About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 816803, 11 pages
http://dx.doi.org/10.1155/2013/816803
Research Article

Numerical Modeling of Fractional-Order Biological Systems

1Department of Mathematical Sciences, College of Science, UAE University, Al Ain 15551, UAE
2Department of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt

Received 17 May 2013; Accepted 23 June 2013

Academic Editor: Ali H. Bhrawy

Copyright © 2013 Fathalla A. Rihan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. Ahmed, A. Hashish, and F. A. Rihan, “On fractional order cancer model,” Journal of Fractional Calculus and Applied Analysis, vol. 3, no. 2, pp. 1–6, 2012.
  2. A. A. M. Arafa, S. Z. Rida, and M. Khalil, “Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection,” Nonlinear Biomedical Physics, vol. 6, no. 1, article 1, 2012. View at Publisher · View at Google Scholar · View at Scopus
  3. K. S. Cole, “Electric conductance of biological systems,” Cold Spring Harbor Symposia on Quantitative Biology, pp. 107–116, 1993.
  4. A. M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka, “On the fractional-order logistic equation,” Applied Mathematics Letters, vol. 20, no. 7, pp. 817–823, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. H. Xu, “Analytical approximations for a population growth model with fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1978–1983, 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. L. Debnath, “Recent applications of fractional calculus to science and engineering,” International Journal of Mathematics and Mathematical Sciences, no. 54, pp. 3413–3442, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. M. A. El-Sayed, “Nonlinear functional-differential equations of arbitrary orders,” Nonlinear Analysis. Theory, Methods & Applications, vol. 33, no. 2, pp. 181–186, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  9. G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport,” Physics Reports, vol. 371, no. 6, pp. 461–580, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. B. Yuste, L. Acedo, and K. Lindenberg, “Reaction front in an A + B → C reaction-subdiffusion process,” Physical Review E, vol. 69, no. 3, Article ID 036126, pp. 1–36126, 2004. View at Publisher · View at Google Scholar · View at Scopus
  11. W. Lin, “Global existence theory and chaos control of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 709–726, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. H. Sheng, Y. Q. Chen, and T. S. Qiu, Fractional Processes and Fractional-Order Signal Processing, Springer, New York, NY, USA, 2012.
  13. K. Assaleh and W. M. Ahmad, “Modeling of speech signals using fractional calculus,” in Proceedings of the 9th International Symposium on Signal Processing and its Applications (ISSPA '07), Sharjah, United Arab Emirates, February 2007. View at Publisher · View at Google Scholar · View at Scopus
  14. Y. Ferdi, “Some applications of fractional order calculus to design digital filters for biomedical signal processing,” Journal of Mechanics in Medicine and Biology, vol. 12, no. 2, Article ID 12400088, 13 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  15. W.-C. Chen, “Nonlinear dynamics and chaos in a fractional-order financial system,” Chaos, Solitons and Fractals, vol. 36, no. 5, pp. 1305–1314, 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. A. Rocco and B. J. West, “Fractional calculus and the evolution of fractal phenomena,” Physica A, vol. 265, no. 3, pp. 535–546, 1999. View at Publisher · View at Google Scholar · View at Scopus
  17. V. D. Djordjević, J. Jarić, B. Fabry, J. J. Fredberg, and D. Stamenović, “Fractional derivatives embody essential features of cell rheological behavior,” Annals of Biomedical Engineering, vol. 31, no. 6, pp. 692–699, 2003. View at Publisher · View at Google Scholar · View at Scopus
  18. K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 3–22, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. C. Li and F. Zeng, “The finite difference methods for fractional ordinary differential equations,” Numerical Functional Analysis and Optimization, vol. 34, no. 2, pp. 149–179, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  20. J. C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup, “State variables and transients of fractional order differential systems,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3117–3140, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  21. X. Zhang and Y. Han, “Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 3, pp. 555–560, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. A. M. A. El-Sayed and M. Gaber, “The Adomian decomposition method for solving partial differential equations of