A Caputo fractional derivative of a function with respect to another function

By Ricardo Almeida. Communications in nonlinear science & numerical simulation, 2016

1,153 citations recorded by the source corpus.

Abstract In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor’s Theorem, Fermat’s Theorem, etc., are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9fbc710121b73690017895d442d0c395167a15bf

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Citations in the corpus

Inequalities by Means of Generalized Proportional Fractional Integral Operators with Respect to Another Function (2019, background): “Almeida [9] expounded Ψ−Caputo derivative in the sense of another function Ψ and Kilbas et al.”
A numerical study of fractional relaxation–oscillation equations involving $$\psi $$ψ-Caputo fractional derivative (2018, positive): “[1,2] introduced the concept of ψ-Caputo fractional derivative as follows.”
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] ”
A Note on Reverse Minkowski Inequality via Generalized Proportional Fractional Integral Operator with respect to Another Function (2020, background): “Almeida [7] proposed a new fractional derivative called Caputo derivative with respect to another function Φ, and Kilbas et al.”
Fractional Differential Equations with Mixed Boundary Conditions (2017, methodology): “In [5], a Caputo-type fractional derivative of a function with respect to another function is presented, and some of its main properties were studied.”
Nonlinear generalized fractional differential equations with generalized fractional integral conditions (2020, background): “problem of fractional differential equations cDα 0+;ψ = f (t, y(t)), t ∈ [a, b], 2 < α < 3, (1) y(a) = ya, y′(a) = ya1 , y(b) = KI a+;ψy(θ), (2) where cDα 0+;ψ is the ψ-Caputo fractional derivative introduced in [20], I 0+;ψ is the ψ-Riemann–Liouville fractional integral presented in [14].”
Fractional differential equations with a Caputo derivative with respect to a Kernel function and their applications (2018, background): “The solution of this problem is the function (see Lemma 2 in Almeida17). x(t) = E𝛼((𝜓(t) − 𝜓(0))1.5).”