MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2016/8409839 8409839 Research Article The New Approximate Analytic Solution for Oxygen Diffusion Problem with Time-Fractional Derivative http://orcid.org/0000-0001-6284-1057 Gülkaç Vildan Sapountzakis Evangelos J. Department of Mathematics Faculty of Arts and Science Kocaeli University Umuttepe Campus Izmit 41380 Kocaeli Turkey kocaeli.edu.tr 2016 1562016 2016 09 03 2016 22 05 2016 2016 Copyright © 2016 Vildan Gülkaç. 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.

Oxygen diffusion into the cells with simultaneous absorption is an important problem and it is of great importance in medical applications. The problem is mathematically formulated in two different stages. At the first stage, the stable case having no oxygen transition in the isolated cell is investigated, whereas at the second stage the moving boundary problem of oxygen absorbed by the tissues in the cell is investigated. In oxygen diffusion problem, a moving boundary is essential feature of the problem. This paper extends a homotopy perturbation method with time-fractional derivatives to obtain solution for oxygen diffusion problem. The method used in dealing with the solution is considered as a power series expansion that rapidly converges to the nonlinear problem. The new approximate analytical process is based on two-iterative levels. The modified method allows approximate solutions in the form of convergent series with simply computable components.

1. Introduction

The diffusion of oxygen into absorbing tissue was first studied in . First the oxygen is allowed to diffuse into a medium, some of the oxygen can be absorbed by the medium, and concentration of oxygen at the surface of the medium is maintained constant. This phase of the problem continues until a steady state is reached in which the oxygen does not penetrate any further and is sealed so that no oxygen passes in or out, the medium continues to absorb the available oxygen already in it, and, as a consequence, the boundary in the steady state starts to recede towards the sealed surface.

Crank and Gupta  also employed uniform space grid moving with the boundary and necessary interpolations are performed with either cube splines or polynomials. Noble  suggested repeated spatial subdivision, Reynolds and Dolton  also developed the heat balance integral method, and Liapis et al.  proposed an orthogonal collocation for solving the partial differential equation of the diffusion of oxygen in absorbing tissue. Gülkaç proposed two numerical methods for solving the oxygen diffusion problem . Mitchell studied the accurate application of the integral method . More references to this problem may be found in .

In recent years, fractional differential equations have drawn much attention. Many important phenomena in physics, engineering, mathematics, finance, transport dynamics, and hydrology are well characterized by differential equations of fractional order. Fractional differential equations play an important role in modelling the so-called anomalous transport phenomena and in the theory of complex systems. These fractional derivatives work more appropriately compared with the standard integer-order models. So, the fractional derivatives are regarded as very dominating and useful tool. For mathematical properties of fractional derivatives and integrals one can consult .

In the present work, we extend a homotopy perturbation method with time-fractional derivatives to obtain solution for oxygen diffusion problem.

We give some basic definitions of fractional derivatives as follows.

Definition 1.

The Riemann-Liouville fractional integral of f C α of the order α 0 is defined as (1) J t α f t = f t , i f α = 0 1 Γ α 0 t t - τ α - 1 f τ d τ , i f α > 0 , where Γ denotes gamma function: (2) Γ z = 0 e - t t z - 1 d t , z C .

Definition 2.

The fractional derivatives of f C α of the order α 0 , in Caputo sense, are defined as (3) D t α f t = J t n - α D t α f t = 1 Γ n - α 0 t t - τ n - α - 1 f n τ d τ for n - 1 < α n , n N , t > 0 , f C α n , α - 1 .

Definition 3.

The Caputo-time-fractional derivative operator of order α > 0 is defined as (4) D t α u x , t = J t n - α u x , t = 1 Γ n - α 0 t t - τ n - α - 1 n u τ n d τ .

Lemma 4.

Let n - 1 < α n , n N , and f C α n , α - 1 ; then (5) D t α J t α f t = f t , J t α D t α = f t - k = 0 n f k 0 + t k k ! , f o r t > 0 .