fractal order in finite domains,” Physics Letters A, vol. 359, no. 3, pp. 175–182, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. H. Jafari and V. Daftardar-Gejji, “Solving a system of nonlinear fractional differential equations using Adomian decomposition,” Journal of Computational and Applied Mathematics, vol. 196, no. 2, pp. 644–651, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. K. Diethelm and G. Walz, “Numerical solution of fractional order differential equations by extrapolation,” Numerical Algorithms, vol. 16, no. 3-4, pp. 231–253, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. L. Galeone and R. Garrappa, “On multistep methods for differential equations of fractional order,” Mediterranean Journal of Mathematics, vol. 3, no. 3-4, pp. 565–580, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. G. Wang, R. P. Agarwal, and A. Cabada, “Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 6, pp. 1019–1024, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. N. J. Ford and A. C. Simpson, “The numerical solution of fractional differential equations: speed versus accuracy,” Numerical Algorithms, vol. 26, no. 4, pp. 333–346, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  28. V. Lakshmikantham, S. Leela, and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge. UK.
  29. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  30. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993. View at MathSciNet
  31. C. Li and Y. Ma, “Fractional dynamical system and its linearization theorem,” Nonlinear Dynamics, vol. 71, no. 4, pp. 621–633, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  32. Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor's formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. N. Bellomo, A. Bellouquid, J. Nieto, and J. Soler, “Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems,” Mathematical Models & Methods in Applied Sciences, vol. 20, no. 7, pp. 1179–1207, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  34. A. Gökdoğan, A. Yildirim, and M. Merdan, “Solving a fractional order model of HIV infection of CD+ T cells,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2132–2138, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  35. D. Kirschner and J. C. Panetta, “Modeling immunotherapy of the tumor—immune interaction,” Journal of Mathematical Biology, vol. 37, no. 3, pp. 235–252, 1998. View at Scopus
  36. F. A. Rihan, M. Safan, M. A. Abdeen, and D. Abdel Rahman, “Qualitative and computational analysis of a mathematical model for tumor-immune interactions,” Journal of Applied Mathematics, vol. 2012, Article ID 475720, 19 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. R. Yafia, “Hopf bifurcation in differential equations with delay for tumor-immune system competition model,” SIAM Journal on Applied Mathematics, vol. 67, no. 6, pp. 1693–1703, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. K. A. Pawelek, S. Liu, F. Pahlevani, and L. Rong, “A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data,” Mathematical Biosciences, vol. 235, no. 1, pp. 98–109, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. A. S. Perelson, D. E. Kirschner, and R. De Boer, “Dynamics of HIV infection of CD4+ T cells,” Mathematical Biosciences, vol. 114, no. 1, pp. 81–125, 1993. View at Publisher · View at Google Scholar · View at Scopus
  40. G. A. Bocharov and F. A. Rihan, “Numerical modelling in biosciences using delay differential equations,” Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 183–199, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. R. V. Culshaw and S. Ruan, “A delay-differential equation model of HIV infection of CD4+ T-cells,” Mathematical Biosciences, vol. 165, no. 1, pp. 27–39, 2000. View at Publisher · View at Google Scholar · View at Scopus
  42. R. Anguelov and J. M.-S. Lubuma, “Nonstandard finite difference method by nonlocal approximation,” Mathematics and Computers in Simulation, vol. 61, no. 3-6, pp. 465–475, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. C. Li and F. Zeng, “Finite difference methods for fractional differential equations,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 22, no. 4, 28 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. F. A. Rihan, “Computational methods for delay parabolic and time-fractional partial differential equations,” Numerical Methods for Partial Differential Equations, vol. 26, no. 6, pp. 1556–1571, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. Z. Odibat and S. Momani, “Numerical methods for nonlinear partial differential equations of fractional order,” Applied Mathematical Modelling, vol. 32, no. 1, pp. 28–39, 2008. View at Publisher · View at Google Scholar · View at Scopus