A Mathematical Analysis of Fractional Fragmentation Dynamics with Growth
We make use of the theory of strongly continuous solution operators for fractional models together with the subordination principle for fractional evolution equations (Bazhlekova (2000) and Prüss (1993)) to analyze and show existence results for a fractional fragmentation model with growth characterized by its growth rate . Indeed, strange phenomena like the phenomenon of shattering (McGrady and Ziff (1987)) and the sudden appearance of infinite number of particles in some systems with initial finite particles number could not be fully explained by classical models of fragmentation or aggregation. Then, there is an increasing volition to try new approaches and extend classical models to fractional ones. In the growth model, one of the major challenges in the analysis occurs when is integrable at , the minimum size of a cell. We restrict our analysis to the case of integrability of at . This case needs more considerations on the boundary condition, which, in this paper, is the McKendrick-von Foerster renewal condition. In the process, some properties of Mittag-Leffler relaxation function Berberan-Santos (2005) are exploited to finally prove that there is a positive solution operator to the full model.
1. Model’s Motivation and Introduction
Despite its three centuries of age, fractional calculus remains lightly unpopular amongst science and engineering community. However, there is a growing interest in extending analysis involving normal calculus with integer orders to noninteger orders (real or complex order) [1–4] because its applications have attracted a great range of attention in the past few years. Most of classical models used are formulated as initial or boundary value problems for dynamical evolution equations taking the form where is the time taken from and is an operator in a Banach space. The problem is to find the state of the system at a time depending on the initial state at the initial time . The normal derivative is seen as the representation of the rate of accumulation or loss in the system, that is, gain rate minus loss rate, at infinitesimal bounded space. However, there are three essential motivations for considering here the fractional fragmentation model with growth.(1)Classical models of clusters’ fission with normal derivative cannot fully explain strange phenomena like the sudden appearance of an infinite number of particles in some systems which contained at the beginning a finite particles number and the phenomenon of shattering . The latter is seen as an explosive or dishonest Markov process (see, for example, [6, 7]) and has been associated with an infinite cascade of breakup events creating a “dust” of particles of zero size which, however, carry nonzero mass.(2)As said above, is seen as the representation of the rate of change (accumulation or loss) in the system, considered at infinitesimal bounded space. But sometime the infinitesimal space contains traps (of various sizes) where the variable under study is temporarily parked then; will the replicate the real picture of accumulation or loss? Similarly these trap pictures could be islands or forbidden zones in the infinitesimal space where the variable (particle, mass, density, flux, etc.) cannot reside; accordingly, the rate of accumulation or loss will be different than . Hence, the fractional differentiation , with or , may give the sub- or superrate of accumulation or loss with index representing the heterogeneity distribution of the infinitesimal space (traps or islands)! However, substituting by the fractional derivative requires some considerations as we will in the following sections.(3)Most of works on fragmentation with normal derivative have been performed under assumption that the ensemble of particles is well mixed so that the particle distribution is uniform in space. However, recent approach using individual based models  yields in a natural way to systems in which splitting particles are distributed in space according to some prescribed probability density, leading, however, to models with explicit space dependence. We note that similar models were also considered earlier in  but with emphasis only on well-posedness.
These reasons are the sources of the increasing volition to try new approaches and extend classical models to models with fractional derivative and investigate them with various and different techniques in order to establish broader outlooks on the real phenomena they describe. For instance, we have differential equations with fractional orders increasingly used to model many problems in applied sciences including engineering, applied mathematics, physics, biology, chemistry, economic, and other domains of applications. In the process, differential equations with fractional derivative have become a useful tool for describing nonlinear phenomena of science and engineering models. In the same way, the process of fragmentation of clusters occurs in many branches of natural sciences ranging from physics to chemistry, engineering, biology, ecology, and numerous domains of applied sciences, such as the depolymerization, the rock fractures, and of break of droplets.
With this in mind, we obtain the following fractional model of fragmentation process with growth: where , , , and is given as with the fractional time evolution, considered as universal attractor semigroups of coarse grained macroscopic time evolutions [10, 11]. It is shown that [10, 12] which is the fractional derivative of in the sense of Michaud; see . The full description of this model is done in the next section.
Recall that fission models of type (2) with “normal” time derivative of order one have been comprehensively analyzed in numerous works (see, e.g., [13–21]). Conservative and nonconservative regimes for fragmentation equations have been thoroughly investigated; see [5, 15], and, in particular, the breach of the mass conservation law (called shattering) has been attributed to a phase transition creating a dust of “zero-size” particles with nonzero mass, which are beyond the model’s resolution. Shattering can be interpreted from the probabilistic point of view as the explosion in the Markov process describing fragmentation  and from an analytic point of view as dishonesty of the semigroup associated with the model. Transport-type models with convection were investigated in  where the author showed that the convection part does not affect the breach of the conservation laws. In , the authors studied the nonlocal fragmentation and showed that the process is conservative if at infinity daughter particles tend to go back into the system with a high known probability.
