حل عددی معادلات دیفرانسیل معمولی منفرد غیرخطی حاصل شده دربیولوژی، ازطریق ماتریس عملیاتی چند جمله ای های زرنیکه شعاعی
محورهای موضوعی : آمار
محمد علی عبادی
1
(دانشجوی گروه ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران)
الهام السادات هاشمی زاده
2
(عضو هیات علمی گروه ریاضی دانشگاه آزاد کرج و معاون آموزشی دانشکده)
امیرحسین رفاهی شیخانی
3
(استاد گروه ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران)
کلید واژه: nonlinear singular differential equations, Zernike radial polynomials, Operational matrix of derivative,
چکیده مقاله :
هدف از این مقاله، ارائه رویکردی عددی جدید، برای حل معادلات دیفرانسیل منفرد غیر خطی که در زمینه ی بیولوژی حاصل می شوند، می باشد. این قبیل معادلات در مسائل متعدد بیولوژی نظیر انتشار اکسیژن در سلول های خونی، انتشار گرما از سر انسان و رشد تومورهای سرطانی ظاهر می شوند. در این مقاله این معادلات به کمک یک روش عددی جدید بر پایه چند جمله های زرنیکه شعاعی حل می‌شوند. در روش ارائه شده برای اولین بار ماتریس های عملیاتی مشتق گیری این توابع به دست آمده و سپس بر اساس ماتریس های عملیاتی برای مشتق توابع زرنیکه شعاعی، معادله ی دیفرانسیل اصلی به یک دستگاه از معادلات غیر خطی جبری تبدیل شود که به راحتی حل پذیرند. پیاده سازی این روش ساده و جذاب است. در پایان، مثال های کاربردی برای نشان دادن پیاده سازی روش ارائه شده و مقایسه جوابهای به دست آمده از این روش با جواب های سایر روش های معروف ارائه و حل شده است، و نتایج حاصل از آن، حاکی از دقت و کارایی این روش عددی است.
The aim of this paper is to provide a new numerical method for solving nonlinear singular differential equations that arise in biology problem. These kind of problems appear in various biology problems like oxygen diffusion in red blood cells, distribution of heat source in human head and cancer tumor growth and etc. In this paper this equations are solved by a new numerical method by using Zernike radial polynomials. In the proposed method for the first time the operational matrix of derivative for Zernike radial polynomials is derived and by using this operational matrices of derivative of Zernike radial functions the differential equation convert to a system of algebraic equations that can be solved easily. The implementation of this proposed method is simple and attractive. Finally some applied models are presented to compare the results by other method results, and they show the accuracy and efficiency of the presented method.
[1] J.A. Adam, A simplified mathematical model of tumor growth, Math. Biosci., vol.81, pp.224–229, 1986
[2] J.A. Adam, A mathematical model of tumor growth II: effect of geometry and spatial non-uniformity on stability, Math. Biosci. vol.86, pp.183–211, 1987
[3] J.A. Adam, S.A. Maggelakis, Mathematical model of tumor growth IV: effect of necrotic core, Math. Biosci., vol. 97, pp. 121–136, 1989
[4] A.C. Burton, Rate of growth of solid tumor as a prob-lem of diffusion, Growth, vol.30 , pp.157–176, 1966
[5] H.P. Greenspan, Models for the growth of solid tumor as a problem by diffusion, Stud. Appl. Math., vol.52, pp.317–340, 197
[6] N.S. Asaithambi, J.B. Goodman, Point wise bounds for a class of singular diffusion problems in physiology, Appl. Math. Comput., vol. 30 , pp.215–222, 1989
[7] H.S. Lin, Oxygen diffusion in a spherical cell with non-linear oxygen uptake kinetics, J. Theor. Biol., vol. 60, pp.449–457, 1976
[8] D.L.S. McElwain, A re-examination of oxygen diffusion in a spherical cell with MichaelisMenten oxygen uptake kinetics, J. Theor. Biol., vol. 71, pp.255–263, 1978
[9] N. Rashevsky, Mathematical Biophysics, vol. 1, Dover, New York, 1960
[10] U. Flesch, The distribution of heat sources in the human head: a theoretical consideration, J. Theor. Biol., vol. 54, pp.
