Research ArticleMathematical Solvability of a Caputo Fractional PolymerDegradation Model Using Further Generalized Functions

Emile Franc Doungmo Goufo and Stella Mugisha

Department of Mathematical Sciences, University of South Africa, Florida Science Campus, Florida, Gauteng 0003, South Africa

Correspondence should be addressed to Emile Franc Doungmo Goufo; dgoufef@unisa.ac.za

Received 12 May 2014; Accepted 10 June 2014; Published 25 June 2014

Academic Editor: Abdon Atangana

Copyright © 2014 E. F. Doungmo Goufo and S. Mugisha. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The continuous fission equation with derivative of fractional order 𝛼, describing the polymer chain degradation, is solved explicitly.We prove that, whether the breakup rate depends on the size of the chain breaking up or not, the evolution of the polymersizes distribution is governed by a combination of higher transcendental functions, namely, Mittag-Leffler function, the furthergeneralized 𝐺-function, and the Pochhammer polynomial. In particular, this shows the existence of an eigenproperty; that is, thesystem describing fractional polymer chain degradation contains replicated and partially replicated fractional poles, whose effectsare given by these functions.

1. Introduction

Polymer degradation is the process where polymers are con-verted into monomers or mixtures of monomers. Polymersrange from familiar synthetic plastics such as polystyrene(also called styrofoam) to natural biopolymers such as DNAand proteins that are fundamental to biological structure andfunction. Historically, products arising from the linkage ofrepeating units by covalent chemical bonds have been theprimary focus of polymer science; emerging important areasof the science now focus on noncovalent links. Polyisopreneof latex rubber and the polystyrene of styrofoam are exam-ples of polymeric natural/biological and synthetic polymers,respectively. In biological contexts, essentially, all biologicalmacromolecules, that is, proteins (polyamides), nucleic acids(polynucleotides), and polysaccharides, are purely polymericand are composed in large part of polymeric components, forinstance, isoprenylated/lipid-modified glycoproteins, wheresmall lipidic molecule and oligosaccharide modificationsoccur on the polyamide backbone of the protein. In thetheory of polymers division, one would expect a conserva-tion of mass, especially when polymers are converted intomonomers or mixtures of monomers, but [1, 2] an infinitecascade of division events creating a “dust” of monomers

of zero size carrying nonzero mass and leading to noncon-servativeness (dishonesty) in the model has been observed.Since this remains partially unexplained by classical modelsof clusters’ fission, extending the analysis to the fractionalversion and expressing the solutions explicitly may bringmore light and a broader outlook about this phenomenonwhich remains a mystery. Thus, we are going to analyzethe fractional fission system describing the polymer chaindegradation and provide explicit expressions of its solutions.Before that, let us have a look at what is known about theclassical kinetics of polymer chain degradation. However,there is a growing interest in extending the normal calculuswith integer orders to noninteger orders (real or complexorder) [3–6] because its applications, like, for example, thetopic of this paper, have caught a great range of considerationin the past decade. For instance, in [3] the authors made useof the homotopy decomposition method (HDM), to solve asystemof fractional nonlinear differential equations that arisein the model for HIV infection of CD4+ T cells and attractorone-dimensional Keller-Segel equations. In [4], two methodsincluding Frobenius and Adomian decomposition methodwere used to generalize the classical Darcy law by regardingthe water flow as a function of a noninteger order derivativeof the piezometric head.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 392792, 5 pageshttp://dx.doi.org/10.1155/2014/392792

