# Special DE Seminar: Evan Sheldon and Isaac Benson

### Undergraduate Students

Location

Mathematics/Psychology : 401

Date & Time

May 8, 2024, 12:00 pm – 1:00 pm

Description

**Title**: Periodic Solutions and Resonance in Heat-Wave Systems

**Speaker**: Evan Sheldon

**Abstract**: Oscillatory behaviors are ubiquitous in physical systems. A key property of these systems is the potential emergence of periodic behaviors. Many natural oscillatory systems couple different dynamics, and an important class can be generalized to a hyperbolic-parabolic (i.e. wave-heat) system of partial differential equations (PDEs), coupled across an interface. In this sense, a heat-wave system can be viewed as an idealized fluid-structure interaction (FSI) model. Some examples of FSI models include aeroelastic systems, organ systems and arterial dynamics in the body, and geophysical poro-elastic flow. However, mathematical challenges for this system arise due to the difference in behavior of the hyperbolic and parabolic components, as well as their complex interaction along the coupling interface. The phenomenon of resonance is an important aspect of coupled wave-heat PDE dynamics. Although we understand resonance and periodic solutions in each component (hyperbolic or parabolic) individually, whether or not resonant behavior in coupled wave-heat systems is possible in two spatial dimensions remains an open problem. In one spatial dimension, we simulated the coupled system using a finite difference scheme and additionally determined the spectral properties of the system. In two spatial dimensions, we similarly modeled the dynamics using a finite difference scheme, verified by the method of forcing. With this code, we can now numerically investigate whether resonance can occur in this coupled system.

**Title**: Resonance and Periodic Solutions for Simple Harmonic Oscillator with Periodic, Bounded, Piecewise Continuous Forcing

**Speaker**: Isaac Benson

**Abstract**: Harmonic oscillator systems have various applications in engineering, physics, and biology. Examples include air flowing through a bridge, mechanical vibration of a wine glass, and blood flowing through an artery. Resonance, which is the phenomenon when the forcing function of the system matches the natural frequency of the harmonic oscillator system has to be known for an understanding of how the system changes. While it is widely known that sinusoidal forcing with the natural frequency of the simple harmonic oscillator causes resonance, it is unknown what kinds of non-sinusoidal forcing functions cause resonance or non-resonance in the simple harmonic oscillator. This paper states the necessary and sufficient conditions for a periodic, bounded, and piecewise continuous forcing function on the simple harmonic oscillator to cause resonance, periodic solutions, or non-periodic solutions. A better understanding of the simple harmonic oscillator leads to a better understanding of the resonance in any harmonic oscillator system. This has implications for the ordinary differential equations and partial differential equations which determine harmonic oscillator systems.

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