New approach to a generalized fractional integral

By Udita N. Katugampola. Applied Mathematics and Computation, 2010

796 citations recorded by the source corpus.

Abstract The paper presents a new fractional integration, which generalizes the Riemann–Liouville and Hadamard fractional integrals into a single form. Conditions are given for such a fractional integration operator to be bounded in an extended Lebesgue measurable space. Semigroup property for the above operator is also proved. We give a general definition of the fractional derivatives and give some examples.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/e151222ff3b6e2bb46ee04d4fa08893b1a9b6a67

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals (2020, positive): “4 for Ψ (θ) = θτ τ , τ > 0, and κ = 1 leads to the Katugampola fractional integrals [24], vi.”
Fractional Hermite-Hadamard Inequalities for some New Classes of Godunova-Levin Functions (2014, neutral): “H1(α ,s)| f (a)|+H2(α ,s)| f ′′(b)|q } 1 q , whereH1(α,s) and H2(α,s) are given by(7) and (8) respectively.”
A New Fractional Derivative with Classical Properties (2014, background): “In [1, 2], the author introduced fractional integrals and derivatives, which generalize the RiemannLiouville and the Hadamard fractional integrals and derivatives to a single form.”
Existence and Uniqueness results for a class of Generalized Fractional Differential Equations (2014, methodology): “In [13], the author derived the Mellin transforms of the generalized fractional integrals and derivatives defined in [11] and [12], respectively.”
More properties of the proportional fractional integrals and derivatives of a function with respect to another function (2020, methodology): “It should be noted that the nonlocal fractional versions of the conformable operators were discussed in [15, 16, 20].”
Generalized fractional derivatives and Laplace transform (2020, background): “After then, the corresponding derivatives are defined (see for example [20, 17, 18, 14, 12, 15, 16]).”
On the generalized fractional derivatives and their Caputo modification (2017, methodology): “The generalized left and right fractional integrals (in the sense of Katugampola) are defined by [13]”
A Gronwall inequality for a general Caputo fractional operator (2017, neutral): “1 in [15], we have I a+ D a+ x(t) = I α,ρ a+ I n−α,ρ a+ x(n)(t) = I n,ρ a+ x(n)(t) = ρ (n− 1)! ∫ t a (t − τ) d dτ x(n−1)(τ)dτ.”
Tempered and Hadamard-Type Fractional Calculus with Respect to Functions (2019, background): “The idea first arose in a 1964 paper [10] in which Erdelyi discussed fractional integration with respect to a power function, defining an operator which Katugampola rediscovered in 2011 [20] and which nowadays is sometimes called Katugampola fractional calculus.”
Inequalities by Means of Generalized Proportional Fractional Integral Operators with Respect to Another Function (2019, background): “([31,37]) A function U (ξ) is said to be in Lq,r[0, ∞] space if”
Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate (2020, methodology): “With singular kernels, we have the Caputo–derivative [2], the Riemann–Liouville derivative [2] and the Katugampola derivative [3].”
On a Caputo-type fractional derivative (2018, methodology): “In order to clarify the fractional integrals and derivatives that will be used, we introduce the Riemann–Liouville and Hadamard fractional integrals, the fractional derivatives proposed by Riemann–Liouville, Caputo, Hadamard, Caputo–Hadamard and, finally, the generalized fractional derivative [4, 9,”