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Arran Fernandez

Publications and source records attributed to Arran Fernandez.

At least 19 recordsLinked to original sources
arXiv2026

Unique axiomatisation of fractional integral operators on the real line

The Cartwright--McMullen theorem (1978) establishes the uniqueness of the Riemann--Liouville family of fractional integral operators on a compact interval $[0,1]$ under a natural set of axioms: inclusion of the classical integral, a semigroup property, positivity, and continuity. By unpacking the proof into its constituent steps and understanding its structure clearly, we provide improvements and extensions of the original result. Firstly, by using some properties of topological groups, we demonstrate that the positivity axiom can be removed with almost no effect on the result. Secondly, we consider a version of the theorem on the whole real line $\mathbb{R}$, with constant of integration $-\infty$ instead of $0$, which was mentioned without proof in the 1978 paper. The theorem can be extended to this setting, but it is not trivial to do so: the theory of Fr\'echet spaces must be used, and we introduced a new shift-commutativity axiom to get the density result that we need to extend the proof to spaces of functions and distributions on $\mathbb{R}$ with left-bounded support.

math.FA↗
arXiv2022

Prabhakar-type linear differential equations with variable coefficients

Linear differential equations with variable coefficients and Prabhakar-type operators featuring Mittag-Leffler kernels are solved. In each case, the unique solution is constructed explicitly as a convergent infinite series involving compositions of Prabhakar fractional integrals. We also extend these results to Prabhakar operators with respect to functions. As an important illustrative example, we consider the case of constant coefficients, and give the solutions in a more closed form by using multivariate Mittag-Leffler functions.

math.CA↗
arXiv2022

On tempered fractional calculus with respect to functions and the associated fractional differential equations

The prime aim of the present paper is to continue developing the theory of tempered fractional integrals and derivatives of a function with respect to another function. This theory combines the tempered fractional calculus with the $Ψ$-fractional calculus, both of which have found applications in topics including continuous time random walks. After studying the basic theory of the $Ψ$-tempered operators, we prove mean value theorems and Taylor's theorems for both Riemann--Liouville type and Caputo type cases of these operators. Furthermore, we study some nonlinear fractional differential equations involving $Ψ$-tempered derivatives, proving existence-uniqueness theorems by using the Banach contraction principle, and proving stability results by using Grönwall type inequalities.

math.CA↗
arXiv2022

Weighted fractional calculus: a general class of operators

The operators of fractional calculus come in many different types, which can be categorised into general classes according to their nature and properties. We conduct a formal study of the class known as weighted fractional calculus and its extension to the larger class known as weighted fractional calculus with respect to functions. These classes contain tempered, Hadamard-type, and Erdélyi--Kober operators as special cases, and in general they can be related to the classical Riemann--Liouville fractional calculus via conjugation relations. Considering the corresponding modifications of the Laplace transform and convolution operations enables differential equations to be solved in the setting of these general classes of operators.

math.CA↗
arXiv2021

A new representation for the solutions of fractional differential equations with variable coefficients

A recent development in the theory of fractional differential equations with variable coefficients has been a method for obtaining an exact solution in the form of an infinite series involving nested fractional integral operators. This solution representation is constructive but difficult to calculate in practice. Here we show a new representation of the solution function, as a convergent series of single fractional integrals, which will be easier to use for computational work and applications. In the particular case of constant coefficients, the solution is given in terms of the Mittag-Leffler function. We also show some applications in Cauchy problems for partial differential equations involving both time-fractional and space-fractional operators and with time-dependent coefficients.

math.CA↗
arXiv2020

On fractional calculus with analytic kernels with respect to functions

Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important classes of fractional-calculus operators are the fractional integrals and derivatives with respect to functions (dating back to the 1970s) and those with general analytic kernels (introduced in 2019). To cover both of these settings in a single study, we can consider fractional integrals and derivatives with analytic kernels with respect to functions, which have never been studied in detail before. Here we establish the basic properties of these general operators, including series formulae, composition relations, function spaces, and Laplace transforms. The tools of convergent series, from fractional calculus with analytic kernels, and of operational calculus, from fractional calculus with respect to functions, are essential ingredients in the analysis of the general class that covers both.

math.CA↗
arXiv2020

Tempered and Hadamard-type fractional calculus with respect to functions

Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the literature are the Hadamard-type fractional calculus and tempered fractional calculus. This paper establishes a connection between these two definitions, writing one in terms of the other by making use of the theory of fractional calculus with respect to functions. By extending this connection in a natural way, a generalisation is developed which unifies several existing fractional operators: Riemann--Liouville, Caputo, classical Hadamard, Hadamard-type, tempered, and all of these taken with respect to functions. The fundamental calculus of these generalised operators is established, including semigroup and reciprocal properties as well as application to some example functions. Function spaces are constructed in which the new operators are defined and bounded. Finally, some formulae are derived for fractional integration by parts with these operators.

