HEMEN DUTTA, PHD, is Faculty Member in the Department of Mathematics at Gauhati University, Guwahati, India.
AHMET OCAK AKDEMIR, PHD, is Associate Professor, Agri Ibrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, Agri, Turkey.
ABDON ATANGANA, PHD, is Professor, Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa.
1. On the fractional derivative and integral operators
2. Generalized conformable fractional operators and their applications
3. Analysis of new trends of fractional differential equations
4. New estimations for exponentially convexity via conformable fractional operators
5. Lyapunov type inequalities for local fractional proportional derivatives
6. Minkowski type inequalities for mixed conformable fractional integrals
7. New estimations for different kinds of convex functions via conformable integrals and Riemann-Liouville fractional integral operators
8. Legendre-Spectral Algorithms for Solving Some Fractional Differential Equations
9. Mathematical Modeling of an Autonomous Nonlinear Dynamical System for Malaria Transmission using Caputo Derivative
10. MHD free convection flow over a vertical plate with ramped wall temperature and chemical reaction in view of non-singular kernel
11. Comparison of the different fractional derivatives for the dynamics of Zika virus
The book covers several new research findings in the area of fractional-order analysis and its applications. Different tools and techniques of fractional-order analysis are presented in the chapters, and several practical applications have also been demonstrated by means of different mathematical methods and models. Readers should find several useful, relevant, and connected topics in the area of fractional-order analysis those are necessary for crucial understanding of various research problems in science and technology. This book should be useful for graduate and PhD students, researchers, and educators interested in fractional-order analysis and its diverse applications. There are 11 chapters in the book, and they are organized as follows:
Chapter "On the Fractional Derivative and Integral Operators" first discussed interesting developments of fractional calculus. Then, it presented the properties of Grünwald-Letnikov, Riemann-Liouville and Caputo fractional derivative and integral operators. Also, comparisons of these operators have been made in detail. The Caputo-Fabrizio derivative operator is obtained by using the exponential function and its features have also been given. Atangana-Baleanu fractional derivative with nonlocal and nonsingular kernel is obtained by using the generalized form of the Mittag-Leffler function. Finally, the Keller-Segel and cancer treatment models were compared by expanding to Caputo, Caputo-Fabrizio, and Atangana-Baleanu fractional derivative operators.
Chapter "Generalized Conformable Fractional Operators and Their Applications" aims to present new generalized fractional integral and derivative operators along with their applications. These operators are known as left-sided and right-sided generalized conformable fractional operators, and they contain several operators of fractional calculus. Then some basic properties such as linearity, continuity, boundedness, etc. of such operators are presented. A nonlinear generalized conformable fractional differential equation is also formed. It is shown that this equation is equivalent to a Volterra integral equation and, then the existence and uniqueness of the solution is demonstrated. Finally, some Hermite-Hadamard-type inequalities for conformable integrals have been presented and discussed their applications for Trapezoidal formula and means.
Chapter "Analysis of New Trends of Fractional Differential Equations" discussed several results related to fractal-fractional derivatives in Caputo sense when the kernels are power law, exponential decay law the generalized Mittag-Leffler functions and presented numerical approximation for each case. It considered partial differential equations with these new differential operators. It also presented numerical analysis in detail and the stability for each case. Numerical simulations have been incorporated to explain the efficiency of the numerical scheme adopted, and also to see the effect of the fractal dimension and fractional order.
Chapter "New Estimations for Exponentially Convexity via Conformable Fractional Operators" discussed different fractional integral operators and their basic properties. It also established some new Hadamard-type integral inequalities for exponentially convex functions via conformable fractional integrals.
Chapter "Lyapunov-type Inequalities for Local Fractional Proportional Derivatives" reviewed some Lyapunov-type inequalities for certain local and nonlocal fractional derivatives and presented a Lyapunov-type inequality for the sequential local fractional proportional derivatives with constant references as the special case a =?2 of the nonlocal fractional proportional derivative aDa,?. An open problem is also presented for a more general sequential local fractional proportional boundary value problem. Then, it presented a higher order extension in order to investigate the ability of proving a Lyapunov inequality, which cannot obtained from the nonlocal one, for the local fractional proportional derivative of order 1?<???=?2.
