On the analytical solution of a Volterra integral equation for Investigation of fractal processes Текст научной статьи по специальности «Математика»

МАТЕМАТИЧНЕ ТА КОМП'ЮТЕРНЕ МОДЕЛЮВАННЯ

МАТЕМАТИЧЕСКОЕ И КОМПЬЮТЕРНОЕ МОДЕЛИРОВАНИЕ

MATHEMATICAL AND COMPUTER MODELING

UDC 517.968.22

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.

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

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, t > 0 ; H is a Hurst exponent; i is a complex unity;

R(t) is a correlation function of the fractal random process;

x(t) is a fractal random process under consideration; r(a, z) is a incomplete Gamma function; r( z) is a Gamma function; ct2 is a process variance;

(a(t)) is a time average of a random process a(t); H(p, k) is a Laplace transform of the function h(t, k);

r (p, k) is a Laplace transform of the function

R(t + k);

r(p) is a Laplace transform of the function R(t);

1F1 (a, p, z) is a confluent hypergeometric function;

B(|, v) is a Beta-function.

INTRODUCTION

This paper is devoted to the obtaining of an analytical solution to a Volterra equation of the first kind which may be applied to the data filtration and forecast of fractal random processes. The kernel of the corresponding integral equation is the correlation function of a fractal random process with a power-law structure function.

The model of the power-law structure function is a very popular model for the description of fractal processes. For example, it is used in the description of plasma fluctuations [1], in the description of the financial market data on the basis of the statistical physics methods [2-4], etc.

Self-similar processes take place in a huge variety of different systems: industry applications, control systems (see, for example, [5, 6]), information-telecommunication systems, financial markets, physical systems (Brownian motion, non-equilibrium fluctuations, etc.), geophysical time series, etc., see [7] and references therein.

In this paper we consider only continuous random processes. The problem of the solution of the Volterra integral equation under consideration was discussed in [8] where this problem was investigated in the framework of the Kolmogorov-Wiener filter. We should stress that, to obtain the weight function and the output of the Kolmogorov-Wiener filter, a Fredholm integral equation of the first kind should be solved rather than the Voterra one (see, for example, [9]). But the Volterra integral equation is of mathematical interest by itself. As is also known, the Volterra integral equation is a special case of the Fredholm integral equation, so it may be applied to practical investigations of fractal processes.

In this paper the idea of the solution of the Volterra integral equation is similar to that of [8], but the results of paper [8] should be refined in some places, see the corresponding discussion in Sec. 2.

The object of study is the Volterra integral equation of the first kind, the kernel of which is the correlation function of a fractal random process with a power-law structure function.

The subject of study is the analytic solution of the system under consideration.

The aim of the work is to obtain an exact analytical solution to the integral equation under consideration and to investigate its asymptotic behavior.

1 PROBLEM STATEMENT

We consider the following Volterra integral equation of the first kind

R (T + k) = J d Th (t, k )R (T - t) ,

where k < T is a finite positive constant,

R (t )=-2 -f t2

(1)

(2)

and h(T, k) is the unknown function. The problem is to obtain an analytical solution to eq. (1) .

2 REVIEW OF THE LITERATURE

The models with a power-law structure function are widely used to describe fractal processes (see, for example, [1-4]). Fractal processes are widely used in investigations of different systems (see [5-7] and references therein).

In paper [8] a continuous random process x(t) is given for t e [0, T]. The process is assumed to be a stationary and ergodic one. The structure function c(t) is assumed to be a power-law function:

c(t) =^(x(t) - x(t -t) )) =a-T2 H.

(3)

where a is a positive constant, and H is the Hurst exponent.

In the model (3) the corresponding correlation function is

R(t) = (( x(t + t) - ( x(t)) t) (x(t) - ( x(t)) t))

2 f 2 H = CT--T .

2

(4)

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 r(2H +1, -Xx) is complex-valued (see eq. (19) in [8]). Besides, a pole residue is not taken into account in [8] either.

2018

In this paper the integral equation (1) is analytically solved and the results of paper [8] are refined.

