doc. Ing. Luděk Nechvátal, Ph.D.
E-mail:   nechvatal@fme.vutbr.cz  WWW:   http://www.mat.fme.vutbr.cz/home/nechvatal Dept.:   Institute of Mathematics Dept. of Mathematical Analysis Position:   Associate Professor Room:   A1/1824

Supervised courses:

• Diploma Project I (N-AIM-A) (SD5-A)
• Diploma Project II (N-AIM-A) (SD6-A)
• Diploma Seminar I (N-AIM-A) (SDP-A)
• Diploma Seminar II (N-AIM-A) (SDQ-A)
• Introduction to the TeX (0TX)
• Mathematical Analysis I (SA1)
• Mathematical Analysis I F (TA1)
• Mathematical Analysis II (SA2)
• Mathematical Analysis II F (TA2)
• Mathematical Methods in Fluid Dynamics (SMM)
• Mathematical Methods in Fluid Dynamics (SMM-A)

Publications:

• ČERMÁK, J.; NECHVÁTAL, L.:
  The Routh–Hurwitz conditions of fractional type in stability analysis of the Lorenz dynamical system, Springer
  journal article in Web of Science
• ČERMÁK, J.; NECHVÁTAL, L.:
  O rozložení kořenů kubického polynomu,
  Kvaternion, Vol.2015, (2015), No.1-2, pp.25-39, ISSN 1805-1324, Ústav matematiky
  journal article - other
• ČERMÁK, J.; NECHVÁTAL, L.; GYŐRI, I.:
  On explicit stability conditions for a linear fractional difference system, Walter de Gruyter GmbH, Berlin/Boston
  journal article in Web of Science
• NECHVÁTAL, L.:
  On asymptotics of discrete Mittag-Leffler function, MÚ AV ČR
  journal article in Scopus
• ČERMÁK, J.; KISELA, T.; NECHVÁTAL, L.:
  Stability regions for linear fractional differential systems and their discretizations,
  APPLIED MATHEMATICS AND COMPUTATION, Vol.219, (2013), No.12, pp.7012-7022, ISSN 0096-3003
  journal article - other
• ČERMÁK, J.; KISELA, T.; NECHVÁTAL, L.:
  Stability and asymptotic properties of a linear fractional difference equation,
  Advances in Difference Equations, Vol.2012, (2012), No.1, pp.1-14, ISSN 1687-1847, Springer Nature
  journal article - other
• FRANCŮ, J.; NECHVÁTAL, L.:
  Homogenization of Monotone Problems with Uncertain Coefficients,
  Mathematical Modelling and Analysis, Vol.16, (2011), No.3, pp.432-441, ISSN 1392-6292, Taylor&Francis a VGTU
  journal article - other
• ČERMÁK, J.; KISELA, T.; NECHVÁTAL, L.:
  Discrete Mittag-Leffler functions in linear fractional difference equations,
  Abstract and Applied Analysis, Vol.2011, (2011), No.2011, pp.1-21, ISSN 1085-3375
  journal article - other
• NECHVÁTAL, L.:
  Homogenization with uncertain input parameters,
  Mathematica Bohemica, Vol.135, (2010), No.4, pp.393-402, ISSN 0862-7959
  journal article - other
• ČERMÁK, J.; NECHVÁTAL, L.:
  On (q,h) - analogue of fractional calculus,
  JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS, Vol.17, (2010), No.1, pp.1-18, ISSN 1402-9251, Atlantic Press
  journal article - other
• KUNDRÁT, P.; NECHVÁTAL, L.:
  A few notes on fractional calculus,
  5. konference o matematice a fyzice na vysokých školách technických s mezinárodní účastí, pp.195-203, ISBN 978-80-7231-274-0, (2007), Univerzita obrany
  conference paper
  akce: 5. konference o matematice a fyzice na vysokých školách technických s mezinárodní účastí, Univerzita obrany, Brno, 13.09.2007-13.09.2007
• NECHVÁTAL, L.:
  On two-scale convergence,
  Mathematics and Computers in Simulation, Vol.61, (2002), pp.489-495, ISSN 0378-4754, ELSEVIER
  journal article - other

List of publications at Portal BUT

Abstracts of most important papers:

• ČERMÁK, J.; NECHVÁTAL, L.:
  The Routh–Hurwitz conditions of fractional type in stability analysis of the Lorenz dynamical system, Springer
  journal article in Web of Science
  This paper discusses stability conditions and a chaotic behavior of the Lorenz dynamical system involving the Caputo fractional derivative of orders between 0 and 1. We study these problems with respect to a general (not specified) value of the Rayleigh number as a varying control parameter. Such a bifurcation analysis is known for the classical Lorenz system; we show that analysis of its fractional extension can yield different conclusions. In particular, we theoretically derive (and numerically illustrate) that nontrivial equilibria of the fractional Lorenz system become locally asymptotically stable for all values of the Rayleigh number large enough, which contradicts the behavior known from the classical case. As a main proof tool, we derive the optimal Routh–Hurwitz conditions of fractional type. Beside it, we perform other bifurcation investigations of the fractional Lorenz system, especially those documenting its transition from stability to chaotic behavior.
• ČERMÁK, J.; NECHVÁTAL, L.; GYŐRI, I.:
  On explicit stability conditions for a linear fractional difference system, Walter de Gruyter GmbH, Berlin/Boston
  journal article in Web of Science
  The paper describes the stability area for an autonomous difference system with the Caputo and Riemann-Liouville forward difference operator whose order is between 0 and 1. Contrary to the existing result on this topic, our stability conditions are fully explicit and involve the decay rate of the solutions. Some comparisons, consequences and illustrated examples are presented as well.
• ČERMÁK, J.; KISELA, T.; NECHVÁTAL, L.:
  Stability regions for linear fractional differential systems and their discretizations,
  APPLIED MATHEMATICS AND COMPUTATION, Vol.219, (2013), No.12, pp.7012-7022, ISSN 0096-3003
  journal article - other
  This paper concerns with basic stability properties of linear autonomous fractional differential and difference systems involving derivative operators of the Riemann-Liouville type. We derive stability regions for special discretizations of the studied fractional differential systems including a precise description of their asymptotics.
• FRANCŮ, J.; NECHVÁTAL, L.:
  Homogenization of Monotone Problems with Uncertain Coefficients,
  Mathematical Modelling and Analysis, Vol.16, (2011), No.3, pp.432-441, ISSN 1392-6292, Taylor&Francis a VGTU
  journal article - other
  The homogenization problem for a nonlinear elliptic equation modelling some physical phenomena set in a periodically heterogeneous medium is studied. Contrary to the usual approach, the coefficients in the equation are supposed to be uncertain functions from a given set of admissible data satisfying suitable monotonicity and continuity conditions. The problem with uncertainties is treated by means of the worst scenario method.
• ČERMÁK, J.; NECHVÁTAL, L.:
  On (q,h) - analogue of fractional calculus,
  JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS, Vol.17, (2010), No.1, pp.1-18, ISSN 1402-9251, Atlantic Press
  journal article - other
  The paper discusses fractional integrals and fractional derivatives appearing in the so-called (q,h)-calculus. There are introduced basic definitions and properties of these notions as well as comparisons with the known definitions and results.