Advertisement

New Solutions of the Functional Equations and Their Possible Application in Treatment of Complex Systems

  • R. R. Nigmatullin
  • B. N. Nougmanov
Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 24)

Abstract

In this paper we want to show some original solutions of the functional equations that can be considered as a main “bridge” connecting the fractional calculus with fractal geometry. This bridge should justify a wide application of the fractional calculus in many important applications and, from another side, increase the possibilities of the fractional geometry, when the fractional calculus is applied as a basic tool. Initially, we justify this solution and then show how it can be applied in the theory of the quasi-reproducible experiments. As an example, we chose the measurements of the corresponding voltammograms (VAGs) in electrochemistry. In the frame of new approach, one can fit the VAGs with high accuracy and prove the wide applicability of the original solution found. In the concluding section, we discuss the basic results obtained in this paper and justify their wide applicability for solution of other nontrivial problems.

Keywords

Functional equations Complex systems Fractal geometry Electrochemistry Prony decomposition Intermediate model Quasy periodic measurenments Functional dispersion Self similar Special functions 

References

  1. 1.
    Rabinerand, L.R., Gold, B.: Theory and Application of Digital Signal Processing. Prentice-Hall, Inc., Englewood Cliffs (1975)Google Scholar
  2. 2.
    Singleton Jr., Royce, A., Straits, B.C., Straits, M.M.: Approaches to Social Research. Oxford University Press, Oxford (1993)Google Scholar
  3. 3.
    Mendel, J.M.: Lessons in Estimation Theory for Signal Processing, Communications, and Control. Pearson Education, Upper Saddle River (1995)zbMATHGoogle Scholar
  4. 4.
    Hagan, M.T., Demuth, H.B., Beale, M.H.: Neural Network Design. PWS Publishers, Boston (1996)Google Scholar
  5. 5.
    Ifeachor, E.C., Jervis, B.W.: Digital Signal Processing: a Practical Approach. Pearson Education, Harlow (2002)Google Scholar
  6. 6.
    Montgomery, D.C., Jennings, C.L., Kulahci, M.: Introduction to Time Series Analysis and Forecasting. Wiley, New York (2011)zbMATHGoogle Scholar
  7. 7.
    Bendat, J.S., Piersol, A.G.: Random Data: Analysis and Measurement Procedures. Wiley, New York (2011)zbMATHGoogle Scholar
  8. 8.
    Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., Rubin, D.B.: Bayesian Data Analysis. CRC Press, London (2013)zbMATHGoogle Scholar
  9. 9.
    Box, G.E.P., Jenkins, G.M., Reinsel, G.C.: Time Series Analysis: Forecasting and Control, Wiley, New York (2013)Google Scholar
  10. 10.
    Chatfield, C.: The Analysis of Time Series: An Introduction. CRC Press, Boca Raton (2013)zbMATHGoogle Scholar
  11. 11.
    Nigmatullin, R.R.: Phys. Wave Phenom. 16, 119 (2008)Google Scholar
  12. 12.
    Nigmatullin, R.R., Khamzin, A.A., Machado, J.T.: Physica Scripta. 89(1), 015201 (2014)Google Scholar
  13. 13.
    Nigmatullin, R.R., Rakhmatullin, R.M.: Commun. Nonlinear Sci. Numer. Simul. 19, 4080 (2014)Google Scholar
  14. 14.
    Osborne, M.R., Smyth, G.K.: SIAM J. Sci. Stat. Comput. 12, 362 (1991)Google Scholar
  15. 15.
    Kahn, M., Mackisack, M.S., Osborne, M.R., Smyth, G.K.: J. Comput. Graph. Stat. 1, 329 (1992)Google Scholar
  16. 16.
    Osborne, M.R., Smyth, G.K.: SIAM J. Sci. Stat. Comput. 16, 119 (1995)Google Scholar
  17. 17.
    Nigmatullin, R.R., Zhang, W., Striccoli, D.: General theory of experiment containing reproducible data: the reduction to an ideal experiment. Commun. Nonlinear Sci. Numer. Simul. 27, 175–192 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nigmatullin, R.R., Evdokimov, Y.K.: The concept of fractal experiments: New possibilities in quantitative description of quasi-reproducible measurements. Chaos. Solitons. Fractals. 91, 319–328 (2016)CrossRefGoogle Scholar
  19. 19.
    Nigmatullin, R.R., Maione, G., Lino, P., Saponaro, F., Zhang, W.: The general theory of the quasi-reproducible experiments: How to describe the measured data of complex systems? Commun. Nonlinear Sci. Numer. Simul.  https://doi.org/10.1016/j.cnsns.2016.05.019

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • R. R. Nigmatullin
    • 1
  • B. N. Nougmanov
    • 2
  1. 1.Radioelectronic and Informative Measurements Technics DepartmentKazan National Research Technical University named by A.V. Tupolev (KNRTU-KAI)KazanRussian Federation
  2. 2.Physical-Mathematical Lyceum № 131KazanRussian Federation

Personalised recommendations