Fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. They are useful to model anomalous diffusion, where a plume of particles spreads in a different manner than the classical diffusion equation predicts. An initial value problem involving a space-fractional diffusion equation is an abstract Cauchy problem, whose analyt...

A new time-fractional derivative with Mittag-Leffler memory kernel, called the generalized Atangana-Baleanu is defined along associated integral operator. Some properties of operators are proved. The operator suitable to generate by particularization known Atangana-Baleanu, Caputo-Fabrizio and Caputo derivatives. mathematical model advection-dispersion process kinetic adsorption formulated cons...

In this article, we define the fractional Mellin transform by using Riemann-Liouville fractional integral operator and Caputo fractional derivative of order [Formula: see text] and study some of their properties. Further, some properties are extended to fractional way for Mellin transform.

In recent years, fractional reaction-diffusion models have been studied due to their usefulness and importance in many areas of mathematics, statistics, physics, and chemistry. In a fractional diffusion equation, the second derivative in the spatial variable is replaced by a fractional derivative. The resulting solutions spread faster than classical solutions and may exhibit asymmetry, dependin...

The fractional Laplacian and the fractional derivative are two different mathematical concepts (Samko et al, 1987). Both are defined through a singular convolution integral, but the former is guaranteed to be the positive definition via the Riesz potential as the standard Laplace operator, while the latter via the Riemann-Liouville integral is not. It is noted that the fractional Laplacian can ...

Fractional diffusion equations are useful for applications where a cloud of particles spreads faster than the classical equation predicts. In a fractional diffusion equation, the second derivative in the spatial variable is replaced by a fractional derivative of order less than two. The resulting solutions spread faster than the classical solutions and may exhibit asymmetry, depending on the fr...

In this article, we present some results on the existence and uniqueness of random solutions to a non-linear implicit fractional differential equation involving generalized Caputo derivative operator supplemented with non-local periodic boundary conditions. We make use fixed point theorems due Banach Krasnoselskii derive desired results. Examples illustrating obtained are also presented.

the purpose of this paper is to study the fuzzy fractional differentialequations. we prove that fuzzy fractional differential equation isequivalent to the fuzzy integral equation and then using this equivalenceexistence and uniqueness result is establish. fuzzy derivative is considerin the goetschel-voxman sense and fractional derivative is consider in theriemann liouville sense. at the end, we...

The author (Appl. Math. Comput. 218(3):860-865, 2011) introduced a new fractional integral operator given by, ( I a+f ) (x) = ρ1−α Γ(α) ∫ x a τρ−1f(τ) (xρ − τρ)1−α dτ, which generalizes the well-known Riemann-Liouville and the Hadamard fractional integrals. In this paper we present a new fractional derivative which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivativ...