doc. Ing. Tomáš Kisela, Ph.D.
E-mail:   kisela@fme.vutbr.cz  Dept.:   Institute of Mathematics Dept. of Mathematical Analysis Position:   Research Worker Room:   A1/1835

Publications:

• KISELA, T.:
  Power functions and essentials of fractional calculus on isolated time scales,
  Advances in Difference Equations, Vol.2013, (2013), No.8, pp.1-18, ISSN 1687-1847, Springer
  journal article - other
• 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
• 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
• KISELA, T.:
  Applications of the fractional calculus: On a discretization of fractional diffusion equation in one dimension,
  Communications, Vol.12, (2010), No.1, pp.5-11, ISSN 1335-4205, EDIS - Publishing Institution of Zilina University
  journal article - other

List of publications at Portal BUT

Abstracts of most important papers:

• KISELA, T.:
  Power functions and essentials of fractional calculus on isolated time scales,
  Advances in Difference Equations, Vol.2013, (2013), No.8, pp.1-18, ISSN 1687-1847, Springer
  journal article - other
  This paper concerns with a recently suggested axiomatic definition of power functions on a general time scale and its consequences to fractional calculus. Besides a discussion of existence and uniqueness of such functions, we derive an efficient formula for the computation of power functions of rational orders on an arbitrary isolated time scale. It can be utilized in the introduction and evaluation of fractional sums and differences. We also deal with the Laplace transform of such fractional operators which, apart from solving of fractional difference equations, enables a more detailed comparison of our results with those in the relevant literature. Some illustrating examples (including special fractional initial value problems) 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.
• 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
  This paper discusses qualitative properties of the two-term linear fractional difference equation with respect to its stability and asymptotics. Some consequences to the theory of Volterra difference equations are presented as well.
• 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
  The paper provides an introductory study to the theory of linear fractional difference equations. It involves, among others, the notion of discrete Mittag-Leffler function and its apllication to this theory.
• KISELA, T.:
  Applications of the fractional calculus: On a discretization of fractional diffusion equation in one dimension,
  Communications, Vol.12, (2010), No.1, pp.5-11, ISSN 1335-4205, EDIS - Publishing Institution of Zilina University
  journal article - other
  The paper discusses the problem of the classical and fractional diffusion models. It is known that the classical one fails in heterogeneous structures with locations where particles move with a large speed for a long distance. If we replace the second derivative in the space variable in the classical diffusion equation by a fractional derivative of order less than two, we obtain the fractional diffusion equation (FDE) which is more useful in this case. In this paper we introduce a discretization of FDE based on the theory of the difference fractional calculus and we sketch a basic numerical scheme of its solving. Finally, we present some examples comparing classical and fractional diffusion models.