H. Yépez-Martínez

Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel

In this paper, the fractional equations of the mass-spring-damper system with Caputo and Caputo–Fabrizio derivatives are presented. The physical units of the system are preserved by introducing an auxiliary parameter σ. The input of the resulting equations is a constant and periodic source; for the Caputo case, we obtain the analytical solution, and the resulting equations are given in terms of the Mittag–Leffler function; for the Caputo–Fabrizio approach, the numerical solutions are obtained by the numerical Laplace transform algorithm. Our results show that the mechanical components exhibit viscoelastic behaviors producing temporal fractality at different scales and demonstrate the existence of Entropy 2015, 17 6290 material heterogeneities in the mechanical components. The Markovian nature of the model is recovered when the order of the fractional derivatives is equal to one.

Feng’s First Integral Method Applied to the ZKBBM and the Generalized Fisher Space-Time Fractional Equations

The fractional derivatives in the sense of the modified Riemann-Liouville derivative and Feng’s first integral method are employed to obtain the exact solutions of the nonlinear space-time fractional ZKBBM equation and the nonlinear space-time fractional generalized Fisher equation. The power of this manageable method is presented by applying it to the above equations. Our approach provides first integrals in polynomial form with high accuracy. Exact analytical solutions are obtained through establishing first integrals. The present method is efficient and reliable, and it can be used as an alternative to establish new solutions of different types of fractional differential equations applied in mathematical physics.

Phenomenological and microscopic cluster models. I. the geometric mapping

The geometrical mapping of algebraic nuclear cluster models is investigated within the coherent state formalism. Two models are considered: the semimicroscopic algebraic cluster model (SACM) and the phenomenological algebraic cluster model (PACM), which is a special limit of the SACM. The SACM strictly observes the Pauli exclusion principle while the PACM does not. The discussion of the SACM is adapted to the coherent state formalism by introducing the new SO(3) dynamical symmetry limit and third-order interaction terms in the Hamiltonian. The potential energy surface is constructed in both models and it is found that the effects of the Pauli principle can be simulated by higher-order interaction terms in the PACM. The present study is also meant to serve as a starting point for investigating phase transitions in the two algebraic cluster models.

Phenomenological and microscopic cluster models. II. Phase transitions

Based on the results of a previous paper (Paper I), by performing the geometrical mapping via coherent states, phase transitions are investigated and compared within two algebraic cluster models. The difference between the Semimicroscopic Algebraic Cluster Model (SACM) and the Phenomenological Algebraic Cluster Model (PACM) is that the former strictly observes the Pauli exclusion principle between the nucleons of the individual clusters, while the latter ignores it. From the technical point of view the SACM is more involved mathematically, while the formalism of the PACM is closer to that of other algebraic models with different physical content. First- and second-order phase transitions are identified in both models, while in the SACM a critical line also appears. Analytical results are complemented with numerical studies on {\alpha}-cluster states of the neon-20 and magnesium-24 nuclei.

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.

Phase transitions for excited states in 16O+α → 20Ne within the SACM

We present the cranking of the Semimicroscopic Algebraic Cluster Model (SACM) for two spherical clusters. A geometrical mapping is applied and a discussion on phase transition as a function of the cranking parameter is given. This parameter can be related to the average angular momentum of the nucleus. The particular cluster system considered is 16O+α → 20Ne. We also investigate the phase transition property when the Pauli exclusion principle is not observed. We show that phase transitions may occur within the same dynamical symmetry limit.

Phases of cluster states

The question of phases and phase transitions of cluster states is reviewed. First some features of the vibron model are recalled, then its extensions are investigated. Preliminary results are also presented from a study on the cluster-shell competition.

Clusters and the quasi-dynamical symmetry

The possible role of the quasi-dynamical symmetry in nuclear clusterization is discussed. Two particular examples are considered: i) the phases and phase-transitions of some algebraic cluster models, and ii) the clusterization in heavy nuclei. The interrelation of exotic (superdeformed, hyperdeformed) nuclear shapes and cluster-configurations are also investigated both for light, and for heavy nuclei, based on the dynamical and quasi-dynamical SU(3) symmetries, respectively.

PHASE-TRANSITIONS AND NUCLEAR CLUSTERIZATION

After reviewing some basic features of the temperature-governed phase-transitions in macroscopic systems and in atomic nuclei we consider non-thermal phase-transitions of nuclear structure in the example of cluster states. Phenomenological and semimicroscopical algebraic cluster models with identical interactions are applied to binary cluster systems of closed and non-closed shell clusters. Phase-transitions are observed in each case between the rotational (rigid molecule-like) and vibrational (shell-like) cluster states. The phase of this finite quantum system shows a quasi-dynamical symmetry.

The concept of nuclear cluster forbiddenness

In nuclear cluster systems, a rigorous structural forbiddenness of virtual nuclear division into unexcited fragments is obtained. We re-analyze the concept of forbiddenness, introduced in Ref. 1 for the understanding of structural effects in nuclear cluster physics. We show that the concept is more involved than the one presented previously, where some errors were committed. Due to its importance, it is reanalyzed here. In the present contribution a simple way for the determination of forbiddenness is given, which may easily be extended to any number of clusters, though in this contribution we discuss only two-cluster systems, for illustrative reasons. A simple rule is obtained for the minimization of the forbiddenness, namely to start from a cluster system with a large SU(3) irrep (lambda,mu), but minimizing (lambda-mu), i.e. the system has to be oblate. The rule can be easily implemented in structural studies, done up to now with an oversimplified definition of forbiddenness. The new method is applied to various systems of light clusters and to some decay channels of 236U and 252Cf.