A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems

ABSTRACT

Recently, Caglar et al. [B-spline method for solving Bratu's problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885–1891] proposed a numerical technique based on cubic B-spline for solving a Bratu-type problem. This method provides a second-order convergent approximation to the solution of the problem. In this paper, we develop a high-order numerical method based on quartic B-spline collocation approach for the Bratu-type and Lane–Emden problems. The error analysis of the quartic B-spline interpolation is carried out. Some numerical examples are provided to demonstrate the efficiency and applicability of the method and to verify its rate of convergence. The numerical results are compared with exact solutions and a numerical method based on cubic B-spline approach. Comparison reveals that our method produces more accurate results than the method proposed by Caglar et al. [B-spline method for solving Bratu's problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885–1891].

Keywords:

• Bratu's problem
• Lane–Emden problems
• B-spline collocation method
• fourth-order convergence
• error analysis

2010 AMS Subject Classifications:

• 65L10
• 65L60
• 34B16

Acknowledgements

The authors are very grateful to anonymous referees for their valuable suggestions and comments which improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.