3 MATERIALS AND METHODS

Let us introduce the following Laplace transforms:

r (p, к) = J dTR (T + к) e"p(T+к) , 0

да

r (p) = J dTR (T)e-pT , (5)

0

да

H (p, к ) = J dTh (T, к )e"

- pT

J dTR (T )<

- pT

The standard definitions and tabulated integrals for The Gamma and incomplete Gamma functions are [11]:

w w

r (a, x) = J dte^ta-1 , r (a) = J dte^ta-1 ,

x 0

w

Jdxx^'e-" = |a-T(v, |au) , (9)

u

w

J dxx^e^ = |a-VT(v).

On the basis of (9) and (2) the integrals in the numerator and the denominator of (8) are calculated:

/ \ T ct2 аГ(2H +1)

f dTR (T )e" pT = ----^ ,

J0 K ' p 2 p +1

J dTR (T + к)e~pT =-

-2 ae^ Г(2Н +1, pk ) 2 p2H+1 '

With account for (10) and (8) the following expression for H (p, k) can be obtained:

, , 2 p2H ct2-aepk r(2H +1, pk)

H (p, k) = —-^-t-f2 . (11)

V ; 2 p2H ct2-ar(2H +1) V 7

Let us investigate the behavior of H (p, k) when p ^w . As is known [11], if x ^w , then T(a, x) can be represented as a series:

Substituting £ = T -t 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

w T

J dTJ d£e"p(T+k5h (T - £, k) R (£) = r (p, k) . (6)

0 0

Multiplying the integrand on the left-hand side of (6) by exp(-p£) exp(p£) and substituting x = T - £, y = £ into (6), with account for (5) we obtain

e" pkH (p, k) r (p ) = r (p, k), (7)

which with account for (5) leads to

w

J dTR (T + k )e~pT

H (p,k)= —w-. (8)

Г (a, x)l

, »(-1)m r(1 -a + m)

= xa-Vx Y^-\-—L. (12)

m=i xm r(1 -a)

On the basis of (12) and (11) we obtain

h ^k )L = 1 -OCT1+0 (p_I). (13)

According to (13), let us split (11) into two parts:

H (p, к ) = 1 -OCT^+H '(p, к),

2ct

H' ( P, к ) p^ = 0.

(14)

As is known [10], the inverse Laplace transform can be calculated as

л с+/да

h (t, к)= 2- J dpH (p, к)ept

с-/да 2 H '

= |!-a2CTrl§(t) + h' (t, к)

(15)

(here we use the fact that the inverse Laplace transform of a constant is the delta-function).

The function h'(t, k) in (15) is the inverse Laplace

transform of the function H' (p, k):

л с+/да

h'(t, к) = 2- J dpH' (p, к)ept

(16)

(10)

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 p = 0 and the poles. The function H' (p, k)

да

satisfies the conditions of the Jordan lemma (see (13)), so the integral (16) is

h '(t, k ) = I (t, k ) + J (t, k ),

(17)

I (t, k) = Ij (t, k) +12 (t, k ),

in œ

Il (t, k) = — J dxH' (xein, k) exp (xeint),

-in 0

I2 (t, k) = — J dxH ' (xe"in, k) exp (xet) .

(18)

As can be seen from (14) and (11), the function H'(p, k) contains the functions p2H, exp(pk) and r(2H +1, pk). Obviously,

e±in = -1, exp (xe±in t ) = exp (-xt ), (xe±in )2H = x2H (cos (2nH) ± i sin (2nH)) .

(19)

As is known [11], the function r(2H +1, pk) can be expanded into a series:

r (a, x) = r (a) - ^

(-1)

n xa+n

n !(a + n)

With account for (20) and (19) one can obtain Re (r (2H +1, kxein)) = r (2H +1) +

+ (kx)2H cos (2nH)f —^-r ;

v ' v 7n=0 n !(2H +1 + n)

Im (r (2H +1, kxe™ )) = (kx )1+n

(20)

(21)

= (kx)2H sin(2nH)£-

(!(2H +1 + n)' Re (r (2H +1, kxe~in)) = Re (r (2H +1, kxe'n)), Im (r (2H +1, kxe~in )) = - Im (r (2H +1, kxein )) .

