The multistep homotopy analysis method for solving fractionalorder model for HIV infection of CD4+T cells
DOI:
https://doi.org/10.4067/S071609172015000400001Keywords:
Fractionalorder, Caputo fractional derivative, Multistep homotopy analysis, HIV infection, Differential equation.Abstract
HIV infection of CD4+T cells is one of the causes of health problems and continues to be one of the significant health challenges. This paper presents approximate analytical solutions to the model of HIV infection of CD4+T cells of fractional order using the multistep homotopy analysis method (MHAM). The proposed scheme is only a simple modification of the homotopy analysis method (HAM), in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding problems. The fractional derivatives are described in the Caputo sense. A comparative study between the new algorithm and the classical RungeKutta method is presented in the case of integerorder derivatives. The solutions obtained are also presented graphically.References
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[2] J. Cang, Y. Tan, H. Xu, S. Liao, Series solutions of nonlinear Riccati differential equations with fractional order, Chaos, Solitons & Fractals, 40 (1), pp. 19, (2009).
[3] R.V. Culshaw, S. Ruan, A delaydifferential equation model of HIV infection of CD4+ Tcells, Mathematical Bioscience 165, pp. 2739, (2000).
[4] Y. Ding, H. Ye, A fractionalorder differential equation model of HIV infection of CD4+ Tcells, Mathematical and Computer Modelling, 50, pp. 386392, (2009).
[5] V. S. Ert¨urk, Z. Odibat, S. Momani, An approximate solution of a fractional order differential equation model of human Tcell lymphotropic virus I (HTLVI) infection of CD4+ Tcells, Computers & Mathematics with Applications, 62, pp. 9921002, (2011).
[6] C.P. Li, W.H. Deng, Remarks on fractional derivatives, Applied Mathematics and Computation 187, pp. 777—784, (2007).
[7] W. Lin, Global existence theory and chaos control of fractional differential equations, JMAA, 332, pp. 709726, (2007).
[8] M. Merdan, homotopy perturbation method for solving a model for HIV infection of Tcells. Istanbul Ticaret Universitesi Fen Bilimleri Dergisi, 12, pp. 3952, (2007).
[9] S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, (1993).
[10] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, (1974).
[11] M. A. Nowak, R. M. May, Virus Dynamics, Oxford University Press, (2000).
[12] A.S. Perelson, Modelling the interaction of the immune system with HIV, in: C. CastilloChavez (Ed.), Mathematical and Statistical Approaches to AIDS Epidemiology, Springer, Berlin, (1989).
[13] A. S. Perelson, D. E. Kirschner, R. De Boer, Dynamics of HIV infection of CD4+ Tcells, Math. Biosci, 114 (1), pp. 81125, (1993).
[14] A.S. Perelson, P.W. Nelson, Mathematical analysis of HIV1, dynamics in vivo, SIAM Rev. 41 (1), pp. 344, (1999).
[15] I. Podlubny, Fractional Differential Equations. Academic Press, New York, (1999).
[16] A. Rafiq, M. Rafiullah, Some multistep iterative methods for solving nonlinear equations, Computers & Mathematics with Applications, 58 (8), pp. 15891597, (2009).
[17] J. Sabatier, O .P. Agrawal, J. A. Tenreiro Machado, Advances in Fractional Calculus; Theoretical Developments and Applications in Physics and Engineering, Springer, (2007).
[18] M. Zurigat, S. Momani, Z. odibat, A. Alawneh, The homotopy analysis method for handling systems of fractional differential equations, Applied Mathematical Modelling, 34 (1), pp. 2435, (2010).
[19] M. Zurigat, S. Momani, A. Alawneh, Analytical approximate solutions of systems of fractional algebraicdifferential equations by homotopy analysis method, Computers and Mathematics with Applications, 59 (3), pp. 12271235, (2010).
How to Cite
[1]
A. H. Handam, A. A. Freihat, and M. Zurigat, “The multistep homotopy analysis method for solving fractionalorder model for HIV infection of CD4+T cells”, Proyecciones (Antofagasta, On line), vol. 34, no. 4, pp. 307322, 1.
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