Margarita Rivero

Fractional differential equations as alternative models to nonlinear differential equations

The main objective of this paper is to demonstrate the possibility of using fractional differential equations to simulate the dynamics of anomalous processes whose analytical representations are continuous but strongly not differentiable, like Weierstrass-type functions. This allows for the possibility of modeling phenomena which traditional differential modeling cannot accomplish. To this end we shall see how some functions of this kind have a fractional derivative at every point in a real interval, and are therefore solutions to fractional differential equations.

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.

Stability of fractional order systems

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled.

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.

Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions

In this paper, we establish the controllability for a class of abstract impulsive mixed-type functional integro-differential equations with finite delay in a Banach space. Some sufficient conditions for controllability are obtained by using the Mönch fixed point theorem via measures of noncompactness and semigroup theory. Particularly, we do not assume the compactness of the evolution system. An example is given to illustrate the effectiveness of our results.MSC:93B05, 34A37, 34G20.

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.

On new existence results for fractional integro-differential equations with impulsive and integral conditions

In this paper, we establish sufficient conditions for the existence of solutions for a class of initial value problems with integral condition for impulsive fractional integro-differential equations. The results are established by the application of the contraction mapping principle and the Krasnoselskii fixed point theorem. An example is provided to illustrate the results.

Linear fractional differential equations with variable coefficients

Abstract This work is devoted to the study of solutions around an α -singular point x 0 ∈ [ a , b ] for linear fractional differential equations of the form [ L n α ( y ) ] ( x ) = g ( x , α ) , where [ L n α ( y ) ] ( x ) = y ( n α ) ( x ) + ∑ k = 0 n − 1 a k ( x ) y ( k α ) ( x ) with α ∈ ( 0 , 1 ] . Here n ∈ N , the real functions g ( x ) and a k ( x ) ( k = 0 , 1 , … , n − 1 ) are defined on the interval [ a , b ] , and y ( n α ) ( x ) represents sequential fractional derivatives of order k α of the function y ( x ) . This study is, in some sense, a generalization of the classical Frobenius method and it has applications, for example, in obtaining generalized special functions. These new special functions permit us to obtain the explicit solution of some fractional modeling of the dynamics of many anomalous phenomena, which until now could only be solved by the application of numerical methods. 1

Existence and stability results for nonlinear fractional order Riemann-Liouville Volterra-Stieltjes quadratic integral equations

Our aim in this paper is to study the existence and the stability of solutions for Riemann-Liouville Volterra-Stieltjes quadratic integral equations of fractional order. Our results are obtained by using some fixed point theorems. Some examples are provided to illustrate the main results.

Controllability of fractional damped dynamical systems

In this paper, we study the controllability of linear and nonlinear fractional damped dynamical systems, which involve fractional Caputo derivatives, with different order in finite dimensional spaces using the Mittag-Leffler matrix function and the iterative technique. A numerical example is provided to illustrate the theory.