Lemma 5.

If n - 1 < α n , n N , and k 0 then one has (6) J t α t k α Γ k α + 1 = t k α + α Γ k α + α + 1 .

The Mittag-Leffler function plays a very important role in the fractional differential equations, in fact introduced by Mittag-Leffler in 1903 . Mittag-Leffler function E α z = n = 0 z n / Γ α n + 1 .

2. Analysis of Homotopy Perturbation Method with Time-Fractional Derivatives

Let us assume that nonlinear fractional differential equation is as follows: (7) D t α u x , t = A u + f x , t , x , t , with the initial condition u ( x , 0 ) = φ , where A is the operator, f is known functions, and u x , t is sough functions. Assume that operator A can be written as A u = L ( u ) + N ( u ) , where L is the linear operator and N is the nonlinear operator. Hence (7) can be written as follows: (8) D t α u x , t = L u + N u + f x , t . For solving (7) by homotopy perturbation method, we construct the homotopy (9) H V , p = 1 - p D t α V - u 0 + p D α V - L V - N V - f x , t = 0 or the equivalent one (10) H V , p = D t α V - u 0 + p u 0 - L V - N V - f x , t = 0 , where p [ 0,1 ] is an embedding or homotopy parameter, H x , t ; p : x [ 0,1 ] R , and u 0 is the initial approximation for solution equation (8).

Clearly, the homotopy equations H V , 0 = 0 and H V , 1 = 1 are equivalent to the equations D t α V - u 0 = 0 and D α V - L V - N V - f x , t = 0 , respectively. Thus, a monotonous change of parameter p from 0 to 1 corresponds to a continuous change of the trivial problem D t α V - u 0 = 0 to the original problem. Now, we assume that the solution of (8) can be written as a power series in embedding parameter p , as follows: (11) V = V 0 + p V 1 , where V 0 and V 1 are functions which should be determined. Now, we can write (11) in the following form: (12) D t α V x , t = u 0 + p - u 0 + L V + N V + f x , t . Apply the inverse operator, J t α , which is the Riemann-Liouville fractional integral of order α > 0 .

On both sides of (12), we have (13) V x , t = V x , 0 + J t α u 0 + p J t α - u 0 + L V + N V + f x , t . Suppose that the initial approximation of solutions equation (8) is in the following form: (14) u 0 = k = 0 a k x t k α Γ k α + 1 , where a k ( x ) for k = 1,2 , are functions which must be computed. Substituting (11) and (14) into (13) we get (15) V 0 + p V 1 = V x , 0 + J t α k = 0 a k x t k α Γ k α + 1 + p J t α - k = 0 a k x t k α Γ k α + 1 + L V 0 + p V 1 + N V 0 + p V 1 + f x , t . Synchronizing the coefficients of the same powers leads to (16) p 0 :   V 0 = V x , 0 + J t α k = 0 a k x t k α Γ k α + 1 , p 1 :   V 1 = J t α - k = 0 a k x t k α Γ k α + 1 + L V 0 + p V 1 + N V 0 + p V 1 + f x , t . Now, we obtain the coefficients a k x , k = 1,2 , . Therefore the exact solution can be obtained as follows: (17) u x , t = V x , t = V x , 0 + J t α k = 0 a k x t k α Γ k α + 1 . Efficiency and reliability of the method are shown.

3. Problem Description and Formulation

Crank and Gupta  were the first researchers to model oxygen diffusion problem mathematically.

The process includes two mathematical levels. At the first level, the stable condition occurs once the oxygen is injected into either the inside or outside of the cell; then the cell surface is isolated.

At the second level, tissues start to absorb the injected oxygen. The moving boundary problem is caused by this level. The aim of this process is to find a balance position and to determine the time-dependent moving boundary position. Writing down the time-fractional derivatives of oxygen diffusion problem in  is adopted, following .

