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E-raamat: Fuzzy Fractional Differential Operators and Equations: Fuzzy Fractional Differential Equations

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Fuzzy Fractional Differential Operators and Equations: Fuzzy Fractional Differential EquationsSuurem pilt
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This book contains new and useful materials concerning fuzzy fractional differential and integral operators and their relationship. As the title of the book suggests, the fuzzy subject matter is one of the most important tools discussed. Therefore, it begins by providing a brief but important and new description of fuzzy sets and the computational calculus they require.





 





Fuzzy fractals and fractional operators have a broad range of applications in the engineering, medical and economic sciences. Although these operators have been addressed briefly in previous papers, this book represents the first comprehensive collection of all relevant explanations.





 





Most of the real problems in the biological and engineering sciences involve dynamic models, which are defined by fuzzy fractional operators in the form of fuzzy fractional initial value problems. Another important goal of this book is to solve these systems and analyze their solutions both theoretically and numerically. Given the content covered, the book will benefit all researchers and students in the mathematical and computer sciences, but also the engineering sciences.

Arvustused

This book can be useful for researchers but also for those who want to initiate in the field of fuzzy fractional differential equations. (Vasile Lupulescu, zbMATH 1482.34001, 2022) In this book, the first of its kind in the literature, the author presents some significant results obtained in the theory of fuzzy fractional differential equations, while emphasizing his own contributions. this reviewer believes that this book can be useful for researchers as well as for those who want an initiation to the field of fuzzy fractional differential equations. (Vasile Lupulescu, Mathematical Reviews, April, 2022)

