Punjab University Journal of Mathematics, Vol 55, No 4 (2023)

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

Fahed Zulfeqarr, Amit Ujlayan, Priyanka Ahuja

Abstract


This paper proposes a generalization to the ordinary derivative, the deformable derivative. For this, we employ a limit approach like theordinary derivative but use a parameter varying over the unit interval. Thedefinition makes the deformable derivative equivalent to the ordinary derivativebecause one’s existence implies another. Its intrinsic property ofcontinuously deforming function to its derivative, together with the graphicalillustration of linear expression of the function and its derivative, renderssufficient substances to name it deformable derivative. We deriveRolle’s, Mean-value and Taylor’s theorems for the deformable derivativeby establishing some of its basic properties. We also define the deformableintegral using the fundamental theorem of calculus and discuss associatedinverse, linearity, and commutativity property. In addition, we establish aconnection between deformable integral and Riemann-Liouville fractionalintegral. As theoretical applications, we solve some fractional differentialequations

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