Extension of Conformal Fractional Derivatives

Abstract

In this work , We have introduced a new conformable fractional derivative operator, 

named as the operator, which depends on two parameters in (0,1] with , 

and based on this operator we propose an extended conformable fractional integrating 

operator which is called . We investigate some essential properties involving the 

commutators, non-commutativity relations when iterating the extended operator, and 

have represented the extended operator as a combination of ordinary differential 

operators and single-parameter conformal derivatives. To show that this methodology 

has some real utility we employ the 

 -integral transform on a Conformal Fractional 

Integro-Differential Equation (CFIDE) which describes a viscoelastic damped system. 

The novel extended fractional method provides an effective analytical and series-based 

technique for the treatment of challenging fractional order integro-differential and 

boundary-value prob