Numerical Methods for Fractional Differentiation

This book discusses numerical methods for solving partial differential and integral equations, as well as ordinary differential and integral equations, involving fractional differential and integral operators. Differential and integral operators presented in the book include those with exponential decay law, known as CaputoFabrizio differential and integral operators, those with power law, known as RiemannLiouville fractional operators, and those for the generalized MittagLeffler function, known as the AtanganaBaleanu fractional operators.

The book reviews existing numerical schemes associated with fractional operators including those with power law, while also highlighting new trends in numerical schemes for recently introduced differential and integral operators. In addition, the initial chapters address useful properties of each differential and integral fractional operator. Methods discussed in the book are subsequently used to solved problems arising in many fields of science, technology, and engineering, including epidemiology, chaos, solitons, fractals, diffusion, groundwater, and fluid mechanics. Given its scope, the book offers a valuable resource for graduate students of mathematics and engineering, and researchers in virtually all fields of science, technology, and engineering, as well as an excellent addition to libraries.

Auteur

KOLADE M. OWOLABI is Senior Lecturer at the Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria. Holding a PhD in Applied (Numerical) Mathematics, he is the author or coauthor of several research papers, and a regular reviewer for several scientific journals. His areas of research interest include numerical methods and scientific computing for the applied problems that arise from interactions between the natural and life sciences and engineering. His research is a well-balanced combination of analytical investigations and numerical experiments using the finite difference technique, spectral method, and exponential time-differencing schemes to address both integer- and non-integer-order ordinary and partial differential equations. The main objective of his work is to use mathematical theories and methodologies to gain insights into the qualitative behavior of non-linear dynamical systems arising from the mathematical modelingof natural phenomena in the applied sciences and engineering, with an emphasis on the transmission and control dynamics of human diseases of public health interest, as well as addressing mathematical models arising in population biology for ecological patterns and processes.
ABDON ATANGANA is Full Professor at the Faculty of Natural and Agricultural Science, Institute for Groundwater Studies, University of the Free State (UFS), Bloemfontein, South Africa. He obtained his honors and master's degrees in Applied Mathematics from the Department of Applied Mathematics at the UFS with distinction. He serves as a reviewer for more than 200 accredited international journals, and has been awarded the world champion of peer review in 2016 and in 2017. He also serves on the editorial boards of more than 20 journals of international repute. Professor Atangana's research interests are in the methods and applications of partial and ordinary differential equations, fractional differential equations, perturbation methods, asymptotic methods, iterative methods, and groundwater modeling. He is the founder of fractional calculus with non-local and non-singular kernels, popular in applied mathematics today. Since 2013, he has published over 225 research articles in several accredited journals of applied mathematics, applied physics, geohydrology, and biomathematics. He is the author of two books, Fractional Operators with Constant and Variable Order with Application to Geo-hydrology; and Derivative with a New Parameter: Theory, Methods and Applications.

Contenu

1. Review of Fractional Differentiation.- 2. Finite Difference Approximations.- 3. Numerical Approximation of Riemann-Liouville Differentiation.- 4. Numerical Approximation of Caputo Differentiation.- 5. Numerical Approximation of Caputo-Fabrizio Differentiation.- 6. Numerical Approximation of Atangana-Baleanu Differentiation.- 7. Application to Ordinary Fractional Differential Equations.- 8. Application to Partial Fractional Differential Equation.