A Generalization to Ordinary Derivative and its Associated Integral with some applications

Abstract

This paper proposes a generalization to the ordinary derivative, the deformable derivative. For this, we employ a limit approach like the ordinary derivative but use a parameter varying over the unit interval. The definition makes the deformable derivative equivalent to the ordinary derivative because one’s existence implies another. Its intrinsic property of continuously deforming function to its derivative, together with the graphical illustration of linear expression of the function and its derivative, renders sufficient substances to name it deformable derivative. We derive Rolle’s, Mean-value and Taylor’s theorems for the deformable derivative by establishing some of its basic properties. We also define the deformable integral using the fundamental theorem of calculus and discuss associated inverse, linearity, and commutativity property. In addition, we establish a connection between deformable integral and Riemann-Liouville fractional integral. As theoretical applications, we solve some fractional differential equations.