Preface
Acknowledgments
1 Introduction
1.1 Literature Review
1.1.1 Interval Analysis
1.1.2 Fuzzy Set Theory
1.1.3 Affine Arithmetic
1.1.4 Uncertain Static Problems
1.1.5 Uncertain Linear Dynamic Problems
1.1.6 Uncertain Nonlinear Dynamic Problems
1.2 References
2 Preliminaries
2.1 Interval
2.2 Basic Terminologies of Interval
2.2.1 Interval Center (xc)
2.2.2 Interval Width (xω)
2.2.3 Interval Radius (X△)
2.3 Equality of Intervals
2.4 Interval Set Operations
2.4.1 Interval Union
2.4.2 Interval Intersection
2.4.3 Interval Hull
2.5 Interval Arithmetic
2.5.1 Interval Addition
2.5.2 Interval Subtraction
2.5.3 Interval Scalar Multiplication
2.5.4 Interval Multiplication
2.5.5 Interval Reciprocal
2.5.6 Interval Division
2.5.7 Interval Power
2.6 Interval Algebraic Properties
2.6.1 Commutative
2.6.2 Associative
2.6.3 Identity
2.6.4 Inverse
2.6.5 Sub-Distributive
2.6.6 Cancellation Law
2.7 Fuzzy Sets
2.8 Basic Terminologies of Fuzzy Set
2.8.1 Boundary (B(F))
2.8.2 Support (S(F))
2.8.3 Core (C(F))
2.8.4 Height (H(F))
2.9 Fuzzy Set Operations
2.9.1 Fuzzy Union
2.9.2 Fuzzy Intersection
2.9.3 Fuzzy Complement
2.10 Arithmetic of Fuzzy Sets
2.10.1 Algebraic Sum
2.10.2 Algebraic Product
2.10.3 Bounded Sum
2.10.4 Bounded Difference
2.10.5 Power
2.11 Fuzzy Numbers
2.12 Different Types of Fuzzy Numbers
2.12.1 Triangular Fuzzy Number (TFN)
2.12.2 Trapezoidal Fuzzy Number (TrFN)
2.13 Exercises
2.14 References
3 Affine Arithmetic
3.1 Interval Dependency Problem
3.2 Definition and Notation
3.3 Conversion of Affine to Interval
3.4 Conversion of Interval to Affine
3.5 Affine Arithmetic Operations
3.5.1 Affine Addition
3.5.2 Affine Subtraction
3.5.3 Affine Scalar Multiplication
3.5.4 Affine Multiplication
3.5.5 Affine Division
3.6 Efficacy of Affine Arithmetic
3.7 Exercises
3.8 References
4 Fuzz-y-Afline Arithmetic
4.1 a-Cut of Fuzzy Number
4.2 Parametric Form of Fuzzy Number
4.2.1 Parametric Form of TFN
4.2.2 Parametric Form of TrFN
4.3 Basic Terminologies of Parametric Fuzzy Number
4.3.1 Parametric Fuzzy Center (Pc)
4.3.2 Parametric Fuzzy Width (Fω)
4.3.3 Parametric Fuzzy Radius (F△)
4.4 Parametric Fuzzy Arithmetic
4.4.1 Parametric Fuzzy Addition
4.4.2 Parametric Fuzzy Subtraction
4.4.3 Parametric Fuzzy Scalar Multiplication
4.4.4 Parametric Fuzzy Multiplication
4.4.5 Parametric Fuzzy Reciprocal
4.4.6 Parametric Fuzzy Division
4.4.7 Parametric Fuzzy Power
4.5 Fuzzy-Affine Form
4.5.1 Fuzzy-Affine Form of TFN
4.5.2 Fuzzy-Affine Form of TrFN
4.6 Fuzzy-Affine Arithmetic Operations
4.7 Efficacy of Fuzzy-A/fine Arithmetic
4.8 Exercises
4.9 References
5 Uncertain Static Problems
5.1 Crisp System of Linear Equations (CSLE)
5.2 Interval System of Linear Equations (ISLE)
5.3 Solution of ISLE
5.4 Fuzzy System of Linear Equations (FSLE)
5.5 Solution of FSLE
5.6 Exercises
5.7 References
6 Uncertain Linear Dynamic Problems
6.1 Linear Eigenvalue Problem (LEP)
6.2 Crisp Generalized Eigenvalue Problem (CGEP)
6.2.1 Crisp Standard Eigenvalue Problem (CSEP)
6.3 Solution of CGEP
6.4 Interval Generalized Eigenvalue Problem (IGEP)
6.4.1 Interval Standard Eigenvalue Problem (ISEP)
6.5 Solution ofIGEP
6.6 Fuzzy Generalized Eigenvalue Problem (FGEP)
6.6.1 Fuzzy Standard Eigenvalue Problem (FSEP)
6.7 Solution of FGEP
6.8 Exercises
6.9 References
7 Uncertain Nonlinear Dynamic Problems
7.1 Crisp Nonlinear Eigenvalue Problem (CNEP)
7.1.1 Crisp Quadratic Eigenvalue Problem (CQ_EP)
7.1.2 Crisp Cub