The existence and unicity of numerical solution of initial value problems by Walsh polynomials approach

TL;DR

This paper extends Walsh polynomial methods to prove the existence, uniqueness, and uniform convergence of numerical solutions for initial value problems with variable coefficients, for any initial point in [0,1[.

Contribution

It generalizes previous results by allowing arbitrary initial points in [0,1[ and establishes the existence and convergence of solutions.

Findings

Existence of numerical solutions for initial value problems with arbitrary initial points.

Uniqueness of the numerical solutions.

Uniform convergence of the Walsh polynomial-based solutions.

Abstract

Chen and Hsiao gave the numerical solution of initial value problems of systems of linear differential equations with constant coefficients by Walsh polynomials approach. This result was improved by G\'at and Toledo for initial value problems of differential equations with variable coefficients on the interval $[0,1[$ and initial value $\xi=0$. In the present paper we discuss the general case while $\xi$ can take any arbitrary value in the interval $[0,1[$. We show the existence and uniform convergence of the numerical solution, as well.