Uncertain calculus is a branch of mathematics that deals with the integral and differential of functions of uncertain processes. This paper first introduces the Liu process as an uncertain process defined by the Liu integral. Some properties of Liu processes are investigated such as sample continuity property, finite variation property, and the fact that a continuously differentiable function of the Liu process is another Liu process, among others. Based on the Liu process, the uncertain integral is extended. Furthermore, some mathematical properties are proved, including the fundamental theorem, change of variable theorem, and integration by parts theorem. Finally, uncertain calculus with respect to multiple Liu processes is discussed.

In recent years, there have been frequent cases of impact on the stable development of supply chain economy caused by uncertain events such as COVID-19 and extreme weather events. The creation, management, and impact coping techniques of the supply chain economy now face wholly novel requirements as a result of the escalating level of global uncertainty. Although a significant literature applies uncertainty analysis and optimization modeling (UAO) to study supply chain management (SCM) under uncertainty, there is a lack of systematic literature review and research classification. Therefore, in this paper, 121 articles published in 44 international academic journals between 2015 and 2022 are extracted from the Web of Science database and reviewed using the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA). Bibliometric analysis and CiteSpace software are used to identify current developments in the field and to summarize research characteristics and hot topics. The selected published articles are classified and analyzed by author name, year of publication, application area, country, research purposes, modeling methods, research gaps and contributions, research results, and journals to comprehensively review and evaluate the SCM in the application of UAO. We find that UAO is widely used in SCM under uncertainty, especially in the field of decision-making, where it is common practice to abstractly model the decision problem to obtain scientific decision results. This study hopes to provide an important and valuable reference for future research on SCM under uncertainty. Future research could combine uncertainty theory with supply chain management segments (e.g., emergency management, resilience management, and security management), behavioral factors, big data technologies, artificial intelligence, etc.

Wave equation is a commonly used tool for describing various kinds of wave phenomena in nature such as sound wave, water wave, electromagnetic wave and string vibration. It is a second-order partial differential equation and describe the wave propagation without noises. However, real world is filled with noises everywhere. So deterministic wave equation is not enough to model some problems with additive noises. As a remedy method, stochastic wave equation driven by Wiener process was presented where the noise is considered random and modeled by using Wiener process. Except for randomness, uncertainty associated belief degrees is another different type of indeterministic phenomenon. For modeling the wave phenomena with uncertain noises, this paper aims at deriving an uncertain wave equation driven by Liu process, which is a type of partial differential equation. Here, Liu process is a Lipschitz continuous uncertain process with stationary and independent increments. Then, we prove the existence and uniqueness of the solution of an uncertain wave equation. Additionally, we give the inverse uncertainty distribution of a solution of an uncertain wave equation.