On the Formalization of the Heat Conduction Problem in HOL

TL;DR

This paper formalizes the heat conduction problem in HOL theorem prover, modeling the heat equation and verifying solutions under various conditions for safety-critical applications.

Contribution

It introduces a formalization of the heat equation in HOL Light, including the heat operator and solution verification using separation of variables.

Findings

Formal model of heat transfer in HOL Light

Verification of heat equation properties

Solution verification under different conditions

Abstract

Partial Differential Equations (PDEs) are widely used for modeling the physical phenomena and analyzing the dynamical behavior of many engineering and physical systems. The heat equation is one of the most well-known PDEs that captures the temperature distribution and diffusion of heat within a body. Due to the wider utility of these equations in various safety-critical applications, such as thermal protection systems, a formal analysis of the heat transfer is of utmost importance. In this paper, we propose to use higher-order-logic (HOL) theorem proving for formally analyzing the heat conduction problem in rectangular coordinates. In particular, we formally model the heat transfer as a one-dimensional heat equation for a rectangular slab using the multivariable calculus theories of the HOL Light theorem prover. This requires the formalization of the heat operator and formal verification of its various properties, such as linearity and scaling. Moreover, we use the separation of variables method for formally verifying the solution of the PDEs, which allows modeling the heat transfer in the slab under various initial and boundary conditions using HOL Light.