Publishing my bachelor's thesis in applied mathematics

About a year ago, I submitted my bachelor’s thesis and completed my degree in mathematics. Now, a second revision has been published by my university’s archiving service and I got my first DOI number: https://doi.org/10.26083/tuprints-00031371

The objective of the thesis was to examine and implement an existing numerical scheme to solve ordinary differential equations called “continuous Petrov-Galerkin method”. It is actually more appropriately described as a class of methods which depend on a parameter (the degree of polynomials used to approximate the solution). In contrast to, say, Runge-Kutta methods, these methods are based on a variational approach which involves “testing” the approximate solution in a suitable sense. These methods can, in theory, reach arbitrarily high convergence order, although the computational cost scales with the convergence order. I’m waving my hands around here and you can find a more technical discussion in the introduction of the thesis.

I tried to write this thesis such that someone with around 1-2 years of mathematical education would be able to read it. This is why the first chapter is quite rigorous in defining the concrete notions such as “numerical scheme” or “convergence order”. Having built this foundation, the second chapter includes the main theoretical contribution. For the most part, I tried to simplify existing proofs from the literature, such as existence, uniqueness and convergence, and make them more accessible. Sometimes this was just rewriting things more clearly, other times I thought of different arguments. I’m particularly proud of the existence proof (Section 2.5) and of the strategy to prove convergence (Section 2.6).

The thesis comes with an implementation in Matlab, which can be found here: https://git.rwth-aachen.de/tim.ktitarev/thesis/. The main function cpgk has a similar syntax to the other ODE solvers available in Matlab (for example ode45) and examples can be found in the experiments/ folder. You can find the implementation details in the fourth chapter of the thesis. The fifth chapter gives some context to the experiments.

While dissertations are required to be published on this archiving platform, it is optional for bachelor’s and master’s theses and in fact you will find only very few in the field of mathematics. I chose to publish my thesis for three reasons:

1. It gave me motivation to do a revision and incorporate the feedback my supervisor gave me.
2. I put a lot of effort and time into this thesis and I would consider it an unnecessary waste if nobody could at least take a look at it.
3. Perhaps most importantly, I was working on a related topic (a modified version of the numerical scheme of my thesis) and it was quite annoying having to repeat what I already wrote in my thesis. Now I can finally just write \begin{proof}See \cite[Prop. 2.8]{Kti2025}.\end{proof}.

If you work on similar methods, feel free to reach out. I’m currently in Italy, finishing a master’s degree and learning Italian, so replies might be slow.