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 Open Access
A numerical approach for the solution of a class of singular boundary value problems arising in physiology
 Mohamadreza Mohsenyzadeh^{1},
 Khosrow Maleknejad^{1}Email author and
 Reza Ezzati^{1}
https://doi.org/10.1186/s136620150572x
© Mohsenyzadeh et al. 2015
 Received: 30 December 2014
 Accepted: 10 July 2015
 Published: 25 July 2015
Abstract
The purpose of this paper is to introduce a novel approach based on the operational matrix of orthonormal Bernoulli polynomial for the numerical solution of the class of singular secondorder boundary value problems that arise in physiology. The main thrust of this approach is to decompose the domain of the problem into two subintervals. The singularity, which lies in the first subinterval, is removed via the application of an operational matrix procedure based on differentiating that is applied to surmount the singularity. Then, in the second subdomain, which is outside the vicinity of the singularity, the resulting problem is treated via employing the proposed basis. The performance of the numerical scheme is assessed and tested on specific test problems. The oxygen diffusion problem in spherical cells and a nonlinear heatconduction model of the human head are discussed as illustrative examples. The numerical outcomes indicate that the method yields highly accurate results and is computationally more efficient than the existing ones.
Keywords
 orthonormal Bernoulli polynomials
 differential operational matrix
 physiology problems
MSC
 34L30
 34L16
1 Introduction
In recent years, finding numerical solutions of singular differential equations, particularly those arising in physiology, has been focused by several authors. Ravi Kanth and Bhattacharya [5] used a quasilinearization technique to reduce a class of nonlinear SBVP arising in physiology to a sequence of linear problems; the resulting set of differential equations is modified at the singular point, then spline collocation is utilized to obtain the numerical solution. Pandey and Singh [6] described a finite difference method based on a uniform mesh for the solution of a class of SBVP arising in physiology; it was shown that the method is of secondorder accuracy under quite general conditions. Caglar et al. [7] used Bspline functions to develop a numerical method for computing approximations to the solution of nonlinear SBVP associated with physiological science. The original differential equation was modified at the singular point, then the boundary value problems were treated by using the Bspline approximation. Asaithambi and Garner [8] presented a numerical technique for obtaining pointwise bounds for the solution of a class of nonlinear boundaryvalue problems appearing in physiology. Gustaffsson [9] presented a numerical method for solving SBVP. Ravi Kanth and Reddy [10] presented a numerical method for solving a twopoint boundary value problem in the interval \([0,1]\) with regular singularity at \(x=0\). Ravi Kanth and Reddy [11] presented a numerical method for singular twopoint boundary value problems via Chebyshev economization. A number of papers discussed the existence of solutions for the given problem, for instance, existence and uniqueness of the solution of SBVP (1)(3) for the special case \(m=2\), \(\alpha_{1}=\alpha_{2}=\gamma\) and \(\beta =1\) were given in [12]. In the past decade, there has been a great deal of interest [13–21] in applying the decomposition method for solving a wide range of nonlinear equations, including algebraic, differential, partialdifferential, differentialdelay and integrodifferential equations.
This paper is organized as follows. In Section 2, we are going to introduce Bernoulli polynomials and their properties; also we will show the operational matrix of derivative for orthonormal Bernoulli polynomials. In Section 3, the operational matrix of derivative of the proposed basis together with collocation method are used to reduce the nonlinear singular ordinary differential equation to a nonlinear algebraic equation that can be solved by Newton’s method. Section 4 illustrates some applied models to show the convergence, accuracy and advantage of the proposed method and compares it with some other existing method. Finally Section 5 concludes the paper.
2 Definition of Bernoulli polynomials
In this section, we introduce Bernoulli polynomials and their properties such as differentiation, integral means conditions, etc. Obviously, Bernoulli polynomials are not orthonormal polynomials, we orthonormal these polynomials by GramSchmidt algorithm. The operational matrix constructed by this new basis is sparser than the operational matrix which is made by Bernoulli polynomial, which can be the advantage of normalization of Bernoulli polynomials. Also, by this new basis, we construct a new approach to solve SBVP, which can get better approximation for numerical solution of these equations in comparison with other methods.
2.1 Bernoulli polynomials

Property 1 (Integral means conditions):$$\int_{0}^{1}OB_{i}(x)\,dx=0,\quad i=1,2,\ldots,n. $$

