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E-raamat: Fuzzy Differential Equations in Various Approaches

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  • Formaat: PDF+DRM
  • Sari: SpringerBriefs in Mathematics
  • Ilmumisaeg: 07-Sep-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319225753
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Fuzzy Differential Equations in Various ApproachesSuurem pilt
  • Formaat: PDF+DRM
  • Sari: SpringerBriefs in Mathematics
  • Ilmumisaeg: 07-Sep-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319225753
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This book may be used as reference for graduate students interested in fuzzy differential equations and researchers working in fuzzy sets and systems, dynamical systems, uncertainty analysis, and applications of uncertain dynamical systems. Beginning with a historical overview and introduction to fundamental notions of fuzzy sets, including different possibilities of fuzzy differentiation and metric spaces, this book moves on to an overview of fuzzy calculus thorough exposition and comparison of different approaches. Innovative theories of fuzzy calculus and fuzzy differential equations using fuzzy bunches of functions are introduced and explored. Launching with a brief review of essential theories, this book investigates both well-known and novel approaches in this field; such as the Hukuhara differentiability and its generalizations as well as differential inclusions and Zadeh’s extension. Through a unique analysis, results of all these theories are examined and compared.

Arvustused

The book under review is aimed at presenting several interpretations of fuzzy initial value problems (FIVPs) for FDEs. Each chapter is supplied with examples that allow for intuition in the theory of FDEs. Also, each chapter ends with its own list of references. This publication is a fine introductory textbook for everyone who is interested in the theory of deterministic differential equations operating in a fuzzy environment. (Marek T. Malinowski, Mathematical Reviews, May, 2016)

The book Fuzzy differential equations in various approaches focuses on fuzzy differential equations (FDEs) and explains the basics of various approaches of FDEs. Sufficient references are given at the end of each chapter and a small index is provided in the book. This precisely written book is concise and includes many examples to illustrate different approaches of FDEs. This book is worthy and can be recommended for graduate students or for researchers who work with uncertain dynamical systems. (Krishnan Balachandran, zbMATH 1338.34005, 2016)

1 Introduction
1(10)
1.1 Initial Value Problems
3(1)
1.2 Fuzzy Initial Value Problem
3(8)
1.2.1 Historical Overview
6(2)
References
8(3)
2 Basic Concepts
11(30)
2.1 Fuzzy Subsets
11(5)
2.2 Extension Principle
16(3)
2.3 Fuzzy Arithmetics for Fuzzy Numbers
19(6)
2.3.1 Standard Interval Arithmetic and Extension Principle
20(1)
2.3.2 Interactive Arithmetic
21(1)
2.3.3 Constraint Interval Arithmetic
22(1)
2.3.4 Hukuhara and Generalized Differences
23(2)
2.4 Fuzzy Metric Spaces
25(3)
2.5 Fuzzy Functions
28(10)
2.6 Summary
38(3)
References
39(2)
3 Fuzzy Calculus
41(28)
3.1 Fuzzy Calculus for Fuzzy-Set-Valued Functions
41(9)
3.1.1 Integrals
41(2)
3.1.2 Derivatives
43(6)
3.1.3 Fundamental Theorem of Calculus
49(1)
3.2 Fuzzy Calculus for Fuzzy Bunches of Functions
50(8)
3.2.1 Integral
50(3)
3.2.2 Derivative
53(4)
3.2.3 Fundamental Theorem of Calculus
57(1)
3.3 Comparison
58(8)
3.4 Summary
66(3)
References
66(3)
4 Fuzzy Differential Equations
69(46)
4.1 Approaches of FIVPs
69(5)
4.1.1 Fuzzy Differential Equations with Fuzzy Derivatives
70(1)
4.1.2 Fuzzy Differential Inclusions
71(1)
4.1.3 Extension of the Solution
72(2)
4.2 Hukuhara Derivative
74(9)
4.3 Strongly Generalized Derivative
83(8)
4.4 Fuzzy Differential Inclusions
91(5)
4.5 Extension of the Solution
96(5)
4.5.1 Autonomous FIVP with Fuzzy Initial Condition
98(1)
4.5.2 FIVP with Fuzzy Initial Condition and Fuzzy Parameter
99(2)
4.6 Extension of the Derivative Operator
101(10)
4.7 Summary
111(4)
References
112(3)
A Mathematical Background
115(2)
A.1 Continuity and Semicontinuity
115(1)
A.2 Spaces of Functions
115(2)
References 117(2)
Index 119
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