Fractional Order Analysis

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

• Preface xiList of Contributors xvAbout the Editors xix1 On the Fractional Derivative and Integral Operators 1Mustafa A. Dokuyucu1.1 Introduction 11.2 Fractional Derivative and Integral Operators 21.2.1 Properties of the Grünwald–Letnikov Fractional Derivative and Integral 21.2.1.1 Integral of Arbitrary Order 61.2.1.2 Derivatives of Arbitrary Order 71.2.2 Properties of Riemann–Liouville Fractional Derivative and Integral 91.2.2.1 Unification of Integer-Order Derivatives and Integrals 101.2.2.2 Integrals of Arbitrary Order 121.2.2.3 Derivatives of Arbitrary Order 141.3 Properties of Caputo Fractional Derivative and Integral 171.4 Properties of the Caputo–Fabrizio Fractional Derivative and Integral 201.5 Properties of the Atangana–Baleanu Fractional Derivative and Integral 241.6 Applications 281.6.1 Keller–Segel Model with Caputo Derivative 281.6.1.1 Existence and Uniqueness Solutions 281.6.1.2 Uniqueness of Solution 311.6.1.3 Keller–Segel Model with Atangana–Baleanu Derivative in Caputo Sense 321.6.1.4 Uniqueness of Solution 331.6.2 Cancer Treatment Model with Caputo-Fabrizio Fractional Derivative 341.6.2.1 Existence Solutions 351.6.2.2 Uniqueness Solutions 381.6.2.3 Conclusion 39Bibliography 402 Generalized Conformable Fractional Operators and Their Applications 43Muhammad Adil Khan and Tahir Ullah Khan2.1 Introduction and Preliminaries 432.2 Generalized Conformable Fractional Integral Operators 462.2.1 Construction of New Integral Operators 472.3 Generalized Conformable Fractional Derivative 522.4 Applications to Integral Equations and Fractional Differential Equations 602.4.1 Equivalence Between the Generalized Nonlinear Problem and the Volterra Integral Equation 612.4.2 Existence and Uniqueness of Solution for the Nonlinear Problem 612.5 Applications to the Field of Inequalities 632.5.1 Inequalities Related to the Left Side of Hermite–Hadamard Inequality 652.5.1.1 Applications to Special Means of Real Numbers 742.5.1.2 Applications to the Midpoint Formula 752.5.2 Inequalities Related to the Right Side of Hermite–Hadamard Inequality 762.5.2.1 Applications to Special Means of Real Numbers 842.5.2.2 Applications to the Trapezoidal Formula 84Bibliography 863 Analysis of New Trends of Fractional Differential Equations 91Abdon Atangana and Ali Akgül3.1 Introduction 913.2 Theory 923.3 Discretization 1013.4 Experiments 1033.5 Stability Analysis 1043.6 Conclusion 110Bibliography 1114 New Estimations for Exponentially Convexity via Conformable Fractional Operators 113Alper Ekinci and Sever S. Dragomir4.1 Introduction 1134.2 Main Results 117Bibliography 1305 Lyapunov-type Inequalities for Local Fractional Proportional Derivatives 133Thabet Abdeljawad5.1 Introduction 1335.2 The Local Fractional Proportional Derivatives and Their Generated Nonlocal Fractional Proportional Integrals and Derivatives 1355.3 Lyapunov-Type Inequalities for Some Nonlocal and Local Fractional Operators 1375.4 The Lyapunov Inequality for the Sequential Local Fractional Proportional Boundary Value Problem 1415.5 A Higher-Order Extension of the Local Fractional Proportional Operators and an Associate Lyapunov Open Problem 1445.6 Conclusion 146Acknowledgement 146Bibliography 1476 Minkowski-Type Inequalities for Mixed Conformable Fractional Integrals 151Erhan Set and Muhamet E. Özdemir6.1 Introduction and Preliminaries 1516.2 Reverse Minkowski Inequality Involving