On the Approximation of Nonlinear Evolution Equations in Particular C*-Algebras of Operators

TL;DR

This paper investigates the stability and convergence of numerical solutions for nonlinear evolution equations using specialized operator algebras, including cases where algebraic identities are not satisfied, to improve numerical analysis techniques.

Contribution

It introduces a novel approach applying particular C*-algebra techniques to analyze and estimate solutions of nonlinear evolution equations, including non-C*-identity cases.

Findings

Established stability and convergence criteria for numerical solutions.

Extended analysis to algebras not satisfying C*-identity.

Applied techniques to discretizable Hilbert spaces.

Abstract

In this article we deal with the stability and convergence of numerical solutions of nonlinear evolution equations of the form $A(u(t))+f(u(t))=u'(t)$, the numerical analysis of solutions to this problems will be performed using some methods from particular algebras of operators which are sometimes represented by unital subalgebras of the unital C*-algebras of operators that are generated by some basic operators say $\mathbf{1},a,\mathcal{D}(\cdot)\in\mathcal{L}(H^m(G))$ that in some suitable sense are related to the operator $A(\cdot)\in\mathcal{L}(H^m(G))$ in the evolution equations, particular cases where the operator algebras do not verify the C*-identity with respect to the norm chosen are also studied, when applicable basic C*-algebra techniques are implemented to perform some estimates of numerical solutions to some types of problems, in all this work expressions like $H^m(G)$ will represent a prescribed discretizable Hilbert space with $G\subset\subset\mathbb{R}^n$.