2 Mathematical Problems in Engineering

1.1. The Classic Polymer Degradation Dynamics. The binaryfission integrodifferential equation,

𝜕

𝜕𝑡

𝑔 (𝑥, 𝑡) = −𝑔 (𝑥, 𝑡) ∫

𝑥

0

𝐻(𝑦, 𝑥 − 𝑦) 𝑑𝑦

+ 2∫

∞

𝑥

𝑔 (𝑦, 𝑡)𝐻 (𝑥, 𝑦 − 𝑥) 𝑑𝑦, 𝑥, 𝑡 > 0,

(1)

is the kinetic equation describing the evolution of the sizesdistribution. Here, 𝑔(𝑥, 𝑡) represents the density of 𝑥-groups(i.e., groups of size 𝑥) at time 𝑡, and𝐻(𝑥, 𝑦) gives the averagefission rate, that is, the average number at which clusters ofsize𝑥+𝑦 undergo splitting to form an𝑥-group and a𝑦-group.This model is applicable inmany branches of natural sciencesranging from physics, through chemistry, engineering, andbiology, to ecology and in numerous domains of appliedsciences, the rock fractures, and break of droplets. Varioustypes of fragmentation equations have been comprehensivelyanalyzed in numerous works (see, e.g., [2, 7–11]). In thedomain of polymer science, the fission dynamics have alsobeen of considerable interest, since degradation of bonds ordepolymerisation results in fragmentation; see [12–14]. In[13], the authors used the statistical arguments to find andanalyse the size distribution of the model.The authors in [12]analysed the model in combination with the inverse process,that is, the coagulation process, and provided a similar resultfor the size distribution.

2. Fractional Fission Differential Problem

We aim to investigate the evolution of the number density ofparticles described by the fractional fission integrodifferentialequation

𝐷𝛼

𝑡𝑔 (𝑥, 𝑡) = −𝑔 (𝑥, 𝑡) ∫

𝑥

0

𝐻𝛼(𝑦, 𝑥 − 𝑦) 𝑑𝑦

+ 2∫

∞

𝑥

𝑔 (𝑦, 𝑡)𝐻𝛼(𝑥, 𝑦 − 𝑥) 𝑑𝑦,

0 ≤ 𝛼 < 1, 𝑥, 𝑡 > 0,

(2)

subject to the following initial condition:

𝑔 (𝑥, 0) = 𝑓 (𝑥) , 𝑥, 𝑡 > 0, (3)

where

𝐶

0𝐷

𝛼

𝑡𝑔 (𝑥, 𝑡) =

𝜕𝛼

𝜕𝑡𝛼𝑔 (𝑥, 𝑡)

=

1

Γ (1 − 𝛼)

∫

𝑡

0

(𝑡 − 𝑟)−𝛼

𝜕

𝜕𝑟

𝑔 (𝑥, 𝑟) 𝑑𝑟,

(4)

where 0 ≤ 𝛼 < 1 is the fractional derivative of 𝑔(𝑥, 𝑡) in thesense of Caputo [15], with Γ the Gama function. For reasonsof simplicity, we note that 𝐶

0𝐷

𝛼

𝑡= 𝐷𝛼

𝑡.

3. Mathematical Analysis

Our analysis consists of two distinct cases: the case where thebreakup rate depends on the size of the chain breaking up and

the case where it does not depend. This will help us compareand analyse the two scenarios.

3.1. The Case 𝐻𝛼(𝑥,𝑦) = 1. Firstly we assume that the rate of

breakup is independent of the length of polymer. Model (2)becomes

𝐷𝛼

𝑡𝑔 (𝑥, 𝑡) = −𝑥𝑔 (𝑥, 𝑡) + 2∫

∞

𝑥

𝑔 (𝑦, 𝑡) 𝑑𝑦, 0 ≤ 𝛼 < 1; (5)

applying the Laplace transform on both sides of the latterequation yields

L (𝐷𝛼

𝑡𝑔 (𝑥, 𝑡) , 𝑠) = L(−𝑥𝑔 (𝑥, 𝑡) + 2∫

∞

𝑥

𝑔 (𝑦, 𝑡) 𝑑𝑦, 𝑠) .

