[Paper Review] Solving General Equations by Order Completion

Elemér E Rosinger|ArXiv.org|Aug 17, 2006

Mathematical and Theoretical Analysis5 references3 citations

TL;DR

This paper introduces an order completion method for solving general equations, including nonlinear systems of PDEs and arbitrary equations of the form $T(A) = F$, by extending the solution space through order-theoretic completion of domains and codomains. The key contribution is a universal existence and explicit solution framework that avoids functional analytic tools and generalized functions, yielding solutions with Hausdorff continuity and type-independent regularity.

ABSTRACT

A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.

Motivation & Objective

• To develop a universal method for solving general equations, including nonlinear PDEs, without relying on distributional or Sobolev-type solution spaces.
• To extend the notion of solution beyond classical functions by using order completion of partially ordered sets.
• To provide necessary and sufficient conditions for solution existence and explicit expressions for solutions in a unified framework.
• To eliminate the need for functional analytic methods in proving existence and regularity of solutions to nonlinear PDEs.
• To demonstrate that the method applies uniformly to both linear and nonlinear operators, without distinction.

Proposed method

• The method uses order completion of the domain $X$ and codomain $Y$ of a mapping $T: X \to Y$, transforming them into order-complete posets $X^\#_T$ and $Y^\#$.
• It constructs a pull-back order $\leq_T$ on the quotient space $X_T = X / \approx_T$, where $u \approx_T v$ iff $T(u) = T(v)$, to ensure injectivity of the induced map $T_\approx$.
• The solution is reformulated as finding $A \in X^\#_T$ such that $T^\#(A) = F$ for a given $F \in Y^\#$, where $T^\#$ is the order completion extension of $T_\approx$.
• Explicit solutions are derived using upper and lower sets: $T^\#(A) = \sup_{Y^\#} \{ \langle T_\approx(U) \rangle \mid U \in A \}$, with $A^{ul}$ denoting the order completion of set $A$.
• The method ensures that solutions can be assimilated with Hausdorff continuous functions, providing a universal regularity property.
• It establishes a commutative diagram involving $X_T$, $Y$, $X^\#_T$, and $Y^\#$, ensuring consistency and order isomorphism in the extended space.

Experimental results

Research questions

• RQ1 Can general equations of the form $T(A) = F$ be solved without assuming continuity or linearity of the operator $T$?
• RQ2 What conditions ensure the existence of solutions in the order completion framework for arbitrary equations?
• RQ3 How can explicit solutions be constructed in the extended solution space using order-theoretic operations?
• RQ4 Can the method unify the treatment of linear and nonlinear PDEs without distinguishing their structure?
• RQ5 Does the order completion approach yield solutions with intrinsic regularity, independent of the equation type?

Key findings

• The method guarantees the existence of solutions for any equation $T(A) = F$ in the order completion framework, provided necessary and sufficient conditions are met.
• Solutions are explicitly expressed via the upper-lower set operation $A^{ul}$, ensuring constructibility and order-theoretic consistency.
• The solutions $U$ obtained for nonlinear PDEs can be assimilated with Hausdorff continuous functions, implying a universal regularity property.
• The approach dispenses with the need for distributional, Sobolev, or hyperfunction spaces, allowing solutions to be represented as standard functions.
• The method is invariant under the nature of the operator: it treats linear and nonlinear operators uniformly, without structural distinction.
• The order completion extension $T^\#$ preserves order isomorphism and allows for the derivation of infima and suprema in the completed spaces, ensuring solution robustness.