Fractional Order Analysis (eBook, ePUB)

Theory, Methods and Applications
Redaktion: Dutta, Hemen; Atangana, Abdon; Akdemir, Ahmet Ocak

• Format: ePub

Produktbeschreibung

A guide to the new research in the field of fractional order analysis Fractional Order Analysis contains the most recent research findings in fractional order analysis and its applications. The authors--noted experts on the topic--offer an examination of the theory, methods, applications, and the modern tools and techniques in the field of fractional order analysis. The information, tools, and applications presented can help develop mathematical methods and models with better accuracy. Comprehensive in scope, the book covers a range of topics including: new fractional operators, fractional derivatives, fractional differential equations, inequalities for different fractional derivatives and fractional integrals, fractional modeling related to transmission of Malaria, and dynamics of Zika virus with various fractional derivatives, and more. Designed to be an accessible text, several useful, relevant and connected topics can be found in one place, which is crucial for an understanding of the research problems of an applied nature. This book: * Contains recent development in fractional calculus * Offers a balance of theory, methods, and applications * Puts the focus on fractional analysis and its interdisciplinary applications, such as fractional models for biological models * Helps make research more relevant to real-life applications Written for researchers, professionals and practitioners, Fractional Order Analysis offers a comprehensive resource to fractional analysis and its many applications as well as information on the newest research.

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Produktdetails

• Verlag: For Dummies
• Seitenzahl: 336
• Erscheinungstermin: 6. August 2020
• Englisch
• ISBN-13: 9781119654230
• Artikelnr.: 59981518

Autorenporträt

HEMEN DUTTA, PHD, is Faculty Member in the Department of Mathematics at Gauhati University, Guwahati, India. AHMET OCAK AKDEMIR, PHD, is Associate Professor, A¿r¿ ¿brahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, A¿r¿, Turkey. ABDON ATANGANA, PHD, is Professor, Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa.

Inhaltsangabe

Preface xi

List of Contributors xv

About the Editors xix

1 On the Fractional Derivative and Integral Operators 1
Mustafa A. Dokuyucu

1.1 Introduction 1

1.2 Fractional Derivative and Integral Operators 2

1.2.1 Properties of the Grünwald-Letnikov Fractional Derivative and Integral 2

1.2.1.1 Integral of Arbitrary Order 6

1.2.1.2 Derivatives of Arbitrary Order 7

1.2.2 Properties of Riemann-Liouville Fractional Derivative and Integral 9

1.2.2.1 Unification of Integer-Order Derivatives and Integrals 10

1.2.2.2 Integrals of Arbitrary Order 12

1.2.2.3 Derivatives of Arbitrary Order 14

1.3 Properties of Caputo Fractional Derivative and Integral 17

1.4 Properties of the Caputo-Fabrizio Fractional Derivative and Integral 20

1.5 Properties of the Atangana-Baleanu Fractional Derivative and Integral 24

1.6 Applications 28

1.6.1 Keller-Segel Model with Caputo Derivative 28

1.6.1.1 Existence and Uniqueness Solutions 28

1.6.1.2 Uniqueness of Solution 31

1.6.1.3 Keller-Segel Model with Atangana-Baleanu Derivative in Caputo Sense 32

1.6.1.4 Uniqueness of Solution 33

1.6.2 Cancer Treatment Model with Caputo-Fabrizio Fractional Derivative 34

1.6.2.1 Existence Solutions 35

1.6.2.2 Uniqueness Solutions 38

1.6.2.3 Conclusion 39

Bibliography 40

2 Generalized Conformable Fractional Operators and Their Applications 43
Muhammad Adil Khan and Tahir Ullah Khan

2.1 Introduction and Preliminaries 43

2.2 Generalized Conformable Fractional Integral Operators 46

2.2.1 Construction of New Integral Operators 47

2.3 Generalized Conformable Fractional Derivative 52

2.4 Applications to Integral Equations and Fractional Differential Equations 60

2.4.1 Equivalence Between the Generalized Nonlinear Problem and the Volterra Integral Equation 61

2.4.2 Existence and Uniqueness of Solution for the Nonlinear Problem 61

2.5 Applications to the Field of Inequalities 63

2.5.1 Inequalities Related to the Left Side of Hermite-Hadamard Inequality 65

2.5.1.1 Applications to Special Means of Real Numbers 74

2.5.1.2 Applications to the Midpoint Formula 75

2.5.2 Inequalities Related to the Right Side of Hermite-Hadamard Inequality 76

2.5.2.1 Applications to Special Means of Real Numbers 84

2.5.2.2 Applications to the Trapezoidal Formula 84

Bibliography 86

3 Analysis of New Trends of Fractional Differential Equations 91
Abdon Atangana and Ali Akgül

3.1 Introduction 91

3.2 Theory 92

3.3 Discretization 101

3.4 Experiments 103

3.5 Stability Analysis 104

3.6 Conclusion 110

Bibliography 111

4 New Estimations for Exponentially Convexity via Conformable Fractional Operators 113
Alper Ekinci and Sever S. Dragomir

4.1 Introduction 113

4.2 Main Results 117

Bibliography 130

5 Lyapunov-type Inequalities for Local Fractional Proportional Derivatives 133
Thabet Abdeljawad

5.1 Introduction 133

5.2 The Local Fractional Proportional Derivatives and Their Generated Nonlocal Fractional Proportional Integrals and Derivatives 135

5.3 Lyapunov-Type Inequalities for Some Nonlocal and Local Fractional Operators 137

5.4 The Lyapunov Inequality for the Sequential Local Fractional Proportional Boundary Value Problem 141

5.5 A Higher-Order Extension of the Local Fractional Proportional Operators and an Associate Lyapunov Open Proble