The pouring of milk into a coffee cup, the collapse of a bridge, and the way air flows around an airplane are all completely different scenarios, yet are modeled using the same differential equations.

This is the focus of Francis Aznaran, postdoctoral researcher at the University of Notre Dame. His work in the Department of Applied and Computational Mathematics and Statistics (ACMS) revolves around the numerical analysis of such equations.

In undergraduate and high school mathematics, students often work to find the exact solutions to problems. Yet, the equations Aznaran deals with are far too complex for that, so he works to create numerical algorithms on computers that allow him to solve them approximately.

“What I do is take these mathematical statements of physical laws and study them in their own right, as pure mathematicians do, but then also move on and design schemes to come up with approximate solutions to them,” said Aznaran. He subsequently proves properties about the resulting schemes and implements them in publicly available code.

Numerical analysis provides the backbone for many other areas of science: “Entire areas of physics revolve around the solutions of their underlying differential equations,” Aznaran said. “It’s particularly important in modeling and predictive contexts.”

These equations have strong pragmatic impacts, helping to predict things like how a disease moves through a population, or how a new roadway might deform under the weight of cars.

While in the abstract they might be approximate, Aznaran said that he can usually prove that his approximate solutions obtained on a computer will actually converge to the true one.

Notre Dame has offered Aznaran a strong community of other people in his field. Many of these researchers are also trained in the same methods as he is, particularly the finite element method, which was the focus of his doctoral degree.

“I was attracted to the excellent research group there is here at Notre Dame, not only in the field of numerical analysis, but even more specifically, the numerical solutions of differential equations, which is my area of focus,” Aznaran said. He was also deeply impressed with the thoroughness of the University’s Catholic mission and identity.

Alongside his supervisor, Martina Bukač, he is now working on more multi-physics applications of such numerical solutions.

As Aznaran pursues his research, he has found an outlet in distance running, which has helped train him for the perseverance that is necessary to have when his work gets challenging.

Aznaran recently won first place in the College of Science Postdocs Lightning Talk Competition, which allowed him to showcase the research he has been conducting and its practical relevance.

“You prove things and see them play out in your computer screens and simulations, but then, you hopefully get to see that mirrored in real-life physics,” Aznaran said. “I love my work because of that deep connection between quite abstract mathematical concepts and very concrete applications.”

Originally published by Anna Salentine at science.nd.edu on October 18, 2024.