A numerical method for solving uncertain wave equation

Abstract

Partial differential equations are usually used to deal with natural or physical problems, and wave equations are an important branch of them. Considering the fluctuation phenomena with uncertain noise in real life, this paper will focus on the study of the uncertain wave equation. The uncertain wave equation is a special kind of partial differential equation excited by the Liu process, which is widely used to describe wave propagation phenomena. However, not all uncertain wave equations can be solved analytically, so numerical solutions are of particular importance. The main objective of this paper is to study the numerical solution of uncertain wave equations. We first introduce the concept of α-path and, based on it, prove an important formula that reveals the connection between the uncertain wave equation and the classical wave equation. Based on this finding, we design a new numerical solution method to solve the uncertain wave equation. To verify the feasibility and accuracy of the method, we conduct numerical experiments. In addition, as an application, we obtain the uncertainty distribution, the inverse uncertainty distribution and the expected value of the solution of the equation based on the important formula. The originality and novelty of the conclusions of this work are demonstrated by comparing the results with those of other researchers.