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<p>Preface xi</p> <p>List of Contributors xv</p> <p>About the Editors xix</p> <p><b>1 On the Fractional Derivative and Integral Operators </b><b>1<br /></b><i>Mustafa A. Dokuyucu</i></p> <p>1.1 Introduction 1</p> <p>1.2 Fractional Derivative and Integral Operators 2</p> <p>1.2.1 Properties of the Grünwald–Letnikov Fractional Derivative and Integral 2</p> <p>1.2.1.1 Integral of Arbitrary Order 6</p> <p>1.2.1.2 Derivatives of Arbitrary Order 7</p> <p>1.2.2 Properties of Riemann–Liouville Fractional Derivative and Integral 9</p> <p>1.2.2.1 Unification of Integer-Order Derivatives and Integrals 10</p> <p>1.2.2.2 Integrals of Arbitrary Order 12</p> <p>1.2.2.3 Derivatives of Arbitrary Order 14</p> <p>1.3 Properties of Caputo Fractional Derivative and Integral 17</p> <p>1.4 Properties of the Caputo–Fabrizio Fractional Derivative and Integral 20</p> <p>1.5 Properties of the Atangana–Baleanu Fractional Derivative and Integral 24</p> <p>1.6 Applications 28</p> <p>1.6.1 Keller–Segel Model with Caputo Derivative 28</p> <p>1.6.1.1 Existence and Uniqueness Solutions 28</p> <p>1.6.1.2 Uniqueness of Solution 31</p> <p>1.6.1.3 Keller–Segel Model with Atangana–Baleanu Derivative in Caputo Sense 32</p> <p>1.6.1.4 Uniqueness of Solution 33</p> <p>1.6.2 Cancer Treatment Model with Caputo-Fabrizio Fractional Derivative 34</p> <p>1.6.2.1 Existence Solutions 35</p> <p>1.6.2.2 Uniqueness Solutions 38</p> <p>1.6.2.3 Conclusion 39</p> <p>Bibliography 40</p> <p><b>2 Generalized Conformable Fractional Operators and Their Applications </b><b>43<br /></b><i>Muhammad Adil Khan and Tahir Ullah Khan</i></p> <p>2.1 Introduction and Preliminaries 43</p> <p>2.2 Generalized Conformable Fractional Integral Operators 46</p> <p>2.2.1 Construction of New Integral Operators 47</p> <p>2.3 Generalized Conformable Fractional Derivative 52</p> <p>2.4 Applications to Integral Equations and Fractional Differential Equations 60</p> <p>2.4.1 Equivalence Between the Generalized Nonlinear Problem and the Volterra Integral Equation 61</p> <p>2.4.2 Existence and Uniqueness of Solution for the Nonlinear Problem 61</p> <p>2.5 Applications to the Field of Inequalities 63</p> <p>2.5.1 Inequalities Related to the Left Side of Hermite–Hadamard Inequality 65</p> <p>2.5.1.1 Applications to Special Means of Real Numbers 74</p> <p>2.5.1.2 Applications to the Midpoint Formula 75</p> <p>2.5.2 Inequalities Related to the Right Side of Hermite–Hadamard Inequality 76</p> <p>2.5.2.1 Applications to Special Means of Real Numbers 84</p> <p>2.5.2.2 Applications to the Trapezoidal Formula 84</p> <p>Bibliography 86</p> <p><b>3 Analysis of New Trends of Fractional Differential Equations </b><b>91<br /></b><i>Abdon Atangana and Ali Akgül</i></p> <p>3.1 Introduction 91</p> <p>3.2 Theory 92</p> <p>3.3 Discretization 101</p> <p>3.4 Experiments 103</p> <p>3.5 Stability Analysis 104</p> <p>3.6 Conclusion 110</p> <p>Bibliography 111</p> <p><b>4 New Estimations for Exponentially Convexity via Conformable Fractional Operators </b><b>113<br /></b><i>Alper Ekinci and Sever S. Dragomir</i></p> <p>4.1 Introduction 113</p> <p>4.2 Main Results 117</p> <p>Bibliography 130</p> <p><b>5 Lyapunov-type Inequalities for Local Fractional Proportional Derivatives </b><b>133<br /></b><i>Thabet Abdeljawad</i></p> <p>5.1 Introduction 133</p> <p>5.2 The Local Fractional Proportional Derivatives and Their Generated Nonlocal Fractional Proportional Integrals and Derivatives 135</p> <p>5.3 Lyapunov-Type Inequalities for Some Nonlocal and Local Fractional Operators 137</p> <p>5.4 The Lyapunov Inequality for the Sequential Local Fractional Proportional Boundary Value Problem 141</p> <p>5.5 A Higher-Order Extension of the Local Fractional Proportional Operators and an Associate Lyapunov Open Problem 144</p> <p>5.6 Conclusion 146</p> <p>Acknowledgement 146</p> <p>Bibliography 147</p> <p><b>6 Minkowski-Type Inequalities for Mixed Conformable Fractional Integrals </b><b>151<br /></b><i>Erhan Set and Muhamet E. Özdemir</i></p> <p>6.1 Introduction and Preliminaries 151</p> <p>6.2 Reverse Minkowski Inequality Involving Mixed Conformable Fractional Integrals 158</p> <p>6.3 Related Inequalities 160</p> <p>Bibliography 167</p> <p><b>7 New Estimations for Different Kinds of Convex Functions via Conformable Integrals and Riemann–Liouville Fractional Integral Operators </b><b>169<br /></b><i>Ahmet Ocak Akdemir and Hemen Dutta</i></p> <p>7.1 Introduction 169</p> <p>7.2 Some Generalizations for Geometrically Convex Functions 172</p> <p>7.3 New Inequalities for Co-ordinated Convex Functions 179</p> <p>Bibliography 191</p> <p><b>8 Legendre-Spectral Algorithms for Solving Some Fractional Differential Equations </b><b>195<br /></b><i>Youssri H. Youssri