## J. Nieto, M. Belmekki, & R. Rodríguez-López on Fractional Differential Equations

#### Fast Breaking Papers Commentary, February 2011

Article: Existence of Periodic Solution for a Nonlinear Fractional Differential Equation
Authors: |

Juan J. Nieto, Mohammed Belmekki, & Rosana Rodríguez-López talk with ScienceWatch.com and answer a few questions about this month's Fast Breaking Paper paper in the field of Mathematics.

**Why do you think your paper is highly
cited?**

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order. The subject is as old as the differential calculus, and goes back to the time when Leibnitz and Newton invented differential calculus. The idea of fractional calculus has been a subject of interest not only among mathematicians but also among physicists and engineers.

Also, fractional-order models are more accurate than integer-order models, that is, there are more degrees of freedom in the fractional-order models. Furthermore, fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a "memory" term in the model. This memory term insures the history and its impact to the present and future.

We combine some elements of classical calculus, fractional calculus and nonlinear analysis to obtain new results. In the paper we made some long calculations to obtain an explicit expression for the solutions. It opens some new directions to deal with similar or related problems.

The number of researches in the area of fractional differential equations has increased significantly in recent years due to the relevance of fractional calculus in many subjects ranging from rheology to electrochemistry, or from viscoelasticity to electromagnetism, and many of them may have been interested in the topic of the paper.

**Does it describe a new discovery, methodology, or
synthesis of knowledge?**

Lead Author

Mohammed Belmekki

Coauthor

Rosana Rodríguez-López

The paper includes significant updates in the theory of fractional differential equations. We hope that our work will contribute, at least partially, to a better knowledge on this field and to spread the interest and applicability of fractional differential models to the prediction of the behavior of real phenomena.

**Would you summarize the significance of your paper
in layman's terms?**

In many real processes, it is relevant to consider the history, not only the present state. Fractional differential equations provide, as indicated before, an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a memory term in the model. This memory term insures the history and its impact to the present and future. In some sense, fractional models are more realistic than classical models.

**How did you become involved in this research, and
how would you describe the particular challenges, setbacks, and
successes that you've encountered along the way?**

As members (J.J. Nieto and R. Rodríguez-López) of the Research Group of Nonlinear Differential Equations of the Department of Mathematical Analysis of the USC, we have been working on boundary value problems for nonlinear differential equations (ordinary, functional, and integro-differential equations) and collaborating with many Research Institutes and Groups around the world such as the group of M. Belmekki in Algeria.

However, the setting of some of our previous works was, in some problems, not the most appropriate to allow the existence of singular solutions, which appear in many real processes, for instance, in the field of Physics. Our interest in studying this type of situation with more detail led us to consider a new concept of periodic boundary value conditions for equations including a Riemann-Liouville fractional derivative. In the paper, some positive results to this problem were deduced by the choice of the appropriate base space and the application of classical Fixed Point Theory.

**Where do you see your research leading in the
future?**

We are interested in the development of theoretical results which specify under which circumstances certain boundary value problems written in terms of differential equations can be solvable, in which case we also try to provide the explicit expression of solutions, if possible, or any of their properties of interest. However, our interest in emphasizing the applications of our achievements is continuously increasing, and our aim is to put the stress on the applications that the study of differential equations has to particular issues corresponding to various fields.

Besides the study of singularities, we would also like to focus our attention on contributing to the development of mathematical models adequate to study phenomena which cannot be analyzed from the classical point of view; we cite, for instance, those equations where the base space is such that it is not possible to define a derivative or there is not even a vector structure.

We also consider international collaboration as beneficial, enriching, and crucial to our research.

**Mohammed Belmekki
Département de Mathématiques
Université de Saïda
Saïda, Algeria**

**Juan J. Nieto
Department of Mathematical Analysis
Faculty of Mathematics
University of Santiago de Compostela
Santiago de Compostela, Spain**

**Rosana Rodríguez-López
Department of Mathematical Analysis
Faculty of Mathematics
University of Santiago de Compostela
Santiago de Compostela, Spain**

ADDITIONAL INFORMATION:

- Read New Hot Paper (May 2010) commentary by Juan J. Nieto & Donal O'Regan

KEYWORDS: PERIODIC SOLUTION, NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION, BOUNDARY VALUE PROBLEMS, INCLUSIONS, GREEN'S FUNCTION, LINEAR CASE.