Existence and Uniqueness Theorem for Uncertain Wave Equation

Rong Gao

doi:10.3390/sym14020191

Abstract

In the real world, the indeterminate phenomenon and determinate phenomenon are symmetric; however, the indeterminate phenomenon absolutely exists. Hence, the indeterminate dynamic phenomenon is studied in this paper by using uncertainty theory, where the indeterminate dynamic phenomenon is associated with the belief degree and called the uncertain dynamic phenomenon. Based on uncertainty theory, the uncertain wave equation derived by the Liu process is constructed to model the propagation of various types of wave with uncertain disturbance in nature, where the Liu process is Lipschitz-continuous and has stationary and independent increments. First important of all, only the equation has solution which can be used to clearly depict the wave propagation influenced by uncertain disturbance. Therefore, the aims of this paper is to propose and prove a theorem of existence and uniqueness with Lipschitz and linear growth conditions.

Highlights

Uncertainty theory was founded by Liu [1] to model the indeterminacy associated with belief degrees
A belief degree is a chance that a possible event may happen and the chance is estimated by a human
The uncertain wave equation involving the Liu process plays a significant role in modeling wave propagation with uncertain disturbances

Summary

Introduction

Uncertainty theory was founded by Liu [1] to model the indeterminacy associated with belief degrees. Gao [15] applied the Yao-Chen formula to design an algorithm and obtain the numerical solution of an uncertain differential equation. It has been used in high numbers of research fields, such as finance [16] and optimal control [17]. To describe phenomena with uncertain disturbance, Yang and Yao [23] introduced uncertainty theory into the partial differential equation and the presented uncertain partial differential equation. They studied the uncertain heat equation, a type of uncertain differential equation.

Preliminaries

Existence and Uniqueness

Discussion and Conclusions