and Waleed M. Abd-Elhameed</i></p> <p>8.1 Introduction 195</p> <p>8.2 Some Properties and Relations Concerned with Shifted Legendre Polynomials 197</p> <p>8.3 Galerkin Approach for Treating Fractional Telegraph Type Equation 200</p> <p>8.4 Discussion of the Convergence and Error Analysis of the Suggested Double Expansion 204</p> <p>8.5 Some Test Problems for Fractional Telegraph Equation 207</p> <p>8.6 Spectral Algorithms for Treating the Space Fractional Diffusion Problem 209</p> <p>8.6.1 Transformation of the Problem 210</p> <p>8.6.2 Basis Functions Selection 211</p> <p>8.6.3 A Collocation Scheme for Solving Eq. 8.44 213</p> <p>8.6.4 An Alternative Spectral Petrov–Galerkin Scheme for Solving Eq. (8.44) 214</p> <p>8.7 Investigation of Convergence and Error Analysis 214</p> <p>8.8 Numerical Results and Comparisons 216</p> <p>8.9 Conclusion 220</p> <p>Bibliography 220</p> <p><b>9 Mathematical Modeling of an Autonomous Nonlinear Dynamical System for Malaria Transmission Using Caputo Derivative </b><b>225<br /></b><i>Abdon Atangana and Sania Qureshi</i></p> <p>9.1 Introduction 225</p> <p>9.2 Mathematical Preliminaries 227</p> <p>9.3 Model Formulation 228</p> <p>9.4 Basic Properties of the Fractional Model 230</p> <p>9.4.1 Reproductive Number 230</p> <p>9.4.2 Existence and Stability of Disease-free Equilibrium Points 231</p> <p>9.4.3 Existence and Stability of Endemic Equilibrium Point 232</p> <p>9.5 Existence and Uniqueness of the Solutions 233</p> <p>9.5.1 Positivity of the Solutions 236</p> <p>9.6 Numerical Simulations 237</p> <p>9.7 Conclusion 247</p> <p>Bibliography 250</p> <p><b>10 MHD-free Convection Flow Over a Vertical Plate with Ramped Wall Temperature and Chemical Reaction in View of Nonsingular Kernel </b><b>253<br /></b><i>Muhammad B. Riaz, Abdon Atangana, and Syed T. Saeed</i></p> <p>10.1 Introduction 253</p> <p>10.2 Mathematical Model 254</p> <p>10.2.1 Preliminaries 256</p> <p>10.3 Solution 256</p> <p>10.3.1 Concentration Fields 257</p> <p>10.3.1.1 Concentration Field with Caputo Time-Fractional Derivative 257</p> <p>10.3.1.2 Concentration Field with Caputo–Fabrizio Time-Fractional Derivative 257</p> <p>10.3.1.3 Concentration Field with Atangana–Baleanu Time-Fractional Derivative 257</p> <p>10.3.2 Temperature Fields 258</p> <p>10.3.2.1 Temperature Field with Caputo Time-Fractional Derivative 258</p> <p>10.3.2.2 Temperature Field with Caputo–Fabrizio Time-Fractional Derivative 258</p> <p>10.3.2.3 Temperature Field with Atangana–Baleanu Time-Fractional Derivative 258</p> <p>10.3.3 Velocity Fields 259</p> <p>10.3.3.1 Velocity Field with Caputo Time-Fractional Derivative 259</p> <p>10.3.3.2 Velocity Field with Caputo–Fabrizio Time-Fractional Derivative 259</p> <p>10.3.3.3 Velocity Field with Atangana–Baleanu Time-Fractional Derivative 262</p> <p>10.4 Results and Discussion 263</p> <p>10.5 Conclusion 263</p> <p>Bibliography 279</p> <p><b>11 Comparison of the Different Fractional Derivatives for the Dynamics of Zika Virus </b><b>283<br /></b><i>Muhammad Altaf Khan</i></p> <p>11.1 Introduction 283</p> <p>11.2 Background of Fractional Operators 284</p> <p>11.3 Model Framework 286</p> <p>11.4 A Fractional Zika Model with Different Fractional Derivatives 287</p> <p>11.5 Numerical Scheme for Caputo–Fabrizio Model 288</p> <p>11.5.1 Solutions Existence for the Atangana–Baleanu Model 289</p> <p>11.5.2 Numerical Scheme for Atangana–Baleanu Model 291</p> <p>11.6 Numerical Results 293</p> <p>11.7 Conclusion 303</p> <p>Bibliography 303</p> <p>Index 307</p>

<p><b>HEMEN DUTTA, P<small>H</small>D,</b> is Faculty Member in the Department of Mathematics at Gauhati University, Guwahati, India. <p><b>AHMET OCAK AKDEMIR, P<small>H</small>D,</b> is Associate Professor, Ağrı İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, Ağrı, Turkey. <p><b>ABDON ATANGANA, P<small>H</small>D,</b> is Professor, Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa.

<p><b>A guide to the new research in the field of fractional order analysis</b> <p><i>Fractional Order Analysis</i> 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. <p>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: <ul> <li>Contains recent development in fractional calculus</li> <li>Offers a balance of theory, methods, and applications</li> <li>Puts the focus on fractional analysis and its interdisciplinary applications, such as fractional models for biological models</li> <li>Helps make research more relevant to real-life applications</li> </ul> <p>Written for researchers, professionals and practitioners, <i>Fractional Order Analysis</i> offers a comprehensive resource to fractional analysis and its many applications as well as information on the newest research.

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