Soft Numerical Computing in Uncertain Dynamic Systems is intended for system specialists interested in dynamic systems that operate at different time scales. The book discusses several types of errors and their propagation, covering numerical methods--including convergence and consistence properties and characteristics--and proving of related theorems within the setting of soft computing. Several types of uncertainty representation like interval, fuzzy, type 2 fuzzy, granular, and combined uncertain sets are discussed in detail. The book can be used by engineering students in control and finite element fields, as well as all engineering, applied mathematics, economics, and computer science students.
One of the important topics in applied science is dynamic systems and their applications. The authors develop these models and deliver solutions with the aid of numerical methods. Since they are inherently uncertain, soft computations are of high relevance here. This is the reason behind investigating soft numerical computing in dynamic systems. If these systems are involved with complex-uncertain data, they will be more practical and important. Real-life problems work with this type of data and most of them cannot be solved exactly and easily--sometimes they are impossible to solve.
Clearly, all the numerical methods need to consider error of approximation. Other important applied topics involving uncertain dynamic systems include image processing and pattern recognition, which can benefit from uncertain dynamic systems as well. In fact, the main objective is to determine the coefficients of a matrix that acts as the frame in the image. One of the effective methods exhibiting high accuracy is to use finite differences to fill the cells of the matrix.
Author(s): Tofigh Allahviranloo; Witold Pedrycz
Publisher: Academic Press
Year: 2020
Language: English
Pages: 388
City: London
Front Matter
Copyright
Dedication
Preface
Introduction
Introduction
Introduction to uncertain dynamic systems
History
Structure of the book
References
Uncertain sets
Short introduction to this chapter
Textual short outline
Measures
Measurable space
Examples
Uncertain sets and variables
Examples
Zigzag uncertain variable
Experimental uncertain variables
Membership function
Fuzzy numbers and their properties
Definition of a fuzzy number
Level-wise form of a fuzzy number
Definition of a fuzzy number in level-wise form
Definition of a fuzzy number in level-wise form
A singleton fuzzy number
Definition of a fuzzy number in parametric form
Nonlinear fuzzy number
Trapezoidal fuzzy number
Triangular fuzzy number
Operations on level-wise form of fuzzy numbers
Summation
Example
Multiplication
Difference
Hukuhara difference
Example
Example
Generalized Hukuhara difference
The level-wise form of generalized difference
Some properties of gH-difference
Example
Partial ordering
Some properties of partial ordering
Absolute value of a fuzzy number
Some properties of partial ordering in gH-difference
Approximately generalized Hukuhara difference
Some properties of g-difference
Example
Example
Generalized division
Some properties of division
Examples
Approximately generalized division
Example
Piece-wise membership function
Some properties of addition and scalar product on fuzzy numbers
Definition-singleton fuzzy number
Advanced uncertainties and their properties
Pseudo-octagonal sets
Z-process
Definition-Z-process
Example
Example
Computations on Z-numbers
Summation of two Z-numbers
Difference of two Z-numbers
Multiplication of two Z-numbers
Division of two Z-numbers
Level-wise form of a Z-number
High membership degree does have high reliability
Definition-Level-wise form of a standard Z-number
Summation in level-wise form
Scalar multiplication in level-wise form
Hukuhara difference in level-wise form
Generalized Hukuhara difference in level-wise form
Some properties of generalized Hukuhara
References
Further reading
Soft computing with uncertain sets
Introduction
Expected value
Distance of two fuzzy numbers
P-distance
Hausdorff Distance
Limit of fuzzy number valued functions
Definition-Fuzzy set valued function
Definition-Fuzzy number valued function
Definition-The limit of Fuzzy number valued function
Theorem-Limit of summation of functions
Theorem-Limit of difference of functions
Theorem-Limit of multiplication
Other properties of limit
Fuzzy Riemann integral operator
Some properties of fuzzy Riemann integral
Differential operator
Definition-gH-differentiability
Example
Example
Definition-gH-differentiability in level-wise form
Definition-Switching points of gH-differentiability
Example
Proposition-Summation in gH-differentiability
Proposition-Difference in gH-differentiability
Proposition-Production in gH-differentiability
Proposition-Composition of gH-differentiability
Proposition-Minimum and maximum
Definition-Continuous fuzzy number valued function
Proposition
Proposition-Cauchy's fuzzy mean value theorem
