On the $ψ$-Hilfer fractional derivative

J. Vanterler da C. Sousa; E. Capelas de Oliveira

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On the $ψ$-Hilfer fractional derivative

J. Vanterler da C. Sousa, E. Capelas de Oliveira

arXiv:1708.05109v1Aug 17, 2017math.CA

TL;DR

The paper addresses the proliferation of fractional operators by introducing a fractional derivative with respect to another function ψ, called the ψ-Hilfer derivative. It develops this operator’s properties and related convergence results, including examples involving the one-parameter Mittag-Leffler function, while identifying extensions for future work.

Problem
Existing fractional operators have broad definitions, but some formulations are restricted in how differentiation and integration are ordered.
Method
The paper defines the ψ-Hilfer fractional derivative with respect to another function ψ and studies its relations and properties.
Results
The paper presents operator results, uniformly convergent sequence results, and examples involving the one-parameter Mittag-Leffler function.
Takeaways & Limitations
The ψ-Hilfer operator provides a class of integrals and fractional derivatives intended to overcome the wide number of existing definitions.
Takeaways & Limitations
Extensions to Gronwall inequalities, Cauchy-type problem existence and uniqueness, and variable-order operators remain future work.

Abstract

from arXiv · show

In this paper we introduce a new fractional derivative with respect to another function the so-called $ψ$-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 $ψ$-Hilfer fractional derivative.

1. Introduction

The introduction frames fractional calculus as a broad and growing field with many operator definitions, motivating a ψ-dependent Hilfer derivative that unifies and generalizes established forms.

Fractional calculus has developed across theory and applications in mathematics, physics, chemistry, engineering, and biology.
Numerous fractional integrals and derivatives have arisen because different definitions use different kernels.
A fractional derivative with respect to another function ψ was introduced using the Riemann-Liouville framework.
The Riemann-Liouville-based definition only covers operators in which differentiation acts on the fractional integral.
The proposed ψ-Hilfer derivative allows the classical differential operator and fractional integral operator to be ordered in either way, enlarging the resulting class.
The paper introduces preliminaries and ψ-based Riemann-Liouville and Caputo derivatives before presenting the ψ-Hilfer operator and its properties.

2. Preliminaries

The preliminaries establish function spaces, weighted spaces, fractional integrals, and ψ-based Riemann-Liouville and Caputo derivatives needed for the later operator theory.

The section introduces function spaces including continuous, absolutely continuous, continuously differentiable, and weighted spaces.
The weighted space Cγ;ψ[a, b] consists of functions for which (ψ(t) − ψ(a))γ f(t) is continuous on [a, b], with 0 ≤ γ < 1.
Riemann-Liouville fractional integrals and derivatives are defined on finite or infinite intervals for suitable orders and function spaces.
Fractional integrals with respect to an increasing function ψ are introduced as a special approach when the kernel is unknown.
The preliminaries state semigroup lemmas and related results used in developing the subsequent fractional-operator theory.
The ψ-Riemann-Liouville and ψ-Caputo derivatives are defined under regularity and monotonicity conditions on f and ψ.

3. ψ-Hilfer fractional derivative

This section defines the ψ-Hilfer fractional derivative and develops its relationships, boundedness, inversion, and semigroup properties under stated regularity conditions.

The ψ-Hilfer derivative is introduced to unify the ψ-Riemann-Liouville and ψ-Caputo approaches through the Hilfer construction.
The section gives the left- and right-sided ψ-Hilfer derivatives for n − 1 < α < n and 0 ≤ β ≤ 1.
For 0 < α < 1 and 0 ≤ β ≤ 1, the operator specializes to a ψ-Riemann-Liouville fractional derivative in a stated case.
The ψ-Hilfer derivatives are bounded operators for n − 1 < α < n and 0 ≤ β ≤ 1.
The paper derives relationships between ψ-Hilfer and ψ-Caputo derivatives and states additional operator identities under suitable regularity assumptions.
Further results address limiting behavior, inversion by fractional integration, semigroup laws, and consequences in weighted spaces.

4. Miscellaneous results and examples

This section develops uniform-convergence results for sequences involving the ψ-Hilfer and fractional integral operators, with examples based on Mittag-Leffler and ψ-power functions. It also records related identities and convergence conditions for these functions.

Uniform convergence: The section presents uniform-convergence results for sequences involving the fractional ψ-Hilfer operator and fractional integral operator.The results are framed on intervals [a + ε, b) or under assumptions involving continuous functions and existing fractional derivatives.
Examples: The section includes examples involving the one-parameter Mittag-Leffler function and ψ-power functions.The Mittag-Leffler examples use functions built from ψ(x) − ψ(a) or ψ(b) − ψ(x), while the power-function results establish corresponding identities.
Operator-limit relations: Under the stated hypotheses, limits can be interchanged with fractional integral or derivative operators.The results concern uniformly convergent sequences of continuous functions and corresponding ψ-fractional operators.
Assumptions: The results assume an increasing ψ with ψ′(x) ≠ 0 and impose existence and continuity conditions on the function sequences and their fractional operators.These conditions appear in the theorem statements governing the convergence results.
Examples: For λ > 0 and n − 1 < α < n, the one-parameter Mittag-Leffler examples yield uniformly convergent function sequences.The supplied theorem statement identifies the parameter range and the uniform-convergence conclusion.

5. A wide class of fractional derivatives and integrals

By choosing the function ψ and parameters in the ψ-Hilfer framework, the paper obtains a wide class of fractional integrals and derivatives, including established operators as particular cases.

Fractional integrals: The fractional integral framework recovers operators including the Hadamard, Erdlyi-Kober, Erdlyi, Kober, Katugampola, Prabhakar, Chen, Riesz, Feller, and Weyl integrals.
Choosing ψ, a, b, and limiting α and β generates a wide class of fractional derivatives from the ψ-Hilfer operator.
Fractional derivatives: Specific choices of ψ and parameter limits recover the ψ-Caputo and ψ-Riemann-Liouville fractional derivatives.
Fractional derivatives: Setting ψ(x)=x, ψ(x)=x^ρ, or ψ(x)=ln x yields classical and generalized derivatives such as Caputo, Katugampola, Riemann-Liouville, Hadamard, and Caputo-Katugampola.
Fractional derivatives: Further parameter choices recover Hilfer-Hadamard, Hilfer-Katugampola, Erdlyi-Kober, Prabhakar, Jumarie, Riesz, Feller, Weyl, Cassar, and Caputo-Riesz derivatives.
The authors conclude that the ψ-Hilfer fractional derivative generalizes numerous fractional derivatives through particular choices of the function and parameters.

6. Concluding remarks

The paper proposes the ψ-Hilfer fractional derivative, develops results and a Mittag-Leffler example, and uses it to present classes of fractional integrals and derivatives. Future work concerns fractional Gronwall inequalities, Cauchy-type problems, and variable-order extensions.

The paper proposes a fractional derivative with respect to another function ψ in the sense of the Hilfer fractional derivative.
It discusses properties and results of the ψ-Hilfer operator, including an example involving Mittag-Leffler functions.
The operator is used to present a class of fractional integrals and derivatives intended to address the wide number of existing definitions.
Future work includes generalizing Gronwall inequalities, studying existence and uniqueness for Cauchy-type problems, and extending the operator to variable order.