David Usero

We present a partial panoramic view of possible contexts and applications of the fractional calculus. In this context, we show some different applications of fractional calculus to different models in ordinary differential equation (ODE) and partial differential equation (PDE) formulations ranging from the basic equations of mechanics to diffusion and Dirac equations.

The dust aerosols have an important effect on the solar radiation in the Martial atmosphere and both surface and atmospheric heating rates, which are also basic drivers of atmospheric dynamics. Aerosols cause an attenuation of the solar radiation traversing the atmosphere and this attenuation is modeled by the Lambert-Beer-Bouguer law, where the aerosol optical thickness plays an important role. Through Angstrom law, the aerosol optical thickness can be approximated and this law allows to model attenuation of the solar radiation traversing the atmosphere by a fractional diffusion equation. The analytical solution is available in the case of one space dimension. When we extend the fractional diffusion equation to the case of two or more space variables, we need large and massive computations to approach numerically the solutions. In this case a suitable strategy is to use the cloud computing to carry out the simulations. We present an introduction to cloud computing applied to the fractional diffusion equation in one dimension.

The dust aerosols floating in the atmosphere of Mars cause an attenuation of the solar radiation traversing the atmosphere that cannot be modeled through the use of classical diffusion processes. However, the definition of a type of fractional diffusion equation offers a more accurate model for this dynamic and the second order moment of this equation allows one to establish a connection between the fractional equation and the Ångstrom law that models the attenuation of the solar radiation. In this work we consider both one and three dimensional wavelength-fractional diffusion equations, and we obtain the analytical solutions and numerical methods using two different approaches of the fractional derivative.

. In this work we propose a new numerical scheme to simulate the motion of a nonrelativistic charge in a general magnetic field. The scheme is very accurate in the sense that it shows a discrete analogous of the underlying continuous energy conservation law, as well as an analogous of the angular momentum variation law.

An approach to model the Martian magnetization can be done using classical source models. Classical models, such as uniformly magnetized spheres and cylinders, allow for the introduction of additional constraints related to the available information of the magnetic field and its sources. The use of a suitable conservative numerical scheme in Cartesian coordinates was carried out for numerical studies. In this work the motion of different charged particles under the influence of different magnetized sources have been analyzed by using the proposed numerical scheme. For that purpose, electron, proton and alpha particles were used. In addition, the impact of the gravitational effect on the particles motion was also studied.

Dark soliton-like solutions are analyzed in the context of a certain nonlocal nonlinear Schrödinger Equation with nonlocal dispersive term of Kac-Baker type. Main purpose is to investigate such solutions with negative nonlinear term and the presence of general integral dispersive terms. First the model is presented and the properties of the fundamental solution, the continuous wave, is studied. Dark solitary waves are perturbations of this plane wave. The study of dark type of solutions is divided in two different cases black and dark solitary waves. The range of existence of such solutions is studied analytically, and also their physical quantities like norm, momentum and energy. Usual behavior of nonlinear systems under nonlocal dispersive terms is found.