4. Solution of Fractional Oxygen Diffusion Problem

We consider the following oxygen diffusion problem: (18) D t α c x , t = c x x - 1 , x , t with the following initial and boundary conditions: (19) c x , 0 = 0.5 1 - x 2 , 0 x 1 , t = 0 , (20) c x = 0 , x = 0 , t 0 , (21) c = c x = 0 , x = s t , t 0 w i t h s 0 = 1 , where 0 < α 1 . To solve (18)–(21) by present method, we construct the following homotopy: (22) H V , p = 1 - p D t α V - c 0 + p D α V - L V - 1 = 0 or (23) H V , p = D t α V - c 0 + p c 0 - L V - 1 = 0 , where p [ 0,1 ] and 0 < α 1 . Consider (24) D t α c x , t = c 0 x , t - p c 0 x , t - c x x x , t - 1 . Assume that the initial approximation of solutions equation (18) is in the following form: (25) c 0 x , t = n = 0 a n x t n α Γ n α + 1 , where a n ( x ) for n = 1,2 , are functions which must be computed.

Applying the inverse operator J t α of D t α on both sides of (24) we obtain (26) c x , t = c x , 0 + J t α c 0 x , t - p J t α c 0 x , t - c 0 x x x , t - 1 . Suppose the solution of (26) has the following form: (27) c x , t = c 0 x , t + p c 1 x , t , where c x , t is functions which should be determined. Substituting (27) into (26), collecting the same powers of p , and equating each coefficient of p to zero yield (28) c 0 x , t = 0.5 1 - x 2 + J t α c 0 x , t , (29) p 0 :   c 0 x , t = 0.5 1 - x 2 + n = 0 a n x t n α + α Γ n α + α + 1 , (30) p 1 :   c 1 x , t = - J t α c 0 x , t - c 0 x x x , t - 1 , where (31) c 0 x x x , t = t α Γ α + 1 + t 2 α Γ 2 α + 1 + + t n α Γ n α + 1 + , c 0 x x x , t = E α t = n = 0 t n α Γ n α + 1 ; then (32) c 1 x , t = - n = 0 a n x t n α + α Γ n α + α + 1 + n = 0 t n α + α Γ n α + α + 1 , c 1 x , t = n = 0 1 - a n x t n α + α Γ n α + α + 1 and from (29) and (30) we obtain (33) a 0 x = 0.5 1 - x 2 , a 0 x = a 1 x = a 2 x = = a n - 1 x = a n x = ; therefore, we obtain solution of (26) for p = 1 : (34) c x , t = 0.5 1 - x 2 + n = 0 a n x t n α + α Γ n α + α + 1 + n = 0 1 - a n x t n α + α Γ n α + α + 1 or (35) c x , t = 0.5 1 - x 2 + n = 0 t n + 1 Γ n + 1 i f n + 1 = m , n , m N , 0 < α 1 , f o r α = 1 and we obtain the following solution: (36) c x , t = 0.5 1 - x 2 + e t . If α = 1 / 2 then (37) c x , t = 0.5 1 - x 2 + e t 2 erf - t . If α = 2 then (38) c x , t = 0.5 1 - x 2 + s i n t . We can now obtain an expression for the location of the moving boundary s ( t ) . We can write (39) d s d t = - c x x = s t a t x = s , s 0 = 1 following : (40) D t α s = - c x x = s t and following initial and boundary conditions (41) s 0 = 1 , c x = 0 , x = 0 , c = c x = 0 , x = s t , t 0 . We construct the following homotopy for moving boundary as (42) H W , p = 1 - p D t α W - c 0 + p D α W - L W = 0 or (43) H W , p = D t α W - c 0 + p c 0 - L W = 0 , where p [ 0,1 ] and 0 < α 1 . Consider (44) D t α s x , t = c 0 x , t - p c 0 x , t - c x x , t ; applying the inverse operator J t α of D t α on both sides of (44), we obtain (45) s x , t = c x , 0 + J t α c 0 x , t - p J t α c 0 x , t - c 0 x x , t ; suppose the solution of (45) has the following form: (46) s x , t = c 0 x , t + p c 1 x , t . Substituting (46) into (45), collecting the same powers of p , and equating each coefficient of p to zero yield (47) p 0 :   c 0 x , t = 1 + n = 0 a n x t n α + α Γ n α + α + 1 , p 1 :   c 1 x , t = - J t α c 0 x , t - c 0 x x x , t - 1 , where (48) c 0 x x , t = a 0 x + n = 1 a n x t n α + α Γ n α + α + 1 , J t α c 0 x x , t = J t α n = 1 t n α Γ n α + 1 = t n α + α Γ n α + α + 1 , so (49) c 1 x , t = - a 0 x t α Γ α + 1 . Then we have from (46) if we let p = 1 (50) c 1 x , t = a 1 x t α Γ α + 1 + a 2 x t 2 α Γ 2 α + 1 + , a 1 x = - a 0 x , a 2 x = a 3 x = = a n x = = 0 . Therefore, we obtain the solutions of moving boundary condition as (51) s x , t = n = 0 1 a n x t n α Γ n α + 1 + a 1 x t α Γ α + 1 or (52) s x , t = 0.5 1 - x 2 1 - t α Γ α + 1 .

