حل عددی مدل کسری عفونت HIV در سلولهای CD4+T
محورهای موضوعی : آمارمحمد رضا دوستدار 1 , طیبه دمرچلی 2 , علیرضا وحیدی 3
1 - گروه ریاضی، واحد یادگار امام خمینی(ره) شهرری، دانشگاه آزاد اسلامی، تهران، ایران
2 - گروه ریاضی، واحد یادگار امام خمینی(ره) شهرری، دانشگاه آزاد اسلامی، تهران، ایران
3 - گروه ریاضی، واحد یادگار امام خمینی(ره) شهرری، دانشگاه آزاد اسلامی، تهران، ایران
کلید واژه: Fractional derivative, Legendre polynomials, Operational matrix, Block-pulse functions, System of differential equations,
چکیده مقاله :
در این مقاله، مدل کسری عفونت HIV در سلولهای CD4+T بررسی قرار میگیرد. در این مدل، مشتقات کسری در مفهوم کاپوتو در نظر گرفته میشوند. در این روش، دستگاه معادلات دیفرانسیل معمولی از مرتبه کسری به یک دستگاه معادلات جبری تبدیل میگردد که میتوان آن را با استفاده از یک روش عددی مناسب حل نمود. همچنین، در بحث آنالیز خطا، کران بالای خطا ارائه شده است. کارایی و دقت روش، با استفاده از یک نمونه عددی برای برخی مشتقات صحیح و کسری بررسی و برخی مقایسه ها و نتایج گزارش شده است. در این مقاله، مدل کسری عفونت HIV در سلولهای CD4+T بررسی قرار میگیرد. در این مدل، مشتقات کسری در مفهوم کاپوتو در نظر گرفته میشوند. در این روش، دستگاه معادلات دیفرانسیل معمولی از مرتبه کسری به یک دستگاه معادلات جبری تبدیل میگردد که میتوان آن را با استفاده از یک روش عددی مناسب حل نمود. همچنین، در بحث آنالیز خطا، کران بالای خطا ارائه شده است. کارایی و دقت روش، با استفاده از یک نمونه عددی برای برخی مشتقات صحیح و کسری بررسی و برخی مقایسه ها و نتایج گزارش شده است.
In this paper, a hybrid function method based on combination of block-pulse functions and Legendre polynomials is used for solving a fractional model of HIV infection of CD4+ T cells in which fractional derivatives are considered in Caputo sense. Using this method, the system of fractional ordinary differential equations which is the mathematical model for the fractional model of HIV infection of CD4+T cells, is reduced into a system of algebraic equations. This system can be solved by a numerical method. Also, convergence analysis of the method is studied and an upper bound of the error is obtained. To show efficiency and accuracy the proposed method, a numerical example is simulated and some comparisons and results are reported. In this paper, a hybrid function method based on combination of block-pulse functions and Legendre polynomials is used for solving a fractional model of HIV infection of CD4+ T cells in which fractional derivatives are considered in Caputo sense. Using this method, the system of fractional ordinary differential equations which is the mathematical model for the fractional model of HIV infection of CD4+T cells, is reduced into a system of algebraic equations. This system can be solved by a numerical method. Also, convergence analysis of the method is studied and an upper bound of the error is obtained. To show efficiency and accuracy the proposed method, a numerical example is simulated and some comparisons and results are reported.
[1] Perelson A. S., Kirschner D. E., Boer R. D., “Dynamics of HIV infection CD4+ T cells”, Mathematical Biosciences, 114 (1993) 81-125.
[2] Culshaw R. V., Ruan S., “A delay-differential equation model of HIV infection of CD4+ T cells”, Mathematical Biosciences, 165 (2000) 27-39.
[3] Tuckwell H. C., Wan F. Y. M., “Nature of equilibria and effects of drug treatments in some simple viral population dynamical models”, IMA Journal of Mathematical Control and Information, 17 (2000) 311–327.
[4] Wang L., Li M. Y., “Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells”, Mathematical Biosciences, 200 (2006) 44–57.
[5] Biazar J., “Solution of the epidemic model by Adomian decomposition method”, Applied Mathematics and Computation, 173 (2006) 1101–1106.
[6] Lv C., Yuan Z., “Stability analysis of delay differential equation models of HIV-1 therapy for fighting a virus with another virus”, Journal of Mathematical Analysis and Applications, 352 (2009) 672–683.
[7] Yan Y., Kou C., “Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay”, Mathematics and Computers in Simulation, 82 (2012) 1572-1585.
