MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2014/392792 392792 Research Article Mathematical Solvability of a Caputo Fractional Polymer Degradation Model Using Further Generalized Functions http://orcid.org/0000-0001-6520-1039 Doungmo Goufo Emile Franc Mugisha Stella Atangana Abdon Department of Mathematical Sciences, University of South Africa, Florida Science Campus, Florida, Gauteng 0003 South Africa unisa.ac.za 2014 2562014 2014 12 05 2014 10 06 2014 25 6 2014 2014 Copyright © 2014 Emile Franc Doungmo Goufo and Stella Mugisha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly 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 polymer sizes distribution is governed by a combination of higher transcendental functions, namely, Mittag-Leffler function, the further generalized G-function, and the Pochhammer polynomial. In particular, this shows the existence of an eigenproperty; that is, the system describing fractional polymer chain degradation contains replicated and partially replicated fractional poles, whose effects are given by these functions.

1. Introduction

1.1. The Classic Polymer Degradation Dynamics

The binary fission integrodifferential equation, (1)tg(x,t)=-g(x,t)0xH(y,x-y)dy+2xg(y,t)H(x,y-x)dy,x,t>0, is the kinetic equation describing the evolution of the sizes distribution. Here, g(x,t) represents the density of x-groups (i.e., groups of size x) at time t, and H(x,y) gives the average fission rate, that is, the average number at which clusters of size x+y undergo splitting to form an x-group and a y-group. This model is applicable in many branches of natural sciences ranging from physics, through chemistry, engineering, and biology, to ecology and in numerous domains of applied sciences, the rock fractures, and break of droplets. Various types of fragmentation equations have been comprehensively analyzed in numerous works (see, e.g., [2, 711]). In the domain of polymer science, the fission dynamics have also been of considerable interest, since degradation of bonds or depolymerisation results in fragmentation; see . In , the authors used the statistical arguments to find and analyse the size distribution of the model. The authors in  analysed the model in combination with the inverse process, that is, the coagulation process, and provided a similar result for the size distribution.

2. Fractional Fission Differential Problem

We aim to investigate the evolution of the number density of particles described by the fractional fission integrodifferential equation (2)Dtαg(x,t)=-g(x,t)0xHα(y,x-y)dy+2xg(y,t)Hα(x,y-x)dy,iiiiiiiiiiiiiiiiiiiiiii0α<1,x,t>0, subject to the following initial condition: (3)g(x,0)=  f(x),x,t>0, where (4)D0Ctαg(x,t)=αtαg(x,t)=1Γ(1-α)0t(t-r)-αrg(x,r)dr, where 0α<1 is the fractional derivative of g(x,t) in the sense of Caputo , with Γ the Gama function. For reasons of simplicity, we note that D0Ctα=Dtα.

3. Mathematical Analysis

Our analysis consists of two distinct cases: the case where the breakup rate depends on the size of the chain breaking up and the case where it does not depend. This will help us compare and analyse the two scenarios.

3.1. The Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M19"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo mathvariant="bold"> </mml:mo><mml:mo mathvariant="bold">=</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>

Firstly we assume that the rate of breakup is independent of the length of polymer. Model (2) becomes (5)Dtαg(x,t)=-xg(x,t)+2xg(y,t)dy,0α<1; applying the Laplace transform on both sides of the latter equation yields (6)L(Dtαg(x,t),s)=L(-xg(x,t)+2xg(y,t)dy,s). Clearly, (7)L(Dtαg(x,t),s)=sαg~(x,s)-sα-1f(x),L(-xg(x,t)+2xg(y,t)dy)=-xg~(x,s)+2xg~(y,s)dy, where g~(x,s) is the Laplace transform L(g(x,t),s). We obtain (8)f(x)=(xsα-1+s)g~(x,s)-2sα-1xg~(y,s)dy.

Remark 1.

We note that by the differential expression (5) we implicitly require that yg(y,t) should be Lebesgue integrable on any [ϵ,) for ϵ>0 and almost every x>0. The same assumption therefore applies to yf(y) and yg~(y,s).

Hence we put Y(x,s)=(2/sα-1)xg~(y,s)dy with the confidence 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 the partial differential equation as follows: (9)f(x)=(xsα-1+s)sα-12xY(x,s)-Y(x,s). Choosing the constant in the general solution so as to have solutions converging to zero at , we obtain its solution which is given as (10)Y(x,s)=2e-ξs,α(x)xeξs,α(η)f(η)η+sαdη, where (11)ξs,α(x)=0x2η+sαdη=ln(x+sαsα)2. This leads to the solution of (8) as follows: (12)g~(x,s)=sα-1f(x)x+sα+2sα-1x+sαe-ξs,α(x)xeξs,α(η)f(η)η+sαdη=sα-1f(x)x+sα+2sα-1(x+sα)3x(η+sα)f(η)dη. Applying the inverse Laplace transform L-1(g~(x,s),t)=g(x,t) to the latter expression yields (13)L-1(sα-1f(x)x+sα,t)=f(x)L-1(sα-1x+sα,t)=f(x)n=0(-x)n(t)nαΓ(nα+1)=f(x)Eα[-xtα], where Eα is the Mittag-Leffler function as follows: (14)Eα[x]=n=0xnΓ(nα+1). We also have (15)L-1(2sα-1(x+sα)3x(η+sα)f(η)dη,t)=2xf(η)L-1(sα-1(η+sα)(x+sα)3,t), and, clearly, (16)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)=ηj=0(-3)(-4)(-j-2)xjt(2+j)αΓ(1+j)Γ({2+j}α+1)+j=0(-3)(-4)(-j-2)xjt(1+j)αΓ(1+j)Γ({1+j}α+1). Thus the solution of fractional model (5) is given by (17)g(x,t)=f(x)Eα[-xtα]+2Gα,α-1,3(-x,t)xηf(η)dη+2Gα,2α-1,3(-x,t)xf(η)dη, where G is the higher transcendental generalized G-function defined by (18)Gq,β,r(x,t)=j=0(-r)(-1-r)(1-j-r)(-x)jt(r+j)q-β-1Γ(1+j)Γ({r+j}q-β) and expressed in terms of the Pochhammer polynomial (19)(x-1)n=(x-1)(x-2)(x-n), as (20)Gq,β,r(x,t)=j=0(r)j(-x)jt(r+j)q-β-1Γ(1+j)Γ({r+j}q-β). (See the appendix for some properties of the generalized G-function.) We see that putting α=1 in solution (17) reduces to the classic first order derivative and corresponds (as well known; see ) to an exponential distribution in x. It also shows that the polymer chains fragment exponentially fast in time. However, in fractional integrodifferential theory, the generalized G-function is of capital importance since it carries increased time domain complexity.

