Accurate relationships between fractals and fractional integrals: New approaches and evaluations

2.1 The exact relationship between temporal fractional integral and the smoothed function averaged over the Cantor set with M bars

In paper [10] the relationship between fractional integral with complex power-law exponent and Cantor set has been established. But this relationship was approximate and obtained in one-mode approximation and it would be desirable to establish the exact relationship between 1D fractals and fractional integrals in time-domain. Attentive analysis shows that one important point in the previous results leading to the desired exact relationship was missed. In order to show it let us reproduce some mathematical expressions that will be helpful for further manipulations and understanding the problem posed. As it has been shown in [10] the Laplace image of the kernel of the Cantor set is described by expression

where z = pT(1 − ξ), p defines the Laplace parameter, T is a period of location of the Cantor set, is the scaling factor. The kernel K(N)(z) can be presented as

Here g(z) describes the structure of the given fractal with asymptotic given below. In particular, the Laplace image of the function g(z) for Cantor set having M bars has the form

In order to satisfy to the functional equation of the type (4) in the limit (N ≪ 1) for the kernel

the function g(z) should have the following decompositions for small and large values of z:

for Re(z) ≪ 1

g(z)=1+c1z+c2z2+…,(5)

for Re(z) ≫ 1

g(z)=g¯+A1z+A2z2+….(6)

It is easy to note that for the function g(z) from (3) having M bars in each self-similar stage these requirements are satisfied (g = 1/M).

The solution of the functional equation (4) has the form (1) with power-law exponent equaled [16, 4]

The log-periodic function π_ν(ln z ± ln ξ) = π_ν(ln z) with period ln(ξ) that figures in 1 can be decomposed to the infinite Fourier series [9, 10]

Here Ω_n = 2πn/ln ξ is a set of frequencies providing a periodicity with ln of product (4). Taking into account this decomposition one can present the limiting solution of (1) in the form

Here the real exponent is defined by expression (7). The presentation of kernel (2) in the form (9) allows reproducing the previous cases (when the sum in (9) is negligible and or it can be presented in one mode approximation [10]) and finding the desired exact relationship. Really, taking into account the well-known relationship [2]

it is easy to find the desired original for the kernel (9)

Here we should take into account that temporal variable t should be dimensionless. Based on the determination of the variable z = pT(1 − ξ) (see expression (1)) it is easy to restore the dimension of the constant t → t/T(1 − ξ). Based on the definition of the fractional integral in the form of the Riemann-Liouville (RL) type one can write the desired relationship

Here we define the RL-integral of the complex order as

From the exact relationship (12), it follows that the averaging procedure of a smooth function over the generalized Cantor set (having M bars) is accompanied always by an infinite set of fractional integrals having complex-conjugated power-law exponents. Only in the partial case when the contribution of log-periodic function becomes negligible relationship (13) restores the RL fractional integral with real power-law exponent [9, 10]. When the total sum in (11) can be replaced approximately by one term

then we restore the basic result of paper [10] in the so-called one-mode approximation. In relationship (14) 〈Ω〉 defines the leading mode, which replaces approximately other modes that figure in the left-hand side of expression (14).

2.2 Numerical verification

For verification purposes one can take time-domain expression for the kernel defined by (11) and the corresponding one-mode approximation expression from (14).

Numerical calculations were realized with the help of the following procedure:

1. Calculation of the right side of expression (14) using one-mode approximation constants for some given M (number of the Cantor columns) and the scaling parameter ξ from [10] (the first 3 rows of Table 1).

   Table 1

   The basic initial (the first 3 rows) and the fitting parameters (rest rows) obtained in the result of numerical verification of expression (11).

   M	ξ	ν	C0	A	〈Ω〉	〈n〉
   2	0.125	0.3333	0.63	0.0082	3.01161	0.9967
   5	0.05	0.5372	0.6117	0.0217	2.09144	0.9972
   15	0.0167	0.6614	0.606	0.0353	1.5331	0.9991
   M	ξ	ν	C0	C1	Ω1	Nm
   2	0.125	0.3333	0.62328	0.00805	3.02157	20
   5	0.05	0.5372	0.61139	0.02157	2.09738	20
   15	0.0167	0.6614	0.60640	0.03523	1.5346	20

2. Calculation of the fitting constants C_n (n = 1, …, N_m), where N_m defines the finite number of modes in the corresponding sum (11) by the linear least-square method (LLSM)).

