Available online at http://ijim.srbiau.ac.ir/ Int. J. Industrial Mathematics (ISSN 2008-5621) Vol. 10, No. 1, 2018 Article ID IJIM-01050, 8 pages Research Article A Numerical Method For Solving Physiology Problems By Shifted Chebyshev Operational Matrix E. Hashemizadeh ∗† , F. Mahmoudi ‡ Received Date: 2017-03-22 Revised Date: 2017-08-18 Accepted Date: 2017-10-22 ————————————————————————————————– Abstract In this study, a new numerical solution of singular nonlinear differential equations, stemming from biology and physiology problems, is proposed. The methodology is primarily based on the shifted Chebyshev polynomials operational matrix of derivative and collocation. Furthermore, the conver- gence analysis on the proposed method is carried out. To assess the accuracy and analysis of perfor- mance of the method, five numerical problems, based on the singular nonlinear differential equations, on different subjects, such as the human head, Oxygen diffusion in a spherical cell and Bessel dif- ferential equation, were solved. The numerical results were compared with other existed methods in tables for verification and further discussions. Keywords : Differential equations; Shifted Chebyshev polynomials; Operational matrix of derivative; Convergence analysis; Physiology problems. —————————————————————————————————– 1 Introduction T he differential equations arise from various ap- plications in fluid mechanics, biology, physics and engineering [1]-[16]. Such equations also ap- pear in electromagnetic and electrodynamic, elas- ticity and dynamic contact, heat and mass trans- fer, fluid mechanic, acoustic, chemical and elec- trochemical processes, molecular physics, popu- lation, medicine and in many other fields [3]-[14]. For numerical solution of the differential equa- tions, there are some well-known numerical meth- ods [11]-[15]. In this paper, we consider the singular prob- ∗ Corresponding author. hashemizadeh@kiau.ac.ir , Tel: +98(26)34182389. † Young Researchers and Elite Club, Karaj Branch, Is- lamic Azad University, Karaj, Iran. ‡ Young Researchers and Elite Club, Karaj Branch, Is- lamic Azad University, Karaj, Iran. lems of the type y ′′ (x)+ p(x)y ′ (x)+ q(x)y(x)= g(x), 0 <x ≤ 1, (1.1) subject to the conditions { α 1 y(0) + β 1 y ′ (0) = γ 1 , α 2 y(1) + β 2 y ′ (1) = γ 2 , (1.2) where x = 0 is a singular point in p(x), also p(x),q(x) and g(x) are continuous functions on (0, 1] and the parameters α 1 ,α 2 ,β 1 ,β 2 ,γ 1 ,γ 2 are real constants. The Chebyshev polynomials have been in exis- tence for over a hundred years and they have been used for solving many different problems [2]. In this paper, we propose a suitable way to approximate the solution of singular nonlinear differential equations with initial or boundary value problems on the interval (0,L), by use of shifted Chebyshev collocation method based on the shifted Chebyshev operational matrix ot 95