High-Degree Polynomial Approximations for Solving Linear Integral, Integro-Differential, and Ordinary Differential Equations

TL;DR

This paper introduces a universal numerical scheme using high-degree polynomial approximations to solve various linear differential and integral equations, demonstrating accuracy, stability, and effective regularization against Runge's phenomenon.

Contribution

The paper develops a novel high-degree polynomial approximation method adaptable for linear integral, integro-differential, and differential equations, including ill-posed problems.

Findings

Accurate solutions demonstrated with noisy data

Stable numerical scheme shown through examples

Regularization effectively prevents Runge's phenomenon

Abstract

This paper presents a universal numerical scheme tailored for tackling linear integral, integro-differential, and both initial and boundary value problems of ordinary differential equations. The numerical scheme is readily adapted for resolving ill-posed problems. Central to our approach is high-degree piecewise-polynomial approximation to the exact solution. We illustrate the accuracy and stability of our numerical solutions in the presence of noise through illustrative examples. Additionally, we demonstrate that proposed regularization being applied to high-degree interpolation, effectively eliminates Runge's phenomenon.