The values of fitting parameters for the given values of M and ξ are collected in last 3 rows of Table 1. One can see that the fitting parameters C₀, A₁, Ω₁, are very close to the corresponding initial ones obtained in one-mode approximation.

The results of further numerical calculations are collected in Figures 1 and 2. There one can see that contribution of each next term in the sum (11) decreases drastically in comparison with the previous one. This observation serves as a good proof of the validity of one-mode approximation approach proposed in [10].

Figure 1

The fitted C_n coefficients presented in log-scale (y-axis) correspond to the successive terms of the sum in (11) (x-axis) for M = 2 (open circles), M = 5 (open squares) and M = 15 (open triangles).

Figure 2

Relative fitting error in % (y-axis) versus number of modes (x-axis) for (a) M = 2, (b) M = 5 and (c) M = 15.

Analyzing these results obtained in this section we found that higher terms n = 1, 2, …, N_m give relatively insignificant contribution in the total result and the value of the fitting error has a plateau with increasing of N_m. One can notice that fitting error mostly depends on 〈n〉 parameter, characterizing the contribution of the main harmonics.

2.3 Physical interpretation of this result and possible generalizations

To understand deeper the results from physical point of view it makes sense to start from the simplest physical examples which can clarify the procedure considered above. For this purpose we consider at first the following mechanical problem. How to calculate the total and averaged path 〈L(t)〉 for some interval of the time t for a set of material points (particles) if every particle is moving in one direction with the constant velocity V coinciding with the points of Cantor set and is idle outside the set? The expression for the path L_N(t) on the N-th stage of Cantor set construction has the form:

Here the value KT,ν(N) determines the Cantor set located on the temporal interval T on the N-th stage of its self-similarity having M bars and dimension defined by expression (7). The value of the normalization interval T for the given case can be found from the condition that during the interval T the body passes the distance L^*. Hence, T = L^*/V. Then, integrating expression (11) we obtain

The sum entering in (16) determines the log-periodic corrections appearing in the result of discretization of the Cantor set. Then, realizing the averaging procedure for the total assembly of particles described in a book [9], we obtain

Here B(ν) determines the averaged value of the sum located in the square brackets (16). For the binary Cantor set it can be evaluated analytically that gives

Actually, the formula (17) expresses the dependence of the averaged path as the function of time for a set of particles if every particle is moving only inside the Cantor set with the fractal dimension and we are interested only by the averaged path of the total assembly. The fractal dimension shows what part of states of the assembly are involved in this movement. We also want to notice that exact intervals where one type of movement is transformed into the state of idleness are averaged and after the performing of the averaging procedure we have the information only about a part of all the states involved into the distribution over Cantor sets. The averaged procedure of a smoothed function over Cantor set described above represents the Cantor’s “filter” which enables to filtrate one type of movement and delete (because of the normalization of the states) another one. The averaged log-periodic states are described by a constant (18). It represents the result of this “filtration” procedure. To stress the filtration properties of the Cantor set we consider more complicated example. Let us assume that it in the intervals between Cantor stripes an assembly of bodies is moved with various and random velocities defined as {U_i}. The index i = 1, 2, …, N, .. refers to the stages of Cantor construction. Two stages of the modified Cantor set are shown on Figure 3.

Figure 3

Here two successive stages of the modified binary set are shown. The set of velocities outside of the Cantor set are random. The widths of Cantor bars are denoted as {Δ_i (i = 0, 1, 2)}, the set of velocities {V_i} defines the movement inside the set.

It is interesting to set up the following question: Is there a condition for the distribution of velocities {U_i} when the influence of movement outside the set becomes negligible?

In this case the recurrence relationship for the density is expressed by formula:

For Laplace-image with the help of retardation theorem

one can obtain the following expression:

The total area on the N-th stage is given by the following expression:

We require that the total area on every stage remains the same (it is equivalent to the condition of conservation of the total number of states):

Let us assume that U_i = U + δU_i, where

is the arithmetic mean of random velocity. The set U_i (i = 1, 2, …, N, …) defines the random deviations of U_i. We assume that these deviations {U_i} satisfy the condition:

Based on this condition we can rewrite (21) as

If we propagate the binary Cantor set on the whole temporal interval (in and out of the given T) we obtain from

The first part of (27) satisfies to the scaling equation (4) with g= 2 and the second part is proportional to t. If we, again, apply the averaging procedure then we obtain

The last expression confirms again the filtration properties of the Cantor set. It divides all motion on two parts: (a) the first motion of the particles located inside the Cantor set, (b) the second part describes the motion that takes place outside the fractal set.

