On Hilfer fractional difference operator

In this article, a new definition of fractional Hilfer difference operator is introduced. Definition based properties are developed and utilized to construct fixed point operator for fractional order Hilfer difference equations with initial condition. We acquire some conditions for existence, uniqueness, Ulam-Hyers and Ulam-Hyers-Rassias stability. Modified Gronwalls inequality is presented for discrete calculus with the delta difference operator.

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.

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 $Psi$-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.

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.