On the ψ-Hilfer fractional derivative

By J. Sousa, E. Capelas de Oliveira. Communications in nonlinear science & numerical simulation, 2017

873 citations recorded by the source corpus.

In this paper we introduce a new fractional derivative with respect to another function the so-called $\psi$-Hilfer fractional derivative. We discuss some properties and important results of the fractional calculus. In this sense, we present some uniformly convergent sequence of function results and examples involving the Mittag-Leffler function with one parameter. Finally, we present a wide class of integrals and fractional derivatives, by means of the fractional integral with respect to another function and the $\psi$-Hilfer fractional derivative.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/875727678f97d82acb227d4dd26e52ee02aa7927

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In Applied Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

Stability Results for Implicit Fractional Pantograph Differential Equations via ϕ-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition (2020, background): “More recently, motivated by classical Riemann-Liouville, Caputo fractional derivative, Hilfer-fractional derivative, ψ-Riemann-Liouville integral and ψ-Caputo fractional derivatives, Sousa and Oliveira [19] initiated an interesting fractional differential operator called ψ-Hilfer fractional derivati”
A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator (2017, methodology): “Recently, Sousa and Oliveira [4], introduced ψ-Hilfer fractional derivative, and presented a class of fractional integrals and fractional derivatives, which can be obtained from such definitions, with the particular choice of the function ψ.”
Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel (2020, neutral): “3 ([17, 44, 56]) Suppose that (ν1,ν2) ⊆ R is a finite or infinite interval of the real numbers R and ψ(x) is an increasing positive monotone function on the interval [ν1,ν2] with ψ ′(x) ∈ L(1)(ν1,ν2).”