# Mathematics Colloquium Series

## Mathematics Department Colloquium Series

The Mathematics Department established the Mathematics Department Colloquium Series through a Faculty Development Grant in 1999 to provide a place to discuss new research in Mathematics. Speakers for the series come from both inside the UTC Department of Mathematics as well as guest speakers from other institutions. The Colloquium Series meets periodically throughout the year.

### Upcoming Colloquium Talks

**Speaker:** Tanner Smith, Ph.D. Student of Mathematics

The University of Tennessee at Chattanooga

**Date: **Tuesday, April 26, 2022

**Time: **9:30AM-10:30AM

**Location:** Lupton 393

**Zoom link:** https://tennessee.zoom.us/j/97275810848

**Title:** Optimization of the Critical Mass Described by a Sturm-Liouville Problem with Spectral Parameter in the Boundary Condition

*Abstract.* We find an optimal design of a structure that is described by a Sturm-Liouville problem with a spectral parameter in the boundary condition. While previous work on the subject focused on a somewhat simplified model with applications in classical mechanics, we focus on finding solutions to a general Sturm-Liouville problem. By virtue of the generality in which the problem is considered diverse applications, to both classical and quantum mechanics, are possible. We use methods of Calculus of Variations. That allows us to reduce the problem of the extremum of an appropriate functional to the explicitly solvable differential equations subject to the boundary conditions. We introduce the notion of pre-design and design. We define pre-design to be a solution to the variational problem without the requirement of positivity, which is physically natural. We de ne design to be a pre-design that is positive on the domain. Using this terminology, we may say that the previous research did not distinguish between these two objects, not to mention a narrow class of the differential equations considered. We present a classification of the set of all parameters of the problem that guarantee the existence of pre-design, as well as design. Furthermore, we present the analytic continuation of these solutions alongside the minimum solvability conditions which guarantee a pre-design to be a design. Examples show that the set of the parameters of the problem has quite a complex structure. Specifically, we find examples of the small variation of the parameters that result in the change of the existence of design to non-existence. Finally, we present an algorithm for testing these conditions, determining the optimal solution, and calculating the corresponding optimal mass and conduct the preliminary numerical experiments.

**Speaker:** J. Rafael Rodríguez Galván

Professor of Mathematics

University of Cadiz

Cadiz, Spain

**Date: **Thursday, April 28, 2022

**Time: **3:30pm

**Location:** Lupton 392

**Title:** Galerkin Schemes for Migration Processes in Biology

*Abstract.* Spatial migration of organisms or substances plays an important role in many biological models. And, for macroscopic PDE models, development of efficient numerical schemes preserving the properties of the continuous equations (bounds, energy law...) is an interesting challenge.

In this talk we show some biological models we are working on: migration of neuron precursor cells, chemotaxis, convective phase fields and tumors. Galerkin FCT numerical schemes are introduced and applied to some of them. Also we show a new discontinuous Galerkin scheme that we have developed very recently, review its properties and compare them to FCT.

**Speaker:** Francisco Ortegón Gallego

Professor of Mathematics and Coordinator of the Ph.D. Program in Mathematics

University of Cadiz

Cadiz, Spain

**Date: **Friday, April 29, 2022

**Time: **3:30pm

**Location:** Lupton 392

**Title:** Capacity solution to the thermistor problem in Sobolev and Orlicz-Sobolev spaces: analysis and numerical simulation

*Abstract.* In this talk we study the existence of a capacity solution to a nonlinear elliptic coupled system in anisotropic Sobolev and Orlicz-Sobolev spaces. The unknowns are the temperature inside a semiconductor material and the electric potential. This system is a generalization of the steady state thermistor problem. The numerical solution is also analyzed by means of a fixed point technique or by using the least squares method combined with a conjugate gradient technique.