Property 2 (Norm):$$\bigl\ OB_{i}(x) \bigr\ _{2}=1,\quad i=1,2,\ldots,n. $$
2.2 Function approximation
2.3 Error bounds
In this section, the error bound and convergence are established by the following lemma.
Theorem 1
[25]
Lemma 1
[26]
Proof
2.4 Operational matrix of derivative
3 Implementation of operational matrix of orthonormal Bernoulli polynomials on physiology problems
4 Illustrative examples
To show the efficiency of the proposed method, we implement it on three nonlinear singular boundary problems that arise in real physiology applications. Our results are compared with the results in Refs. [28, 29]. The austerity of our method in implementation in analogy to other existing methods and its trusty answers is considerable.
Example 1
Approximate solutions for Example 1
x  Present method with n = 14  Method in [28] with n = 20  Method in [30] with n = 60 

0.0  0.82848328186193  0.82848329481355  0.82848327295802 
0.1  0.82970609243390  0.82970609688790  0.82970607521884 
0.2  0.83337473359110  0.83337473804308  0.83337471691089 
0.3  0.83948991395381  0.83948991833986  0.83948989814383 
0.4  0.84805278499617  0.84805278876051  0.84805277036165 
0.5  0.85906492716933  0.85906492753032  0.85906491397434 
0.6  0.87252831995828  0.87252831569855  0.87252830841853 
0.7  0.88844530562329  0.88844529949702  0.88844529589927 
0.8  0.90681854806690  0.90681854179965  0.90681854026297 
0.9  0.92765098836568  0.92765098305256  0.92765098252660 
1.0  0.95094579849657  0.95094579480523  0.95094579461056 
Example 2
Approximate solutions for Example 2
x  Present method with n = 14  Method in [27] with forthorder  Method in [29] 

0.0  0.3675152742  0.3675181074  0.3675169710 
0.1  0.3663623292  0.3663637561  0.3663623697 
0.2  0.3628940661  0.3628959378  0.3628941066 
0.3  0.3570975457  0.3570991429  0.3570975842 
0.4  0.3489484206  0.3489499903  0.3489484612 
0.5  0.3384121487  0.3384136581  0.3384121893 
0.6  0.3254435224  0.3254450019  0.3254435631 
0.7  0.3099860402  0.3099878567  0.3099860810 
0.8  0.2919711030  0.2919789654  0.2919711440 
0.9  0.2713170101  0.2713185637  0.2713170512 
1.0  0.2479277233  0.2479292837  0.2479277646 
Example 3
Numerical errors for Example 3
x  Present method with n = 10  Present method with n = 14  Approach II [28] with n = 20 

0.0  3.77 × 10^{−8}  6.72 × 10^{−8}  2.00 × 10^{−6} 
0.1  1.05 × 10^{−7}  6.69 × 10^{−8}  1.99 × 10^{−6} 
0.2  6.33 × 10^{−9}  7.87 × 10^{−9}  1.97 × 10^{−6} 
0.3  5.91 × 10^{−8}  6.92 × 10^{−9}  1.94 × 10^{−6} 
0.4  2.12 × 10^{−7}  2.87 × 10^{−8}  1.83 × 10^{−6} 
0.5  1.00 × 10^{−8}  7.40 × 10^{−10}  1.78 × 10^{−6} 
0.6  5.36 × 10^{−7}  6.32 × 10^{−8}  1.67 × 10^{−6} 
0.7  4.25 × 10^{−8}  6.95 × 10^{−8}  1.34 × 10^{−6} 
0.8  8.32 × 10^{−7}  3.38 × 10^{−9}  9.20 × 10^{−7} 
0.9  4.67 × 10^{−8}  7.85 × 10^{−8}  4.57 × 10^{−7} 
1.0  6.42 × 10^{−8}  6.63 × 10^{−8}  0 
5 Conclusion
This paper presents a new approach based on the operational matrix of derivative of the orthonormal Bernoulli polynomials for the numerical solution of a class of singular boundary value problems arising in physiology. By use of orthonormal Bernoulli polynomials as basis and the operational matrix of derivative of these functions, we convert such problems to an algebraic system. The implementation of current approach in analogy to existing methods is more convenient and the accuracy is high, and we can execute this method in a computer in a speedy manner with minimum CPU time used. The numerical applied models that have been presented in the paper are compared with other methods.
Declarations
Acknowledgements
The authors are grateful to the anonymous referees for their constructive comments and helpful suggestions to improve this paper greatly.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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