The order completion method for systems of nonlinear PDEs: Pseudo-topological perspectives

TL;DR

This paper reformulates the Order Completion Method for nonlinear PDEs using uniform convergence structures, enabling more topological approaches and establishing existence and uniqueness of solutions for continuous nonlinear PDEs.

Contribution

It introduces a pseudo-topological framework for the Order Completion Method, enhancing the theoretical foundation for solving nonlinear PDEs.

Findings

Existence of solutions for arbitrary continuous nonlinear PDEs.

Uniqueness of solutions under the new framework.

Reformulation aligns the method with classical topological PDE approaches.

Abstract

By setting up appropriate uniform convergence structures, we are able to reformulate the Order Completion Method of Oberguggenberger and Rosinger in a setting that more closely resembles the usual topological constructions for solving PDEs. As an application, we obtain existence and uniqueness results for the solutions of arbitrary continuous, nonlinear PDEs.