M. Pilar Velasco

Fractional dynamics of populations

Abstract Nature often presents complex dynamics, which cannot be explained by means of ordinary models. In this paper, we establish an approach to certain fractional dynamic systems using only deterministic arguments. The behavior of the trajectories of fractional non-linear autonomous systems around the corresponding critical points in the phase space is studied. In this work we arrive to several interesting conclusions; for example, we conclude that the order of fractional derivation is an excellent controller of the velocity how the mentioned trajectories approach to (or away from) the critical point. Such property could contribute to faithfully represent the anomalous reality of the competition among some species (in cellular populations as Cancer or HIV). We use classical models, which describe dynamics of certain populations in competition, to give a justification of the possible interest of the corresponding fractional models in biological areas of research.

Fractional models, non-locality, and complex systems

Abstract In this paper, a new approach to the deterministic modelling of dynamics of certain processes in an anomalous environment is proposed. To this end, the standard assumptions that are usually justified by the experiments and led to the classical dynamics models are rewritten in the way that takes into consideration the non-local features of the anomalous environment. The new class of models obtained in this way is characterized by the memory functions that have to be properly determined for a concrete process. In particular, the so-called fractional dynamics models described in terms of the fractional differential equations are among particular cases of the general model. When a concrete process is observed and its characteristics are measured within a certain time interval, the memory functions that characterize the non-locality of the medium can be found by solving an inverse problem for a system of the Volterra integral equations. Special attention is given to the population dynamics examples to highlight the advantages of the new way to focus the model of the dynamics of complex processes compared with the classical ones.

A fractional approach to the Sturm-Liouville problem

The objective of this paper is to show an approach to the fractional version of the Sturm-Liouville problem, by using different fractional operators that return to the ordinary operator for integer order. For each fractional operator we study some of the basic properties of the Sturm-Liouville theory. We analyze a particular example that evidences the applicability of the fractional Sturm-Liouville theory.

Anomalous NMR relaxation in cartilage matrix components and native cartilage: Fractional-order models

We present a fractional-order extension of the Bloch equations to describe anomalous NMR relaxation phenomena (T(1) and T(2)). The model has solutions in the form of Mittag-Leffler and stretched exponential functions that generalize conventional exponential relaxation. Such functions have been shown by others to be useful for describing dielectric and viscoelastic relaxation in complex, heterogeneous materials. Here, we apply these fractional-order T(1) and T(2) relaxation models to experiments performed at 9.4 and 11.7 Tesla on type I collagen gels, chondroitin sulfate mixtures, and to bovine nasal cartilage (BNC), a largely isotropic and homogeneous form of cartilage. The results show that the fractional-order analysis captures important features of NMR relaxation that are typically described by multi-exponential decay models. We find that the T(2) relaxation of BNC can be described in a unique way by a single fractional-order parameter (α), in contrast to the lack of uniqueness of multi-exponential fits in the realistic setting of a finite signal-to-noise ratio. No anomalous behavior of T(1) was observed in BNC. In the single-component gels, for T(2) measurements, increasing the concentration of the largest components of cartilage matrix, collagen and chondroitin sulfate, results in a decrease in α, reflecting a more restricted aqueous environment. The quality of the curve fits obtained using Mittag-Leffler and stretched exponential functions are in some cases superior to those obtained using mono- and bi-exponential models. In both gels and BNC, α appears to account for micro-structural complexity in the setting of an altered distribution of relaxation times. This work suggests the utility of fractional-order models to describe T(2) NMR relaxation processes in biological tissues.

Fractional heat equation and the second law of thermodynamics

In the framework of second law of thermodynamics, we analyze a set of fractional generalized heat equations. The second law ensures that the heat flows from hot to cold regions, and this condition is analyzed in the context of the Fractional Calculus.

Anomalous T2 relaxation in normal and degraded cartilage.

To compare the ordinary monoexponential model with three anomalous relaxation models—the stretched Mittag‐Leffler, stretched exponential, and biexponential functions—using both simulated and experimental cartilage relaxation data.

A real regularised fractional derivative

A real regularised integral formulation of the fractional derivative is obtained from the generalised Grünwald–Letnikov derivative without using the Cauchy derivative. This new approach is based on the properties of the Mellin transform. The usual Riemann–Liouville and Caputo derivatives are expressed in a similar way emphasising their regularising capabilities. Some examples involving the Heaviside unit step function are presented in the last section of the paper.

New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian

Abstract The non local fractional Laplacian plays a relevant role when modeling the dynamics of many processes through complex media. From 1933 to 1949, within the framework of potential theory, the Hungarian mathematician Marcel Riesz discovered the well known Riesz potential operators, a generalization of the Riemann-Liouville fractional integral to dimension higher than one. The scope of this note is to highlight that in the above mentioned works, Riesz also gave the necessary tools to introduce several new definitions of the generalized coupled fractional Laplacian which can be applied to much wider domains of functions than those given in the literature, which are based in both the theory of fractional power of operators or in certain hyper-singular integrals. Moreover, we will introduce the corresponding fractional hyperbolic differential operator also called fractional Lorentzian Laplacian.

On Deterministic Fractional Models

This chapter is dedicated to presenting some aspects of the so-called Ordinary and/or Partial Fractional Differential Equations. During last 20 years the main underground reason that explain the interest of the applied researchers in the fractional models have been the known close link that exists between such kind of models and the so-called “Jump” stochastic models, such as the CTRW (Continuous Time Random Walk). During the second half of the twentieth century (until the 1990s), the CTRW method was practically the main tool available to describe subdiffusive and/or superdiffusive phenomena associated with Complex Systems for the researches that work in applied fields. The fractional operators are non-local, while the ordinary derivative is a local operator, and on the other hand, the dynamics of many anomalous processes depend of certain memory of its own dynamics. Therefore, the fractional models linear and/or non-linear look as a good alternative to the ordinary models. Note that fractional operators also provide an alternative method to the classical models including dilate terms. The main of this chapter is dedicated to do a first approach to show how introduce Fractional Models only under a deterministic basement. We will considerate Fractional Dynamics Systems with application to study anomalous growing of populations, and on the other hand, the Fractional Diffusive Equation.

Fractional operators and some special functions

This paper considers the Riemann-Liouville fractional operator as a tool to reduce linear ordinary equations with variable coefficients to simpler problems, avoiding the singularities of the original equation. The main result is that this technique allow us to obtain an extension of the classical integral representation of the special functions related with the original differential equations. In particular, we will use as examples the cases of the well-known Generalized, Gauss and Confluent Hypergeometric equations, Laguerre equation, Hermite equation, Legendre equation and Airy equation.