Analytical solution to the nonlinear singular boundary value problem arising in biology
 Abdollah Dinmohammadi^{1},
 Abdolrahman Razani^{2} and
 Elyas Shivanian^{2}Email author
Received: 1 September 2016
Accepted: 19 April 2017
Published: 27 April 2017
Abstract
This paper proposes an analytical procedure for the nonlinear singular boundary value problem that arises in biology and in the study of some diseases. As a first step, we present a constructive proof of the existence and uniqueness of solution. Then, we apply the Picard iterative sequence by constructing an integral equation whose Green’s function is not negative. The convergence of this iterative sequence is then controlled by an embedded parameter so that it tends to the unique solution.
Keywords
1 Introduction
On the other hand, boundary value problem of the form (1.1)(1.2) involving the governing ordinary differential equation (1.1) or slight generalizations of it has been investigated by Gatica et al. [12], Fink et al. [13], Baxley [14], Baxley and Gersdorff [15], Wang and Li [16], Tinio [17], Wang [18], Ebaid [19], Agarwal and O’Regan [20] and also by the authors of [21, 22]. In these works, the fixed point theory or approximation theory was prevalently used, and their studies included assumptions that restrict \(f(x,y)\) to be of one sign and usually continuous for \(y\geq0\). Later, in Refs [23–25], authors started to allow signchanging nonlinearities but still they have required \(f(x,y)\) to be continuous for \(y\geq0\). We notice that, in all these works, it has been assumed \(n=0\) and they have considered some particular types of boundary conditions (1.2) or some specific forms of governing differential equation (1.1).
The aim of the present work is to provide some constructive existence and uniqueness theorems for problem (1.1)(1.2) of the same type as those provided by Ford and Pennline in [26] by the same assumptions considered by them, i.e., \(f (x,y(x) )\) is not only allowed to be signchanging but it also can be singular with respect to y. Also, we apply the Picard iterative sequence by constructing an integral equation whose Green’s function is not negative. The convergence of this iterative sequence is then controlled by an embedded parameter so that it converges to the unique solution.
2 Constructing the integral equation
3 Some properties of the Green’s function
In this section, we present some properties of the Green’s function as theorems which are extremely important in the analysis of (1.1)(1.2).
Lemma 3.1

The function \(v(x)\) is a positive decreasing function of \(x\in [0,1]\) and unbounded at the origin.

Assuming that m, n and k are such that \(W(1)\) is positive, then \(u(x)\) is a positive increasing function of \(x\in[0,1]\).

Assuming that m, n and k are such that \(W(1)\) is negative, then \(u(x)\) is a negative decreasing function of \(x\in[0,1]\).
Theorem 3.2
Proof
First, suppose that m, n and k are such that \(W(1)\) is positive, then from Lemma 3.1 it follows that \(u(x)\) is a positive function of \(x\in[0,1]\). On the other hand, \(v(x)\) is a positive decreasing function of \(x\in[0,1]\) from Lemma 3.1. Therefore, we conclude that \(G(x,t)\geq0\).
Second, suppose that m, n and k are such that \(W(1)\) is negative, then from Lemma 3.1 it follows that \(u(x)\) is a negative function of \(x\in[0,1]\). On the other hands, \(v(x)\) is a positive decreasing function of \(x\in[0,1]\) from Lemma 3.1. Therefore, we conclude again that \(G(x,t)\geq0\). □
Theorem 3.3
Proof
4 The region of existence and uniqueness
Theorem 4.1
 (a)
\(\frac{\partial f}{\partial y}\) is continuous in \(\mathbf {D}: [0,1]\times[y_{L}(x),y_{U}(x)]\) and satisfies \(0\leq\frac{\partial f}{\partial y}\leq N_{D}\) within it.
 (b)
\(y_{0}(x)=\frac{1}{2}[y_{L}(x)+y_{U}(x)]\) and \(y_{L}(x)\leq y_{n}(x)\leq y_{U}(x), n=1,2,3,\ldots \) .
 (c)
The value of \(k^{2}\) satisfies \(k^{2}\geq\frac{N_{D}}{2}\).
Proof
5 Solution procedure and illustrative physiology models
Example 5.1
Example 5.2
Declarations
Acknowledgements
The authors are grateful to anonymous reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper.
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
References
 Garner, J, Shivaji, R: Diffusion problems with a mixed nonlinear boundary condition. J. Math. Anal. Appl. 148(2), 422430 (1990) MathSciNetView ArticleMATHGoogle Scholar
 Adam, JA: A simplified mathematical model of tumor growth. Math. Biosci. 81(2), 229244 (1986) View ArticleMATHGoogle Scholar
 Adam, JA: A mathematical model of tumor growth. II. Effects of geometry and spatial nonuniformity on stability. Math. Biosci. 86(2), 183211 (1987) View ArticleMATHGoogle Scholar
 Adam, JA, Maggelakis, S: Mathematical models of tumor growth. IV. Effects of a necrotic core. Math. Biosci. 97(1), 121136 (1989) View ArticleMATHGoogle Scholar
 Burton, AC: Rate of growth of solid tumours as a problem of diffusion. Growth 30(2), 157176 (1966) Google Scholar
 Greenspan, H: Models for the growth of a solid tumor by diffusion. Stud. Appl. Math. 51(4), 317340 (1972) View ArticleMATHGoogle Scholar
 Asaithambi, N, Garner, J: Pointwise solution bounds for a class of singular diffusion problems in physiology. Appl. Math. Comput. 30(3), 215222 (1989) MathSciNetMATHGoogle Scholar
 McElwain, D: A reexamination of oxygen diffusion in a spherical cell with MichaelisMenten oxygen uptake kinetics. J. Theor. Biol. 71(2), 255263 (1978) View ArticleGoogle Scholar
 Flesch, U: The distribution of heat sources in the human head: a theoretical consideration. J. Theor. Biol. 54(2), 285287 (1975) View ArticleGoogle Scholar
 Gray, B: The distribution of heat sources in the human head  theoretical considerations. J. Theor. Biol. 82(3), 473476 (1980) View ArticleGoogle Scholar
 Duggan, R, Goodman, A: Pointwise bounds for a nonlinear heat conduction model of the human head. Bull. Math. Biol. 48(2), 229236 (1986) View ArticleMATHGoogle Scholar
 Gatica, J, Oliker, V, Waltman, P: Singular nonlinear boundary value problems for secondorder ordinary differential equations. J. Differ. Equ. 79(1), 6278 (1989) MathSciNetView ArticleMATHGoogle Scholar
 Fink, A, Gatica, JA, Hernandez, GE, Waltman, P: Approximation of solutions of singular second order boundary value problems. SIAM J. Math. Anal. 22(2), 440462 (1991) MathSciNetView ArticleMATHGoogle Scholar
 Baxley, JV: Some singular nonlinear boundary value problems. SIAM J. Math. Anal. 22(2), 463479 (1991) MathSciNetView ArticleMATHGoogle Scholar
 Baxley, JV, Gersdorff, GS: Singular reactiondiffusion boundary value problems. J. Differ. Equ. 115(2), 441457 (1995) MathSciNetView ArticleMATHGoogle Scholar
 Wang, H, Li, Y: Existence and uniqueness of solutions to two point boundary value problems for ordinary differential equations. Z. Angew. Math. Phys. 47(3), 373384 (1996) MathSciNetView ArticleMATHGoogle Scholar
 Tineo, A: On a class of singular boundary value problems which contains the boundary conditions. J. Differ. Equ. 113, 116 (1994) MathSciNetView ArticleMATHGoogle Scholar
 Wang, J: On positive solutions of singular nonlinear twopoint boundary value problems. J. Differ. Equ. 107(1), 163174 (1994) MathSciNetView ArticleMATHGoogle Scholar
 Ebaid, A: A new analytical and numerical treatment for singular twopoint boundary value problems via the Adomian decomposition method. J. Comput. Appl. Math. 235(8), 19141924 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Agarwal, RP, O’Regan, D: Nonlinear superlinear singular and nonsingular second order boundary value problems. J. Differ. Equ. 143(1), 6095 (1998) MathSciNetView ArticleMATHGoogle Scholar
 Jia, R, Shao, J: Existence and uniqueness of periodic solutions of secondorder nonlinear differential equations. J. Inequal. Appl. 2013, 115 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Zhou, J: The existence and uniqueness of the solution for nonlinear elliptic equations in Hilbert spaces. J. Inequal. Appl. 2015, 250 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Agarwal, RP, O’Regan, D: An upper and lower solution approach for singular boundary value problems with sign changing nonlinearities. Math. Methods Appl. Sci. 25(6), 491506 (2002) MathSciNetView ArticleMATHGoogle Scholar
 Kannan, R, O’Regan, D: Singular and nonsingular boundary value problems with sign changing nonlinearities. J. Inequal. Appl. 5, 621637 (2000) MathSciNetMATHGoogle Scholar
 Lü, H, Bai, Z: Positive radial solutions of a singular elliptic equation with sign changing nonlinearities. Appl. Math. Lett. 19(6), 555567 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Ford, WF, Pennline, JA: Singular nonlinear twopoint boundary value problems: existence and uniqueness. Nonlinear Anal. 71(3), 10591072 (2009) MathSciNetView ArticleMATHGoogle Scholar