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Fractional calculus

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New: On orthogonal regression (total least squares)

Not so new: two articles in the field of scientometrics.

Selected papers on:

Fractional calculus

  1. Skovranek T., Podlubny I., Petras I.: Modeling of the national economies in state-space: a fractional calculus approach. Economic Modelling, Volume 29, Issue 4, July 2012, Pages 1322–1327, doi:10.1016/j.econmod.2012.03.019
  2. Sierociuk D., Podlubny I., Petras I.: Experimental evidence of variable-order behavior of ladders and nested ladders. IEEE Transactions on Control Systems Technology, 15 February 2012 (Early Access), doi:10.1109/TCST.2012.2185932. — [Pre-print: http://arxiv.org/abs/1107.2575]
  3. Petras I., Sierociuk D., Podlubny, I.: Identification of parameters of a half-order system. IEEE Transactions on Signal Processing, Volume 60, Issue 10, 2012, Pages: 5561-5566, doi:10.1109/TSP.2012.2205920.
  4. Magin R., Ortigueira M., Podlubny I., Trujillo J.: On the fractional signals and systems. Signal Processing, Volume 91, Issue 3, March 2011, Pages 350-371, doi:10.1016/j.sigpro.2010.08.003
  5. Li Y., Chen YQ, Podlubny I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Computers & Mathematics with Applications, Volume 59, Issue 5, March 2010, Pages 1810-1821, doi:10.1016/j.camwa.2009.08.019
  6. Li Y., Chen YQ, Podlubny I.: Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica, Volume 45, Issue 8, August 2009, Pages 1965-1969, doi:10.1016/j.automatica.2009.04.003.
  7. Podlubny I., Chechkin A.V., Skovranek T., Chen YQ, Vinagre B.: Matrix approach to discrete fractional calculus II: partial fractional differential equations. Journal of Computational Physics, vol. 228, no. 8, 1 May 2009, pp. 3137-3153, doi:10.1016/j.jcp.2009.01.014 — [Pre-print: http://arxiv.org/abs/0811.1355]
  8. Chen YQ, Ahn HS, Podlubny I.: Robust stability check of fractional order linear time invariant systems with interval uncertainties. Signal Processing, vol. 86, no. 10, pp.2611-2618, October 2006.
  9. Heymans, N., and Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheologica Acta, vol. 45, no. 5, June 2006, pp. 765–772 — [Pre-print: math-ph/0512028]
  10. Chen, YQ.; Vinagre, B.M.; and Podlubny, I.: Continued Fraction Expansion Approaches to Discretizing Fractional Order Derivatives—an Expository Review. Nonlinear Dynamics, vol. 38 (December 2004), no. 1–2, pp. 155–170.
  11. Chen, YQ.; Vinagre, B.M.; and Podlubny, I.: Fractional Order Disturbance Observer for Robust Vibration Suppression. Nonlinear Dynamics, vol. 38 (December 2004), no. 1–2, pp. 355–367.
  12. Podlubny, I.: Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation. Fractional Calculus and Applied Analysis, vol. 5, no. 4, 2002, pp. 367–386. [Original article: PDF (220 kB). — Preprint: math.CA/0110241 – Postscript (93 kB) – PDF (222 kB)] [Review in Zentralblatt für Mathematik / Mathematics Abstracts]
  13. Vinagre, B.M., Petras, I., Podlubny, I., Chen, YQ.: Using Fractional Order Adjustment Rules Fractional Order Reference Models in Model-Reference Adaptive Control. Nonlinear Dynamics, vol. 29, no. 1–4, 2002, pp. 269–279.
  14. Podlubny, I., Petras, I., Vinagre, B.M., O'Leary P., Dorcak L.: Analogue Realizations of Fractional-order Controllers. Nonlinear Dynamics, vol. 29, no. 1–4, 2002, pp. 281–296.
  15. Podlubny, I.: Fractional-Order Systems and PIλDμ -Controllers, IEEE Transactions on Automatic Control, vol. 44, no. 1, January 1999, pp. 208–213. [Abstract] – [PDF]