Chapter "Minkowski-type Inequalities for Mixed Conformable Fractional Integrals" first presented some fractional integrals and Minkowski-type inequalities obtained for these integrals. Then, the reverse Minkowski inequality and related inequalities for mixed conformable fractional integrals have been presented.
Chapter "New Estimations for Different Kinds of Convex Functions via Conformable Integrals and Riemann-Liouville Fractional Integral Operators" discussed several new fractional bounds involving the functions having geometrical convexity and co-ordinated convexity properties via conformable integrals and Riemann-Liouville fractional integrals. In order to obtain main results, it also derived new fractional integral identities. Then, it generalized some new integral inequalities for GG-convex functions whose second derivative at certain powers are established via conformable integrals. Several new results were further derived by choosing different values of n and a.
Chapter "Legendre-Spectral Algorithms for Solving Some Fractional Differential Equations" discussed some algorithms for treating some kinds of fractional differential equations by utilizing suitable spectral methods. A Galerkin method is employed for solving time-fractional telegraph equation, and a double shifted Legendre expansion is proposed as an approximating polynomial. Further, the spectral methods Petrov-Galerkin and collocation, respectively, are applied for obtaining spectral solutions of space fractional linear diffusion problem. The two suggested algorithms have been built by using a certain double shifted Legendre basis. Investigation for the convergence and error analysis of the two suggested approximate double expansions have also been performed. Numerical results were provided to justify the efficiency of the proposed algorithms.
Chapter "Mathematical Modeling of an Autonomous Nonlinear Dynamical System for Malaria Transmission Using Caputo Derivative" proposed a seven-dimensional Caputo-type fractional-order model describing the dynamics for the spread of malaria virus transmitted to humans (host) by the bite of mosquito (vector), affecting the latter itself. Considering that both underlying populations may not behave exactly the same, the first group of population (human) is assigned the fractional-order e whereas the fractional-order ? is for the second group (mosquito). The model is shown to have globally asymptotically stable steady-state solution with R0?<?1 (disease cannot spread) and an unstable endemic equilibrium point for R0?>?1 (disease can spread), where R0 is the basic reproductive number. Fixed point theory is used for discussing the existence and uniqueness of the solution of the model. Further, it is proved that the non-negative hyperoctant is a positively invariant region for the model. Numerical simulation is also presented to show that the model under Caputo differentiation is more accurate than its classical version to describe the complexity of the dynamics of the disease's transmission.
Chapter "MHD-free Convection Flow Over a Vertical Plate with Ramped Wall Temperature and Chemical Reaction in view of Nonsingular Kernel" aims to study the unsteady MHD-free convection flow of an electrically conducting incompressible generalized Maxwell fluid over an infinite vertical plate with ramped temperature and constant concentration. Fractional-order Caputo, Caputo-Fabrizio, and Atangana-Baleanu time derivatives are used to study the effect of fractional parameters on the dynamics of fluid. The motion of plate is rectilinear translation with an arbitrary time-dependent velocity. It observed that fractional-order model is best to explain the memory effect and flow behavior of the fluid. The influence of transverse magnetic fields is also studied. Moreover, the effects of system parameters on the filed velocity are analyzed through numerical simulation and graphs.
Chapter "Comparison of the Different Fractional Derivatives for the Dynamics of Zika Virus" presented a comparative study of the Zika model with two different operators, viz., the Caputo-Fabrizio (CF) and the Atangana-Baleanu (AB) derivative. It considered a latest mathematical model that considers Zika dynamics with mutation. Then, it presented some basic mathematical results for the model and then applied the CF and AB derivative, and also presented their analysis. Some key results related to fractional order are further incorporated for the appropriateness of CF and AB for modeling purposes.
The editors are grateful to the contributors for their contribution and co-operation throughout the whole process of editing the book. The editors have benefited from the remarks and comments of several experts on the topics of this book. The editors would also like to thank the editors at Wiley and production staff for their support and help. Finally, the editors offer sincere thanks to all those who contributed in some way to complete this...