Skip to main content

Advertisement

SpringerLink
Log in

On the Numerical Method for Solving a System of Nonlinear Fractional Ordinary Differential Equations Arising in HIV Infection of CD4\(^{+}\) T Cells

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions A: Science Aims and scope Submit manuscript

Abstract

In this study, shifted Legendre collocation method is used to provide an approximate solution of a fractional order model of HIV infection of CD4\(^{+}\) T cells. This model corresponds to a nonlinear system of fractional ordinary differential equation. By using the present method, the solution of this nonlinear system converts to the solution of nonlinear system of algebraic equations which can be solved by using a suitable numerical method such as Newton’s method. Furthermore, convergence and error analysis of the proposed method are discussed and an upper error bound is provided under weak assumptions. Finally, numerical results for solving fractional model of HIV infection of CD4\(^{+}\) T cells with various parameters are reported in tables. Also, we survey the effect of changing the parameter \(\alpha \) on the present model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Akrami MH, Atabakzadeh MH, Erjaee GH (2013) The operational matrix of fractional integration for shifted Legendre polynomials. Iran J Sci Technol (Sci) 37(4):439–444

    MathSciNet  MATH  Google Scholar 

  • Bellucci MA (2014) On the explicit representation of shifted orthonormal Bernstein polynomials. ArXiv Preprint arXiv:1404.2293

  • Culshaw RV, Ruan S (2000) A delay-differential equation model of HIV infection of CD4\(^{+}\) T-cells. Math Biosci 165(1):27–39

    Article  MATH  Google Scholar 

  • Ding Y, Ye H (2009) A fractional-order differential equation model of HIV infection of CD4\(^{+}\) T cells. Math Comput Model 50(3–4):386–392

    Article  MathSciNet  MATH  Google Scholar 

  • Ghoreishi M, Ismail AIBMd, Alomari AK (2011) Application of the homotopy analysis method for solving a model for HIV infection of CD4\(^{+}\) T-cells. Math Comput Model 54(11–12):3007–3015

    Article  MathSciNet  MATH  Google Scholar 

  • Heydari MH, Hooshmandasl MR, Ghaini FM, Fereidouni F (2013) Two-dimensional Legendre wavelets for solving fractional Poisson equation with Dirichlet boundary conditions. Eng Anal Bound Elem 37(11):1331–1338

    Article  MathSciNet  MATH  Google Scholar 

  • Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier Science B.V, Amsterdam

    MATH  Google Scholar 

  • Merdan M (2007) Homotopy perturbation method for solving a model for HIV infection of CD4\(^{+}\) T cells. Istanb Commer Univ 12:39–52

    Google Scholar 

  • Merdan M, Gökdoğan A, Yildirim A (2011) On the numerical solution of the model for HIV infection of CD4\(^{+}\) T cells. Comput Math Appl 62(1):118–123

    Article  MathSciNet  MATH  Google Scholar 

  • Mirzaee F, Samadyar N (2017) Application of orthonormal Bernstein polynomials to construct a efficient scheme for solving fractional stochastic integro-differential equation. Optik Int J Light Electron Opt 132:262–273

    Article  Google Scholar 

  • Mirzaee F, Samadyar N (2017) Parameters estimation of HIV infection model of CD4\(^{+}\) T cells by applying orthonormal Bernstein collocation method. Int J Biomath (in press)

  • Ongun MY (2011) The laplace adomian decomposition method for solving a model for HIV infection of CD4\(^{+}\) T cells. Math Comput Model 53(5–6):597–603

    Article  MathSciNet  MATH  Google Scholar 

  • Perelson AS (1989) Modeling the interaction of the immune system with HIV, Mathematical and statistical approaches to AIDS epidemiology. Springer, Berlin, pp 350–370

    Book  Google Scholar 

  • Quarteroni A, Sacco R, Saleri F (2010) Numerical mathematics, vol 37. Springer Science and Business Media, Berlin

    MATH  Google Scholar 

  • Rudin W (1976) Principles of mathematical analysis, 3d edn. McGraw-Hill Book Company, New York

    MATH  Google Scholar 

  • Srivastava VK, Awasthi MK, Kumar S (2014) Numerical approximation for HIV infection of CD4\(^{+}\) T cells mathematical model. Ain Shams Eng J 5(2):625–629

    Article  Google Scholar 

  • Venkatesh SG, Balachandar SR, Ayyaswamy SK, Balasubramanian K (2016) A new approach for solving a model for HIV infection of CD4\(^{+}\) T cells arising in mathematical chemistry using wavelets. J Math Chem 54(5):1072–1082

    Article  MathSciNet  MATH  Google Scholar 

  • Yüzbaşı Ş (2012) A numerical approach to solve the model for HIV infection of CD4\(^{+}\) T cells. Appl Math Model 36(12):5876–5890

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express our very great appreciation to reviewers for their valuable comments and suggestions which have helped to improve the quality and presentation of this paper.

Author information

Authors and Affiliations

  1. Faculty of Mathematical Sciences and Statistics, Malayer University, P. O. Box 65719-95863, Malayer, Iran

    Farshid Mirzaee & Nasrin Samadyar

Authors
  1. Farshid Mirzaee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nasrin Samadyar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farshid Mirzaee.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mirzaee, F., Samadyar, N. On the Numerical Method for Solving a System of Nonlinear Fractional Ordinary Differential Equations Arising in HIV Infection of CD4\(^{+}\) T Cells. Iran J Sci Technol Trans Sci 43, 1127–1138 (2019). https://doi.org/10.1007/s40995-018-0560-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40995-018-0560-6

Keywords

Mathematics Subject Classification

Search

Navigation