3.2. The Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M56"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo mathvariant="bold">(</mml:mo><mml:mi>x</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>y</mml:mi><mml:mo mathvariant="bold">)</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mo mathvariant="bold">=</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>x</mml:mi><mml:mo mathvariant="bold">+</mml:mo><mml:mi>y</mml:mi></mml:math></inline-formula>

This case represents a process where the rate of fission increases with size. Such a process can occur when the polymers are under tenseness or in a destructive force field such as ultrasound. Model (2) becomes (21)Dtαg(x,t)=-x2g(x,t)+2xyg(y,t)dy,0α<1. As done previously, we apply the Laplace transform to have (22)f(x)=(x2sα-1+s)g~(x,s)-2sα-1xyg~(y,s)dy. By Remark 1, we can also put Y(x,s)=(2/sα-1)xyg~(y,s)dy and then (23)f(x)=(x2sα-1+s)sα-12xxY(x,s)-Y(x,s), leading to (24)Y(x,s)=2e-ξs,α(x)xeξs,α(η)ηf(η)η2+sαdη, with ξs,α(x)=0x(2η/(η2+sα))dη=ln((x2+sα)/sα). The solution of (22) reads as (25)g~(x,s)=sα-1f(x)x2+sα+2sα-1(x2+sα)2x(η)f(η)dη. Applying the inverse Laplace transform and following the same steps as in the previous section finally yield the solution of fractional model (21) which is given by (26)g(x,t)=f(x)Eα[-x2tα]+2Gα,α-1,2(-x2,t)xηf(η)dη, where E and G are, respectively, defined in (14) and (18).

If we take f(x)=δ(x-l), then the latter solution becomes (27)g(x,t)={0  for  x>lδ(x-l)Eα[-l2tα]  for  x=l2lGα,α-1,2(-x2,t)  for  x<l, giving the compact form (28)g(x,t)=δ(x-l)Eα[-x2tα]+2lϑ(l-x)Gα,α-1,2(-x2,t), 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, we see that the second model shows a much slower production of daughter particles due to fission. This is an expected outcome given the relative behaviour of the two breakup speeds.

4. Concluding Remarks

We have used the model of fractional αth order describing the polymer chain degradation to express the solutions explicitly. We first considered the case where the rate of breakup is independent of the length of the polymer before investigating the case where the rate of fission is a function of the size of the polymer. In both cases, we found that the solutions are given by a combination of higher transcendental functions, the Mittag-Leffler function, the further generalized G-function, and the Pochhammer polynomial, showing the existence in the system of repeated and partially replicated fractional poles, whose effects are given by these functions. Moreover, it is of significant usefulness to obtain here a generalized function which when fractionally differentiated or integrated (differintegrated) by any order returns itself. Like exponential, trigonometric, and hyperbolic functions of integer order calculus, the definitions of such generalized functions are important in fractional calculus, especially to describe real phenomena like the polymer chain degradation. Therefore this work extends the preceding ones, with the inclusion of fractional differentiation which was not considered before, and the results we got here, especially the further generalized G-function which is of capital importance since it carries increased time domain complexity.

Appendix Relevant Properties of the <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M74"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math></inline-formula>-Function: (Partially) Replicated Fractional Poles

It is obvious to see that taking r=1 reduces the G-function into the following generalized function, R-function : (A.1)Rq,β(x,t)=j=0(x)jt(1+j)q-β-1Γ({1+j}q-β), subject to some proportional coefficients 1/Γ(1+j)=1/(1+j)!. Using the fractional derivative Dtα (0<α<1), defined in (4), the R-function is proven to return itself under αth order differentiation. In fact we know (see [17, page 67]) that (A.2)Dtα(x)p=Γ(p+1)(x)p-αΓ(p-α+1),p>-1, yields (A.3)DtαRα,0(x,t)=j=0(x)jtjα-1Γ(jα). Taking j=k+1 leads to (A.4)DtαRα,0(x,t)=k=-1(x)k+1tk+1α-1Γ(k+1α). Thus, (A.5)DtαRα,0(x,t)=xRα,β(x,t),1>α>0 for t>0. Hence for x=1, the function is proven to replicate, showing clearly its eigenproperty. In general, αth order differintegration of the R-function Rq,β(x,t) returns another R-function, namely, Rq,β+α(x,t) . Special cases of the G-function also include the exponential function, the sine, cosine, hyperbolic sine, and hyperbolic cosine functions, and the Mittag-Leffler function, Agarwal’s function, Erdelyi’s function, Hartley’s F-function, and Miller and Ross’s function.

Conflict of Interests

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

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