2.4 New relationships connecting a fractal process in time with the smoothed function by means of the fractional integral

It is natural to consider another self-similar fractal process, which can be presented in the form of an additive summation [9, 10]. These sums are appeared naturally when the self-similar RLC elements connected in parallel or in-series are considered [9, 10]

Here z is dimensionless Laplace parameter, b and ξ are some constant scaling factors. Each term in (29) can be associated, for example, with the scaled resistor (R_n = R₀bⁿ), capacitance (C_n = R/zξⁿ, z = jωRC) or inductance (L_n = Rzξⁿ, z = jωL₀/R) that can form a self-similar fractal circuit [9] or with additive contribution of a bar belonging to some Cantor set. In this case, an additive contribution of each bar is expressed by the function

It is easy to note that sum (29) satisfies to the following equation

We suppose that the contributions of the last two terms on the ends of the finite interval are negligible

Below we will discuss these suppositions in detail. Equation (31) at conditions (32) satisfies the following functional equation that formally coincides with equation (4) considered above

So, based on the results obtained in the previous section one can write

In spite of the formal coincidence of equation (33) with (4), in order to satisfy to conditions (32) the function f(z) in (31) should have another asymptotic decompositions

for Re(z) ≪ 1

f(z)=c1z+c2z2+…,(35)

for Re(z) ≫ 1

f(z)=A1z+A2z2+…,(36)

and condition

As it follows from (34) the last condition provides in addition the obvious inequality

So, the results obtained for the sum (31) coincides with the results obtained earlier for the product (2) and the exact relationship in time-domain (11) with subsequent convolution (12) of the kernel (11) with a smooth function remains invariant. Is it possible to consider more complicated self-similar process that is presented by an additive combination of two sums of the type?

If we assume, by analogy with requirement (32) that the contribution of each sum on the end of the intervals are negligible

(here the number of elements N and the scaling factor for both sums are kept the same values) then from (39) we have a system of equations

Excluding from (39) and the first row of (41) the unknown sums S_1,2(z) (defined above by expressions (39)) and inserting them to the second row of (41) we obtain the generalized scaling equation of the second order

Solutions of the scaling equations of the second order was considered in paper [11] and, therefore, we can write

where the power-law exponents are found from the equation

In solution (43) we have now two log-periodic functions Pr_{1, 2} (ln z ± ln ξ) = Pr_{1, 2}(ln z) that can be decomposed to the Fourier series relatively variable ln (z). Therefore, one can present the solution (43) in the form

where we introduced new definitions of the power-law exponents and discrete frequencies reflecting the discrete structure of the fractal process considered.

Moreover, these results can be further generalized. Let us suppose that we have three additive combination of self-similar processes

The functions f_k(z) (k = 1, 2, 3) are chosen in a such way that their contributions on the ends of the interval remain negligible

We suppose again that for each function f_k(z) (k = 1, 2, 3) the asymptotic behavior similar to expressions (35) and (36) is conserved. Therefore, the suppositions (47) are satisfied if the constants b_k and ξ follow the requirements

Therefore, we obtain the following system of equations for the finding of unknown S_k(z):

Excluding from the system (49) the unknown sums S_k(z) and inserting them into the relationship

after simple algebraic manipulations we obtain the following functional equation with respect to the function S(z)

The solution of the functional equation (51) can be written as

The power-law exponents entering into (52) are found from the cubic equation

and can expressed as

Therefore, expression (45) conserves its form and for this case the solution (52) can be rewritten as

It is quite obvious that this result admits the further generalization for the case of k = 1, 2, …, K sums

Using the mathematical induction method one can write the final result for this case

The desired roots are found from the relationships

So, the kernel connecting the given fractal process with a smooth function in time-domain accept the following general form

To the best of our knowledge, this is the most general result that can be obtained for a set of additive self-similar processes in time. As one notice from the calculations, all dynamical functions describe a self-similar process with the same scale ξ. If the scales are different and the discrete boundaries of the processes are different also N_k ≠ N then these results are becoming questionable and require the further research. But nevertheless, one can show the situation taking place for different scales ξ_i ≠ ξ when the result can be reduced to the case considered above. Let us suppose that we have a product combining a random combination of different scales

but the deviations from some averaged scale (considered as the dominant scale) are small. We have for this case

For the condition δ_s/ 〈ξ〉 ≪ 1 we have approximately

Therefore, this result allows to consider different scales if they are not strongly deviated from each other. All the previous results are valid if we simply replace