arXiv subjects

Hafiz Muhammad Fahad

Publications and source records attributed to Hafiz Muhammad Fahad.

6 recordsLinked to original sources

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↗

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↗

arXiv2019

Generalized substantial fractional operators and well-posedness of Cauchy problem

In this work we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional substantial derivatives are also introduced both in Riemann-Liouville and Caputo sense. Furthermore, we analyze fundamental properties of these operators. Finally, we consider a class of generalized substantial fractional differential equations and discuss the existence, uniqueness and continuous dependence of solutions on initial data.

math.CA↗