- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 816803, 11 pages
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.
- 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.
- 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.
- K. S. Cole, “Electric conductance of biological systems,” Cold Spring Harbor Symposia on Quantitative Biology, pp. 107–116, 1993.
- 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.
- 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.
- L. Debnath, “Recent applications of fractional calculus to science and engineering,” International Journal of Mathematics and Mathematical Sciences, no. 54, pp. 3413–3442, 2003.
- 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.
- R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000.
- G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport,” Physics Reports, vol. 371, no. 6, pp. 461–580, 2002.
- 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.
- 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.
- H. Sheng, Y. Q. Chen, and T. S. Qiu, Fractional Processes and Fractional-Order Signal Processing, Springer, New York, NY, USA, 2012.
- 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.
- 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.
- 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.
- A. Rocco and B. J. West, “Fractional calculus and the evolution of fractal phenomena,” Physica A, vol. 265, no. 3, pp. 535–546, 1999.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- V. Lakshmikantham, S. Leela, and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge. UK.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.
- C. Li and Y. Ma, “Fractional dynamical system and its linearization theorem,” Nonlinear Dynamics, vol. 71, no. 4, pp. 621–633, 2013.
- Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor's formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293, 2007.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.