(6)

Clearly,

L (𝐷𝛼

𝑡𝑔 (𝑥, 𝑡) , 𝑠) = 𝑠

𝛼

𝑔 (𝑥, 𝑠) − 𝑠𝛼−1

𝑓 (𝑥) ,

L(−𝑥𝑔 (𝑥, 𝑡) + 2∫

∞

𝑥

𝑔 (𝑦, 𝑡) 𝑑𝑦) = −𝑥𝑔 (𝑥, 𝑠)

+ 2∫

∞

𝑥

𝑔 (𝑦, 𝑠) 𝑑𝑦,

(7)

where 𝑔(𝑥, 𝑠) is the Laplace transform L(𝑔(𝑥, 𝑡), 𝑠). Weobtain

𝑓 (𝑥) = (

𝑥

𝑠𝛼−1

+ 𝑠) 𝑔 (𝑥, 𝑠) −

2

𝑠𝛼−1

∫

∞

𝑥

𝑔 (𝑦, 𝑠) 𝑑𝑦. (8)

Remark 1. We note that by the differential expression (5) weimplicitly require that 𝑦 → 𝑔(𝑦, 𝑡) should be Lebesgueintegrable on any [𝜖,∞) for 𝜖 > 0 and almost every 𝑥 > 0.The same assumption therefore applies to 𝑦 → 𝑓(𝑦) and𝑦 → 𝑔(𝑦, 𝑠).

Hence we put 𝑌(𝑥, 𝑠) = (2/𝑠𝛼−1

) ∫

∞

𝑥

𝑔(𝑦, 𝑠)𝑑𝑦 with theconfidence that the integrand is integrable over any interval[𝜖,∞) so that the integral is absolutely continuous at each 𝑥 >

0 and we can thus differentiate so as to convert (8) into thepartial differential equation as follows:

𝑓 (𝑥) = (

𝑥

𝑠𝛼−1

+ 𝑠)

𝑠𝛼−1

2

𝜕𝑥𝑌 (𝑥, 𝑠) − 𝑌 (𝑥, 𝑠) . (9)

Choosing the constant in the general solution so as to havesolutions converging to zero at ∞, we obtain its solutionwhich is given as

𝑌 (𝑥, 𝑠) = 2𝑒−𝜉𝑠,𝛼(𝑥)

∫

∞

𝑥

𝑒𝜉𝑠,𝛼(𝜂)

𝑓 (𝜂)

𝜂 + 𝑠𝛼𝑑𝜂, (10)

where

𝜉𝑠,𝛼

(𝑥) = ∫

𝑥

0

2

𝜂 + 𝑠𝛼𝑑𝜂 = ln(

𝑥 + 𝑠𝛼

𝑠𝛼

)

2

. (11)

Mathematical Problems in Engineering 3

This leads to the solution of (8) as follows:

𝑔 (𝑥, 𝑠) =

𝑠𝛼−1

𝑓 (𝑥)

𝑥 + 𝑠𝛼

+

2𝑠𝛼−1

𝑥 + 𝑠𝛼𝑒−𝜉𝑠,𝛼(𝑥)

∫

∞

𝑥

𝑒𝜉𝑠,𝛼(𝜂)

𝑓 (𝜂)

𝜂 + 𝑠𝛼𝑑𝜂

=

𝑠𝛼−1

𝑓 (𝑥)

𝑥 + 𝑠𝛼

+ 2

𝑠𝛼−1

(𝑥 + 𝑠𝛼)3∫

∞

𝑥

(𝜂 + 𝑠𝛼

) 𝑓 (𝜂) 𝑑𝜂.

