Extension of rate of change concept: From local to nonlocal operators with applications

Abstract

The concept of rate of change gave birth to numerous important theories and applications in mathematics, applied mathematics and other related academic disciplines. For example, differential calculus which is linked to integral calculus via the fundamental theorem of calculus. On one hand, differential calculus provides information regarding the instantaneous speed of change and the inclines of the curvatures. While integral calculus covers the study of accumulations of quantities and regions under or amid curvatures. The founders of the concept of derivative defined the derivative of a function y with a variable x and called it derivative of y with respect to x. In this paper, we define, with the concept of rate of change, a derivative of a function f with respect of a function g. The concept is more general as several newly defined differential operators based on the rate of change can be recovered with an appropriate choice of the function g. Several important properties of this concept are presented in details. Before the presentation of the associated integral, we present a discussion underpinning difference between the Caputo and the Riemann-Liouville derivative. We argue that, it is mathematically incorrect to expect the Caputo derivative to satisify the fundamental theorem of calculus with the Riemann-Liouville since this integral is not the anti-derivative. To overcome this misunderstanding, we introduce a fractional integral operator in Caputo sense and verify the fundamental theorem of calculus with each derivative from Caputo derivative to Atangana-Baleanu derivative in Caputo sense. We present in details the existence and uniqueness of the Cauchy problem with Caputo and Atangana-Baleanu derivatives. We derive the associated integral of the suggested derivative which happens to be the Riemann-Stieltjes integral. This turns to be a clear indication that the conformable and its variants and fractal are derivatives and have as anti-derivative the Riemann-Stieltjes integral. We extend the concept to fractional calculus and present several important properties. Numerical approximations are presented for each case and applications to some real world problems.