  16. Podlubny, I.: Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, pp. 359–386. [Abstract – PDF (560 kB)]
  17. Podlubny, I., Kostial, I., Kacenak, M., Terpak, J.: Modeling of the Distribution of the Liquid Iron Level in the Blast Furnace Hearth. METALURGIJA, vol. 40, no. 2, 2001, pp. 107–109 (ISSN 0543-5846).
  18. Podlubny, I.: "Application of orthogonal polynomials to solution of fractional integral equations." In the book: Transform Methods and Special Functions, Varna'96. (Eds.: P.Rusev, I.Dimovski, V.Kiryakova), IMI–BAS, Sofia, 1998, pp. 350–359. [Review in Zentralblatt für Mathematik / Mathematics Abstracts]
  19. Podlubny, I.: Fractional-order models: a new stage in modelling and control. Preprints of the IFAC Conference on System Structure and Control, July 8–10, 1998, Nantes, France, vol.2, pp. 231–235.
  20. Podlubny, I.; Dorcak, L., Kostial, I.: "On fractional derivatives, fractional-order dynamic systems and PIλDμ controllers". Proceeedings of the 36th IEEE Conference on Decision and Control, December 10–12, 1997, San Diego, pp. 4985–4990. ISBN 0780341902. [PDF]
  21. Podlubny, I.: "Riesz potential and Riemann-Liouville fractional integrals and derivatives of Jacobi polynomials". Applied Mathematics Letters, vol.10, no.1, 1997, pp. 103–108. [Review in Zentralblatt für Mathematik / Mathematics Abstracts]
  22. Podlubny, I.: "Numerical solution of ordinary fractional differential equations by the fractional difference method." In the book: Advances in Difference Equations, (eds.: S.Elaydi, I.Gyori, G.Ladas), Gordon and Breach Sci. Publ., Amsterdam, 1997, pp. 507–516. [Review in Zentralblatt für Mathematik / Mathematics Abstracts]
  23. Podlubny, I., and El-Sayed, A.M.A.: On Two Definitions of Fractional Derivatives. UEF-03-96, Inst. Exp. Phys, Slovak Acad. Sci., Kosice, 21 pp., ISBN 80-7099-252-2.
  24. Podlubny, I.: "Solution of linear fractional differential equations with constant coefficients". In the book: P. Rusev, I. Dimovski and V. Kiryakova (eds.) Transform Methods and Special Functions, SCT Publ., Singapore, 1995, pp. 217–228. [Review in Zentralblatt für Mathematik / Mathematics Abstracts]
  25. Podlubny, I.: Fractional Differential Equations: Methods of Solution and Applications to Modelling and Control , TU Kosice, 1995, 94 pp.
  26. Podlubny, I., Dorcak, L., and Misanek, J.: "Application of fractional-order derivatives to calculation of heat load intensity changes in blast furnace walls". Trans. of the TU of Kosice, vol. 5(1995), no. 2, pp. 137–144.
  27. Podlubny, I.: Fractional-Order Systems and Fractional-Order Controllers, UEF-03-94, Inst. Exp. Phys, Slovak Acad. Sci., Kosice, 1994, 21 pp. [PDF]
  28. Podlubny, I.: The Laplace Transform Method for Linear Differential Equations of the Fractional Order, UEF-02-94, Inst. Exp. Phys, Slovak Acad. Sci., Kosice, 1994, 32 pp.
    (There is an e-copy at Arxiv.org, paper funct-an/9710005, in which only the misprints found in the 1994 printed version are corrected.)
  29. Podlubny, I.: "Analytical solution of linear differential equations of the fractional order". Proceedings of the 14th World Congress on Computation and Applied Mathematics, July 11–15, 1994, Atlanta, Georgia, USA. Late Papers volume (ed.: W.F.Ames) pp. 102–106
  30. Podlubny, I.: "Numerical solution of initial-value problems for ordinary fractional-order differential equations." Proceedings of the 14th World Congress on Computation and Applied Mathematics, July 11–15, 1994, Atlanta, Georgia, USA. Late Papers volume (ed.: W.F.Ames) pp. 107–111.

