Fractional Calculus Based Models for Photocatalytic Reactive

Systems: Deterministic and Stochastic Approaches

Vicente Rico-Ramireza, Rasiel Toledo-Hernándeza, Sergio Damian-Vazqueza and Urmila

M. Diwekarb

aInstituto Tecnologico de Celaya, Departamento de Ingenieria Quimica, Av. Tecnologico y

Garcia Cubas S/N, Celaya, Guanajuato, Mexico 38010

bVishwamitra Research Institute, 368 56th Street, Clarendon Hills, IL, 60514, USA vicente@iqcelaya.itc.mx, gais@iqcelaya.itc.mx, rth199@hotmail.com, urmila@vri- custom.org

Fractional calculus is a generalization of ordinary calculus which introduces derivatives and integrals of fractional order. Reports of successful modeling applications of fractional calculus include a number of mechanical and electrical dynamic systems (Magin, 2006; Sabatier et al, 2007). In the area of chemical engineering, literature suggests that diffusion processes are accurately represented by fractional differential equations (FDE) (Sokolov et al, 2009; San Jose Martinez et al, 2007). It is generally accepted that physical considerations, such as memory and hereditary effects, favor the use of fractional derivative-based models.
The goal of this work is twofold. First, we focus on the use of fractional calculus as a novel mathematical tool with potential applications to chemical engineering; in particular, to the area of chemical reaction engineering. To illustrate this idea, we consider a reactive system involving the photocatalytic degradation of phenol, which exhibits anomalous kinetics and does not follow the classical mass-action form. Anticipating a potential memory effect on the dynamics of such system, we show that it can be represented by a set of fractional order differential equations. To obtain the model parameters and the orders of the fractional differential equations, a non-liner fitting approach is coupled to a numerical solution technique for systems of FDE´s; in this work we use the predictor-corrector method proposed by Diethelm et al (2002). Such a method is a generalization of the Adams- Bashforth-Moulton technique suitable for FDE. The fractional model dynamics is compared against experimental values to validate the suitability of the proposed representation approach for the photocatalytic degradation of phenol.
Besides the use of fractional calculus as a modeling tool, recent literature reports significant theoretical developments in fractional calculus in order to consolidate the fundamentals and provide the basis for a more extensive use of this tool in science and engineering. Therefore, the second part of this work is in that direction; it involves the implementation of a numerical algorithm for the integration of stochastic fractional differential equations (SFDE).
To define the SFDE's, we combine the idea of a FDE with the concept of an Ito process. An Ito process is a stochastic process on which the increment of the stochastic variable includes a deterministic term and a stochastic term defined by a Wiener process. A significant property of a Wiener process is that it satisfies the Markov property; that is, the probability distribution for all future values of the process depends only on its current value. This concept seems to be inconsistent with the inherent memory effects of a fractional derivative. Our main assumption (and the main limitation of our approach),
however, is that a FSDE consists of two terms. The first term of the FSDE is deterministic and it represents the memory effects (the current value of the state variable depends on its whole history) affecting the state variable dynamics. The second term of the FSDE, the stochastic term, is a Wiener process, so that the future values of the uncertainties involved in the formulation depend only on the current value of the state variable. To illustrate this idea, we have incorporated time dependent uncertainties to the FDE of the deterministic model for the photocatalytic degradation of phenol described above, which results in a set of SFDE´s. Further, we have extended the predictor-corrector algorithm proposed by Diethelm et al (2002) so that SFDE (whose stochastic term is based on Wiener processes) can be integrated. This stochastic integrator is then used to evaluate the impact of the uncertainties in the fractional model for the photocatalytic degradation of phenol. Results indicate that time dependent uncertainties might have a significant effect on the dynamic behavior of fractional calculus based models.

Keywords: Fractional calculus, fractional kinetics, photocatalytic degradation, stochastic fractional model.

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