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Numerical solution of an influenza model with vaccination and antiviral treatment by the Newton-Chebyshev polynomial method | ||
| Journal of Mathematical Modeling | ||
| دوره 11، شماره 1، خرداد 2023، صفحه 103-116 اصل مقاله (492.23 K) | ||
| نوع مقاله: Research Article | ||
| شناسه دیجیتال (DOI): 10.22124/jmm.2022.21989.1933 | ||
| نویسنده | ||
| Bahman Babayar-Razlighi* | ||
| Department of Mathematics, Qom University of Technology, P.O.Box 1519-37195, Qom, Iran | ||
| چکیده | ||
| We consider a mathematical model of an influenza disease with vaccination and antiviral treatment. This model is expressed by a system of nonlinear ordinary differential equations. We linearize this system by the Newton's method and obtain a sequence of linear systems. The linear systems can be solved by the Chebyshev polynomial solutions, which is a convergence method for numerical solution of linear systems. We solve the problem on a union of many partial intervals. In each partial interval, we first obtain a crude approximation for starting the Newton's method, then solve the problem on current interval by using the lag intervals. An illustration of procedures, we give an algorithm for the initial guess and apply this algorithm for obtaining the total algorithm of the method. We investigate the convergence conditions of the Newton's method for the presented model. In the numerical examples section, we provide some numerical examples to illustrate of the accuracy of the method, and see that the main criterion of the convergence is true for such problems. | ||
| کلیدواژهها | ||
| The Newton's method؛ influenza model؛ Chebyshev polynomial solutions؛ long time؛ nonlinear nonaotonomous ODE | ||
| مراجع | ||
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[1] K. Atkinson. W. Han, Theoritical Numerical Analysis, Springer, New York, 2001. [2] A. Aky¨uz, M. Sezer, Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients, Appl. Math. Comput. 144 (2003) 237–247. [3] B. Babayar-Razlighi, Extrapolation method for Numerical solution of a model for endemic infec- tious diseases, Mathematical Researches 5 (2019) 29–38 (In Persian). [4] B. Babayar-Razlighi, Newton-Taylor polynomial solutions of systems of nonlinear differential equa- tions with variable coefficients, Int. J. Nonlinear Anal. Appl. 12 (2021) 237–248. [5] B. Babayar-Razlighi, Numerical solution of nonlinear system of ordinary differential equations by the Newton-Taylor polynomial and extrapolation with application from a Corona virus model, Intern. J. Math. Model. Comput. 11 (2021) 1–15. [6] B. Babayar-Razlighi, Extrapolation method on a long interval for a class of nonlinear system of Volterra integral equations of the second kind, Int. J. Nonlinear Anal. Appl., 2022, In Press. [7] J. P. Boyd, Solving Transcendental Equations, The Chebyshev Polynomial Proxy and Other Numer- ical Rootfinders, Perturbation Series, and Oracles, University of Michigan Ann Arbor, Michigan, 2014. [8] M. Lipsitch, T. Cohen, M. Muray, B.R. Levin, Antiviral resistance and the control of pandemic influenza, PLoS Med., 4 (2007) 0111–0120. [9] Z. Qiu, Z. Feng, Transmission dynamics of an influenza model with vaccination and antiviral treat- ment, Bull. Math. Biol. 72 (2010) 1–33. [10] M. Sezer, M. Kaynak, Chebyshev polynomial solutions of linear differential equations, Int. J. Math. Educ. Sci. Technol. 27 (1996) 607–618. [11] E. Zeidler, Nonlinear Functional Analysis and its Applications. I: Fixed Point Theorems, Springer- Verlog, New York Inc., 1986. | ||
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