Zernike radial polynomials method for solving nonlinear singular boundary value problems arising in physiology

Document Type: research paper

Authors

1 Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran

2 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Abstract

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.

Keywords


Article Title [Persian]

حل عددی معادلات دیفرانسیل معمولی منفرد غیرخطی حاصل شده در بیولوژی، ازطریق ماتریس عملیاتی چند جمله ای های زرنیکه شعاعی

Authors [Persian]

  • محمد علی عبادی 1
  • الهام السادات هاشمی زاده 2
  • امیرحسین رفاهی شیخانی 1
1 دانشجوی گروه ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران
2 عضو هیات علمی گروه ریاضی دانشگاه آزاد کرج و معاون آموزشی دانشکده
Abstract [Persian]

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.

Keywords [Persian]

  • مسائل مقدار مرزی منفرد غیر خطی
  • چند جمله ای های زرنیکه شعاعی
  • ماتریس عملیاتی مشتق

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