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

Gorev V. N. – PhD, Assistant Lecturer of the Department of Information Security and Telecommunications, National Technical University Dnipro Polytechnic, Dnipro, Ukraine. Gusev A. Yu. – PhD, Associate Professor, Associate Professor of the Department of Information Security and Telecommunications, National Technical University Dnipro Polytechnic, Dnipro, Ukraine. Korniienko V. I. – Doctor of Science, Associate Professor, Head of the Department of Information Security and Telecommunications, National Technical University Dnipro Polytechnic, Dnipro, Ukraine. 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.


NOMENCLATURE
( ) c t is a structure function of the fractal random process; ( , ) h t k is a unknown function for which the solution of the integral equation is obtained, 0 t ≥ ; H is a Hurst exponent; i is a complex unity; ( ) R t is a correlation function of the fractal random process; ( ) where k T < is a finite positive constant, and ( , ) h k τ is the unknown function.The problem is to obtain an analytical solution to eq. ( 1) .

REVIEW OF THE LITERATURE
The models with a power-law structure function are widely used to describe fractal processes (see, for example, [1][2][3][4]).Fractal processes are widely used in investigations of different systems (see [5][6][7] and references therein).
In paper [8] a continuous random process ( ) . The process is assumed to be a stationary and ergodic one.The structure function ( ) c t is assumed to be a power-law function: ( ) ( ) ( ) where α is a positive constant, and H is the Hurst exponent.
In the model (3) the corresponding correlation function is The Voterra integral equation of the first kind (1) is considered in [8] in the framework of the Kolmogorov-Wiener filter.Of course, it should be stressed that, in order to obtain the Kolmogorov-Wiener filter output, the Fredholm integral equation should be solved rather than the Volterra one.Nevertheless, the Volterra integral equation discussed in [8] is of mathematical interest by itself.The Volterra integral equation is also a special case of the Fredholm integral equation, so it may be applied to data filtration and forecast in some cases (maybe even not necessarily in the framework of the Kolmogorov-Wiener problem).
The problem of the solution of the integral equation under consideration is investigated in [8] with the help of the standard Laplace transform method [10].The authors of [8] carefully divided the solution into two parts, one of which contains the Dirac delta-function.However, the results of paper [8] should be significantly refined.First of all, eq. ( 19) in [8] contains a complex function as a result because the incomplete Gamma-function (2 1, ) H x Γ + −λ is complex-valued (see eq. ( 19) in [8]).
Besides, a pole residue is not taken into account in [8] either.In this paper the integral equation ( 1) is analytically solved and the results of paper [8] are refined.

MATERIALS AND METHODS
Let us introduce the following Laplace transforms: Substituting T ξ = − τ into the right-hand side of (1), multiplying the both sides of (1) by exp( ( )) p t k − + and taking the integral over T , with account for (5) we obtain ( ) ( ) ( ) Multiplying the integrand on the left-hand side of (6) by exp( ) exp( ) p p − ξ ξ and substituting x T = −ξ, y = ξ into (6), with account for (5) we obtain which with account for (5) leads to The standard definitions and tabulated integrals for The Gamma and incomplete Gamma functions are [11]: , ( ) On the basis of ( 9) and (2) the integrals in the numerator and the denominator of (8) are calculated: With account for (10) and ( 8) the following expression for ( , ) H p k can be obtained: Let us investigate the behavior of ( , ) H p k when p → ∞ .As is known [11], if x → ∞ , then ( , ) x Γ α can be represented as a series: On the basis of ( 12) and ( 11) we obtain ( ) ( ) According to (13), let us split (11) into two parts: As is known [10], the inverse Laplace transform can be calculated as (here we use the fact that the inverse Laplace transform of a constant is the delta-function).The function ( ) and in what follows we calculate it.It should be stressed that up to this point all the results coincide with [8], but the following result for ( , ) h t k ′ significantly differs from [8].The singular points of the function ( , ) H p k ′ are the branch point 0 p = and the poles.The function ( , ) H p k ′ satisfies the conditions of the Jordan lemma (see ( 13)), so the integral ( 16) is where ( , ) I t k is the sum of the integrals over the banks of cut and ( , ) J t k is the pole residue part (see, for example, [12]).
The following banks of cut should be chosen: , , e x p 2 , , e x p 2 As can be seen from ( 14) and ( 11), the function ( , ) cos 2 sin 2 As is known [11], the function ( 21, ) H pk Γ + can be expanded into a series: With account for (20) and ( 19) one can obtain It should be noticed that we consider processes with fractal properties, i.e. we consider cases where (0.5;1) H ∈ . In this range of parameters we have On the basis of ( 18)-( 21) the following result for ( , ) I t k is obtained: , 2 cos 2 , , 2 sin 2 , We should stress that in contrast to [8] our result ( 22) is a real-valued function.Let us investigate the convergence of the integral in (22).On the basis of ( 22), (20) and the property ( 1) ( ) ( , ) from which it follows that (0, )  (11) to zero because of (14).The solutions of the corresponding equations are ( ) According to [12], only the poles with arg( ) [ , ] z ∈ −π π contribute to ( , ) J t k in (17).We consider the case where (0.5;1) H ∈ , so from (24) we can see that the only pole that contributes to ( , ) J t k is ( ) On the basis of ( 11), ( 14), ( 25) and ( 26) one can obtain that in the vicinity of the point 0 p p = (i.e. in the vicinity of the point As can be seen from ( 27), the expansion of ( , ) pt H p k e ′ into a Laurent series of 0 p p − begins with the minus first term, so 0 p is a simple pole and ( ) ( ) it should be noticed that ( , ) J t k is not taken into account in [8].
So, the following solution is obtained: where the explicit expressions for ( , ) I t k and ( , ) J t k are given in ( 22) and (28); the expression for 0 p from (28) is given in (25).

