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.

The diffusion of oxygen into absorbing tissue was first studied in [

Crank and Gupta [

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.

The Riemann-Liouville fractional integral of

The fractional derivatives of

The Caputo-time-fractional derivative operator of order

Let

If

The Mittag-Leffler function plays a very important role in the fractional differential equations, in fact introduced by Mittag-Leffler in 1903 [

Let us assume that nonlinear fractional differential equation is as follows:

Clearly, the homotopy equations

On both sides of (

Crank and Gupta [

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 [

We consider the following oxygen diffusion problem:

Applying the inverse operator

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

Surface concentration

Surface concentration

Surface concentration

Position of moving boundary

Position of moving boundary

Position of moving boundary

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.

The author declares that she has no competing interests.