Cosinor rhythmometry in physiology

  1. Svorc, P., Podlubny, I.: Modulatory influences of ventilatory disorders on electrical stability of the rat heart. Biomedicine & Pharmacotherapy, vol. 56, 2002, Supplement II, pp. 327s–332s.
  2. Svorc, P., Bracokova, I., Podlubny, I., Kassayova, K., Dorko, E.: Circadian rhythm of the ventricular fibrillation threshold in the hypoventilation-reoxygenation model in female Wistar rats. Bratislava Medical Journal, vol. 101, no. 7, 2000, pp. 371–382. [Get full text article in PDF]
  3. Svorc, P., Bracokova, I., Podlubny, I.: Relation of ventricular fibrillation threshold to heart rate during normal ventilation and hypoventilation in female Wistar rats: chronophysiological study. Physiological Research, vol. 49, no. 6, 2000, pp. 711–719.
  4. Svorc, P., Podlubny, I., Kujanik, S., Bracokova, I.: "24h rhythm of the ventricular fibrillation threshold during normal and hypoventilation in female Wistar rats." Chronobiology International, vol. 14, no. 4, 1997, pp. 363–370.
  5. Svorc, P., Wilk, P., Murar, J., Podlubny, I., Kujanik S., Bracokova, I., Murin, M.: "Circadian rhythm of the ventricular fibrillation threshold in female Wistar rats". Physiological Research, vol. 43, 1994, pp.355–358.

Parallel algorithms for ODEs

  1. Podlubny, I.: "Parallel algorithms for initial and boundary value problems for linear ordinary differential equations and their systems." Kybernetika, vol.32, no.3, 1996, pp. 251–260. [Review in Zentralblatt für Mathematik / Mathematics Abstracts]
  2. Pavlus, M., Podlubny, I.: "Parallel method for initial value problems for linear ordinary differential equations: Speed up estimation". Kybernetika, vol.33, no.3, 1997, pp. 279–290. [Review in Zentralblatt für Mathematik / Mathematics Abstracts]

Integral transforms

  1. Podlubny I: Orthogonal with non-integrable weight function Jacobi polynomials and their application to singular integral equations in elasticity and heat conduction problems. In the book: W.F.Ames and P. Van der Houwen (eds.), Computational and applied mathematics, II, Elsevier, 1992, pp. 207–215. [Review in Zentralblatt für Mathematik / Mathematics Abstracts]
  2. Podlubny, I., Srivastava H.M.: "Some properties of finite Fourier transformations and their application to summation of a class of trigonometric series". Integral Transforms and Special Functions, vol. 8, 1999, pp. 261–274. .
  3. Podlubny, I.: The Laplace Transform Method for Linear Differential Equations of the Fractional Order, UEF-02-94, Inst. Exp. Phys, Slovak Acad. Sci., Kosice, 32 pp.
    (this is the same paper as listed in the Fractional Calculus section of this list of selected publication)

TeX, LaTeX, word processing

  1. Podlubny, I.: 4TeX–LaTeX–Slovak Language. Bulletin CSTUG, vol. 7, no. 1–2, 1997, pp. 20-26, ISSN 1211-6661 (in Slovak). [Get the entire issue in PDF]
  2. Podlubny, I., Kassayova, K.: New standard for text editors for Windows? – II. Bulletin CSTUG, vol. 9, no. 3, 1999, pp. 132–136, ISSN 1211-6661 (in Slovak). [Get the entire issue in PDF]
  3. Podlubny, I.: LaTeX2eps – a macro library for the NoteTab tex editor. Bulletin CSTUG, vol. 9, no. 3, 1999, pp. 137–145, ISSN 1211-6661 (in Slovak). [Get the entire issue in PDF]

Scientometrics

  1. Podlubny, I., Kassayova, K.: Towards a better list of citation superstars: compiling a multidisciplinary list of highly cited researchers. Research Evaluation, vol. 15, no. 3, December 2006, pp. 154–162. Pre-print is available at: math.ST/0603024)
  2. Podlubny, I.: Comparison of Scientific Impact Expressed by the Number of Citations in Different Fields of Science. Scientometrics, vol. 64, no. 1, July 2005, pp. 95–99. Pre-print is available at: math.ST/0410574)

Orthogonal regression (total least squares)

  1. Petras I., Podlubny I.: Least squares or least circles? CHANCE, vol. 23, no. 2, pp. 38–42, doi: 10.1007/s00144-010-0021-2.
  2. Petras, I., Podlubny, I.: State space description of national economies: the V4 countries. Computational Statistics & Data Analysis, vol 52, no. 2, 2007, pp. 1223-1233, doi:10.1016/j.csda.2007.05.014. Pre-print is available at: math.ST/0609789