[8] Podlubny I., “Fractional Differential Equations”, Academic Press, New York, 1999.
[9] Kilbas A. A., Srivastava H. H., Trojillo J. J., “Theory and Applications of Fractional Differential Equations”, Elsevier, New York, 2006.
[10] Hilfer R., “Applications of Fractional Calculus in Physics”, World Scientific, Singapore, 2000.
[11] Baleanu D., Diethelm K., Scalas E., Trojillo J. J., “Fractional Calculus Models and Numerical Methods”, World Scientific, Berlin, 2012.
[12] Tatom F. B., “The relationship between fractional calculus and fractals", Fractals, 3 (1995) 217-229.
[13] Arafa A. A. M., Rida S. Z., Khalil M., “The effect of antiviral drug treatment of human immunodeficiency virus type 1 (HIV-1) described by a fractional order model”, Applied Mathematical Modelling, 37 (2013) 2189-2196.
[14] Vahidi A. R., Jalalvand B., “Improving the accuracy of the Adomian decomposition method for solving nonlinear equations”, Applied Mathematical Sciences, 6 (2012), 487-497.
[15] Hi L., Yi L., Tang P., “Numerical scheme and dynamic analysis for variable-order fractional van der pol model of nonlinear economic cycle”, Advances in Difference Equations, 195 (2016) 1–11.
[16] Sun H. G., Zhang Y., Baleanu W., Chen W., Chen Y. Q., “A new collection of real world applications of fractional calculus in science and engineering”, Communications in Nonlinear Science and Numerical Simulation, 64 (2018) 213–231.
[17] Vahidi A. R., Javadi S., Khorasani S. M., “Solving System of Nonlinear Equations by restarted Adomain’s method”, Journal of Applied Mathematical and Computation, 6 (2012) 509-516.
[18] Vahidi A. R., Damercheli T., “A modified ADM for solving systems of linear Fredholm integral equations of the second kind”, Applied Mathematical Sciences, 6 (2012), 1267-1273
[19] Vahidi A. R., Azimzadeh Z., Didgar M., “An efficient method for solving Riccati equation using homotopy perturbation method”, Mathematical Sciences, 11 (2017) 113-118.
[20] Ongun M. Y., “The Laplace Adomian decomposition method for solving a model for HIV infection of CD4+ T cells”, Mathematical and Computer Modelling, 53 (2011) 597–603.
[21] Ghoreishi M., Ismail A. M., Alomari A., “Application of the homotopy analysis method for solving a model for HIV infection of CD4+ T-cells”, Mathematical and Computer Modelling, 54 (2011) 3007–3015.
[22] Khan Y., Vazquez-Leal H., Wu Q., “An efficient iterated method for mathematical biology model”, Neural Computing and Applications, 23 (2013) 677–682.
[23] Arafa A. A. M., Rida S., Khalil M., “A fractional-order model of HIV infection with drug therapy effect”, Journal of the Egyptian Mathematical Society, 22 (2014) 538–543.
[24] Lichae B.H., Biazar J., Ayati Z., “The fractional differential model of hiv-1 infection of CD4+ T-cells with description of the effect of antiviral drug treatment”, Computational and Mathematical Methods in Medicine, Article ID 4059549, 2019.
[25] Kumar S., Kumar R., Singh J., Nisar K.S., Kumar D., “An efficient numerical scheme for a fractional model of HIV-1 infection of CD4+ T-cells with the effect of antiviral drug therapy”, Alexandria Engineering Journal, 59 (2020) 2053-2064.
[26] Doostdar M. R., Vahidi A. R., Damercheli T., Babolian E., “A Hybrid Functions Method for Solving Linear and Non-linear Systems of Ordinary Differential Equations”, Mathematical Communications, 26 (2021), 197-213.
[27] Maleknejad K., Basirat B., Hashemizadeh E., “Hybrid Legendre polynomials and Block-pulse functions approach for nonlinear VolterraFredholm integro-differential equations”, Computers & Mathematics with Applications, 61 (2011) 2821–2828.
[28] Kilicman A., Al Zhour Z. A., “Kronecker operational matrices for fractional calculus and some applications”, Applied Mathematics and Computation, 187 (2007) 250–265.
[29] Canuto C., Hussaini M. Y., Quarteroni A., Zang T. A., “Spectral methods in fluid dynamics”, Springer Verlag, Berlin Heidelberg, 1988.