A modification to the conformable fractional calculus with some applications

Abstract

In the conformable fractional calculus, TαTβ≠TβTα and IαIβ≠IβIα, where Tα and Iα are conformable fractional differential and integral operators, respectively. Also, Tβ≠Tnα and Iβ≠Inα, where β=nα for some n∈N. In this work, we introduce a modification to the definition of conformable fractional derivative and conformable fractional integral. Our definition characterizes by the realization of the commutative property between the differential operators and the integral operators. The higher derivatives in the sense of the modified conformable fractional derivative are coinciding with the sequential derivatives. Finally, we present the solutions of the Cauchy-Euler and linear (with constant coefficients) fractional differential equations to show our new approach.