Partial Differential Equations

Mathematical inquiry into partial differential equations (PDEs) explores the behavior of functions of multiple variables that satisfy equations involving their partial derivatives. This field addresses fundamental questions about existence, uniqueness, and regularity of solutions to these equations, which model a wide array of natural phenomena. Research activities include developing new analytical techniques, such as those from harmonic analysis and functional analysis, to study properties of solutions. Investigations also encompass numerical methods for approximating solutions to PDEs, often involving finite element or finite difference schemes, and their implementation using computational tools. Specific areas of focus include the study of elliptic, parabolic, and hyperbolic equations, as well as nonlinear PDEs.

The mathematical understanding derived from PDE research has direct relevance to Arkansas's economy and environment. For instance, modeling fluid dynamics and turbulent flows using PDEs is crucial for optimizing agricultural irrigation systems, managing water resources in river basins like the Arkansas and Mississippi, and understanding atmospheric phenomena relevant to the state's weather patterns. Furthermore, PDEs are employed in modeling processes in materials science and engineering, which can support the state's manufacturing and aerospace sectors. The development of computational methods for solving these equations also aligns with the growing demand for data science and computational expertise across various industries.

This research area benefits from strong interdisciplinary connections, particularly with operator theory, complex analysis, mathematical physics, and differential geometry. Expertise in these areas is applied to tackle complex problems in nonlinear dynamics, fluid dynamics, and computational physics. The engagement with machine learning applications further broadens the scope of inquiry, enabling novel approaches to solving and analyzing PDEs across multiple institutions within the state.