(12)

Applying the inverse Laplace transform L−1(𝑔(𝑥, 𝑠), 𝑡) =

𝑔(𝑥, 𝑡) to the latter expression yields

L−1

(

𝑠𝛼−1

𝑓 (𝑥)

𝑥 + 𝑠𝛼

, 𝑡) = 𝑓 (𝑥)L−1

(

𝑠𝛼−1

𝑥 + 𝑠𝛼, 𝑡)

= 𝑓 (𝑥)

∞

∑

𝑛=0

(−𝑥)𝑛

(𝑡)𝑛𝛼

Γ (𝑛𝛼 + 1)

= 𝑓 (𝑥) 𝐸𝛼[−𝑥𝑡𝛼

] ,

(13)

where 𝐸𝛼is the Mittag-Leffler function as follows:

𝐸𝛼[𝑥] =

∞

∑

𝑛=0

𝑥𝑛

Γ (𝑛𝛼 + 1)

. (14)

We also have

L−1

(2

𝑠𝛼−1

(𝑥 + 𝑠𝛼)3∫

∞

𝑥

(𝜂 + 𝑠𝛼

) 𝑓 (𝜂) 𝑑𝜂, 𝑡)

= 2∫

∞

𝑥

𝑓 (𝜂)L−1

(

𝑠𝛼−1

(𝜂 + 𝑠𝛼

)

(𝑥 + 𝑠𝛼)3

, 𝑡) ,

(15)

and, clearly,

L−1

(

𝑠𝛼−1

(𝜂 + 𝑠𝛼

)

(𝑥 + 𝑠𝛼)3

, 𝑡)

= L−1

(

𝜂𝑠𝛼−1

(𝑥 + 𝑠𝛼)3, 𝑡) +L

−1

(

𝑠2𝛼−1

(𝑥 + 𝑠𝛼)3, 𝑡)

= 𝜂

∞

∑

𝑗=0

(−3) (−4) ⋅ ⋅ ⋅ (−𝑗 − 2) 𝑥𝑗

𝑡(2+𝑗)𝛼

Γ (1 + 𝑗) Γ ({2 + 𝑗} 𝛼 + 1)

+

∞

∑

𝑗=0

(−3) (−4) ⋅ ⋅ ⋅ (−𝑗 − 2) 𝑥𝑗

𝑡(1+𝑗)𝛼

Γ (1 + 𝑗) Γ ({1 + 𝑗} 𝛼 + 1)

.

(16)

Thus the solution of fractional model (5) is given by

𝑔 (𝑥, 𝑡) = 𝑓 (𝑥) 𝐸𝛼[−𝑥𝑡𝛼

] + 2𝐺𝛼,𝛼−1,3

(−𝑥, 𝑡) ∫

∞

𝑥

𝜂𝑓 (𝜂) 𝑑𝜂

+ 2𝐺𝛼,2𝛼−1,3

(−𝑥, 𝑡) ∫

∞

𝑥

𝑓 (𝜂) 𝑑𝜂,

(17)

where 𝐺 is the higher transcendental generalized 𝐺-functiondefined by

𝐺𝑞,𝛽,𝑟

(𝑥, 𝑡) =

∞

∑

𝑗=0

(−𝑟) (−1 − 𝑟) ⋅ ⋅ ⋅ (1 − 𝑗 − 𝑟) (−𝑥)𝑗

𝑡(𝑟+𝑗)𝑞−𝛽−1

Γ (1 + 𝑗) Γ ({𝑟 + 𝑗} 𝑞 − 𝛽)

(18)

and expressed in terms of the Pochhammer polynomial

(𝑥 − 1)𝑛= (𝑥 − 1) (𝑥 − 2) ⋅ ⋅ ⋅ (𝑥 − 𝑛) , (19)

as

𝐺𝑞,𝛽,𝑟

(𝑥, 𝑡) =

∞

∑

𝑗=0

(𝑟)𝑗(−𝑥)𝑗

𝑡(𝑟+𝑗)𝑞−𝛽−1

Γ (1 + 𝑗) Γ ({𝑟 + 𝑗} 𝑞 − 𝛽)

. (20)

(See the appendix for some properties of the generalized𝐺-function.) We see that putting 𝛼 = 1 in solution (17)reduces to the classic first order derivative and corresponds(as well known; see [14]) to an exponential distribution in 𝑥.It also shows that the polymer chains fragment exponentiallyfast in time. However, in fractional integrodifferential theory,the generalized 𝐺-function is of capital importance since itcarries increased time domain complexity.

