V. N. Gorev, A. Yu. Gusev, V. I. Korniienko


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.


Volterra equation of the first kind; Hurst exponent; Laplace transform; fractal process; exact analytical solution.


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.

GOST Style Citations

Copyright (c) 2019 V. N. Gorev, A. Yu. Gusev, V. I. Korniienko

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Address of the journal editorial office:
Editorial office of the journal «Radio Electronics, Computer Science, Control»,
Zaporizhzhya National Technical University, 
Zhukovskiy street, 64, Zaporizhzhya, 69063, Ukraine. 
Telephone: +38-061-769-82-96 – the Editing and Publishing Department.

The reference to the journal is obligatory in the cases of complete or partial use of its materials.