I am on my fourth year studying in a bachelor program in applied mathematics and computer science and plan to write a term paper on mathematical modeling using differential equations. This will be the basis for my bachelor thesis.
However, I am not sure where to start with. I read about some models of dynamic systems but still cannot make up my mind. What I am looking for is a model of a certain phenomenon which requires a minimum of initial knowledge in disciplines other than mathematics. It has to be something that has already been studied extensively with tangible results, yet something that contains open problems which I can think on by myself after a certain point. The term paper will most probably be an introduction of a mathematical model with a focus on the available results while the final thesis could deal with modifying the model to make it more complex and accurate. As mentioned above, the model has to deal mostly with differential equations and preferably with the use of eigenvalues in differential operators.
Any kind of advice on this matter, like book recommendations, references or personal knowledge and experience would be very much appreciated.
My recommendation would be Burgers equation. Don't know the specifics, but it arises in fluid mechanics. It is so called conservation law, and it means that for a invicid flow in fluid total energy of a flow is conserved. Fluid dynamics needs some learning, but much less than termodynamics, quantum mechanics etc
In a inviscid case it looks like this (in one dimension): $$ u_t + \left( \frac{u^2}2 \right)_x = 0$$ This is simple equation, can be solved for reasonable initial value function $f(x)$ by method of characteristics. But it has one remarkable property: some solutions "break down" after finite time. Meaning that for some $C^1$ initial value function the solution becomes not continious in finite time.Shock singularities This kind of behaviour is called a schockwave and ilustrates the need to seek for more general, "weak" definition of soution to the PDE.
Leter on You could consider more complicated form for a vicious fluid: $$ u_t + \left( \frac{u^2}2 \right)_x = \kappa u_{xx}$$ As far as I am aware, the Burgers equation is not very accurate model, it can be derived from Navier-Stokes equations by neglecting some important terms.Burgers equation So the next natural thing would be to turn to N-S. Many open problems there, as well as along the way to get there.