3.2. The Case 𝐻𝛼(𝑥,𝑦) = 𝑥+𝑦. This case represents a process

where the rate of fission increases with size. Such a processcan occur when the polymers are under tenseness or in adestructive force field such as ultrasound.Model (2) becomes

𝐷𝛼

𝑡𝑔 (𝑥, 𝑡) = −𝑥

2

𝑔 (𝑥, 𝑡) + 2∫

∞

𝑥

𝑦𝑔 (𝑦, 𝑡) 𝑑𝑦, 0 ≤ 𝛼 < 1.

(21)

As done previously, we apply the Laplace transform to have

𝑓 (𝑥) = (

𝑥2

𝑠𝛼−1

+ 𝑠)𝑔 (𝑥, 𝑠) −

2

𝑠𝛼−1

∫

∞

𝑥

𝑦𝑔 (𝑦, 𝑠) 𝑑𝑦. (22)

By Remark 1, we can also put 𝑌(𝑥, 𝑠) =

(2/𝑠𝛼−1

) ∫

∞

𝑥

𝑦𝑔(𝑦, 𝑠)𝑑𝑦 and then

𝑓 (𝑥) = (

𝑥2

𝑠𝛼−1

+ 𝑠)

𝑠𝛼−1

2𝑥

𝜕𝑥𝑌 (𝑥, 𝑠) − 𝑌 (𝑥, 𝑠) , (23)

leading to

𝑌 (𝑥, 𝑠) = 2𝑒−𝜉𝑠,𝛼(𝑥)

∫

∞

𝑥

𝑒𝜉𝑠,𝛼(𝜂)

𝜂𝑓 (𝜂)

𝜂2+ 𝑠𝛼𝑑𝜂, (24)

with 𝜉𝑠,𝛼(𝑥) = ∫

𝑥

0

(2𝜂/(𝜂2

+ 𝑠𝛼

))𝑑𝜂 = ln((𝑥2 + 𝑠𝛼

)/𝑠𝛼

). Thesolution of (22) reads as

𝑔 (𝑥, 𝑠) =

𝑠𝛼−1

𝑓 (𝑥)

𝑥2+ 𝑠𝛼

+

2𝑠𝛼−1

(𝑥2+ 𝑠𝛼)2∫

∞

𝑥

(𝜂) 𝑓 (𝜂) 𝑑𝜂. (25)

Applying the inverse Laplace transform and following thesame steps as in the previous section finally yield the solutionof fractional model (21) which is given by

𝑔 (𝑥, 𝑡) = 𝑓 (𝑥) 𝐸𝛼[−𝑥2

𝑡𝛼

]

+ 2𝐺𝛼,𝛼−1,2

(−𝑥2

, 𝑡) ∫

∞

𝑥

𝜂𝑓 (𝜂) 𝑑𝜂,

(26)

where 𝐸 and 𝐺 are, respectively, defined in (14) and (18).

4 Mathematical Problems in Engineering

If we take𝑓(𝑥) = 𝛿(𝑥−𝑙), then the latter solution becomes

𝑔 (𝑥, 𝑡) =

{{

{{

{

0 for 𝑥 > 𝑙

𝛿 (𝑥 − 𝑙) 𝐸𝛼[−𝑙2

𝑡𝛼

] for 𝑥 = 𝑙

2𝑙𝐺𝛼,𝛼−1,2

(−𝑥2

, 𝑡) for 𝑥 < 𝑙,

(27)

giving the compact form

𝑔 (𝑥, 𝑡) = 𝛿 (𝑥 − 𝑙) 𝐸𝛼[−𝑥2

𝑡𝛼

] + 2𝑙𝜗 (𝑙 − 𝑥) 𝐺𝛼,𝛼−1,2

(−𝑥2

, 𝑡) ,

(28)

where 𝜗 is the step function.If we compare this distribution to the previous case where

the breakup rate is independent of the length of polymer, wesee that the secondmodel shows amuch slower production ofdaughter particles due to fission.This is an expected outcomegiven the relative behaviour of the two breakup speeds.

