Applied theory of differential equations of real orders
The group has long been dealing with qualitative and numerical analysis of ordinary differential equations of integer and non-integer orders, including equations involving the time delay factor. Key topics include linear and nonlinear differential equations, dynamic equations on time scales, stability, oscillations and asymptotics of their solutions, regular variations, dynamic systems and their control (stabilization, destabilization, chaotification, synchronization), numerical discretization.
Formal outputs
Members of the group participate in solving GAČR projects as part of long-term cooperation with Masaryk University (the currently solved project ends this year, an application for a new grant competition was submitted). They publish the results in professional journals with a large readership, or in monographs, primarily in the form of co-authorship (within the group, or as part of collaboration with foreign experts). They also participate in the development of regular academic life (organization of seminars, conferences, serving on the editorial boards of professional journals).
Foreign cooperation
The University of Florence (S. Matucci), The University of Modena (V. Taddei, L. Malaguti), Osaka Metropolitan University (H. Matsunaga, N. Yamaoka).
Cooperation with other departments/groups at ÚM
This group is open to any professional cooperation within the institute (and outside it). The nature of our research offers contact topics in the field of mathematical analysis (especially complex and functional), numerical mathematics (including programming), (linear) algebra, probability theory (non-Gaussian probability distributions when solving fractional models, asymptotically scale invariance), fuzzy logic (fuzzy management of fractional systems), or in the issue of fractals (graphical visualization of the solution of chaotic dynamic systems). This naturally opens up space for cooperation with experts in the mentioned areas, while such cooperation has already been implemented within the ÚM.
Scientific preparation of bachelor's and master's students
The writing of theses topics is to some extent coordinated within the group. Topics are listed either in direct connection to the research being conducted (usually numerical simulations of obtained theoretical results, or their other support), or from the basic areas of this research. However, the group also offers original topics from other areas of mathematics, the main purpose of which is to support students' interest in the field of mathematical engineering.
Researchers – employees:
Jan Čermák, Tomáš Kisela, Luděk Nechvátal, Pavel Řehák, Petr Tomášek
Researchers – doctoral students:
Lucie Fedorková
Ordinary differential equations – boundary value problems and asymptotic theory
We focus on the qualitative properties of solutions to ordinary differential equations. In particular, we study the existence and uniqueness of a solution to various boundary value problems both in linear and non-linear cases as well as dependence of solutions on parameters leading to bifurcation problems. We are also interested in the asymptotic properties of solutions to ordinary differential equations, for instance, stability and oscillations of solutions. The equations considered in our research are motivated by the problems appearing in mathematical models of the processes studied in mechanics and other technical disciplines, for example, Duffing type equations, equations with a relativistic differential operator, equations with the so-called pp-laplacian, etc. The main tool in our research is methods of mathematical analysis, which are adapted for the study of the particular problems.
Collaboration Opportunities:
The group is open to collaboration in all of the above-mentioned areas. Several tasks we have been working on recently are listed below.
Currently, there is a collaboration with the Viktor Kaplan Department of Fluid Engineering of the Energy Institute, BUT.
Some Areas of Research, Outcomes:
The scientific results of the team members are published in the international scientific journals and are presented at scientific conferences and seminars both in the Czech Republic and abroad.
Examples of research areas, outputs can be found here.
Members:
Alexander Lomtatidze, Zdeněk Opluštil, Jiří Šremr