math.CA↗
arXiv2020

On Laplace transforms with respect to functions and their applications to fractional differential equations

An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $Ψ$-fractional calculus. The operational calculus approach has proved useful for understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in practice. We prove several important properties of these generalised Laplace transforms, including an inversion formula, and apply it to solve some fractional differential equations, using the operational calculus approach for efficient solving.

math.CA↗
arXiv2020

Fractional differential relations for the Lerch zeta function

Starting from a recent result expressing the Lerch zeta function as a fractional derivative, we consider further fractional derivatives of the Lerch zeta function with respect to different variables. We establish a partial differential equation, involving an infinite series of fractional derivatives, which is satisfied by the Lerch zeta function.

math.NT↗
arXiv2020

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

We define an analogue of the classical Mittag-Leffler function which is applied to two variables, and establish its basic properties. Using a corresponding single-variable function with fractional powers, we define an associated fractional integral operator which has many interesting properties. The motivation for these definitions is twofold: firstly their link with some fundamental fractional differential equations involving two independent fractional orders, and secondly the fact that they emerge naturally from certain applications in bioengineering.

math.CA↗
arXiv2019

On some analytic properties of tempered fractional calculus

We consider the integral and derivative operators of tempered fractional calculus, and examine their analytic properties. We discover connections with the classical Riemann-Liouville fractional calculus and demonstrate how the operators may be used to obtain special functions such as hypergeometric and Appell's functions. We also prove an analogue of Taylor's theorem and some integral inequalities to enrich the mathematical theory of tempered fractional calculus.

math.CA↗
arXiv2019

A complex analysis approach to Atangana-Baleanu fractional calculus

The standard definition for the Atangana-Baleanu fractional derivative involves an integral transform with a Mittag-Leffler function in the kernel. We show that this integral can be rewritten as a complex contour integral which can be used to provide an analytic continuation of the definition to complex orders of differentiation. We discuss the implications and consequences of this extension, including a more natural formula for the Atangana-Baleanu fractional integral and for iterated Atangana-Baleanu fractional differintegrals.

math.CV↗
arXiv2019

Analytical Development of Incomplete Riemann-Liouville Fractional Calculus

The theory of fractional calculus has developed in a number of directions over the years, including: the formulation of multiple different definitions of fractional differintegration; the extension of various properties of standard calculus into the fractional scenario; the application of fractional differintegrals to assorted special functions. Recently, a new variant of fractional calculus has arisen, namely incomplete fractional calculus. In two very recent papers, incomplete versions of the Riemann-Liouville and Caputo fractional differintegrals have been formulated and applied to several important special functions. In the current work, we develop the theory of incomplete fractional calculus in more depth, investigating further properties of the incomplete Riemann-Liouville fractional differintegrals and answering some fundamental questions about these operators. By considering appropriate function spaces, we formulate rigorously the definitions of incomplete Riemann-Liouville fractional integration, and justify how this model may be used to analyse a wider class of functions than classical fractional calculus. By using analytic continuation, we formulate definitions for incomplete Riemann-Liouville fractional differentiation, hence extending the incomplete integrals to a fully-fledged model of fractional calculus. We also investigate and analyse these operators further, in order to prove new properties. These include a Leibniz rule for incomplete differintegrals of products, and composition properties of incomplete differintegrals with classical calculus operations. These are natural and expected issues to investigate in any new model of fractional calculus, and in the incomplete Riemann-Liouville model the results emerge naturally from the definition previously proposed.

math.CA↗
arXiv2019

On fractional calculus with general analytic kernels

Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann-Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators.

math.CA↗
arXiv2019

On a new class of fractional difference-sum operators based on discrete Atangana-Baleanu sums

We formulate a new class of fractional difference and sum operators, study their fundamental properties, and find their discrete Laplace transforms. The method depends on iterating the fractional sum operators corresponding to fractional differences with discrete Mittag-Leffler kernels. The iteration process depends on the binomial theorem. We note in particular the fact that the iterated fractional sums have a certain semigroup property and hence the new introduced iterated fractional difference-sum operators have this semigroup property as well.

math.CA↗
arXiv2018

Solving PDEs of fractional order using the unified transform method

We consider the unified transform method, also known as the Fokas method, for solving partial differential equations. We adapt and modify the methodology, incorporating new ideas where necessary, in order to apply it to solve a large class of partial differential equations of fractional order. We demonstrate the applicability of the method by implementing it to solve a model fractional problem.

math.AP↗