Numerical Solution of Nonlinear Singular Ordinary Differential Equations Arising in Biology Via Operational Matrix of Shifted Legendre Polynomials

This paper proposed a numerical method for nonlinear singular ordinary differential equations, that arises in biology and some diseases. We solved these nonlinear problems by a new method based on shifted Legendre polynomials. Operational matrices of derivatives for this function are presented to reduce the nonlinear singular boundary value problems to a system of nonlinear algebraic equations. The method is computationally very simple and attractive, and applications are demonstrated through illustrative examples. The results obtained are compared by the known results.


Introduction
The aim of this paper is to introduce a new method for the numerical solution of the following class of singular boundary value problems 1, 0 ), , x y x f y x m a x y (1) , = (0) (0) head, see [10][11][12][13] Existence-uniqueness results for such problems have been established by several researchers [14][15][16]. In recent years, finding numerical solutions of singular differential equations, particularly those arising in physiology, has been the focus of a number of authors, which you can see some of them in [17][18][19][20].
The purpose of this paper is to introduce a novel method based on operational matrices of derivatives of shifted Legandre polynomials that have been introduced recently in Saadatmandi and Dehghan work's [21] for the numerical solution of the class of singular second-order boundary value problems given in the (1-3) that arise in physiology. In this work by use of shifted Legendre polynomials as basis and operational matrices of derivatives of them we convert these kinds of equations to algebraic equations. The advantage of this method analogy to other existed method for these problems is its trusty and simply in implementation, we compared our results with some existed results to prove this claim. This paper is organized as follows: Section 2 represents preliminaries, in this section we introduced shifted Legendre polynomials, and some properties of them, specially the operational matrices of derivatives, in Section 3 we implemented them on physiology problems. In Section 4, a number of applied models in physiology are discussed to show the efficiency and accuracy of the proposed method, the results obtained are compared by the known results. Finally, Section 5 includes a conclusion for the paper.

Shifted Legendre Polynomials
Consider the Legendre polynomials ) (z L m on the in- [22,23]. In order to use these polynomials on the interval we define the so-called shifted Legendre polynomials by introducing the change of variable can be obtained as follows:

Function Approximation
Any function where the shifted Legendre coefficient vector C and the shifted Legendre vector B are given by: .

Operational Matrix of Derivative
The derivative of the vector where (1) D is the operational matrix of derivative given by [21] otherwise, 0, where   n and the superscript, in (1)

Implementation of Shifted Legendre Polynomials Method on Physiology Problems
In this section we solve nonlinear singular boundary value problem of the form Eq.(1) with the mixed conditions (2) and (3) by using shifted Legendre polynomials.

Illustrative Examples and Applied Models
To show the efficiency of the proposed numerical method, we implement it on three nonlinear singular boundary problems that arise in real physiology applications. Our results are compared with result in Refs. [17][18][19][20]. The austerity of our method in implementation in analogy to other existed methods and its trusty answers is considerable.

Example 1
Consider the following oxygen diffusion problem , 0.03119 with the boundary conditions:

Example 2
Consider the following singular two point boundary value problem: with the exact solution ) 1 . Table 2 shows numerical errors of this example in analogy to errors for this example in [17].

Example 3
Consider this problem that is coincide by heat conduction model of the human head, we consider the solution of this problem with conditions as follows: Table 3 illustrates results for this example by proposed method alongside numerical solutions for this example that have been given in Refs [19][20].  in Table 4 and for 1  h in Table 5, which show the accuracy of proposed method and these results in analogy to exhibited results for this example in [19][20] show advantage of this method.

Conclusions
This paper present a new approach, based on shifted Legendre polynomials for the numerical solution of a class of singular boundary value problems arising in biology and physiology problems. By use of shifted Legendre polynomials as basis and operational matrices of derivatives of these functions we convert such problems to an algebraic system. The implementation of current approach in analogy to existed methods is more convenient and the accuracy is high and we can execute this method in a computer speedy with minimum CPU time used. The numerical applied models that have been presented in the paper and the compared results support our claim.