New concept in calculus: Piecewise differential and integral operators

Abstract

• Piecewise derivative with classical and global derivative is presented. • Piecewise derivative with singular and non-singular kernel is presented. • Some properties of piecewise derivative is introduced. • Numerical algorithm based on the Newton polynomial is constructed for Cauch problem with such derivative. • Applications of this derivative to some chaotic and epidemiological models are given and numerical simulations are depicted for different values of fractional orders and fractal dimensions. In the last decades, many methodologies have been suggested to depict behaviors of some complex world’s problems arising in many academic fields. One of these problems is the multi-steps behavior displayed by some problems. A concept of piecewise derivative is introduced in this paper with the aim to model real world problems following multiples processes. We have presented some important properties of these definitions. We considered different scenarios and presented numerical schemes that could be used to solve such problems. Illustrative examples, including chaotic and epidemiological models are presented to see the effectiveness of the suggested concept.