Use Residual Correction Method and Monotone Iterative Technique to Calculate the Upper and Lower Approximate Solutions of Singularly Perturbed Non-linear Boundary Value Problems

Abstract

This paper seeks to use the proposed residual correction method in coordination with the monotone iterative technique to obtain upper and lower approximate solutions of singularly perturbed non-linear boundary value problems. First, the monotonicity of a non-linear differential equation is reinforced using the monotone iterative technique, then the cubic-spline method is applied to discretize and convert the differential equation into the mathematical programming problems of an inequation, and finally based on the residual correction concept, complex constraint solution problems are transformed into simpler questions of equational iteration. As verified by the four examples given in this paper, the method proposed hereof can be utilized to fast obtain the upper and lower solutions of questions of this kind, and to easily identify the error range between mean approximate solutions and exact solutions.

KEYWORDS:

• Singular perturbation
• Cubic spline
• Residual
• Upper and lower solutions
• Error bounds