OTI, V. Numerické metody pro řešení počátečních úloh zlomkových diferenciálních rovnic [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2021.

Posudek vedoucího

Tomášek, Petr

The thesis deals with Caputo fractional ordinary differential equations, namely numerical methods for initial value problems. The thesis consists of four main parts. There comes a section with necessary theoretical fundamentals like special functions, fractional differential operators and their properties, Caputo fractional initial value problem etc. next to the introduction section. Then the numerical part presents a survey of numerical approaches which are used for solving initial value problems to Caputo fractional differential equations. Conclusion section finishes the thesis main text. Appendix contains original matlab codes for two modifications of the Euler method. The student worked mostly individually. He was pressed for time at the end of the semester, which concluded to incomplete numerical analysis in cases of Euler methods. The original aim to analyse more then the presented two numerical schemes was not reached. Improving of text and its corrections were scanty for a lack of time, which influenced the final quality of the thesis. The thesis contains quite a lot of mistakes and language style imperfections. The theoretical background is introduced a bit carelessly with respect to several objects specification in definitions and theorems, e.g. J in definition 2.2.2 is not specified; p. 14 and p.15.: function f is insufficiently specified in the statements about power function and Taylor expansion and in Proposition 2.3.1. The initial term t_0=0 is not specified in several places, but the considerations are made under this assumption. The notation is not unified throughout the thesis, e.g. I and J are used for fractional integrals whereas J also denotes interval in other part of the thesis. end of p. 9: The bounds should be considered just to the last term, not for the whole term in brackets. p. 18: Statement/description next to Fig 2.3 is incorrect. p. 20: Caption of Fig. 2.5 is incorrect - interval of alpha. p. 36, the first paragraph: the observation is made without appropriate figures introduced in the thesis. There are just figures with outputs for h=0.01 in the thesis. Several conclusions throughout the thesis come mainly from graphical output observations than by formula provided statements. Presented numerical methods should be introduced in more detail, e.g. method accuracy, stability etc. At p. 28 a negotiation of the “second order” term (containing h powered by 2*alpha) significantly influences the error, particularly considering alpha close to zero. Although the intended goals were achieved partially, I appreciate the student's self-reliance, his insight into considered methods and his Matlab programming abilities.

Posudek oponenta

Nechvátal, Luděk

The topic of the thesis belongs to the area of fractional differential equations with emphasis put on numerical solving these equations. Modeling via fractional ODEs has become quite popular during the last decades as fractional models often reflect the reality better than the traditional (integer-order) models, hence, the topic surely deserves an attention. The thesis in its first part recalls basic concepts related to fractional calculus and fractional ODEs while the second part is devoted tu numerical methods. In particular, the forward fractional Euler and improved forward Euler methods are discussed (although several other methods are briefly described as well). The presented two methods are rather standard, and, as such, I can imagine a more thorough theoretical background here (e.g, the issues of convergence and its speed, stability, etc. could be somehow discussed). On the other hand, the information provided in the first part could be shorter (several concepts are not utilized at all). In general, I feel that presented text could be somewhat more consistent. The practical part consists of MATLAB's implementation of the two studied Euler methods (the codes are decent from a programming point of view) and several numerical simulations on selected initial value problems (the figures contain only simulations with stepsize h=0.01, at least some other stepsize could be taken into account for comparative reasons). The formal side of the thesis could also be better. I have noticed quite a number of errors (e.g., in formula (2.15)), missing assumptions (e.g., in Thm. 2.2.1), language or typographical imperfections (e.g., in the proof of Lemma 2.2.2 or the Caputo type IVP in Sect. 2.3), the notation is not unified, etc.

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