ON THE ANALYTICAL SOLUTION OF A VOLTERRA INTEGRAL EQUATION FOR INVESTIGATION OF FRACTAL PROCESSES
Keywords:Volterra equation of the first kind, Hurst exponent, Laplace transform, fractal process, exact analytical solution.
Context. We consider a Volterra integral equation of the first kind which may be applied to the data filtration and forecast of fractal random processes, for example, in information-telecommunication systems and in control of complex technological processes.
Objective. The aim of the work is to obtain an exact analytical solution to a Volterra integral equation of the first kind. The kernel of the corresponding integral equation is the correlation function of a fractal random process with a power-law structure function.
Method. The Volterra integral equation of the first kind is solved with the help of the standard Laplace transform method. The inverse Laplace transform leads to the calculation of the line integral of the function of complex variable. This integral is calculated
as a sum of a residue part and integrals over the banks of cut. The corresponding integrals are obtained on the basis of the known expansions of special functions.
Results. We obtained an exact analytical solution of the Volterra integral equation the kernel of which is the correlation function of a fractal random process. The paper is based on a model where the structure function of the corresponding process is a power-law function. It is shown that the part of the solution that does not contain delta-function is convergent at any point if the Hurst exponent is larger than 0.5, i.e. if the process has fractal properties. It is shown that the obtained solution is a real-valued function. The
obtained solution is verified numerically; it is also shown that our solution gives the correct asymptotic behavior. Although the solution contains an exponentially growing function of time, at large times the integral of the obtained solution asymptotically behaves as a power-law function.
Conclusions. It is important to stress that we obtained an exact solution of the Volterra integral equation under consideration rather than an approximate one. The obtained solution may be applied to the data filtration and forecast of fractal random processes. As is known, fractal processes take place in a huge variety of different systems, so the results of this paper may have a wide field of application.
Gilmore M., Yu C. X., Rhodes T. L., W. A. Peebles
Investigation of rescaled range analysis, the Hurst exponent,
and long-time correlations in plasma turbulence, Physics of
Plasmas, 2002, Vol. 9, pp. 1312–1317. DOI:10.1063/1.1459707
Gorski A. Z., Drozdz S., Spethc J. Financial multifractality
and its subtleties: an example of DAX, Physica A, 2002,
Vol. 316, pp. 496–510. DOI: 10.1016/S0378-
Preis T., Virnau P., Paul W., Schneider J. Accelerated
fluctuation analysis by graphic cards and complex pattern
formation in financial markets, New Journal of Physics,
, Vol. 11, 093024 (21 pages). DOI:10.1088/1367-
Preis T., Paul W., Schneider J. Fluctuation patterns in highfrequency
financial asset returns, Europhysics Letters, 2008,
Vol. 82, 68005 (6 pages). DOI: 10.1209/0295-
Gusev O., Kornienko V., Gerasina O., Aleksieiev O. Fractal
analysis for forecasting chemical composition of cast iron,
In book “Energy Efficiency Improvement of Geotechnical
Systems”, Taylor & Francis Group, London, 2016, pp. 225–
Kornienko V., Gerasina A., Gusev A. Methods and
principles of control over the complex objects of mining and
metallurgical production, In book “Energy Efficiency
Improvement of Geotechnical Systems”, Taylor & Francis
Group. London, 2013, pp. 183–192. ISBN 978-1-138-
Pipiras V., Taqqu M. Long-Range Dependence and Self-
Similarity. Cambridge University Press, 2017, 668p. DOI:
Bagmanov V. Kh., Komissarov A. M., Sultanov A. Kh.
Prognozirovanie teletraffika na osnove fraktalnykh filtrov,
Vestnik Ufimskogo gosudarstvennogo aviatsionnogo
universiteta, 2007, Vol. 9, No. 6 (24), pp. 217–222.
Miller S., Childers D. Probability and Random Processes
With Applications to Signal Processing and
Communications. Second edition. Amsterdam,
Elseiver/Academic Press, 2012, 598 p. DOI:
Polyanin A. D., Manzhirov A. V. Handbook of the integral
equations. Second edition. Boca Raton, Chapman &
Hall/CRC Press. Taylor & Francis Group, 2008, 1143 p.
Gradshteyn I. S. and Ryzhik I. M. Table of Integrals, Series,
and Products. Seventh edition, Translated from the Russian,
Translation edited and with a preface by A. Jeffrey and
D. Zwillinger. Amsterdam, Elsevier/Academic Press, 2007,
Angot A. Matematika dlya elektro- i radioingenerov.
Moscow, Nauka, 1967, 780 p.
Oliver F., Lozier D., Boisvert R., Clark C. NIST Handbook
of Mathematical Functions. New York, Cambridge
University Press, 2010, 951 p.
How to Cite
Copyright (c) 2019 V. N. Gorev, A. Yu. Gusev, V. I. Korniienko
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Creative Commons Licensing Notifications in the Copyright Notices
The journal allows the authors to hold the copyright without restrictions and to retain publishing rights without restrictions.
The journal allows readers to read, download, copy, distribute, print, search, or link to the full texts of its articles.
The journal allows to reuse and remixing of its content, in accordance with a Creative Commons license СС BY -SA.
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License CC BY-SA that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.