Numerical solutions of ordinary differential equations using Spline-Integral Operator

TL;DR

This paper presents a new numerical method for solving initial value problems of differential equations using spline approximation and integral formulation, with proven order and stability analysis, validated through examples.

Contribution

Introduces a novel spline-integral based numerical method for differential equations, including proof of order and stability analysis, and comparative examples.

Findings

Method achieves expected order of accuracy.

Stable under specified conditions.

Performs comparably or better than Taylor's methods.

Abstract

In this work, we introduce a novel numerical method for solving initial value problems associated with a given differential. Our approach utilizes a spline approximation of the theoretical solution alongside the integral formulation of the analytical solution. Furthermore, we offer a rigorous proof of the method's order and provide a comprehensive stability analysis. Additionally, we showcase the effectiveness method through some examples, comparing with Taylor's methods of same order.