We present and prove a new generalisation of the Malgrange-Ehrenpreis theorem to fractional partial differential equations, which can be used to construct fundamental solutions to all partial differential operators of ...

In this paper, we have investigated the Langevin and Brownian equations on fractal time sets using F α-calculus and shown that the mean square displacement is not varied linearly with time. We have also generalized the ...

This corrigendum corrects two equations presented in the paper “Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions” [Commun. Nonlinear Sci. Numer. Simulat. 67 (2019) ...

The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function <mml:semantics>f(t)</mml:semantics>, ...

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 ...

We establish a new formula for the fractional derivative with Mittag-Leffler kernel, in the form of a series of Riemann-Liouville fractional integrals, which brings out more clearly the non-locality of fractional derivatives ...

We consider an integral transform introduced by Prabhakar, involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We ...

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 ...

We establish analogues of the mean value theorem and Taylor's theorem for fractional differential operators defined using a Mittag-Leffler kernel. We formulate a new model for the fractional Boussinesq equation by using ...