Corollary-Fuzzy mean value theorem
Proposition-Increasing and decreasing function
Proposition-Integral of gH-differentiability
Proposition-Switching points in integration
High order differentiability
Extended integral relation
Part-by-part integration
Taylor expansion
Example
gH-partial differentiability
Example
Another simple example
Level-wise form of gH-partial differentiability
Switching point in gH-partial differentiability
Example
Higher order of gH-partial differentiability
Integral relation in gH-partial differentiability
Multivariate fuzzy chain rule in gH-partial differentiability
The fuzzy Laplace transform operator
Example
Definition-Absolutely convergence
First translation theorem
Second translation theorem
Laplace transform on the derivative
Derivative theorem
High order derivation theorem
Fuzzy improper integral
Definition-Uniform convergence
Theorem-Interchanging integrals
Theorem-Integral and derivative
Fourier transform operator
Definition-Fuzzy Fourier transform
Example-Fuzzy Fourier transform
Definition-Fuzzy inverse Fourier transform
Theorem-Existence
Theorem-Linearity property
Theorem-Fourier transform of gH-derivative
References
Continuous numerical solutions of uncertain differential equations
Introduction
Uncertain differential equations
Definition-Uncertain process as a canonical Liu process
Definition-Liu integral of an uncertain process
Theorem-Chain rule
Theorem-Integration by parts
Definition-Uncertain differential equation
Remark
Fuzzy differential equations
Theorem-Existence and uniqueness
Fuzzy differential equations-Variation of constants
Theorem-Existence of the solution
Length function
Definition-Length function
Nonlinear property
Theorem-Nonlinear property of fuzzy functions
Remark
Remark-Differentiability and length
Theorem-Nonlinear property of fuzzy functions
Theorem-Derivative of integral equation
The length function-Fuzzy differential equations
Fuzzy differential equations-Laplace transform
Fuzzy differential equations-Second order
Fuzzy differential equations-Variational iteration method
Fuzzy differential equations-Legendre differential equation
Definition-Power series with fuzzy coefficients
Some properties of fuzzy series
Fuzzy calculated operations
Fuzzy power series method for solving Legendre's equation
Linear systems of fuzzy differential equations
Homogeneous fuzzy linear differential systems
Nonhomogeneous fuzzy linear differential systems
Reduction of a second order fuzzy differential equations to a system of first order equations
Z-differential equations
References
Discrete numerical solutions of uncertain differential equations
Introduction
Fuzzy Euler method
Analysis of the fuzzy Euler method
Local truncation error and consistency
Global truncation error and convergence
Theorem-Convergence
Stability
Fuzzy modified Euler method
Analysis of the fuzzy modified Euler method
Local truncation error and consistency
Global truncation error and convergence
Theorem-Convergence
Stability of the modified fuzzy Euler method
Fuzzy Euler method for fuzzy hybrid differential equations
Fuzzy Euler method for fuzzy impulsive differential equations
Error analysis
Stability
Fuzzy predictor and corrector methods
Definition-Fuzzy explicit method
Definition-Fuzzy implicit method
Fuzzy explicit three steps method
Fuzzy implicit two steps method
Fuzzy predictor and corrector three steps methods
Numerical solution of fuzzy nth-order differential equations
References
Further reading
Numerical solutions of uncertain fractional differential equations
Introduction
Fuzzy Riemann-Liouville Derivative-Fuzzy RL Derivative
Note-Combination Property
Level-Wise form of Fuzzy Riemann-Liouville Integral Operators
The RL Fractional Integral Operator
The Fuzzy Riemann-Liouville Derivative Operators
Fuzzy Caputo Fractional Derivative
Caputo gH-Differentiability
Caputo-Katugampola gH-Fractional Derivative
Fuzzy Fractional Differential Equations-Caputo-Katugampola Derivative
Definition-Fuzzy Fractional Differential Equations
Existence and Uniqueness of the Solution
Theorem-Existence and Uniqueness in Real Fractional Differential Equation
Theorem-Existence and Uniqueness in Fuzzy Fractional Differential Equation
Some Properties of the Mittag-Leffler Function
Fuzzy Generalized TaylorS Expansion
Fuzzy Fractional Euler Method
References
Numerical solutions of uncertain partial differential equations
Introduction
Partial ordering
Continuity
Minimum and maximum
Production in partial gH-differentiability
Fuzzy integrating factor
The fuzzy heat equation
Theorem-Fuzzy maximum principle
Theorem-Existence
Analytical solution of the fuzzy heat equation
The fundamental solution of the fuzzy heat equation
Fuzzy Fourier transform
Fuzzy inverse Fourier transform
Fourier transform of gH-derivative
Fuzzy finite difference method for solving the fuzzy Poisson's equation
Theorem-Uniqueness
Error analysis
References
Index
A
B
C
D
E
F
G
H
I
L
M
N
P
R
S
T
U
V
W
Z
