The Asymptotic Expansion And Chebyshev Collocation Method To Nonlinear Singular Two Point Boundary Value Probleme

Nonlinear singular two point boundary value problem is an important model equation,which has wide applications in many branches of mathematics and physics.Since the equation involves a singular factor,its solution usually behaves like singular features about derivative at one endpoint of the interval,which leads to obvious decreasing of computational accuracy when standard algo-rithms are used to solve the problem.This paper aims to accurately describe the singular features of the solution at the singularity,from which a high accuracy algorithm can be designed.Firstly the nonlinear singular two point boundary value problem is transformed to Fredholm integral equa-tion of the second kind using Greens function.Secondly,the finite-term-truncation of the Puiseux series of the solution at the singularity is obtained by using Picard iteration and series expansion for the Fredholm integral equation from which the singular degree of the solution is accurately described.It is noted that the asymptotic expansion includes an undetermined parameter.Thirdly a smoothing variable transformation is constructed such that the solution of the transformed two point boundary value problem is sufficiently smooth by using the known singular information of the solution.Finally,the Chebyshev collocation method is used to solve the deduced equation to obtain high accuracy numerical solutions,from which an approximation of Chebyshev interpolating polynomial over the whole interval is constructed,and then the undetermined parameter in the Puiseux expansion is determined accurately.Numerical examples illustrate the effectiveness of the algorithm.The computational accuracy improves extensively compared with the straightforward using of the Chebyshev collocation method.