New fractional integral unifying six existing fractional integrals
arXiv:1612.08596
Abstract
In this paper we introduce a new fractional integral that generalizes six existing fractional integrals, namely, Riemann-Liouville, Hadamard, Erdélyi-Kober, Katugampola, Weyl and Liouville fractional integrals in to one form. Such a generalization takes the form \[ \left({}^ρ\mathcal{I}^{α, β}_{a+;η, κ}f\right)(x)=\frac{ρ^{1-β}x^κ}{Γ(α)}\int_a^x \frac{τ^{ρη+ρ-1}}{(x^ρ-τ^ρ)^{1-α}}f(τ)\text{d}τ, \quad 0\leq a < x < b \leq \infty. \] A similar generalization is not possible with the Erdélyi-Kober operator though there is a close resemblance with the operator in question. We also give semigroup, boundedness, shift and integration-by-parts formulas for completeness.
arXiv commentPreprint submitted for publication
References in corpus (7)
- New Approach To A Generalized Fractional Integral
- A New Approach to Generalized Fractional Derivatives
- Hermite-Hadamard and Hermite-Hadamard-Fejér type Inequalities for Generalized Fractional Integrals
- Review of Some Promising Fractional Physical Models
- Existence and Uniqueness results for a class of Generalized Fractional Differential Equations
- An approximation formula for the Katugampola integral
- Variational Problems Involving a Caputo-Type Fractional Derivative