Accurate method of verified computing for solutions of semilinear heat equations

TL;DR

This paper introduces a highly accurate verification method for solutions of semilinear heat equations, enabling longer time interval validation and improved error control compared to previous approaches.

Contribution

The authors develop a novel verification technique based on a fixed-point formulation and iterative scheme, extending solution validation duration for semilinear heat equations.

Findings

The method successfully verifies solutions over longer time intervals.

Numerical examples demonstrate improved efficiency and accuracy.

The approach avoids error overestimation propagation.

Abstract

We provide an accurate verification method for solutions of heat equations with a superlinear nonlinearity. The verification method numerically proves the existence and local uniqueness of the exact solution in a neighborhood of a numerically computed approximate solution. Our method is based on a fixed-point formulation using the evolution operator, an iterative numerical verification scheme to extend a time interval in which the validity of the solution can be verified, and rearranged error estimates for avoiding the propagation of an overestimate. As a result, compared with the previous verification method using the analytic semigroup, our method can enclose the solution for a longer time. Some numerical examples are presented to illustrate the efficiency of our verification method.