The rubric for grading projects is here. Some possible projects are listed below. If you have an idea for a different project based on work you are doing in another class or with a professor, please let me know! These can make great topics, but we will want to make sure they are appropriate for this assignment and that you will have the necessary background to complete them.
Here is a sample project writeup.
Here are some possible topics for the first project:
- Smart Euler’s Method
- Adapt Euler’s method to run faster and give more accurate solutions when faced with differential equations which can cause the basic Euler’s method algorithm trouble. This project involves some programming.
- Population Models With Harvesting
- How does adding harvesting to population growth models change their solutions, equilibria, etc?
- Heating and Cooling models and applications
- Apply Newton’s Law of Heating and Cooling to warming a house and baking a cake.
- Peak Oil Predictions
- Geologist M.K. Hubbert used differential equations to predict when worldwide oil production would peak. Repeat his analysis and see how shifting some assumptions in the model can give different predictions.
- Cooperative and Competitive Species
- Model the population dynamics of two species which compete or cooperate with each other.
- The Tsiolkovsky Rocket Equation
- This DE controls the flight of a rocket as it burns through its fuel.
- Insights into managing a pandemic from the SIR model
- In this project, you will adapt the SIR model to study a pandemic affecting a population divided into low- and high-risk groups, and study two strategies for managing that epidemic.
Here are some possible topics for the second project:
- Modified Damping
- Adapt the damped harmonic oscillator for some other types of damping.
- Radioactive Decay Chains
- Study how radioactive elements decay by modeling their decay chains with a system of DEs.
- The Precession of Mercury
- See how general relativity predicts that the orbits of planets like Mercury are slightly different from what Kepler and Newton predicted.
- The Robertson-Walker model from cosmology
- This model for the entirety of space-time in general relativity describes the long-term and large-scale evolution of the universe.
- A DEs model for circadian rhythms
- This model describes how our bodies experience circadian rhythms.