It should be noticed that we consider processes with fractal properties, i.e. we consider cases where H e (0.5;1). In this range of parameters we have

sin(2nH) < 0, Im(r(2H +1, kxe~i71)) > 0 . On the basis of (18)-(21) the following result for I (t, k) is obtained:

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:

p = xe'n and p = xe-m, so

I (t, k )=- J dxxftx f (x, k )

f (x, k ) =

Y (x) © (x, k) - A (x) Q (x, k)

A2 (x) + Y2 (x)

©(x, k) = 2ct2 x2H cos (2nH )-ae-kxA (x, k) , Q(x, k) = 2ct2x2H sin (2nH) + ae-kxB (x, k), (22) A(x) = 2ct2x2H cos(2nH)-ar(2H +1) , x) = 2ct2 x2H sin (2nH), A (x, k) = Re (r (2H +1, xke-n)), B (x, k) = Im (r (2H +1, xke-n )) > 0 .

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 r(a + 1) = ar(a) one can obtain

f (x, k )|x

ar( 2H +1) sin ( 2nH )

4a4

;(2a2 -ak2H )• x"2H + O (x"2H-1 )

(23)

so f (x, k)|~ x 2H from which it follows that I(0, k) is convergent if H e (0.5;1). Obviously, I (t, k ) is convergent for t > 0. So I (t, k) is convergent for any t > 0 if the process has fractal properties.

Let us calculate the poles of the function H '(p, k)ept. Obviously, to calculate the poles, we should equate the denominator of (11) to zero because of (14). The solutions of the corresponding equations are

2a'

-r(2H +1)

2 H i—

e H ,

(24)

According to [12], only the poles with arg(z) e [-n, n] contribute to J(t, k) in (17). We

consider the case where H e (0.5; 1), so from (24) we can

see that the only pole that contributes to J (t, k) is

P0 =1^ r(2 H + !)

(25)

Let us investigate the function H'(p, k)ept in the vicinity of the point p = p0. Let us introduce the

parameter Ç = p - p0. As is known [11], if | y |<| x |, then the following expansion is true:

-IS

r (a, x + y ) = r (a, x ) -(-1)"r(1 -a+ m) )-^yh/' xm r(1 -a) [ ^ h l !

(26)

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

On the basis of (11), (14), (25) and (26) one can obtain that in the vicinity of the point p = p0 (i.e. in the vicinity of the point \ = 0)

H ' (p, k) ept = H ' (p0 +Ç, k) e

( Po+y< _

= pepot r(2H +1)-epkr(2H +1 Pok)^ +

+0 C ) =

2 H r(2H +1)

r (2H +1) - ePok r (2H +1, p0 k )

(27)

J (t, k) = Res H ' (p, k) ept = eP°'p0 >

P=p0

r (2H +1) - ep°k r (2H +1, p0 k ) > 2 H r(2H +1)

(28)

it should be noticed that J (t, k) is not taken into account in [8].

So, the following solution is obtained:

ak2

2a2

h (t, k) = l 1--T ls(t) +1 (t, k) + J (t, k)

(29)

where the explicit expressions for I (t, k) and J (t, k) are given in (22) and (28); the expression for p0 from (28) is given in (25).

4 EXPERIMENTS

Numerical calculations for some parameters are made in order to verify the solution (29). The integral

J dTh (t, k) (T -T) = -Ok-rJ R (T) +

T T

+J dtI (t, k)R (T - t) + J dTJ (t, k)R (T - t)

(30)

r(2H +1, z ) = J dxe-xx2H + J dxe~ xx2

0 z

A (x, k) = Re (r (2H +1, -xk)), B ( x, k ) = - Im (r ( 2 H +1, - xk )),

(31)

2 H r(2 H +1)

xp0ept (p - p0 )-1 + 0 ((p - p0 )0 ) .

As can be seen from (27), the expansion of H '(p, k)ept into a Laurent series of p - p0 begins with the minus first term, so p0 is a simple pole and

the sign "-" in B(x, k) in (31) is due to the fact that Mathcad interprets +1, - xk) from (31) as

r(2H +1, xkein); see (21).

