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 Gr¼nwald¢"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 Akg¼l
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 Problem 144
5.6 Conclusion 146
Acknowledgement 146
Bibliography 147
6 Minkowski-Type Inequalities for Mixed Conformable Fractional Integrals 151
Erhan Set and Muhamet E. -zdemir
6.1 Introduction and Preliminaries 151
6.2 Reverse Minkowski Inequality Involving Mixed Conformable Fractional Integrals 158
6.3 Related Inequalities 160
Bibliography 167
7 New Estimations for Different Kinds of Convex Functions via Conformable Integrals and Riemann¢"Liouville Fractional Integral Operators 169
Ahmet Ocak Akdemir and Hemen Dutta
7.1 Introduction 169
7.2 Some Generalizations for Geometrically Convex Functions 172
7.3 New Inequalities for Co-ordinated Convex Functions 179
Bibliography 191
8 Legendre-Spectral Algorithms for Solving Some Fractional Differential Equations 195
Youssri H. Youssri and Waleed M. Abd-Elhameed
8.1 Introduction 195
8.2 Some Properties and Relations Concerned with Shifted Legendre Polynomials 197
8.3 Galerkin Approach for Treating Fractional Telegraph Type Equation 200
8.4 Discussion of the Convergence and Error Analysis of the Suggested Double Expansion 204
8.5 Some Test Problems for Fractional Telegraph Equation 207
8.6 Spectral Algorithms for Treating the Space Fractional Diffusion Problem 209
8.6.1 Transformation of the Problem 210
8.6.2 Basis Functions Selection 211
8.6.3 A Collocation Scheme for Solving Eq. 8.44 213
8.6.4 An Alternative Spectral Petrov¢"Galerkin Scheme for Solving Eq. (8.44) 214
8.7 Investigation of Convergence and Error Analysis 214
8.8 Numerical Results and Comparisons 216
8.9 Conclusion 220
Bibliography 220
9 Mathematical Modeling of an Autonomous Nonlinear Dynamical System for Malaria Transmission Using Caputo Derivative 225
Abdon Atangana and Sania Qureshi
9.1 Introduction 225
9.2 Mathematical Preliminaries 227
9.3 Model Formulation 228
9.4 Basic Properties of the Fractional Model 230
9.4.1 Reproductive Number 230
9.4.2 Existence and Stability of Disease-free Equilibrium Points 231
9.4.3 Existence and Stability of Endemic Equilibrium Point 232
9.5 Existence and Uniqueness of the Solutions 233
9.5.1 Positivity of the Solutions 236
9.6 Numerical Simulations 237
9.7 Conclusion 247
Bibliography 250
10 MHD-free Convection Flow Over a Vertical Plate with Ramped Wall Temperature and Chemical Reaction in View of Nonsingular Kernel 253
Muhammad B. Riaz, Abdon Atangana, and Syed T. Saeed
10.1 Introduction 253
10.2 Mathematical Model 254
10.2.1 Preliminaries 256
10.3 Solution 256
10.3.1 Concentration Fields 257
10.3.1.1 Concentration Field with Caputo Time-Fractional Derivative 257
10.3.1.2 Concentration Field with Caputo¢"Fabrizio Time-Fractional Derivative 257
10.3.1.3 Concentration Field with Atangana¢"Baleanu Time-Fractional Derivative 257
10.3.2 Temperature Fields 258
10.3.2.1 Temperature Field with Caputo Time-Fractional Derivative 258
10.3.2.2 Temperature Field with Caputo¢"Fabrizio Time-Fractional Derivative 258
10.3.2.3 Temperature Field with Atangana¢"Baleanu Time-Fractional Derivative 258
10.3.3 Velocity Fields 259
10.3.3.1 Velocity Field with Caputo Time-Fractional Derivative 259
10.3.3.2 Velocity Field with Caputo¢"Fabrizio Time-Fractional Derivative 259
10.3.3.3 Velocity Field with Atangana¢"Baleanu Time-Fractional Derivative 262
10.4 Results and Discussion 263
10.5 Conclusion 263
Bibliography 279
11 Comparison of the Different Fractional Derivatives for the Dynamics of Zika Virus 283
Muhammad Altaf Khan
11.1 Introduction 283
11.2 Background of Fractional Operators 284
11.3 Model Framework 286
11.4 A Fractional Zika Model with Different Fractional Derivatives 287
11.5 Numerical Scheme for Caputo¢"Fabrizio Model 288
11.5.1 Solutions Existence for the Atangana¢"Baleanu Model 289
11.5.2 Numerical Scheme for Atangana¢"Baleanu Model 291
11.6 Numerical Results 293
11.7 Conclusion 303
Bibliography 303
Index 307
