KHAN, A. Haar wavelet kolokační metoda vyššího řádu pro systémy integro-diferenciálních rovnic [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2025.

Posudek vedoucího

Tomášek, Petr

The thesis deals with numerical solution of initial value problems to differential and integro-differential equations which is based on the Haar wavelets. The Haar wavelet approach offers interesting alternative to classical methods in numerical computation of studied class of problems. Section 2 introduces fundamentals about the considered solved problems, Sections 3 and 4 introduce Haar wavelets and their collocation method. Section 5 mentions two results about methods convergence order. Finally, Section 6 presents computational part of the thesis, where Haar wavelet methods are applied to various test initial value problems. This part also contains comments to the enclosed Matlab scripts developed within the thesis. The student suggested the studied topic which follows from his previous research. Student worked independently and the practical part of the aims was finished quite soon. Therefore, it is a bit of a shame that the thesis contains some stylistic, grammar and logical mistakes, e.g. The section counter of references to equations in Section 2 is shifted to 3, which is confusing. The examples in section 2 are not well introduced - just list of equations. The comments are in nearby text. The functions should be also more precisely introduced with respect to their variables in the examples. Equation (2.21) is not of the form (2.20). The maximum error norm is not clearly introduced on page 18 (max in which sense?). The reference "[26] S. Lakhotia. Theory of Integro-Differential Equations. Narosa Publishing House, 2019." does not exist. A question arisis if it is a product of AI. The conclusion regarding (at least) the test problem 6.2 is incorrect since the introduced error is lower for HWCM instead of H-HWCM - see Table 6.2. Also the results for the second variable $v$ of the solved system should be introduced. The Matlab codes should be more commented within m-files and appropriate "help" header of files should be introduced. A recommended list of attached files with a short description is also missing, which would significantly improve the reader's orientation in the attached files and their use. On the other hand, the thesis introduces and illustrates usage of the Haar wavelet approach to various classes of tasks and it gives the reader a template for numerical computation of another initial value problems using these methods. The aims of the thesis were met and I suggest the thesis for defense.

Posudek oponenta

Rozehnalová, Petra

The thesis consists of three main parts. The first part summarizes basic concepts. The second part introduces the HWCM and HHWCM methods. The third part contains numerical experiments. The thesis cites current literature. However, the thesis's structure has some shortcomings. For example, the initial and boundary problems for integro-differential equations are defined before these equations are introduced. There are many errors in the theoretical section. For example, there is an inappropriate definition of a linear differential equation (p. 14), an equation that is not linear is given as an example of a homogeneous differential equation (p. 15), and there are unexplained symbols (e.g., R(u), J(u), p. 18). The next two sections clearly describe the HWCM and HHWCM methods and include numerical experiments. The attached codes are clear.

eVSKP id 165633