We should notice that Mathcad fails to calculate the function I (t, k) as the integral from 0 to &, so I (t, k) is treated as

dxe

I 233 I

I (t, k) « - J dxe-txf (x, k) + - J

n 0 n 233

C =ar(2H +1)sin(2nH) (2-ak2H)

(32)

i.e. if x > 233, 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

k = 3, H = 0.8, a = a = 1

R(T + k ) the integral (30)

T = 4 -10.24934 -10.24935

T = 5 -12.92881 -12.92884

T = 10 -29.28861 -29.28809

T = 30 -133.4598 -133.4534

k = 3, H = 0.8, a = n/ 2, a = 0.8

R(T + k ) the integral (30)

T = 4 -17.03041 -17.03042

T = 10 -46.93724 -46.93725

T = 15 -79.43844 -79.44034

T = 20 -117.89713 -117.89911

k = 3, H = 0.7, a = n/ 2, a = 1.2

R(T + k ) the integral (30)

T = 4 -10.53367 -10.53367

T = 10 -27.04465 -27.04475

T = 20 -61.87533 -61.87372

T = 30 -103.51651 -103.51531

T = 40 -150.5975 -150.61689

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 T = 103, 104, etc. In order to verify the solution (29) for large values of T , we seek the asymptotics of the integrals in (30) if T ^œ .

Let us denote

r ( 2H +1) - eP0k r ( 2H +1, p0 k ) = 2H r(2H +1) P°

then

T

J d tJ (t, k )R (T - t) = A J dtep0' la2 - - (T -1 )2

0 0 V 2

As is known [11],

Jxv- (u - x) 1eP'dx =

0

= B (|, v)uv+|-1 • jF (v, | + v, Pu) . On the basis of (35) and (34) we have

J d tJ (t, k )R (T-t) = A [— (eP0T -1)-0 V p0

(35)

a T 2H+1 2

r(2H +1)

(36)

, F (1,2H + 2, pT)

r(2H + 2)

As is known [13], the function lF (a, P, z) has the following asymptotics:

1F (a, P, z)| n 3n ~

1 1 v >\\z\ ^»,--<argz <—

r(P)ezza-P -a)j (p-a) ^ +

~ - / -z +

r (a) -=0 5 !

+4:eV(P)ia - (a-P+1)- (-z )-,

r(p-a) s ! V 7 '

as = a (a + 1)(a + 2 )... (a + s -1) , a0 = 1.

On the basis on (37) it can be seen that

1F1 (1, P, z )|, z|

(37)

< 3n

< 2

= r(P)

z 1-p

e z -

r(P) -+Ol 1

zr(P-1)

(38)

On the basis of (38), (36) and (25) we obtain

a A

J dtR (T -1)J (t, k)

,__T2 2 p0

(39)

It should be noticed that for T ^ro the integral in (39) behaves as a power-law rather than an exponent function! The integral of I (t, k) is as follows:

(33)

(34)

J dtR (T -1 )I (, k )= I1 -12,

0

T T

I =CT2JdtI(t,k), i2 = aJdtI(,k)(t-tfh.

0 2 0

After substituting % = T -1 into I2 we obtain

1 ro T

I2 = -lJdxf (x,k)e-T'Jd%2H . 2 n 0 0

With the help of the tabulated integral

u

J dxxv"1e"1 =|-vy(v, |u ),

0

Y (a, x) = r (a) - r (a, x) and eqs. (41) and (20) we obtain

(40)

(41)

(42)

12 T2H Jdx^^g (x,T)

/IT "" V

f ( x, k )

2 n

r(x, T ) = e~Tx X-

(xT Y

(43)

n !(2H +1 + n) * It should be noticed that

< e- X

(x, T ) = e- X

n=0

(xT )n+' = e ^

(n +1)!

(xT )n

n !(2H +1 + n)

(eTx -1) = 1 - e~Tx < 1,

(44)

so with account for (23) the integral in (43) is convergent; and the asymptotics of (43) for T ^ro is not larger than aT2H where a is a constant. We assume that I2 T~ T2H, this assumption is confirmed numerically in what follows.

As for Ij, we have

I = a2 J dtI (t, k)

œ

= a2 J dxf (x, k)

1 - xi

; 1 - e-xT < 1,

(45)

so obviously Ij is bounded by a constant and Il = o ( I2 ) if T ^œ. Obviously

1 I * (T)

2ct2

R (T + k)|

a ( ak2H

_ ~2 I -lO1"

__t 2 H

lT—ro 2

(46)

So, the asymptotic behavior of the left-hand and right-hand sides of (1) on the basis of (29) for T — ro are

__t 2 H

2

R (T + k))

T

J dth (t, k)R (T -t)

0 T —ro

- lim — J dx~f( , )

T—ro tt J V

A_

P0

(47)

_I 1 -

ak

2ct2

g (x, T)

see also (39), (43) and (46). So if

f (x, k)

A-(1 _ ak2H

P0

2ct2

i- 1 fd -lim — I dx-

T—ro TT J

T—ro n = _1 .

g (x, T) =

(48)

then our solution (29) is true for T — ro .

