## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### Modelling the oxygen diffusion equation within the scope of fractional calculus

**ABSTRACT**

The diffusion of oxygen into human body with simultaneous absorption is an important problem and it is of great importance in medical applications. This problem can be formulated in two stages; At the first stage, the absorption of oxygen at the surface of the medium is constant and an another stage considering the moving boundary problem of oxygen absorbed by the human body. In this paper we obtain analytical solutions for the oxygen diffusion equation considering the Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in the Liouville-Caputo sense and Atangana-Koca-Caputo fractionalorder derivatives. Numerical simulations were obtained for different values of the fractional order.

**KEYWORDS**

PAPER SUBMITTED: 2018-01-08

PAPER REVISED: 2018-08-31

PAPER ACCEPTED: 2018-01-31

PUBLISHED ONLINE: 2018-03-04

- S. Kumar, M.M. Rashidi. New analytical method for gas dynamics equation arising in shock fronts. Computer Physics Communications, 185(7), (2014), pp. 1947-1954.
- A. Atangana, D. Baleanu. Modelling the advancement of the impurities and the melted oxygen concentration within the scope of fractional calculus. International Journal of Non-Linear Mechanics, 67, (2014), pp. 278-284.
- X.J. Yang, D. Baleanu. Fractal heat conduction problem solved by local fractional variation iteration method. Thermal Science, 17(2), (2013), pp. 625-628.
- Z. Rahimi, W. Sumelka, X.J. Yang. A new fractional nonlocal model and its application in free vibration of Timoshenko and Euler-Bernoulli beams. The European Physical Journal Plus, 132(11), (2017), pp. 1-21.
- X.J. Yang, F. Gao, H.M. Srivastava. New rheological models within local fractional derivative. Rom. Rep. Phys, 69(3), (2017), pp. 1-12.
- J. Crank, R.S. Gupta. A moving boundary problem arising from the diffusion of oxygen in absorbing tissue. J. Inst. Math. Appl., 10, (1972), pp. 19-33.
- V. Gulkac. The New Approximate Analytic Solution for Oxygen Diffusion Problem with Time-Fractional Derivative. Mathematical Problems in Engineering, 1, (2016), pp. 1-8.
- A.I. Liapis, G.G. Lipscomb, O.K. Crosser, E. Tsiroyianni-Liapis, A model of oxygen diffusion in absorbing tissue. Mathematical Modelling, 3(1), (1982), pp. 83-92.
- B.S. Alkahtani, O.J. Algahtani, R.S. Dubey, P. Goswami. Solution of fractional oxygen diffusion problem having without singular kernel. Journal of Nonlinear Sciences & Applications (JNSA), 10(1), (2017), pp. 1-9.
- S.L. Mitchell. An accurate application of the integral method applied to the diffusion of oxygen in absorbing tissue. Applied Mathematical Modelling, 38(17), (2014), pp. 4396-4408.
- V. Gulkac. Comparative study between two numerical methods for oxygen diffusion problem. International Journal for Numerical Methods in Biomedical Engineering, 25(8), (2009), pp. 855-863.
- S.G. Ahmed. A numerical method for oxygen diffusion and absorption in a sike cell. Applied Mathematics and Computation, 173(1), (2006), pp. 668-682.
- I. Podlubny. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic press, New York, 198, (1998).
- A. Atangana, D. Baleanu. New Fractional Derivatives with Nonlocal and Non-Singular Kernel. Theory and Application to Heat Transfer Model. Therm Sci. 20(2), (2016), pp. 763-769.
- F. Jarad, E. Ugurlu, T. Abdeljawad, D. Baleanu. On a new class of fractional operators. Advances in Difference Equations, 2017(1), (2017), pp. 1-16.
- A. Atangana, I. Koca. New direction in fractional differentiation. Math. Nat. Sci., 1, (2017), pp. 18-25
- B.S.T. Alkahtani, I. Koca, A. Atangana. A novel approach of variable order derivative: Theory and Methods. J. Nonlinear Sci. Appl., 9, (2016), pp. 4867-4876.