4. Concluding Remarks

Wehave used themodel of fractional𝛼th order describing thepolymer chain degradation to express the solutions explicitly.We first considered the case where the rate of breakup isindependent of the length of the polymer before investigatingthe case where the rate of fission is a function of the size of thepolymer. In both cases, we found that the solutions are givenby a combination of higher transcendental functions, theMittag-Leffler function, the further generalized 𝐺-function,and the Pochhammer polynomial, showing the existence inthe system of repeated and partially replicated fractionalpoles, whose effects are given by these functions. Moreover,it is of significant usefulness to obtain here a generalizedfunction which when fractionally differentiated or integrated(differintegrated) by any order returns itself. Like exponen-tial, trigonometric, and hyperbolic functions of integer ordercalculus, the definitions of such generalized functions areimportant in fractional calculus, especially to describe realphenomena like the polymer chain degradation. Thereforethis work extends the preceding ones, with the inclusion offractional differentiation which was not considered before,and the results we got here, especially the further generalized𝐺-function which is of capital importance since it carriesincreased time domain complexity.

Appendix

Relevant Properties of the 𝐺-Function:(Partially) Replicated Fractional Poles

It is obvious to see that taking 𝑟 = 1 reduces the 𝐺-functioninto the following generalized function, 𝑅-function [16]:

𝑅𝑞,𝛽

(𝑥, 𝑡) =

∞

∑

𝑗=0

(𝑥)𝑗

𝑡(1+𝑗)𝑞−𝛽−1

Γ ({1 + 𝑗} 𝑞 − 𝛽)

, (A.1)

subject to some proportional coefficients 1/Γ(1 + 𝑗) = 1/(1 +

𝑗)!. Using the fractional derivative𝐷𝛼𝑡(0 < 𝛼 < 1), defined in

(4), the 𝑅-function is proven to return itself under 𝛼th orderdifferentiation. In fact we know (see [17, page 67]) that

𝐷𝛼

𝑡(𝑥)𝑝

=

Γ (𝑝 + 1) (𝑥)𝑝−𝛼

Γ (𝑝 − 𝛼 + 1)

, 𝑝 > −1, (A.2)

yields

𝐷𝛼

𝑡𝑅𝛼,0

(𝑥, 𝑡) =

∞

∑

𝑗=0

(𝑥)𝑗

𝑡𝑗𝛼−1

Γ (𝑗𝛼)

. (A.3)

Taking 𝑗 = 𝑘 + 1 leads to

𝐷𝛼

𝑡𝑅𝛼,0

(𝑥, 𝑡) =

∞

∑

𝑘=−1

(𝑥)𝑘+1

𝑡𝑘+1𝛼−1

Γ (𝑘 + 1𝛼)

. (A.4)

Thus,

𝐷𝛼

𝑡𝑅𝛼,0

(𝑥, 𝑡) = 𝑥𝑅𝛼,𝛽

(𝑥, 𝑡) , 1 > 𝛼 > 0 (A.5)

for 𝑡 > 0. Hence for 𝑥 = 1, the function is proven toreplicate, showing clearly its eigenproperty. In general, 𝛼thorder differintegration of the 𝑅-function 𝑅

𝑞,𝛽(𝑥, 𝑡) returns

another 𝑅-function, namely, 𝑅𝑞,𝛽+𝛼

(𝑥, 𝑡) [18]. Special casesof the 𝐺-function also include the exponential function, thesine, cosine, hyperbolic sine, and hyperbolic cosine functions,and the Mittag-Leffler function, Agarwal’s function, Erdelyi’sfunction, Hartley’s 𝐹-function, and Miller and Ross’s func-tion.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208–214, 1999.

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