A new high-accuracy difference method for a class of weakly nonlinear singular boundary-value problems

Abstract

A fourth-order finite difference method based on a uniform mesh is described by using Chawla’s identity [Sakai, M. and Usmani, R.A., 1990, An application of Chawla’s identity to a different scheme for singular problems. BIT Numerical Mathematics, 30, 566–568] for a class of singular two point boundary-value problems (p(x)y′)′=p(x)f(x, y), 0<x≤1; y(0)=A, α y(1)+β y′(1)=γ with p(x)>0 on (0, 1], p(x)∈C ¹(0, 1] and , for all x in (0, 1], 0≤b ₀<1. Fourth-order convergence has been established under quite general conditions on f(x, y). The method reduces to Numerov’s method for p(x)=1 and boundary conditions y(0)=A, y (1)=B. This method provides better results than some existing fourth-order methods ford , 0≤b ₀<1 and boundary conditions y(0)=A, y(1)=B, which is verified by two examples and also the order of the method is verified for general functions p(x).

Keywords:

• Two point singular BV problems
• Finite difference method
• Chawla’s identity

Acknowledgements

The authors thank the referees for their valuable suggestions. This work is supported by CSIR, New Delhi, India, and National Board for Higher Mathematics, Department of Atomic Energy, Mumbai, India.