises, that facilitates the understanding of the concepts and applications.

· A general methodology for solving functional equations is provided in Chapter 2.
· It deals with functional networks, a powerful generalization of neural networks.
· Many examples, coming from different fields, as geometry, science, engineering, economics, probability, statistics, etc, illustrate the concept of functional equation.
· Functional equations are presented as a powerful alternative to differential equations.
· The book contains end of chapter exercises.

✦ Table of Contents

Cover
Note from the Editor
Note from the Publisher
Contents
Preface
Functional Equations
Introduction and Motivation
Introduction
Some Examples of Functional Equations
First Example: Area of a Rectangle (Legendre (1791))
Second Example: Simple Interest
Third Example: Sum of the Internal Angles of a Polygon
Fourth Example: The Associativity Equation
Basic Concepts and Definitions
Exercises
Some Methods for Solving Functional Equations
Introduction
Replacement of Variables by Given Values
Transforming One or Several Variables
Transforming One or Several Functions
Using a More General Equation
Treating Some Variables as Constants
Inductive Methods
Iterative Methods
Separation of Variables
Reduction by Means of Analytical Techniques
Mixed Methods
Exercises
Equations for One Function of One Variable
Introduction
Homogeneous Functions
A General Type of Equation
Cauchy's Equations
Jensen's Equation
Generalizations of Cauchy's Equations
D'Alembert's Functional Equation
Linear Difference Equations
Solution of the Homogeneous Equation
Particular Solution of the Complete Equation
Exercises
Equations with Several Functions in One Variable
Introduction
Pexider's Equations
The Sum of Products Equation
Other Generalizations
Exercises
Equation for One Function of Several Variables
Introduction
Generalized Cauchy and Jensen Equations
Other Equations
Application to Iterative Methods
Some Examples
Exercises
Equations with Functions of Several Variables
Introduction
Generalized Pexider and Jensen Equations
Generalized Sincov Equation
A General Equation
The Associativity Equation
The Transitivity Equation
The Bisymmetry Equation
The Transformation Equation
Exercises
Functional Equations and Differential Equations
Introduction
A Motivating Example
From Functional to Differential Equations
Reduction to Ordinary Differential Equations
Reduction to Partial Differential Equations
From Difference to Differential Equations
From Differential to Functional Equations
Exact Approach
Taylor Series Approach
From Functional to Difference Equations
A New Approach to Physical and Engineering Problems
An Illustrative Example: The Case of Static Beams
Classical Approach: Differential Equations
New Approach: Functional Equations
Exercises
Vector and Matrix Equations
Introduction
Cauchy's Equation
Pexider's Equation
Sincov's Equation and Generalizations
Exercises
Applications of Functional Equations
Functional Networks
Introduction
Motivating Functional Networks
Elements of a Functional Network
Differences between Neural and Functional Networks
Working with Functional Networks
Model Selection in Functional Networks
Some Examples of the Functional Network Methodology
The Associative Example
The Uniqueness Model
The Separable Model
Serial Functional Model
Some Applications of Functional Networks
A Real Economical Example
Modelling Chaotic Maps
The Cubic Holmes Map
Higher Dimensional Separable Models: The Lorenz System
Noise Reduction
Retrieval of Masked Information
The Beam Example
First Alternative
Second Alternative
Exercises
Applications to Science and Engineering
Introduction
A Motivating Example
Laws of Science
A Statistical Model for Lifetime Analysis
Derivation of the Fatigue Model
Statistical Models for Fatigue Life of Longitudinal Elements
Model 1
Model 2
Model 3
Reaching a Consensus
Differential, Functional and Difference Equations
A Motivating Example
Equally Spaced Data
Unequally Spaced Data
Model 1
Model 2
Exercises
Applications to Geometry and CAGD
Introduction
Fundamental Formula for Polyhedra
Two Interesting Functions in Computer Graphics
Absolute Value Function
Integral Part Function
Geometric Invariants Given by Functional Equations
The Preserve Distance Functional Equation
The Functional Equation of Preserve Area
The Functional Equation of Angle Preserve
Using Functional Equations for CAGD
Preliminary Results
Families of Implicit Surfaces
Families of Explicit Surfaces
Some Other Surfaces
Gordon—Coons-Surfaces
Uniqueness Theorem
A Particular Case
Tensor-Product Surfaces
Application of Functional Networks to Fitting Surfaces
The Case of Parametric Surfaces
Exercises
Applications to Economics
Introduction
Price and Quantity Levels
Price Indices
Interest Rates
Demand Function. Price and Advertising Policies
The Monopoly Model
Eichhorn's Model
Model A4-A5
A Modified Eichhorn Model
Model A4-A5a
Duopoly Models
Duopoly Model I
Duopoly Model II
Taxation Functions
Restrictions to be Imposed on Taxation Functions
Compatible Tax Functions
Case 1: Same Taxes for Joint and Separate Statements
Case 2: Joint Statement Tax Amount Smaller than that for Separate Statements
Exercises
Applications to Probability and Statistics
Introduction
Bivariate Distributions with Normal Conditionals
Case of Independence
Bivariate Distributions with Gamma Conditionals
Other Equations
One Characterization of the Normal Distribution
Linear Regressions with Conditionals in Location- Scale Families
Estimation of a Multinomial Model
Sum of a Random Number of Discrete Random Variables
Particular Case
Bayesian Conjugate Distributions
Maximum Stability
Reproductivity
Reproductivity in Single Parameter Families
Reproductivity in Two-Parameter Families
Exercises
Bibliography
Index