285– 287, 1975
[11] J.B. Garner, R. Shivaji, Diffusion problems with mixed non-linear boundary condition, J. Math. Anal. Appl., vol. 148, pp. 422–430, 1990
[12] B.F. Gray, The distribution of heat sources in the hu-man head: a theoretical consideration, J. Theor. Biol., vol. 82 pp. 473– 476, 1980
[13] R.C. Duggan, A.M. Goodman, Point wise bounds for nonlinear heat conduction model for the human head, Bull. Math. Biol., vol. 48 (2), pp. 229–236, 1989
[14] R.K. Pandy, On a class of weakly regular singular two point boundary value problems II, J, Differential Equa-tions, vol. 127, pp. 110-123, 1996
[15] M.M. Chawla, P.N. Shivkumar, On the existence of so-lution of a class of singular two--point nonlinear boun-dary value problems, J. Comput. Appl. Math., vol. 19, pp. 379–388, 1987
[16] R.D. Russell, L.F. Shampine, Numerical methods for singular boundary value problems, SIAM J. Numer. Anal., vol. 12, pp. 13-36, 1975
[17] S.A. Khuri, A. Sayfy, A novel approach for the solution of a class of singular boundary value problems arising in physi- ology, J. Math. Comput. Model., vol. 52, pp. 626–636, 2010
[18] Hikmet Caglar, Nazan Caglar, Mehmet Ozer, B-spline solu-tion of non-linear singular boundary valueproblems arising in physiology, Chaos Solitons Fractals, vol. 39, pp. 1232-1237, 2009
[19] J. Rashidinia, R. Mohammadi, R. Jalilian, The numerical solution of non-linear singular boundary value problems arising in physiology, J. Appl. Math. Comput., vol. 185, pp. 360–367, 2007
[20] R.K. Pandey, Arvind K. Singh, On the convergence of a finite difference method for a class of singular boundary value problems arising in physiology, J. Comput. Appl. Math., vol. 166, pp. 553–564, 2004
[21] A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional-order differential equations, J. Comput. Math. Appl., vol. 59, pp. 1326–1336, 2010
[22] K. Maleknejad, B. Basirat, E. Hashemizadeh, Hybrid Legendre polynomials and Block-Pulse functions approach for nonlinear Volterra–Fredholm integro-differential eq-uations, Comput. Math. Appl.,vol. 61, pp. 2821–2828, 2011
[23] K. Maleknejad, E. Hashemizadeh , A numerical approach for Hammerstein integral equations of mixed type using opera- tional matrices of hybrid functions, Scientific Bulle-tin, Se-ries A: Applied Mathematics and Physics, vol .73(3), pp. 95-104, 2011
[24] K. Maleknejad, S. Sohrabi, H. Derili, A new computa-tional method for solution of non-linear Volter-ra- Fredholm inte-gro- differential equations, Int. J. Appl. Math., vol. 87(2), pp. 327–338, 2010
[25] K. Maleknejad, E. Hashemizadeh, B. Basirat, Computational method based on Bernestein operational matrices for nonli- near Volterra- Fredholm- Hammerstein integral equations, Commun. Nonlinear. Sci. Numer. Simulat., vol. 17, pp.52–61, 2012
[26] Mahajan VN. Zernike annular polynomials for imaging systems with annular pupils. JOSA. 1981 Jan 1; 71 (1): 75-85
[27] Noll RJ. Zernike polynomials and atmospheric turbulence. JOsA. 1976 Mar 1; 66(3):207-11.
[28] Kintner EC. On the mathematical properties of the Zernike polynomials. Journal of Modern Optics. 1976 Aug 1; 23(8):679-80.
[29] Frieden BR. VIII Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions. Progress in optics. 1971 Dec 31; 9:311-407.
[30] K. Maleknejad, E. Hashemizadeh , Numerical Solution of Nonlinear Singular Ordinary Differential Equations Arising in Biology Via Operational Matrix of Shifted Legendre Polynomials, Scientific Bulle-tin, Se-ries A: Applied Mathematics and Physics, vol .73(3), pp. 95-104, 2011
