The study of Partial Differential Equations (PDEs) is an interdisciplinary field within mathematics and various applied sciences, such as physics, engineering, biology, finance, and beyond. PDEs serve as indispensable tools for modeling fundamental physical and biological phenomena, such as diffusion, transportation and flows, wave propagation, vibration, among others. In engineering, PDEs play a crucial role in assessing structural integrity, analyzing fluid dynamics, and understanding the deformation and response of materials subject to external force, among other applications. Additionally, PDEs find application in finance, where they are employed to capture the intricacies of market dynamics.
The formulation of models is grounded in first principles, often drawing from conservation laws and conducted by skilled practitioners. However, a significant challenge arises due to the inherent complexity of the resulting PDEs, rendering them challenging to solve analytically. Consequently, sophisticated numerical algorithms become essential for solving these equations effectively as well as examining the resulting solutions. Developing suitable numerical schemes, in turn, necessitates a clear understanding of the underlying analytical properties of the equations and potential solutions.
Here at UTK, we have a dedicated cohort actively engaged in exploring both the numerical and analytical aspects of the study of PDEs. Faculty members examine the existence, uniqueness, and regularity of solutions of PDEs, as well as integral equations that serve as models across diverse scientific fields. Faculty have ongoing research in multiscale analysis, homogenization, calculus of variations, and optimal control, to name a few. The research activities we conduct reflect our effort to advance the understanding and application of PDEs in multifaceted scientific contexts.