A NEW APPROACH TO GENERALIZED FRACTIONAL DERIVATIVES

By Udita N. Katugampola. 2011

578 citations recorded by the source corpus.

The author (Appl. Math. Comput. 218:860-865, 2011) introduced a new fractional integral operator given by, � I � a+f � (x) = � 1 � which generalizes the well-known Riemann-Liouville and the Hadamard fractional integrals. In this paper we present a new fractional derivative which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivatives to a single form. We also obtain two representations of the generalized derivative in question. An example is given to illustrate the results.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/14f0950f2629d519c44ddd07d5b1a225785d52ee

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

On more general forms of proportional fractional operators (2019, background): “While if one considers ν(x) = x σ σ , the fractional operators in the settings of Katugampola [12, 13] are derived.”
On the generalized fractional derivatives and their Caputo modification (2017, background): “The left and right generalized fractional derivatives of order α are defined by [14]”
Analysis of some generalized ABC – Fractional logistic models (2019, background): “It can be realized that starting from the turn of this century researchers have proposed a variety of fractional operators [10, 11, 12, 13, 14, 15, 16] and added variety of fractional operators with different approaches to the field of fractional calculus.”
On the ψ-Hilfer fractional derivative (2017, background): “These integrals and fractional derivatives have a different kernel and this makes the number of definitions [1, 2, 15, 16, 17, 18, 19] wide.”
New properties of conformable derivative (2015, background): “We have noticed that recently a new derivative has been suggested in [1, 17, 21, 22] and it seems to satisfy all the requirements of the standard derivative.”
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].”
On generalized fractional operators and a gronwall type inequality with applications (2017, methodology): “Other types of inequalities were considered in [20, 21] In this paper, we generalize the Gronwall inequality to differential equations involving a two-parameter generalized fractional derivative [22, 23] which provides the Riemann-Liouville and Hadamard fractional derivatives when one of the paramet”
On the Nonlinear $\Psi$-Hilfer Hybrid Fractional Differential Equations (2020, background): “The importance of the Ψ-Hilfer fractional derivative lies in the fact that it incorporates several well recognized fractional derivative operators as it’s particular cases, for example, RL[17], Caputo[17], Ψ-RL[17], Hadmard[17], Riesz[17], Erdély-Kober[17], ΨCaputo[18], Katugampola[19], Hilfer[20] ”
On the existence of positive solutions for generalized fractional boundary value problems (2019, unknown): “In [17], however, the author introduced Katugampola fractional derivative.”
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]).”
Fractional differential equations with dependence on the Caputo-Katugampola derivative (2016, background): “Then, in [10], it appeared a generalization to the Riemann–Liouville and Hadamard fractional derivatives, called the Katugampola fractional derivatives: D a+ x(t) = ρ Γ(1− α) t d dt ∫ t a τ (tρ − τρ)α x(τ)dτ, and D b− x(t) = −ρ Γ(1− α) t d dt ∫ b t τ (τρ − tρ)α x(τ)dτ.”