Differential Equations

Differential equations form the mathematical language for describing change in dynamic systems. Researchers in this area investigate the existence, uniqueness, and behavior of solutions to various types of differential equations, including ordinary, partial, and stochastic differential equations. Work encompasses theoretical analysis, such as proving properties of solutions and developing new analytical techniques, as well as computational approaches through numerical analysis to approximate solutions when exact forms are unattainable. Specific areas of study include the behavior of traveling wave solutions, analysis of reaction-diffusion systems, and the formulation and study of boundary value problems.

The mathematical modeling of phenomena relevant to Arkansas is a key application of differential equations research. This includes understanding the spread of infectious diseases for public health initiatives, modeling the dynamics of natural resources like forests or water systems, and analyzing the complex interactions within engineered systems. For example, research may focus on developing models for agricultural processes or understanding the spread of information and its implications for cybersecurity, including the security of Internet of Things devices.

This research area involves significant interdisciplinary connections, drawing upon mathematical analysis and numerical analysis. Engagement spans multiple institutions within Arkansas, fostering a collaborative environment for addressing complex scientific and engineering challenges.