Affine Arithmetic Based Solution of Uncertain Static and Dynamic Problems (Synthesis Lectures on Mathematics and Statistics)

✍ Scribed by Snehashish Chakraverty, Saudamini Rout

• Publisher: Morgan & Claypool Publishers
• Year: 2020
• Tongue: English
• Leaves: 172
• Category: Library

✦ Synopsis

Uncertainty is an inseparable component of almost every measurement and occurrence when dealing with real-world problems. Finding solutions to real-life problems in an uncertain environment is a difficult and challenging task. As such, this book addresses the solution of uncertain static and dynamic problems based on affine arithmetic approaches. Affine arithmetic is one of the recent developments designed to handle such uncertainties in a different manner which may be useful for overcoming the dependency problem and may compute better enclosures of the solutions. Further, uncertain static and dynamic problems turn into interval and/or fuzzy linear/nonlinear systems of equations and eigenvalue problems, respectively. Accordingly, this book includes newly developed efficient methods to handle the said problems based on the affine and interval/fuzzy approach. Various illustrative examples concerning static and dynamic problems of structures have been investigated in order to show the reliability and efficacy of the developed approaches.

✦ Table of Contents

Preface
Acknowledgments
Introduction
Literature Review
Interval Analysis
Fuzzy Set Theory
Affine Arithmetic
Uncertain Static Problems
Uncertain Linear Dynamic Problems
Uncertain Nonlinear Dynamic Problems
References
Preliminaries
Interval
Basic Terminologies of Interval
Interval Center (x_c)
Interval Width (x_w)
Interval Radius (x_)
Equality of Intervals
Interval Set Operations
Interval Union
Interval Intersection
Interval Hull
Interval Arithmetic
Interval Addition
Interval Subtraction
Interval Scalar Multiplication
Interval Multiplication
Interval Reciprocal
Interval Division
Interval Power
Interval Algebraic Properties
Commutative
Associative
Identity
Inverse
Sub-Distributive
Cancellation Law
Fuzzy Sets
Basic Terminologies of Fuzzy Set
Boundary (B())
Support (S())
Core (C())
Height (H())
Fuzzy Set Operations
Fuzzy Union
Fuzzy Intersection
Fuzzy Complement
Arithmetic of Fuzzy Sets
Algebraic Sum
Algebraic Product
Bounded Sum
Bounded Difference
Power
Fuzzy Numbers
Different Types of Fuzzy Numbers
Triangular Fuzzy Number (TFN)
Trapezoidal Fuzzy Number (TrFN)
Exercises
References
Affine Arithmetic
Interval Dependency Problem
Definition and Notation
Conversion of Affine to Interval
Conversion of Interval to Affine
Affine Arithmetic Operations
Affine Addition
Affine Subtraction
Affine Scalar Multiplication
Affine Multiplication
Affine Division
Efficacy of Affine Arithmetic
Exercises
References
Fuzzy-Affine Arithmetic
a-Cut of Fuzzy Number
Parametric Form of Fuzzy Number
Parametric Form of TFN
Parametric Form of TrFN
Basic Terminologies of Parametric Fuzzy Number
Parametric Fuzzy Center (c)
Parametric Fuzzy Width (_w)
Parametric Fuzzy Radius ()
Parametric Fuzzy Arithmetic
Parametric Fuzzy Addition
Parametric Fuzzy Subtraction
Parametric Fuzzy Scalar Multiplication
Parametric Fuzzy Multiplication
Parametric Fuzzy Reciprocal
Parametric Fuzzy Division
Parametric Fuzzy Power
Fuzzy-Affine Form
Fuzzy-Affine Form of TFN
Fuzzy-Affine Form of TrFN
Fuzzy-Affine Arithmetic Operations
Efficacy of Fuzzy-Affine Arithmetic
Exercises
References
Uncertain Static Problems
Crisp System of Linear Equations (CSLE)
Interval System of Linear Equations (ISLE)
Solution of ISLE
Fuzzy System of Linear Equations (FSLE)
Solution of FSLE
Exercises
References
Uncertain Linear Dynamic Problems
Linear Eigenvalue Problem (LEP)
Crisp Generalized Eigenvalue Problem (CGEP)
Crisp Standard Eigenvalue Problem (CSEP)
Solution of CGEP
Interval Generalized Eigenvalue Problem (IGEP)
Interval Standard Eigenvalue Problem (ISEP)
Solution of IGEP
Fuzzy Generalized Eigenvalue Problem (FGEP)
Fuzzy Standard Eigenvalue Problem (FSEP)
Solution of FGEP
Exercises
References
Uncertain Nonlinear Dynamic Problems
Crisp Nonlinear Eigenvalue Problem (CNEP)
Crisp Quadratic Eigenvalue Problem (CQEP)
Crisp Cubic Eigenvalue Problem (CCEP)
Linearization of CNEP
Solution of CNEP
Interval Nonlinear Eigenvalue Problem (INEP)
Interval Quadratic Eigenvalue Problem (IQEP)
Interval Cubic Eigenvalue Problem (ICEP)
Solution of INEP
Fuzzy Nonlinear Eigenvalue Problem (FNEP)
Fuzzy Quadratic Eigenvalue Problem (FQEP)
Fuzzy Cubic Eigenvalue Problem (FCEP)
Solution of FNEP
Exercises
References
Authors' Biographies
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