New fractional integral unifying six existing fractional integrals

By Udita N. Katugampola. 2016

54 citations recorded by the source corpus.

In this paper we introduce a new fractional integral that generalizes six existing fractional integrals, namely, Riemann-Liouville, Hadamard, Erd\'elyi-Kober, Katugampola, Weyl and Liouville fractional integrals in to one form. Such a generalization takes the form \[ \left({}^{\rho}\mathcal{I}^{\alpha, \beta}_{a+;\eta, \kappa}f\right)(x)=\frac{\rho^{1-\beta}x^{\kappa}}{\Gamma(\alpha)}\int_a^x \frac{\tau^{\rho \eta +\rho-1}}{(x^\rho-\tau^\rho)^{1-\alpha}}f(\tau)\text{d}\tau, \quad 0\leq a < x < b \leq \infty. \] A similar generalization is not possible with the Erd\'elyi-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.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/517fbeba8cd94ffda163c629961f3aa8512a6752

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

On the ψ-Hilfer fractional derivative (2017, background): “his section, we present a class of fractional integrals and fractional derivatives, based on the choice of the arbitrary ψ function. For the class of integrals that will be presented next, we suggest [1, 2, 15, 16, 18, 20, 22, 24]. (1) If we consider ψ(x) = xin Eq.(2.3), we have Iα;x a+ f(x) = 1 Γ(α”
Ostrowski type inequalities in the sense of generalized $\mathcal{K}$-fractional integral operator for exponentially convex functions (2020, background): “(5) They degenerate to the both-sided Katugampola fractional integral operators [33] if Φ ( z ) = z β β ( β > 0) and K = 1.”
A new truncated $M$-fractional derivative type unifying some fractional derivative types with classical properties (2017, background): “[8] U. N. Katugampola, A new fractional derivative with classical properties, arXiv:1410.6535v2, (2014).”