EXPERIMENTS
Numerical calculations for some parameters are made in order to verify the solution (29).The integral is compared to ( ) R t k + , i.e. the right-hand side of ( 1) is compared to the left-hand side of (1).The calculations are made on the basis of Mathcad 14 package.The incomplete Gamma function, which is a built-in Mathcad function, is not defined for a negative second argument.So the functions ( , ) A x k and ( , ) B x k in (22) are introduced as ( )  ( , )  I t k as the integral from 0 to ∞ , so ( , ) i.e. if 233 x > , then ( , ) f x k is replaced with its asymptotics for x → ∞ ; see (32) and (23).Mathcad is able to calculate the first integral on the right-hand side of (32) in the range of parameters which is given in table 1.
The following results were obtained.
Table 1 -Verification of the obtained solution As can be seen from the table 1, ( ) R T k + is in good agreement with the integral (30), so the solution (29) is true.In our opinion, the slight difference of the second and the third columns in table 1 is due to machine errors.
Of course, Mathcad could not adequately calculate the integral (30) at large values of T , i.e. at 3 10 T = , 4  10 , etc.In order to verify the solution (29) for large values of T , we seek the asymptotics of the integrals in (30) if T → ∞ .
Let us denote As is known [11], ( ) On the basis of ( 35) and (34) we have As is known [13], the function , , F z α β has the following asymptotics: ( ) , , On the basis on (37) it can be seen that ( ) 1, , On the basis of (38), ( 36) and (25) we obtain ( ) It should be noticed that for T → ∞ the integral in (39) behaves as a power-law rather than an exponent function!The integral of ( , ) I t k is as follows: After substituting T t ξ = − into 2 I we obtain ( ) With the help of the tabulated integral ( ) and eqs.( 41) and (20) we obtain ( It should be noticed that so with account for (23) the integral in (43) is convergent; and the asymptotics of (43) for T → ∞ is not larger than where a is a constant.We assume that 2 2 , this assumption is confirmed numerically in what follows.
As for 1 I , we have ( ) so obviously 1 I is bounded by a constant and ( ) ( ) So, the asymptotic behavior of the left-hand and righthand sides of (1) on the basis of (29) for T → ∞ are ( ) see also (39), ( 43) and ( 46).So if then our solution (29) is true for T → ∞ .
The validity of ( 48) is checked numerically with the help of the Wolfram Mathematica 11 package, which is able to calculate the integral on the left-hand side of (48).The following results are obtained.As can be seen from table 2, eq.( 48) is valid, which justifies our solution.

RESULTS
An exact analytical solution to the Volterra integral equation ( 1) is obtained, see (29).The kernel of the corresponding Volterra integral equation is the correlation function of the continuous fractal process with the powerlaw structure function (3).Only the cases where the Hurst exponent (0.5;1) H ∈ are considered.The obtained solution (29) is verified numerically.The asymptotic behavior of both sides of (1) for T → ∞ is investigated.It is shown that our solution gives the correct asymptotic behavior.

DISCUSSION
The corresponding Volterra integral equation was discussed in [8] in the framework of the Kolmogorov-Wiener filter for the rather popular model with the structure function (3).It seems that the use of the Volterra integral equation in the framework of the Kolmogorov-Wiener filter is in some sense inadequate because the Fredholm integral equation of the first kind should be solved in order to obtain the Kolmogorov-Wiener filter output.
Nevertheless, the Volterra integral equation is of mathematical interest.It should also be noted that the Volterra integral equation is a special case of the Fredholm integral equation, which is rather popular in investigations of fractal processes, so the Volterra integral equation may be applied to some investigations of fractal processes.
An exact analytical solution to eq. ( 1) is obtained.It is shown that the term ( , )  I t k in (29) which comes from the integrals over the banks of cut is convergent for any 0 t ≥ if the Hurst exponent (0.5;1) H ∈ . In contrast to [8], ( , ) I t k is a real-valued function.Also in contrast to [8], the residue part of ( 29) is taken into account.The obtained solution (29) is verified numerically on the basis of Mathcad 14 package.
The asymptotic behavior of both sides of (1) for T → ∞ is also investigated on the basis of (29).It is shown that our solution gives the correct asymptotic behavior; the corresponding integral in (48) was taken numerically with the help of Wolfram Mathematica 11 package.

CONCLUSIONS
The Volterra integral equation (1) of the first kind the kernel of which is the correlation function ( 4) is solved.
The scientific novelty of the obtained results is that an exact analytic solution to the corresponding integral equation is obtained.The solution is verified numerically, it is also shown that our solution the gives correct asymptotic behavior.The results of the previous papers devoted to the integral equation under consideration are refined.
The practical significance is that the obtained results may be applied to investigations of fractal random processes.