Theory and Methods of Piecewise Defined Fractional Operators

Theory and Methods of Piecewise Defined Fractional Operators introduces new mathematical methods to derive complex modeling solutions with stability, consistency, and convergence. These tools include new types of non-local derivatives and integrals, such as fractal-fractional derivatives and integrals. Drs. Atangana and Araz present the theoretical and numerical analyses of the newly introduced piecewise differential and integral operators where crossover behaviors are observed, as well as their applications to real-world problems. The book contains foundational concepts that will help readers better understand piecewise differential and integral calculus and their applications to modeling processes with crossover behaviors. Several Cauchy problems with piecewise differential operators are considered, and their existence and uniqueness under some conditions are presented; in particular, the Carathéodory principle is used to ensure the existence and uniqueness of these new Cauchy problems. New numerical schemes are introduced to derive numerical solutions to these new equations, and the stability, consistency, and convergence analysis of these new numerical approaches are also presented.

1. Piecewise differential operators and their properties
2. Piecewise integral operators and their properties
3. Existence and uniqueness of Cauchy problems with nonlocal operators under the framework of Carathéodory’s
4. Extension of second-order parametrized Runge-Kutta method to ODE with nonlocal operators
5. Existence, uniqueness, and numerical analysis of piecewise IVP with classical and global differentiation
6. Theoretical and numerical analysis of piecewise IVP with classical and fractional derivative
7. Analysis of piecewise deterministic and stochastic Cauchy problems
8. Numerical analysis of piecewise Cauchy problem with fractional and global derivative
9. Analysis of a deterministic-fractional stochastic Cauchy problem
10. Analysis of piecewise Cauchy problem with global and fractal-fractional derivative
11. Analysis of piecewise Cauchy problem with global and stochastic
12. Piecewise fractal-fractional: Theoretical and numerical analysis

Dr. Abdon Atangana is Academic Head of Department and Professor of Applied Mathematics at the University of the Free State, Bloemfontein, Republic of South Africa. He obtained his honours and master’s degrees from the Department of Applied Mathematics at the UFS with distinction. He obtained his PhD in applied mathematics from the Institute for Groundwater Studies. He was included in the 2019 (Maths), 2020 (Cross-field) and the 2021 (Maths) Clarivate Web of Science lists of the World's top 1% scientists, and he was awarded The World Academy of Sciences (TWAS) inaugural Mohammed A. Hamdan award for contributions to science in developing countries. In 2018 Dr. Atangana was elected as a member of the African Academy of Sciences and in 2021 a member of The World Academy of Sciences. He also ranked number one in the world in mathematics, number 186 in the world in all fields, and number one in Africa in all fields, according to the Stanford University list of top 2% scientists in the world. He was one of the first recipients of the Obada Award in 2018. Dr. Atangana published a paper that was ranked by Clarivate in 2017 as the most cited mathematics paper in the world. Dr. Atangana serves as an editor for 20 international journals, lead guest editor for 10 journals, and is also a reviewer of more than 200 international accredited journals. His research interests include methods and applications of partial and ordinary differential equations, fractional differential equations, perturbation methods, asymptotic methods, iterative methods, and groundwater modelling. Dr. Atangana is a pioneer in research on fractional calculus with non-local and non-singular kernels popular in applied mathematics today. He is the author of numerous books, including Integral Transforms and Engineering: Theory, Methods, and Applications, Taylor and Francis/CRC Press; Numerical Methods for Fractal-Fractional Differential Equations and Engineering: Simulations and Modeling, Taylor and Francis/CRC