Solution of fractional oxygen diffusion problem having without singular kernel

Volume 10, Issue 1, pp 299--307 http://dx.doi.org/10.22436/jnsa.010.01.28

Publication Date: January 27, 2017 Submission Date: July 19, 2016

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Authors

Badr S. Alkahtani - Mathematics Department, College of Science, King Saud University, Riyadh 11989, Saudi Arabia. Obaid J. Algahtani - Mathematics Department, College of Science, King Saud University, Riyadh 11989, Saudi Arabia. Ravi Shanker Dubey - Department of Mathematics, Yagyavalkya Institute of Technology, Jaipur-302022, India. Pranay Goswami - School of Liberal Studies, Ambedkar University Delhi, Delhi-11006, India.

Abstract

In the present paper, we use an efficient approach to solve fractional differential equation, oxygen diffusion problem which is used to describe oxygen absorption in human body. The oxygen diffusion problem is considered in new Caputo derivative of fractional order in this paper. Using an iterative approach, we derive the solutions of the modified system.

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Keywords

• Oxygen diffusion problem
• Caputo-Fabrizio fractional derivative
• fractional differential equation
• Laplace transform
• fixed-point theorem.

MSC

•  26A33
•  35A22
•  33E12
•  35R11
•  65L10

References

• [1] A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation, Appl. Math. Comput., 273 (2016), 948–956.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] A. Atangana, E. Alabaraoye, Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equations, Adv. Difference Equ., 2013 (2013 ), 14 pages.
  • View Article
  • Google Scholar
  • MATH


• [3] A. Atangana, B. S. T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 17 (2015), 4439–4453.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] A. Atangana, S. T. A. Badr, Extension of the RLC electrical circuit to fractional derivative without singular kernel, Adv. Mech. Eng., 7 (2015), 1–6.
  • View Article
  • Google Scholar


• [5] F. B. M. Belgacem, Diffusion and drift models for population dispersal from stochastic and continuum views, Int. J. Appl. Math., 5 (2001), 85–106.
  • MathSciNet
  • Google Scholar
  • MATH


• [6] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, II, Reprinted from Geophys, J. R. Astr. Soc., 13 (1967), 529–539, Fract. Calc. Appl. Anal., 11 (2008), 4–14.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85.
  • View Article
  • Google Scholar


• [8] V. B. L. Chaurasia, R. S. Dubey, Analytical solution for the generalized time-fractional telegraph equation, Fract. Differ. Calc., 3 (2013), 21–29.
  • View Article
  • MathSciNet
  • Google Scholar


• [9] V. B. L. Chaurasia, R. S. Dubey, F. B. M. Belgacem, Fractional radial diffusion equation analytical solution via Hankel and Sumudu transforms, Int. J. Math. Eng. Sci. Aero., 3 (2012), 1–10.
  • Google Scholar
  • MATH


• [10] J. Crank, R. S. Gupta, A method for solving moving boundary problems in heat-flow using cubic splines or polynomials, J. Inst. Math. Appl., 10 (1972), 296–304.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] J. Crank, R. S. Gupta, A moving boundary problem arising from the diffusion of oxygen in absorbing tissue, IMA J. Appl. Math., 10 (1972), 19–23.
  • View Article
  • Google Scholar
  • MATH


• [12] R. S. Dubey, B. S. T. Alkahtani, A. Atangana, Analytical solution of space-time fractional Fokker-Planck equation by homotopy perturbation Sumudu transform method, Math. Probl. Eng., 2014 (2015 ), 7 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [13] R. S. Dubey, P. Goswami, F. B. M. Belgacem, Generalized time-fractional telegraph equation analytical solution by Sumudu and Fourier transforms, J. Fract. Calc. Appl., 5 (2014), 52–58.
  • MathSciNet
  • Google Scholar


• [14] V. Gülkaç, Comparative study between two numerical methods for oxygen diffusion problem, Comm. Numer. Methods Engrg., 25 (2009), 855–863.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)
  • MathSciNet


• [16] A. I. Liapis, G. G. Lipscomb, O. K. Crosser, E. Tsiroyianni-Liapis, A model of oxygen diffusion in absorbing tissue, Math. Modelling, 3 (1982), 83–92.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87–92.
  • View Article
  • Google Scholar


• [18] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993)
  • MathSciNet
  • Google Scholar


• [19] S. L. Mitchell, An accurate application of the integral method applied to the diffusion of oxygen in absorbing tissue, Appl. Math. Model., 38 (2014), 4396–4408.
  • View Article
  • MathSciNet
  • Google Scholar


• [20] B. Noble, Heat balance methods in melting problems, J. R. Ockendon, W. R. Hodgkins (Eds.), Moving boundary problems in heat flow and diffusion, Clarendon Press, Oxford, (1975), 208–209.
  • Google Scholar


• [21] I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)
  • Google Scholar


• [22] W. C. Reynolds, T. A. Dolton, The use of integral methods in transient heat transfer analysis, ASME paper, (1958)
  • Google Scholar