SciELO - Scientific Electronic Library Online

 
vol.34 issue4Inequalities of Hermite-Hadamard Type for h-Convex Functions on Linear Spaces author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.34 no.4 Antofagasta Dec. 2015

http://dx.doi.org/10.4067/S0716-09172015000400001 

Proyecciones Journal of Mathematics Vol. 34, No 4, pp. 307-322, December 2015. Universidad Católica del Norte Antofagasta - Chile

The multi-step homotopy analysis method for solving fractional-order model for HIV infection of CD4+T cells

AH H. Handam

Al Al-Bayt University Jordan 

Asad A. Freihat

Ajloun College Al-Balqa Applied University

M. Zurigat

Al Al-Bayt University

Jordan 


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 multi-step ho-motopy 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 Runge-Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically.

Subjclass [2000] : 26A33, 34A34.

Keywords: Fractional-order, Caputo fractional derivative, multistep homotopy analysis, HIV infection, Differential equation.


REFERENCES

[1]    A. K. Alomari, M. S. M. Noorani, R. Nazar, C. P. Li, Homotopy analysis method for solving fractional Lorenz system, Commun Nonliear Sci Numer Simult, 15 (7) , pp. 1864-1872, (2010).

[2]    J. Cang, Y. Tan, H. Xu, S. Liao, Series solutions of non-linear Riccati differential equations with fractional order, Chaos, Solitons & Fractals, 40 (1), pp. 1-9, (2009).

[3]    R.V. Culshaw, S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Mathematical Bioscience 165, pp. 27-39, (2000).

[4]    Y. Ding, H. Ye, A fractional-order differential equation model of HIV infection of CD4+ T-cells, Mathematical and Computer Modelling, 50, pp. 386-392, (2009).

[5]    V. S. Erturk, Z. Odibat, S. Momani, An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells, Computers & Mathematics with Applications, 62, pp. 992-1002, (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. 709-726, (2007).

[8]    M. Merdan, homotopy perturbation method for solving a model for HIV infection of T-cells. Istanbul Ticaret Universitesi Fen Bilimleri Dergisi, 12, pp. 39-52, (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. Castillo-Chavez (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+ T-cells, Math. Biosci, 114 (1), pp. 81-125, (1993).

[14]    A.S. Perelson, P.W. Nelson, Mathematical analysis of HIV-1, dynamics in vivo, SIAM Rev. 41 (1), pp. 3-44, (1999).

[15]    I. Podlubny, Fractional Differential Equations. Academic Press, New York, (1999).

[16]    A. Rafiq, M. Rafiullah, Some multi-step iterative methods for solving nonlinear equations, Computers & Mathematics with Applications, 58 (8), pp. 1589-1597, (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. 24-35, (2010).

[19]    M. Zurigat, S. Momani, A. Alawneh, Analytical approximate solutions of systems of fractional algebraic-differential equations by homotopy analysis method, Computers and Mathematics with Applications, 59 (3), pp. 1227-1235, (2010).

Ali H. Handam

Department of Mathematics

Al al-Bayt University P. O. Box: 130095

Al Mafraq

Jordan

e-mail : ali.handam@windowslive.com

Asad A. Freihat

Applied Science Department

Ajloun College Al-Balqa Applied University Ajloun 26816

Jordan

e-mail : asadfreihat@yahoo.com

M. Zurigat

Department of Mathematics,

Al al-Bayt University P. O. Box: 130095

Al Mafraq

Jordan

e-mail : moh_zur@hotmail.com

Received : March 2015. Accepted : October 2015

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License