5. Numerical Simulations

In this section numerical results for the solution of the oxygen diffusion problem using the constructed homotopy perturbation method with the time-fractional derivative are presented. These proposed homotopy perturbation methods are applied and figures present solutions are presented using different values for the derivative order α . Figures 1(a) and 1(b) show the surface concentration c ( x , t ) for α = 1 / 2 , Figures 2(a) and 2(b) show the surface concentration c ( x , t ) for α = 1 , and Figures 3(a) and 3(b) show the surface concentration c ( x , t ) for α = 2 . Figures 4(a) and 4(b) show position of moving boundary s ( x , t ) for α = 1 , Figures 5(a) and 5(b) show position of moving boundary s ( x , t ) for α = 2 , and finally Figures 6(a) and 6(b) show position of moving boundary s ( x , t ) for α = 3 .

Surface concentration c ( x , t ) for α = 1 / 2 .

Surface concentration c ( x , t ) for α = 1 .

Surface concentration c ( x , t ) for α = 2 .

Position of moving boundary s ( x , t ) for α = 1 .

Position of moving boundary s ( x , t ) for α = 2 .

Position of moving boundary s ( x , t ) for α = 3 .

6. Conclusion

In this study, we extended homotopy perturbation method with time-fractional derivative to find the exact solution of oxygen diffusion problem with moving boundary. It is effortless and also easy to apply and we can say that the present method is an effective method and has appropriate technique to find the exact solution to many complex problems.

Competing Interests

The author declares that she has no competing interests.

