In this paper, we introduce some fundamental notions related to the so-called stochastic derivatives with respect to a given σ-field Q. In our framework, we recall well-known results about Markov Wiener diffusions. We afterwards mainly focus on the case where X is a fractional diffusion and where Q is the past, the future or the present of X . We treat some crucial examples and our main result ...

Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow a modeling of attenuation and dispersion that better describes sound propagation in biological tissues. Traditional constitutive equations from solid mechanics and heat conduction are modified using fractional derivatives. They are used to derive a nonlin...

The growing number of applications of fractional derivatives in various fields of science and engineering indicates that there is a significant demand for better mathematical algorithms for models with real objects and processes. Currently, most algorithms are designed for 1D problems due to the memory effect in fractional derivatives. In this work, the 2D fractional subdiffusion problems are s...

This paper develops appropriate boundary conditions for the two-sided fractional diffusion equation, where the usual second derivative in space is replaced by a weighted average of positive and negative fractional derivatives. Mass preserving, reflecting boundary conditions for two-sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary. Stable, con...

In some real world problems, fractional-order models are found to be more adequate than integer-order models. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. The mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electro dynamics of complex medium, polymer...

This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences similarities. The derivative is closely connected to Haussdorff's concepts dimension geometry. paper distinguishes between the a function on domain function, where image space. Different continuous approximations for are discussed, it shown that $q$-calculus approximation function. A s...

It has been shown that anomalous relaxation in dielectrics can be described in terms of equations with fractional derivatives. The solutions of the resulting equation with fractional derivatives are expressed by the Mittag –Leffler function and the Fox function. The conditions of a change from the Debye relaxation to “slow” (anomalous) relaxation with a power time dependence have been examined ...

It has been recognized recently that fractional calculus is useful for handling scaling structures and processes. We begin this survey by pointing out the relevance of the subject to physical situations. Then the essential definitions and formulae from fractional calculus are summarized and their immediate use in the study of scaling in physical systems is given. This is followed by a brief sum...

The fractional derivatives in the sense of Caputo, and the homotopy analysis method (HAM) are used to construct the approximate solutions for nonlinear fractional dispersive long wave equation with reaspect to time fractional derivative. The HAM contains a certain auxiliary parameter which provides us with a simple way to adjust and control the convergence region and rate of convergence of the ...

Fractional centred differences and derivatives definitions are proposed generalising to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalisations of the Cauchy integral derivative. To compute this integral, a special two straight line path was use...