1 Introduction to Fuzzy Fractional Operators and Equations
1(6)
1.1 Introduction
1(3)
1.2 Introduction to Fuzzy Fractional Differential Equations
4(2)
1.3 Structure of the Book
6(1)
2 Fuzzy Sets
7(66)
2.1 Introduction
7(1)
2.2 Fuzzy Sets and Variables
7(5)
2.2.1 Membership Function
12(1)
2.3 Fuzzy Numbers and Their Properties
12(29)
2.3.1 Definition of a Fuzzy Number
12(1)
2.3.2 Level-Wise Form of a Fuzzy Number
13(1)
2.3.3 Definition of a Fuzzy Number in Level-Wise Form
14(1)
2.3.4 Definition of a Fuzzy Number in Level-Wise Form
15(1)
2.3.5 Definition of a Fuzzy Number in Parametric Form
16(2)
2.3.6 Non-linear Fuzzy Number
18(1)
2.3.7 Trapezoidal Fuzzy Number
18(1)
2.3.8 Triangular Fuzzy Number
19(2)
2.3.9 Operations on Level-Wise Form of Fuzzy Numbers
21(18)
2.3.10 Piece Wise Membership Function
39(1)
2.3.11 Some Properties of Addition and Scalar Product on Fuzzy Numbers
39(2)
2.4 Some Operators on Fuzzy Numbers
41(32)
2.4.1 Distance
41(2)
2.4.2 Limit of Fuzzy Number Valued Functions
43(4)
2.4.3 Fuzzy Riemann Integral Operator
47(2)
2.4.4 Some Additional Properties of gH-Difference
49(5)
2.4.5 Proposition--Minimum and Maximum
54(1)
2.4.6 Proposition--Cauchy's Fuzzy Mean Value Theorem
54(1)
2.4.7 Corollary--Fuzzy Mean Value Theorem
55(1)
2.4.8 Integral Relation
56(3)
2.4.9 Fuzzy Taylor Expansion of Order One
59(5)
2.4.10 Integration by Part
64(1)
2.4.11 Definition--gH-Partial Differentiability
65(3)
2.4.12 Fuzzy Fubini--Theorem
68(1)
2.4.13 First Order Fuzzy Taylor Expansion for Two Variables Function
69(4)
3 Fuzzy Fractional Operators
73(54)
3.1 Introduction
73(1)
3.2 Fuzzy Grunwald-Letnikov Derivative--Fuzzy GL Derivative
73(15)
3.2.1 Definition--Length of Fuzzy Function
75(7)
3.2.2 Level-Wise Form of Grunwald-Letnikov GL-Derivative
82(5)
3.2.3 Remark--Fuzzy Fractional Integral Operator
87(1)
3.3 Fuzzy Riemann-Liouville Derivative--Fuzzy RL Derivative
88(11)
3.3.1 Level-Wise Form of Fuzzy Riemann-Liouville Integral Operators
92(1)
3.3.2 The Fuzzy Riemann-Liouville Derivative Operators
93(5)
3.3.3 RL--Fractional Derivative for m = 1
98(1)
3.4 Fuzzy Caputo Fractional Derivative
99(4)
3.4.1 Caputo--Fuzzy Fractional Derivative for m = 1
100(1)
3.4.2 Caputo gH Differentiability
101(2)
3.5 Fuzzy Riemann-Liouville Generalized Fractional Derivative
103(6)
3.5.1 Fuzzy Riemann-Liouville Generalized Fractional Integral
103(2)
3.5.2 Riemann Liouville-Katugampola gH--Fractional Derivative
105(4)
3.6 Caputo-Katugampola gH-Fractional Derivative
109(6)
3.7 Riemann-Liouville gH-Fractional Derivative of at C (1,2)
115(9)
3.7.1 Definition--Fuzzy Caputo Fractional Derivative of Order a C (1.2)
121(3)
3.8 Generalized Fuzzy ABC Fractional Derivative
124(3)
4 Fuzzy Fractional Differential Equations
127(66)
4.1 Introduction
127(1)
4.2 Fuzzy Fractional Differential Equations Caputo-Katugampola Derivative
128(19)
4.2.1 Existence and Uniqueness of the Solution
137(8)
4.2.2 Some Properties of Mittag-Lefflcr Function
145(2)
4.3 Fuzzy Fractional Differential Equations--Laplace Transforms
147(22)
4.3.1 Definition--Absolutely Convergence
150(1)
4.3.2 Definition--Exponential Order
150(1)
4.3.3 Some Properties of Laplace
151(5)
4.3.4 Convolution Theorem
156(2)
4.3.5 First Translation Theorem
158(1)
4.3.6 Second Translation Theorem
158(1)
4.3.7 Remark--Laplace Forms of Fractional Derivatives
159(7)
4.3.8 Fuzzy Fourier Transform Operator
166(1)
4.3.9 Existence of Fourier Transform
166(3)
4.4 Fuzzy Solutions of Time-Fractional Problems
169(5)
4.4.1 Fuzzy Explicit Solution of the Time-Fractional Problem
170(4)
4.5 Fuzzy Impulsive Fractional Differential Equations
174(7)
4.6 Concrete Solution of Fractional Differential Equations
181(12)
4.6.1 Fractional Hyperbolic Functions
181(1)
4.6.2 Some Derivation Rules for the Fractional Hyperbolic Functions
182(11)
5 Numerical Solution of Fuzzy Fractional Differential Equations
193(66)
5.1 Introduction
193(1)
5.2 Preliminaries
193(3)
5.2.1 Fuzzy Mean Value Theorem for Riemann-Liouville Integral
196(1)
5.3 Fuzzy Fractional Taylor's Expansion with Caputo gH-Derivative
196(16)
5.3.1 Fuzzy Fractional Taylor Expansion
201(8)
5.3.2 Fuzzy Generalized Taylor's Expansion
209(3)
5.4 Fuzzy Fractional Euler with Caputo gH-Derivative
212(11)
5.5 ABC-PI Numerical Method with ABC gH-Derivative
223(13)
5.5.1 Definition--ABCgh Fractional Derivative in the Sense of Caputo Derivative
223(3)
5.5.2 Fuzzy Time Fractional Ordinary Differential Equation
226(2)
5.5.3 Remark--Uniqueness
228(1)
5.5.4 An Efficient Numerical Method for ABC Fractional Problems
229(7)
5.6 Numerical Method for Fuzzy Fractional Impulsive Differential Equations
236(23)
5.6.1 Remark--Uniqueness
238(1)
5.6.2 Reproducing Kernel Hilbert Space Method (RKHSM)
239(2)
5.6.3 Numerical Solving Fuzzy Fractional Impulsive Differential Equation in W22 [ 0, 1]
241(8)
5.6.4 Combination of RKHM and FDTM
249(3)
5.6.5 Algorithm
252(7)
6 Applications of Fuzzy Fractional Differential Equations
259(32)
6.1 Introduction
259(1)
6.2 Fuzzy Fractional Calculus--Preliminaries for Control Problem
259(8)
6.2.1 Theorem--Interchanging Operators
264(1)
6.2.2 Theorem--Fuzzy Fractional Integration by Part 1
265(1)
6.2.3 Theorem--Fuzzy Fractional Integration by Part II
266(1)
6.3 Fuzzy Optimal Control Problem
267(11)
6.3.1 Definition--Relative Extremum of Fuzzy Functional Function
268(1)
6.3.2 Theorem--Fuzzy Fundamental Theorem of Calculus of Variation
268(1)
6.3.3 Necessary Optimality Conditions
269(4)
6.3.4 Sufficient Optimality Conditions
273(5)
6.4 Fuzzy Fractional Diffusion Equations
278(13)
6.4.1 Remark--Laplace Transform of Caputo Derivative with Order 0 < α < 2
283(1)
6.4.2 Fundamental Solution of Fuzzy Fractional Diffusion Equation
283(2)
6.4.3 Theorem--Fundamental Solution
285(3)
6.4.4 Application of Fuzzy Fractional Diffusion in Drug Release
288(3)
Bibliography 291
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