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.

Table 2 - Verification of the obtained assymptotics

k = 3, H = 0.7, a = n/2, a = 1.2

T A -(1 - ak 7 I-If xf ( x,k ) g (x, T ) Pc I 2a2 J nJ x '

103 -0.999798

104 -0.9998

105 -0.999998

k = 4, H = 0.8, a = n/ 2, ct = 0.8

T A -(1 - ak 7 U J dxf ( x,k ) g (x, T ) P0 I 2a2 J nJ x ^ ' ^

103 -0.996715

104 -0.999671

105 -0.999967

As can be seen from table 2, eq. (48) is valid, which justifies our solution.

5 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 power-law structure function (3). Only the cases where the Hurst exponent H e (0.5;1) are considered. The obtained solution (29) is verified numerically. The asymptotic

© Gorev V. N., Gusev A. Yu., Korniienko V. I., 2018 DOI 10.15588/1607-3274-2018-4-4

behavior of both sides of (1) for T — ro is investigated. It is shown that our solution gives the correct asymptotic behavior.

6 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 t > 0 if the Hurst exponent H e (0.5;1). 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 —^ ro 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.

Prospects for further research are to apply the obtained results to practical problems.

REFERENCES

1. 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

2. 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/S0378-4371(02)01021-X

3. 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, 2009, Vol. 11, 093024 (21 pages). D0I:10.1088/1367-2630/11/9/093024

4. Preis T., Paul W., Schneider J. Fluctuation patterns in high-frequency financial asset returns, Europhysics Letters, 2008, Vol. 82, 68005 (6 pages). DOI: 10.1209/02955075/82/68005

5. 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. 225231.

6. 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 978-1-13800126-8.

7. Pipiras V., Taqqu M. Long-Range Dependence and Self-Similarity. Cambridge University Press, 2017, 668p. DOI: 10.1017/CBO9781139600347

УДК 517.968.22

8. Bagmanov V. Kh., Komissarov A. M., Sultanov A. Kh. Prognozirovanie teletraffika na osnove fraktalnykh filtrov, Vestnik Ufmskogo gosudarstvennogo aviatsionnogo universiteta, 2007, Vol. 9, No. 6 (24), pp. 217-222.

9. 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: doi.org/10.1016/B978-0-12-386981-4.50001-1

10. 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.

11. 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, 1200 p.

12. Angot A. Matematika dlya elektro- i radioingenerov. Moscow, Nauka, 1967, 780 p.

13. Oliver F., Lozier D., Boisvert R., Clark C. NIST Handbook of Mathematical Functions. New York, Cambridge University Press, 2010, 951 p.

Received 22.05.2018.

Accepted 03.06.2018.

ДО АНАЛ1ТИЧНОГО РОЗВ'ЯЗКУ ШТЕГРАЛЬНОГО Р1ВНЯННЯ ВОЛЬТЕРИ ДЛЯ ДОСЛ1ДЖЕННЯ

ФРАКТАЛЬНИХ ПРОЦЕС1В

Горев В. М. - канд. физ.-мат. наук, асистент кафедри безпеки шформаци та телекомунжацш, Нацюнальний техшчний ушверситет Дншровська Полгтехшка, Дншро, Украша.

Гусев О. Ю. - канд. физ.-мат. наук, доцент, доцент кафедри безпеки шформаци та телекомушкацш, Нацюнальний техшчний унверситет Дншровська Полгтехшка, Дншро, Украша.

Коршенко В. I. - д-р техн. наук, доцент, завщувач кафедри безпеки шформаци та телекомушкацш, Нацюнальний технчний ушверситет Дншровська Полгтехшка, Дншро, Украша.

AНОТАЦIЯ

Актуальшсть. Розглянуто штегральне р1вняння Вольтери першого роду, яке може бути застосовним до фшьтраци та прогнозування випадкових фрактальних процесгв, наприклад, у шформацшно-телекомушкацшних мережах та при керуванш складними технолопчними процесами.

