C. Calderón-Ramón

Nonlocal electrical diffusion equation

In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is 0<β≤1 and for the time domain is 0<γ≤2. We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes.

Experimental evaluation of viscous damping coefficient in the fractional underdamped oscillator

In this article, we obtain the viscous damping coefficient β theoretically and experimentally in the spring–mass–viscodamper system. The calculation is performed to obtain the quasi-period τ. The influence of the viscosity of the fluid and the damping coefficient is analyzed using three fluids, water, edible oil, and gasoline engine oil SAE 10W-40. These processes exhibit temporal fractality and non-local behaviors. Our general non-local damping model incorporates fractional derivatives of Caputo type in the range ( 0 , 1 ] , and the viscous damping coefficients are determined in terms of the inverse Mittag-Leffler function. The classical models are recovered when the order of the fractional derivatives is equal to 1.

Fractional electromagnetic waves in conducting media

We present the fractional wave equation in a conducting material. We used a Maxwell’s equations with the assumptions that the charge density and current density J were zero, and that the permeability and permittivity were constants. The fractional wave equation will be examined separately; with fractional spatial derivative and fractional temporal derivative, finally, consider a Dirichlet conditions, the Fourier method was used to find the full solution of the fractional equation in analytic way. Two auxiliary parameters and are introduced; these parameters characterize consistently the existence of the fractional space-time derivatives into the fractional wave equation. A physical relation between these parameters is reported. The fractional derivative of Caputo type is considered and the corresponding solutions are given in terms of the Mittag-Leffler function show fractal space-time geometry different from the classical integer-order model.

Equivalent Circuits Applied in Electrochemical Impedance Spectroscopy and Fractional Derivatives with and without Singular Kernel

We present an alternative representation of integer and fractional electrical elements in the Laplace domain for modeling electrochemical systems represented by equivalent electrical circuits. The fractional derivatives considered are of Caputo and Caputo-Fabrizio type. This representation includes distributed elements of the Cole model type. In addition to maintaining consistency in adjusted electrical parameters, a detailed methodology is proposed to build the equivalent circuits. Illustrative examples are given and the Nyquist and Bode graphs are obtained from the numerical simulation of the corresponding transfer functions using arbitrary electrical parameters in order to illustrate the methodology. The advantage of our representation appears according to the comparison between our model and models presented in the paper, which are not physically acceptable due to the dimensional incompatibility. The Markovian nature of the models is recovered when the order of the fractional derivatives is equal to 1.

Enhancing ultrasound images using the recursive p-step unbiased FIR filter

In this paper, we present a computational scheme for the implementation of p-step UFIR filter ultrasound image processing. The recursive scheme is focused both signal to noise ratio and root mean-square-error on filter improved performance. In this way, quantitative evaluations are computed to verify the performance carried out by the new recursive computational scheme under metrics. On the other hand, qualitative evaluations are showed for both cases: signals (one dimensional) and images (two dimensional). Finally, this efficient computational algorithm manifests an outstanding signal/image enhancing with strong metric convergence despite the computational complexity.

A mathematical model of circadian rhythms synchronization using fractional differential equations system of coupled van der Pol oscillators

This paper presents an alternative representation of a system of differential equations qualitatively showing the behavior of the biological rhythm of a crayfish during their transition from juveni...

Numerical simulation of metallic nanostructures interacting with electromagnetic fields using the Lorentz–Drude model and FDTD method

In this paper, we present an analysis and modeling of the interaction of electromagnetic fields with metallic nanostructures using computational tools that allow us to study the phenomena that are produced as an electromagnetic field interacts with the nanostructure. The analysis of dielectric systems including metals can be very complicated because of the metal parameters. For this reason, we propose to integrate the dielectric function of the Lorentz-Drude model with the finite difference time-domain (FDTD) method, which will permit to study the surface and internal effects within the metal nanostructure system added to the dielectric system, and the interaction of electromagnetic fields with atoms, ions or molecules excited up to their resonant frequency, which causes transitions among atomic energy levels. We solved the system and showed the results of the simulation for the following case studies, silver nanosphere of 100nm in diameter, gold nanorod of 12nm in thickness and 30nm in length and gold nanogroove of 70nm.

Integer Wavelet Transform Based Watermarking System for PCM Signals

In this paper, a watermarking system based on the integer wavelet transform is proposed. A set of different wavelets was chosen for this propose so their performance as watermark transmission channel can be analyzed. The watermark is embedded in the second decomposition level of the wavelet transform of a PCM signal in a block wise approach, the coefficients of the wavelet transform are assumed to be accurately modeled as a Laplacian channel. A set of attacks were performed in order to test system’s robustness, we found that this approach is robust against several attacks such as additive white noise, low pass filtering, cropping, among other attacks; the best suited wavelet class for developing watermarking systems was identified too.