Theory and Methods of Piecewise Defined Fractional Operators

The mathematical, computer science, and computational modeling community is striving to develop new tools that provide better analysis of complex phenomena. Piecewise Defined Fractional Operators- Volume 1: Theory and Methods 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 to better understand piecewise differential and integral calculus and their applications to modeling processes with crossover behaviors. Volume 1 starts off with a presentation of why piecewise calculus is needed. Then definitions of derivatives and integrals are presented with their different properties. Several Cauchy problems with piecewise differential operators are considered, and their existence and uniqueness under some conditions are presented; in particular, the Carath廩dory 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 presented. In particular, the authors introduce a modified parametrized method and show that their version is far more accurate for solving classical and fractional differential equations. Several important theoretical concepts are presented and proven, then these concepts are applied to several fields including chaos, epidemiological modeling, biological modeling, and others in the case of ordinary differential equations. This concept is then finally adapted to partial differential equations where novel numerical schemes are presented. Important concepts such as energy methods are discussed for some equations. In Volume 2, the concepts are then applied to a wide variety of problems arising from heat transfer, groundwater transport, groundwater flow, telegraph dynamics, and others.