ON THE ANALYTICAL SOLUTION OF A VOLTERRA INTEGRAL EQUATION FOR INVESTIGATION OF FRACTAL PROCESSES

Abstract

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.