Reliability analysis of the uncertain heat conduction model

Abstract

Uncertainty theory has been demonstrated as a rigorous mathematical system to measure the reliability of products when there are few or no samples. Meanwhile, first hitting time, which is the first type of undetermined time appeared in history, has been extensively used in reliability analysis. Based upon it, this article mainly focuses on the reliability analysis of the uncertain heat equation through the first hitting time theorem and uncertain partial differential equation. First of all, first hitting time theorems of uncertain heat equation are provided via the inverse uncertainty distribution of the solution. Secondly, two novel belief reliability indexes (belief reliability function and belief reliable lifetime) are presented based on the proposed first hitting time theorem. Furthermore, analytic expressions and numerical calculations of belief reliability indexes are derived from uncertain heat conduction model with fixed heat source and periodic heat source, respectively. Lastly, numerical methods are designed, and some numerical examples are provided to demonstrate the rationality of the belief reliability indexes.