Investigation of mathematical model of human liver by Caputo fractional derivative approach

Document Type : Original Paper

Authors

1 Department of Mathematics, Mehran Branch, Islamic Azad University, Ilam, Iran

2 Department of Mathematics, Tabriz Branch, Islamic Azad University,Tabriz, Iran

3 Department of Mathematics, Miandoab Branch, Islamic Azad University, Miandoab, Iran

4 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz,, Iran

Abstract

The study of the function of vital organs of the body by mathematical models is one of the most important and interesting topics for researchers. In this work, we intend to study the mathematical model of human liver function using the fractional order derivative with Caputo approach. The Adomian Analytical Analysis (ADM) method will be used to solve the system of fractional-order differential equations obtained in the new model of the liver. We also provide a numerical simulation for the results obtained from the fractional order system and the integer order system using the available clinical data.

Keywords

Main Subjects


[1] Abbaoui K., Cherruault Y., New ideas for proving convergence of decomposition methods, Computers
and Mathematics with Applications, 29(7) (1995), 103–108.
[2] Almeida R., Bastos N. R. O., Monteiro M. T. T., Modelling some real phenomena by fractional differential
equations, Mathematical Methods in the Applied Sciences, 39(16) (2016), 4846–4855.
[3] Atangana A., Alkahtani B. S. T., Analysis of the Keller-Segel model with a fractional derivative without
singular kernel, Entropy, 17(6) (2015), 4439–4453.
[4] Atangana A., Secer A., A note on fractional order derivatives and table of fractional derivatives of
some special functions, In Abstract and applied analysis, Vol. 2013, Article ID 279681 (2013), 1–8.
[5] Atangana A., Bildik N., The use of fractional order derivative to predict the groundwater flow, Mathematical
Problems in Engineering, Vol. 2013, Article ID 543026 (2013), 1–9.
[6] Baleanu D., Mustafa O. G., On the global existence of solutions to a class of fractional differential
equations, Computers and Mathematics with Applications, 59(5) (2010), 1835–1841.
[7] Baleanu D., Guvenc Z. B., Teneiro Machado J. A., New Trends in Nano-Technology and Fractional
Calculus Applications, Springer, Dordrecht, (2010).
[8] Celechovska L., A simple mathematical model of the human liver, Applications of Mathematics, 49
(2004), 227–246.
[9] Dholkawala Z. F., Sarma H.K., Kam S. I.,Application of fractional flow theory to foams in porous
media, Petroleum Science and Engineering, 57(2007), 152–165.
[10] Erturk V. S., Zaman G., Momani S., A numeric analytic method for approximating a giving up smoking
model containing fractional derivatives, Computers and Mathematics with Applications, 64(10)
(2012), 3065–3074.
[11] Haq F., Shah K., Ur-Rahman G., Shahzad N., Numerical analysis of fractional order model of HIV-1
infection of CD4+ T-cells, Computational Methods for Differential Equations, 5(1) (2017), 1–11.
[12] Jesus I. S., Tenreiro Machado J. A., Implementation of fractional-order electromagnetic potential
through a genetic algorithm, Communications in Nonlinear Science and Numerical Simulation, 14(5)
(2009), 1838–1843.
[13] Jafari H., Daftardar-Gejji V., Solving a system of nonlinear fractional differential equations using
Adomian decomposition, Journal of Computational and Applied Mathematics, 196(2) (2006), 644–
651.
[14] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential
Equations, Elsevier, (2006).
[15] Koca I., Analysis of rubella disease model with non-local and non-singular fractional derivatives, An
International Journal of Optimization and Control: Theories and Applications (IJOCTA), 8(1), (2018),
17–25.
[16] Miller K. S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equation,
John Wiley, (1993).
[17] Oldham K., Spanier J., The Fractional Calculus, Theory and Applications of Differentiation and
Integration to Arbitrary Order, Academic Press, New York, (1974).
[18] Podlubny I., Fractional Differential Equations, Academic Press, (1999).
[19] Samko S. G., Kilbas A. A., Marichev O. I., Fractional Integral and Derivative, Theory and Applications,
Gordon and Breach, (1993).
[20] Sottinen T., Fractional Brownian motion, random walks and binary market models, Finance and
Stochastics, 5 (2001), 343–355.