A High-Order B-Spline Collocation Method for Solving Nonlinear Singular Boundary Value Problems Arising in Engineering and Applied Science

Abstract

In this paper, a high-order B-spline collocation method on a uniform mesh is presented for solving nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions: $$\begin{aligned} (p(x)y')'= & {} p(x)f(x,y), \quad 0<x\le 1, \\ y'(0)= & {} 0,\quad ay(1)+by'(1)=c, \end{aligned}$$ where $$p(x)=x^{\alpha }g(x),\alpha \ge 0$$ is a general class of non-negative function. The error analysis for the quartic B-spline interpolation is discussed. To demonstrate the applicability and efficiency of our method we consider eight numerical examples, seven of which arise in various branches of applied science and engineering: (1) equilibrium of isothermal gas sphere; (2) thermal explosion; (3) thermal distribution in the human head; (4) oxygen diffusion in a spherical cell; (5) stress distribution on shallow membrane cap; (6) reaction diffusion process in a spherical permeable catalyst; (7) heat and mass transfer in a spherical catalyst. It is shown that our method has fourth-order convergence and is more accurate than finite difference methods (Chawla et al., in BIT 28:88–97, 1988; Pandey et al. in J Comput Appl Math 224:734–742, 2009) and B-spline collocation methods (Abukhaled et al. in Int J Numer Anal Model 8:353–363, 2011; Khuri and Sayfy in Int J Comput Methods 11(1):1350052, 2014).