Метою роботи е отримати точний аналгтичний розв'язок штегрального ргвняння Вольтери першого роду. Ядром вщповщного штегрального ргвняння е кореляцшна функцiя фрактального випадкового процесу, структурна функцiя якого е степеневою.

Метод. 1нтегральне ргвняння Вольтери першого роду розв'язано за допомогою стандартного методу перетворення Лапласа. Зворотне перетворення Лапласа приводить до контурного штегралу в1д функцп комплексно! змшно!. Цей штеграл обчислено як суму частини, що мютить лишок, та штегралгв вздовж берега розр1зу. В1дпов1дш штеграли пораховано за допомогою вщомих розвинень спещальних функцш.

Результати. Нами отримано точний аналгтичний розв'язок штегрального р1вняння Вольтери, ядром якого е кореляцшна функцiя фрактального випадкового процесу. Робота базуеться на модел1, в якш структурна функцiя вщповщного фрактального процесу е степеневою функщею. Показано, що та частина розв'язку, яка не мютить дельта-функци, е зб1жною в будь-якш точщ, якщо показник Херста е бшьшим за 0,5, тобто якщо процес мае фрактальш властивоста. Показано, що отриманий розв'язок е дшсною функщею. Отриманий розв'язок перев1рено чисельно; також показано, що наш розв'язок дае правильну асимптотичну поведшку. Хоча отриманий розв'язок мютить експоненцшно зростаючу функцта часу, при великих часах штеграл в1д отриманого розв'язку асимптотично веде себе як степенева функцш.

Висновки. Важливо шдкреслити, що нами отримано точний, а не наближений розв'язок штегрального ргвняння Вольтери, яке дослвджуеться. Отриманий розв'язок може бути застосовним до фшьтраци та прогнозування даних випадкового фрактального процесу. Як ввдомо, фрактальн процеси мають м1сце у величезнш кшькосп р1зномангтних систем, тому результати ще! статт можуть мати широку область застосувань.

КЛЮЧОВ1 СЛОВА: штегральне ргвняння Вольтери першого роду, показник Херста, перетворення Лапласа, фрактальний процес, точний аналгтичний розв'язок.

УДК 517.968.22

К АНАЛИТИЧЕСКОМУ РЕШЕНИЮ ИНТЕГРАЛЬНОГО УРАВНЕНИЯ ВОЛЬТЕРРЫ ДЛЯ ИССЛЕДОВАНИЯ

ФРАКТАЛЬНЫХ ПРОЦЕССОВ

Горев В. Н. - канд. фiз.-мат. наук, ассистент кафедры безопасности информации и телекоммуникаций, Национальный технический университет Днепровская Политехника, Днепр, Украина.

Гусев А. Ю. - канд. фiз.-мат. наук, доцент, доцент кафедры безопасности информации и телекоммуникаций, Национальный технический университет Днепровская Политехника, Днепр, Украина.

Корниенко В. И. - д-р техн. наук, доцент, заведующий кафедры безопасности информации и телекоммуникаций, Национальный технический университет Днепровская Политехника, Днепр, Украина.

АННОТАЦИЯ

Актуальность. Рассмотрено интегральное уравнение Вольтерры первого рода, которое может быть применено к фильтрации и прогнозированию случайных фрактальных процессов, например, в информационно-телекоммуникационных сетях и при управлении сложными технологическими процессами.

Целью работы является точное аналитическое решение интегрального уравнения Вольтерры первого рода. Ядром соответствующего интегрального уравнения является корреляционная функция фрактального случайного процесса, структурная функция которого является степенной.

Метод. Интегральное уравнение Вольтерры первого рода решено с помощью стандартного метода преобразования Лапласа. Обратное преобразование Лапласа приводит к контурному интегралу от функции комплексной переменной. Этот интеграл посчитан как сумма части, содержащей вычет, и интегралов вдоль берегов разреза. Соответствующие интегралы получены на основе известных разложений специальных функций.

Результаты. Нами получено точное аналитическое решение интегрального уравнения Вольтерры, ядром которого есть корреляционная функция фрактального случайного процесса. Работа основывается на модели, в которой структурная функция соответствующего фрактального процесса является степенной функцией. Показано, что та часть решения, которая не содержит дельта-функции, сходится в любой точке, если показатель Херста больше 0,5, то есть если процесс имеет фрактальные свойства. Показано, что полученное решение является действительной функцией. Полученное решение проверено численно; также показано, что наше решение дает правильное асимптотическое поведение. Хотя полученное решение содержит экспоненциально возрастающую функцию времени, при больших временах интеграл от полученного решения асимптотически ведет себя как степенная функция.