As said earlier, in the model with growth, one of the major problems in the analysis occurs when is integrable at which is the minimum size of a cell. In this paper, the analysis is restricted to the case where is integrable at . This case needs more considerations on the boundary condition, namely, the McKendrick-von Foerster renewal condition. In the investigation, some properties of Mittag-Leffler relaxation function are exploited to show that there is a positive solution operator to the full model.
2. Model Description and Settings
2.1. Classical Fragmentation Differential Equation with Growth
To describe the dynamics here, we use the cluster (or group) density function . Here is a variable that represents the size of the cluster, is the minimum single cell size, the variable represents time, and is the concentration of groups of size at time . We assume that for each the function is from the space This space is the most appropriate one because is the total number of cells in the population.
Clusters grow as a result of divisions of phytoplankton cells. The growth rate is denoted by . We assume that is a nonnegative function, differentiable at and where is the dual space of and means that is absolutely continuous in the standard sense on each compact subinterval of . We denote by the norm of and we recall that and the duality pairing is the normal integral
During a small time interval , a fraction of the clusters of size are undergoing breakup, that is, is the fragmentation rate. We assume that it is a nonnegative function and The size distribution of daughter particles after fragmentation is denoted by . We assume that which accounts for mass conservation after any fragmentation event. Thus, classical model of the dynamics of fragmentation with growth is given by the integrodifferential equation:
The boundary condition considered here is the McKendrick-von Foerster renewal condition: where . The function describes the number of single cells that fall off a cluster of size and join the single cell population . The boundary condition represents the addition of newborn single cells to the single cell population.
2.2. Abstract Reformulation
The analysis is performed in the space . We denote by and the expressions appearing on the right-hand side of (11); that is, and are defined on measurable and finite almost everywhere functions for which they make pointwise (almost everywhere) sense. For each fixed , we define a function of the “mass” variable by Hence, is the function from into the space . Since is a Banach space of type , can be thought of as the fractional derivative with respect to of the function defined by (15). For fixed , we can write the right-hand side of (11) as defined on its maximal domain. The initial condition can therefore be written as instead of .
A vital role in the analysis of the model is played by the integrability of at . Indeed, if is integrable at , the characteristics do reach the line and therefore the boundary condition becomes crucial for the uniqueness investigation. If not, the characteristics do not reach the line and the prescription of a boundary condition is of no use. Therefore, we consider only the case of integrability of .
3. Solution Operator for the Fractional Growth Problem with Integrable at
We are interested in investigating the fractional differential model given by where is the realization of (defined via (13)) on , as the realization of (see (14)) on the domain and is given as with , the fractional time evolution, considered as universal attractor semigroups of coarse grained macroscopic time evolutions [10, 11]. It is shown that [10, 12] which is the fractional derivative of in the sense of Michaud; see . However, substituting by in (11) is justified in the sense that the presence of in (11) reflects a basic symmetry of the time translation invariance and the basic principle of locality. Indeed, from the relation we see that is identified as the infinitesimal generator of time translation . Hence, this considers as the expression of the general time evolution, which is the same consideration done in the definitions (18) and (19) of . Thus, the derivative of fractional order , was found to be, in general, infinitesimal generator of coarse grained macroscopic time evolution. It is shown  that all macroscopic time evolutions have fractional derivatives, with order less than unity, as their infinitesimal generators. Therefore, in Proposition 4 below, we provide a relation between the generator , of the macroscopic time evolution and .
Since is integrable at , the characteristics do reach the line and we have to restrict the operator to a domain in which the boundary condition (12) is satisfied. In this respect, we introduce as the restriction of to Considering now as the restriction (via (14)) of to , the model (16) becomes with and conserving the same definition as in (16).
Lemma 1. is a well-defined operator.
Proof. It is obvious to see that . Indeed, let ; then by definition of , it is clear that . It follows that where we used (10). Therefore, .
To proceed we need the following definition.
Definition 2. Consider an operator applying in the fractional model subject to the initial condition and defined in the Banach space . A family of bounded operators on is called a solution operator of the fractional Cauchy problem (24)-(25) if (i); (ii) is strongly continuous for every ;(iii) for all ;(iv); (v) is a (classical) solution of the model (24)-(25) for all , .
It is well known [14, 17] that an operator means generates a -semigroup so that there exist and such that Whence, by analogy if the fractional Cauchy problem (24)-(25) has a solution operator verifying (26), then we say that . The solution operator is contractive if and we say .
Note that if we have a contraction solution operator, we can use Definition 2 to identify the fractional Cauchy problem of which it is a solution. Usually, however, we are interested in the reverse question, that is, in finding the solution operator, that is, contractive, for a given fractional model. The answer is given by the following theorem (seen as an analogue of Hille-Yosida theorem).