Mixed Conformable Fractional Integrals 1586.3 Related Inequalities 160Bibliography 1677 New Estimations for Different Kinds of Convex Functions via Conformable Integrals and Riemann–Liouville Fractional Integral Operators 169Ahmet Ocak Akdemir and Hemen Dutta7.1 Introduction 1697.2 Some Generalizations for Geometrically Convex Functions 1727.3 New Inequalities for Co-ordinated Convex Functions 179Bibliography 1918 Legendre-Spectral Algorithms for Solving Some Fractional Differential Equations 195Youssri H. Youssri and Waleed M. Abd-Elhameed8.1 Introduction 1958.2 Some Properties and Relations Concerned with Shifted Legendre Polynomials 1978.3 Galerkin Approach for Treating Fractional Telegraph Type Equation 2008.4 Discussion of the Convergence and Error Analysis of the Suggested Double Expansion 2048.5 Some Test Problems for Fractional Telegraph Equation 2078.6 Spectral Algorithms for Treating the Space Fractional Diffusion Problem 2098.6.1 Transformation of the Problem 2108.6.2 Basis Functions Selection 2118.6.3 A Collocation Scheme for Solving Eq. 8.44 2138.6.4 An Alternative Spectral Petrov–Galerkin Scheme for Solving Eq. (8.44) 2148.7 Investigation of Convergence and Error Analysis 2148.8 Numerical Results and Comparisons 2168.9 Conclusion 220Bibliography 2209 Mathematical Modeling of an Autonomous Nonlinear Dynamical System for Malaria Transmission Using Caputo Derivative 225Abdon Atangana and Sania Qureshi9.1 Introduction 2259.2 Mathematical Preliminaries 2279.3 Model Formulation 2289.4 Basic Properties of the Fractional Model 2309.4.1 Reproductive Number 2309.4.2 Existence and Stability of Disease-free Equilibrium Points 2319.4.3 Existence and Stability of Endemic Equilibrium Point 2329.5 Existence and Uniqueness of the Solutions 2339.5.1 Positivity of the Solutions 2369.6 Numerical Simulations 2379.7 Conclusion 247Bibliography 25010 MHD-free Convection Flow Over a Vertical Plate with Ramped Wall Temperature and Chemical Reaction in View of Nonsingular Kernel 253Muhammad B. Riaz, Abdon Atangana, and Syed T. Saeed10.1 Introduction 25310.2 Mathematical Model 25410.2.1 Preliminaries 25610.3 Solution 25610.3.1 Concentration Fields 25710.3.1.1 Concentration Field with Caputo Time-Fractional Derivative 25710.3.1.2 Concentration Field with Caputo–Fabrizio Time-Fractional Derivative 25710.3.1.3 Concentration Field with Atangana–Baleanu Time-Fractional Derivative 25710.3.2 Temperature Fields 25810.3.2.1 Temperature Field with Caputo Time-Fractional Derivative 25810.3.2.2 Temperature Field with Caputo–Fabrizio Time-Fractional Derivative 25810.3.2.3 Temperature Field with Atangana–Baleanu Time-Fractional Derivative 25810.3.3 Velocity Fields 25910.3.3.1 Velocity Field with Caputo Time-Fractional Derivative 25910.3.3.2 Velocity Field with Caputo–Fabrizio Time-Fractional Derivative 25910.3.3.3 Velocity Field with Atangana–Baleanu Time-Fractional Derivative 26210.4 Results and Discussion 26310.5 Conclusion 263Bibliography 27911 Comparison of the Different Fractional Derivatives for the Dynamics of Zika Virus 283Muhammad Altaf Khan11.1 Introduction 28311.2 Background of Fractional Operators 28411.3 Model Framework 28611.4 A Fractional Zika Model with Different Fractional Derivatives 28711.5 Numerical Scheme for Caputo–Fabrizio Model 28811.5.1 Solutions Existence for the Atangana–Baleanu Model 28911.5.2 Numerical Scheme for Atangana–Baleanu Model 29111.6 Numerical Results 29311.7 Conclusion 303Bibliography 303Index 307