The series expansion and Chebyshev collocation method for nonlinear singular two-point boundary value problems

Abstract

The solution of singular two-point boundary value problem is usually not sufficiently smooth at one or two endpoints of the interval, which leads to a great difficulty when the problem is solved numerically. In this paper, an algorithm is designed to recognize the singular behavior of the solution and then solve the equation efficiently. First, the singular problem is transformed to a Fredholm integral equation of the second kind via Green’s function. Second, the truncated fractional series of the solution about the singularity is formulated by using Picard iteration and implementing series expansion for the nonlinear function. Third, a suitable variable transformation is performed by using the known singular information of the solution such that the solution of the transformed equation is sufficiently smooth. Fourth, the Chebyshev collocation method is used to solve the deduced equation to obtain approximate solution with high precision. Fifth, the convergence analysis of the collocation method is conducted in weighted Sobolev spaces for linear singular equations. Sixth, numerical examples confirm the effectiveness of the algorithm. Finally, the Thomas–Fermi equation and the Emden–Fowler equation as some applications are accurately solved by the method.