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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.

Fractional Order Analysis - Theory, Methods and Applications

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 Grunwald-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 Akgul





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. OEzdemir





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