Crank J. Gupta R. S. A moving boundary problem arising from the diffusion of oxygen in absorbing tissue IMA Journal of Applied Mathematics 1972 10 1 19 33 10.1093/imamat/10.1.19 2-s2.0-77958399004 Crank J. Gupta R. S. A method for solving moving boundary problems in heat flow using cubic splines or polynomials Journal of the Institute of Mathematics and Its Applications 1972 10 296 304 10.1093/imamat/10.3.296 MR0347105 2-s2.0-0004869782 Noble B. Ockendon J. R. Hodkins W. R. Heat balance method in melting problems Moving Boundary Problems in Heat Flow and Diffusion 1975 Oxford, UK Clarendon Press 208 209 Reynolds W. C. Dolton T. A. Use of integral methods in transient heat transfer analysis ASME Paper 1958 58-A-248 Liapis A. I. Lipscomb G. G. Crosser O. K. Tsiroyianni-Liapis E. A model of oxygen diffusion in absorbing tissue Mathematical Modelling 1982 3 1 83 92 10.1016/0270-0255(82)90014-8 ZBL0516.92004 2-s2.0-0019986194 Gülkaç V. Comparative study between two numerical methods for oxygen diffusion problem Communications in Numerical Methods in Engineering with Biomedical Applications 2009 25 8 855 863 10.1002/cnm.1127 MR2555749 2-s2.0-77449122645 Mitchell S. L. An accurate application of the integral method applied to the diffusion of oxygen in absorbing tissue Applied Mathematical Modelling 2014 38 17-18 4396 4408 10.1016/j.apm.2014.02.021 MR3247276 2-s2.0-84906044205 Hansen E. Hougaard P. On a moving boundary problem from biomechanics Journal of Institute of Mathematical and its Applications 1974 13 3 385 398 10.1093/imamat/13.3.385 Gupta R. S. Kumar A. Variable time-step method with coordinate transformation Computer Methods in Applied Mechanics and Engineering 1984 44 1 91 103 10.1016/0045-7825(84)90121-X ZBL0526.65085 2-s2.0-0021446816 Miller J. V. Morton K. W. Baines M. J. A finite element moving boundary computation with an adaptive mesh Journal of the Institute of Mathematics and Its Applications 1978 22 4 467 477 10.1093/imamat/22.4.467 MR517294 2-s2.0-0041396070 Ahmed S. G. A numerical method for oxygen diffusion and absorption in a sike cell Applied Mathematics and Computation 2006 173 1 668 682 10.1016/j.amc.2005.04.010 ZBL1090.65115 2-s2.0-32144443989 Özis T. Gülkaç V. Application of variable interchange method for solution of two-dimensional fusion problem with convective boundary conditions Numerical Heat Transfer; Part A: Applications 2003 44 1 85 95 10.1080/713838172 2-s2.0-0242579355 Gülkaç V. Özis T. Erratum to ‘On a lod method for solution of two dimensional fusion problem with convective boundary conditions’ International Communications in Heat and Mass Transfer 2004 31 4 597 606 Gülkaç V. On the finite differences schemes for the numerical solution of two-dimensional moving boundary problem Applied Mathematics and Computation 2005 168 1 549 556 10.1016/j.amc.2004.09.039 MR2170850 2-s2.0-25644454229 Gülkaç V. A numerical solution of the two-dimensional fusion problem with convective boundary conditions International Journal for Computational Methods in Engineering Science and Mechanics 2010 11 1 20 26 10.1080/15502280903446853 MR2578439 2-s2.0-75149153782 Furzeland R. M. A survey of the formulation and solution of free and moving boundary (Stefan) problems Brunel University Technical Report 1977 TR/76 Uxbridge, UK Brunel University Crank J. Free and Moving Boundary Problems 1984 Oxford, UK The Clarendon Press Oxford Science Publications MR776227 Voller V. R. Fractional stefan problems International Journal of Heat and Mass Transfer 2014 74 269 277 10.1016/j.ijheatmasstransfer.2014.03.008 2-s2.0-84898445121 Wang X. Liu F. Chen X. Novel second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations Advances in Mathematical Physics 2015 2015 14 590435 10.1155/2015/590435 MR3431500 Voller V. R. An exact solution of a limit case Stefan problem governed by a fractional diffusion equation International Journal of Heat and Mass Transfer 2010 53 23-24 5622 5625 10.1016/j.ijheatmasstransfer.2010.07.038 2-s2.0-77956056756 Shao Y. Ma W. Finite difference approximations for the two-side space-time fractional advection-diffusion equations Journal Computational Analysis and Applications 2016 21 2 369 379 Zhou Y. Xia L.-J. Exact solution for Stefan problem with general power-type latent heat using Kummer function International Journal of Heat and Mass Transfer 2015 84 114 118 10.1016/j.ijheatmasstransfer.2015.01.001 2-s2.0-84921391871 Arenas A. J. Gonzalas-Parra G. Chen-Charpentier B. M. Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order Mathematics and Computers in Simulation 2016 121 48 63 10.1016/j.matcom.2015.09.001 MR3425284 Mittag-Leffler M. G. Sur la nouvelle fonction Eα(x), Comptes Rendus Acad. Sci. Paris 1903 137 554 558