Quasi-collocation based on CCC-Schoenberg operators and collocation methods

TL;DR

This paper introduces collocation and quasi-collocation methods using CCC-Schoenberg operators for solving second order boundary value problems, especially singular and perturbed cases, with error analysis.

Contribution

It develops new collocation and quasi-collocation techniques based on CCC-Schoenberg operators for challenging boundary value problems, including error bounds.

Findings

Effective for singular and perturbed boundary value problems

Provides error bounds for the proposed methods

Utilizes Green's function for calculations

Abstract

We propose a collocation and quasi-collocation method for solving second order boundary value problems $L_2 y=f$, in which the differential operator $L_2$ can be represented in the product formulation, aiming mostly on singular and singularly perturbed boundary value problems. Seeking an approximating Canonical Complete Chebyshev spline $s$ by a collocation method leads to demand that $L_2s$ interpolates the function $f$. On the other hand, in quasi-collocation method we require that $L_2 s$ is equal to an approximation of $f$ by the Schoenberg operator. We offer the calculation of both methods based on the Green's function, and give their error bounds.