Theorem 3. An operator for if and only if(a) (b) with , , where is a family of strongly continuous operators satisfying (27) and is the resolvent set of the operator
Proof. This theorem is a particular version of [24, Theorem 1.3] and the proof follows the same steps.
In our analysis we will need some interesting properties of the Mittag-Leffler relaxation function which arises in the description of complex relaxation processes and that corresponds to a relaxation function when is a positive real number like the time variable. We have that (see [25, 26]) where is the Gamma function. Using the Laplace transform and its inverse, it can be proved (see [27–29]) that where is a contour domain which encircles the origin counterclockwise, going from to .
Now, we consider the full operator in (22) as a perturbation of the following Cauchy problem representing the loss part of the fractional fission process: Take we can use the point of view of -semigroup theory to say that the operator is the infinitesimal generator of a -semigroup , from the original semigroup (the multiplication semigroup) generated by the differentiation operator and obtained via a simple integration as The existence of comes from assumption that the fission rate satisfies (9). Therefore, the infinitesimal generators of may be interpreted  as the distributions, evaluated on the right time translation group, which leads to the following proposition.
Proposition 4. Consider , ; let . The infinitesimal generators of -semigroups are related to by the representations
Proof. We exploit the Marchaud type representation (19) of infinitesimal generators (see also [12, 31]). Let be the Dirac measure; then if we take as the set of test functions on , we know that (see ) the fractional derivative distribution , , of can be expressed by and approximated by the family of finite Borel measure on given as The family generates the bounded linear operators defined by with defined in (35). Using Laplace transform , we can show (see ) that which yields, by uniqueness theorems for Laplace transformation and well known properties of convolution operator , and then Now taking , the assertion follows by using the fact that is a closed operator  and
Theorem 5. There is an extension of the operator , say , such that there is a solution operator for the fractional model (22) that is positive and then satisfies
Proof. We make use of the subordination principle developed in [24, 35]. Consider the classical model where and are defined via (21) and (14), respectively. Then, by [36, Theorem 2], there is an extension of the operator which generates a positive semigroup satisfying Now, using the Definition 2, we say . Exploiting the relaxation relation (32), we set where We aim to show that is the solution operator for the fractional model (22) and is positive. The second last relation of (32) and monotonicity of imply that is positive. Finally, by the subordination principle developed in [24, 35], we have since . This shows that given by (47) verifies . Thus by Theorem 3, is the solution operator for the fractional model (22), that is, positive, and satisfies (44).
4. Concluding Remarks
Under the McKendrick-von Foerster renewal condition (12), we have set conditions allowing us to analyze and show the existence of a positive solution operator to the fission model (22) with growth and derivative of fractional order , with . We have made use of the theory of strongly continuous solution operators, properties of Mittag-Leffler relaxation function, and the subordination principle. Growth fragmentation equations with fractional order dynamics has never been studied before and this work improves the preceding ones with the inclusion of the transport (growth) and the derivative of order less than unity. Whence, the theory analyzed in this paper is viewed as an opening step that will lead to the full characterization of the operator under which the fractional model admits solution operators and, maybe, leads to a better understanding of strange phenomena like the phenomenon of shattering and the sudden appearance of an infinite number of particles in some systems with initial finite particles number.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
M. Caputo, “Linear models of dissipation whose Q is almost frequency independent: part II,” Journal of Royal Astronomical Society, vol. 13, pp. 529–539, 1967.View at: Google Scholar
K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1999.View at: MathSciNet
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.View at: MathSciNet
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.View at: MathSciNet
J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, UK, 1998.View at: MathSciNet
K. Yosida, Fonctional Analysis, Springer, Berlin, Germany, 6th edition, 1980.View at: MathSciNet
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Franctional Integrals and Derivatives. Theory and Application, Gordon and Breach, Amsterdam, The Netherlands, 1993.
J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer, 2006.View at: MathSciNet
R. Hilfer, “On new class of phase transitions,” in Random Magnetism High-Temperature Superconductivity, p. 85, World Scientific, Singapore, 1994.View at: Google Scholar
G. M. Mittag-Leffler, “Sur la nouvelle fontion ,” Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, vol. 137, pp. 554–558, 1903.View at: Google Scholar
A. Érdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. 3, McGraw-Hill, New York, NY, USA, 1955.View at: MathSciNet
I. Gel'fand and G. Shilov, Generalized Functions, vol. 1, Academic Press, New York, NY, USA, 1964.
J. L. Lions and J. Peetre, “Sur une classe d'espace d'interpolation,” Publications Mathématiques de l'Institut des Hautes Études Scientifiques, vol. 19, pp. 5–68, 1964.View at: Google Scholar
B. Rubin, Fractional Integrals and Potentials, Addison Wesley Longman, Harlow, UK, 1996.View at: MathSciNet
K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2000.View at: MathSciNet
E. G. Bazhlekova, “Subordination principle for fractional evolution equations,” Fractional Calculus & Applied Analysis, vol. 3, no. 3, pp. 213–230, 2000.View at: Google Scholar