Выводы. Следует подчеркнуть, что нами получено точное, а не приближенное решение исследуемого интегрального уравнения Вольтерры. Полученное аналитическое решение может быть применено к фильтрации и прогнозированию данных случайного фрактального процесса. Как известно, фрактальные процессы имеют место в огромном количестве разных систем, поэтому результаты этой статьи могут иметь широкую область применения.

КЛЮЧЕВЫЕ СЛОВА: интегральное уравнение Вольтерры первого рода, показатель Херста, преобразование Лапласа, фрактальный процесс, точное аналитическое решение.

Л1ТЕРАТУРА / ЛИТЕРАТУРА

1. Investigation of rescaled range analysis, the Hurst exponent, and long-time correlations in plasma turbulence / [M. Gilmore, C. X. Yu, T. L. Rhodes, W. A. Peebles] // Physics of Plasmas. - 2002. - Vol. 9. - P. 1312-1317. DOI: 10.1063/1.1459707

2. Gorski A. Z. Financial multifractality and its subtleties: an example of DAX / A. Z. Gorski, S. Drozdz, J. Spethc // Physica A. - 2002. - Vol. 316. - P. 496-510. DOI: 10.1016/S0378-4371(02)01021-X

3. Accelerated fluctuation analysis by graphic cards and complex pattern formation in financial markets / [T. Preis, P. Virnau, W. Paul, J. Schneider] // New Journal of Physics. - 2009. - Vol. 11. - 093024 (21 pages). D0I:10.1088/1367-2630/11/9/093024

4. Preis T. Fluctuation patterns in high-frequency financial asset returns / T. Preis, W. Paul, J. Schneider // Europhysics Letters. - 2008. - Vol. 82. - 68005 (6 pages). DOI: 10.1209/0295-5075/82/68005

5. Fractal analysis for forecasting chemical composition of cast iron / [O. Gusev, V. Kornienko, O. Gerasina, O. Aleksieiev] // In book "Energy Efficiency Improvement of Geotechnical Systems". - Taylor & Francis Group, London, 2016. -P. 225-231.

6. Kornienko V. Methods and principles of control over the complex objects of mining and metallurgical production / V. Kornienko, A. Gerasina, A. Gusev // In book "Energy Efficiency Improvement of Geotechnical Systems". - Taylor

& Francis Group, London, 2013. - P. 183-192. ISBN 978-1138-00126-8.

7. Pipiras V. Long-Range Dependence and Self-Similarity / V. Pipiras, M. Taqqu. - Cambridge University Press, 2017. - 668 p. DOI: 10.1017/CB09781139600347

8. Bagmanov V. Kh. Prognozirovanie teletraffika na osnove fraktalnykh filtrov / V. Kh. Bagmanov, A. M. Komissarov, A. Kh. Sultanov // Vestnik Ufimskogo gosudarstvennogo aviatsionnogo universiteta. - 2007. - Vol. 9, No. 6 (24). -P. 217-222.

9. Miller S. Probability and Random Processes With Applications to Signal Processing and Communications. Second edition / S. Miller, D. Childers. - Amsterdam: Elseiver/Academic Press, 2012. - 598 p. DOI: doi.org/10.1016/B978-0-12-386981-4.50001-1

10. Polyanin A. D. Handbook of the integral equations. Second edition / A. D. Polyanin, A. V. Manzhirov. - Boca Raton : Chapman & Hall/CRC Press, Taylor & Francis Group, 2008. - 1143 p.

11. Gradshteyn I. S. Table of Integrals, Series, and Products. Seventh edition / I. S. Gradshteyn and I. M. Ryzhik // Translated from the Russian, Translation edited and with a preface by A. Jeffrey and D. Zwillinger. - Amsterdam: Elsevier/Academic Press, 2007. - 1200p.

12. Angot A. Matematika dlya elektro- i radioingenerov / A. Agnot. - Moscow : Nauka, 1967. - 780 p.

13. NIST Handbook of Mathematical Functions / [F. Oliver, D. Lozier, R. Boisvert, C. Clark]. - New York : Cambridge University Press, 2010. - 951 p.