First, I would like to note that this work does not fully reflect what the student is capable of. It shows only what she managed to do in the given time, forced to interrupt her work in the middle of the process. By my opinion, seven months was not sufficient for this project, regarding the circumstances and the state of the student’s knowledge at the beginning. She was consulting regularly, learning new things, working with enthusiasm; however, it was not within our capacity to accelerate her progress.

At the beginning of the thesis, the author presents the basic theory of solving second order linear difference equations with constant coefficients. This is later used to determine the eigenvalues (and the inverse) of tridiagonal matrix. The student discovered that the version presented in the referenced literature is incorrect, therefore, the correct formula is derived in detail in her work.

In the studied system, representing the discrete heat equation, a symmetric tridiagonal matrix appears, and it is important to determine whether all its eigenvalues have absolute values less than one, so the solution is stable. The referenced literature incorrectly states that this is always true. The student in her work derives the condition, when this really holds. Theoretical conclusions were then tested through simple experiments in MATLAB.

There was not much time left to work on the main objective, that is the study of the two-dimensional case with different grids. These parts are not completed and I have several comments regarding them.

I appreciate student’s decision to create all figures independently. These she learned to generate directly by programming in LaTeX code, and it also took her a lot of time (some figures are still only provisional).

In general, the student could improve in making her own conclusions and in working more independently.

The thesis represents the heat equation as a system of linear difference equations on one-dimensional, rectangular, triangular and hexagonal grids, and uses the spectrum of the system matrix to discuss stability and the limit temperature distribution. Several parts deserves to be pointed out as interesting in particular: the Toeplitz eigenvalue derivation (Sec. 2.1, 3.2); the correctly and proved stability bound in Theorem 3.7 and the closed-form steady state in Lemma 3.5; and that the theory is validated through the MATLAB experiments reproduce the analytical limits to several decimal places.

The work is held back by uneven depth and several inaccuracies. The derivation of continuous PDE from the discrete idea (Eqs. (3.2)-(3.6)) takes a redundant step through (3.3) and mixed $\alpha$ as a fixed parameter with $\alpha$ as a ratio depending on the solution. The proof of Lemma 2.2 reaches the right result, but it omits the step exposing the common (X Y) factor and the orthogonality needed to arrive at the eigenvalues. The spectral analysis is carried out only for the rectangular grids, while for the triangular and hexagonal ones divergence is merely asserted (Ex. 5.5, 5.8) without computing the eigenvalues. The comparison in Sec. 5.4 is internally contradictory as the hexagonal grid is called six-neighbour, though Fig. 5.6 and (5.3) show three.

For the thesis of this length, the conclusion is quite brief and does not connect the obtained bound (which tends to 1/2 as k tends to infinity) with the classical limit of the explicit finite-difference scheme ($\alpha\leq1/2$). The formal layer also needs more care: broken cross-references, a duplicated equation number, several typos, and a seven-item bibliography limited to textbooks with generic in-text citing.

For future work I recommend keeping the derivations logically and dimensionally clean, extending the stability analysis to every grid that is plotted, and relating the results to the standard numerical-analysis literature.

Overall the thesis meets the goals of the assignment and contains needed results together with working code, but it is limited by the issues above. I recommend the thesis for defence with the grade C (good). Topics for thesis defence:

1. The simulations show that the final, steady temperature distribution depends only on the boundary values and not on the initial temperatures and not on $\alpha$. What is the explanation of this and what role $\alpha$ actually plays in the computation?
2. In Example 3.6 you find the one-dimensional bar is stable for $\alpha$ up to about 0.59, while in Example 4.1 the rectangular plate, where each interior point has four neighbours instead of two, is stable only up to about 0.31. Can you explain intuitively why having more neighbours per point forces you to use a smaller $\alpha$? What would you expect for the triangular grid, where each point has even more neighbours?