=Paper=
{{Paper
|id=Vol-2830/paper19
|storemode=property
|title=On One Model of Information Distribution in a Spatially Distributed Environment
|pdfUrl=https://ceur-ws.org/Vol-2830/paper19.pdf
|volume=Vol-2830
|authors=Andrey Nesterov,Alexander Zaborskiy
}}
==On One Model of Information Distribution in a Spatially Distributed Environment==
https://ceur-ws.org/Vol-2830/paper19.pdf
On One Model of Information Distribution in a Spatially
Distributed Environment
Andrey V. Nesterov 1[0000-0002-4702-4777] and Alexander V. Zaborskiy 2[0000-0003-2480-7215]
1
Plekhanov Russian University of Economics, 36 Stremyanny lane, Moscow, 115998, Russia
andrenesterov@yandex.ru
2
LLC RPE RADICO, 14a Marks Ave., Obninsk, 249035, Kaluga's region, Russia
alexander.zaborskiy@mail.ru
Abstract. We consider one of the models of information (or other substance)
propagation in a spatially distributed multiphase environment, when the ex-
change rate between phases is much higher than the transfer rate, and the num-
ber of phases is large. The transfer is described by a singularly perturbed partial
differential operator equation in the critical case. An asymptotic expansion for a
small parameter of the initial problem solution is constructed. From the ob-
tained formulas, it follows that in the first approximation, the initial perturba-
tion propagates at a certain average speed with simultaneous diffusion spread-
ing. Formulas for the average transfer rate and pseudodiffusion coefficient are
obtained. The obtained formulas can be used both for qualitative analysis of
problem solutions and for creating economical difference schemes that require
significantly less (by orders of magnitude) computational resources.
Keywords: multiphase media, distributed systems, differential operator equa-
tions, small parameter, singular perturbations, asymptotic decomposition of the
solution.
1 Introduction
In the description of a number of transport processes of various substances in a
spatially-distributed multi-phase environments as mathematical models of phenomena
described by systems of partial differential equations. In the case of one spatial varia-
ble x this system of equations has the form
Here the vector-function, the dimension of which is determined by the
number of phases, are the trasfer rates of each component of the
solution, the matrix A describes the exchange processes between phases, the elements
of the matrix aij make sense of the exchange rate between the phases i and j . Spatial
variable x can make sense of the real spatial variable or to describe some characteris-
tics of the environment, "along" which the transfer substance.
Proceedings of the 10th International Scientific and Practical Conference named after A. I. Kitov
"Information Technologies and Mathematical Methods in Economics and Management
(IT&MM-2020)", October 15-16, 2020, Moscow, Russia
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
In some cases, the exchange processes between phases have significantly higher
speeds than the transfer processes (which, for example, may correspond to the pro-
cesses of information transfer in a social environment, where the exchange "horizon-
tally" (between phases) occurs much faster than the exchange "vertically" - by the
variable x).Then when moving to dimensionless variables in the task matrix A takes
the form A=ΕA1 ,where Ε >>1 is a large positive parameter and A1 has elements of
order O(1). It is convenient to rewrite A in A=A1/ ε where 0<ε<<1 is a small positive
parameter. Below we denote a small positive parameter as ε2 (second degree is intro-
duced for convenience and to reduce the record below), and the lower index in the
matrix A1 down. In this case the system of equations takes the form
The presence of a small parameter at higher derivative makes the problem singular-
ly perturbed [16], numerical calculations solution of which is quite time consuming.
However, to simplify numerical calculations of solutions and identify some of the
hidden regularities of the behavior of the solution (and consequently, the simulated
system), it is advisable to try to construct the asymptotic expansion (AE) of the solu-
tion in powers of the small parameterter ε. If the number of phases is very large or
tends to infinity ( in case n>>1 ), then the discrete index i, 1≤i≤n, becomes a continu-
ous parameter p, p1≤p≤ p2 . In this case the system of equations becomes one differen-
tial operator equation generalizing the above system to the case in which the matrix A
is replaced by a linear operator acting on the parameter p . In this case, the vector
function becomes a function , depending on the parameter p, and
the system of equations is transformed into a differential-operator partial differential
equation
Similar equations (possible, without a small parameter) appear when modeling
various processes, for example, coagulation processes [1]-[4], (Boltzmann and
Smolukhovsky equations), turbulence processes [5]-[7], modeling social processes
[8]-[12], and others [13]-[14]. Differential-operator equations have been studied in
many works, for example [19]-[23]. To construct an asymptotic expansion of the
solution, the technique developed in the works [16]-[18] is used.
2 Statement of the problem
Consider the initial problem for a singularly perturbed differential operator equation
(1)
(2)
here: – solution ,
, is a small positive parameter; continuous function on
; is a linear operator acting in the space of continuous by
functions with the scalar product ; the initial condition
satisfies the inequality together with their derivatives at up to
order N+3 ,here N - some natural number .
Let the operator has a single eigenvalue , - corresponding
eigenfunction, a eigenfunction of the adjoint operator , corresponding .
.
It follows from the condition that in the evolution of the "generalized
quantity" of a substance does not change. Indeed, multiplying (1) on scalar and
integrating the result on x from -∞ to +∞, we get
I. We require that the remaining eigenvalues of the operator have negative
real parts .
II. . In this condition it is possible to choose these functions so that
.
3 Algorithm for constructing an AE solution
AE of solutions is constructed as the sum of the functions of the surge , concen-
trated in the neighborhood of a line – "Pseudocharacteristic" of equations
and boundary layer functions concentrated in the neighborhood of the boundary
and the remainder term :
(3)
here is a variable, which describes the function of the
surg ; the stretched variables, which describe the boundary layer
function . The algorithm for constructing a AE similar to the algorithm described in
[3].
3.1 Building a surge function
Function S must satisfy the original equation (1):
(4)
Move on to equation (1) from variables to new variables
(5)
where
(6)
Function S is searched in the form:
(7)
Substituting (7) into (5), in a standard way [4] we get the system of equations for
the terms of the expansion :
,
,
,
…
,
is:
(8)
where - as yet unknown function. Write conditions for the solvability of the
equations for and [4]:
(9)
(10)
The condition (9) is true due to the choice of the variable ζ, therefore, can be
written as:
(11)
where – pseudo-inverse to operator . Substituting (8), (11) in (10), and elimi-
nating φ1 derived equation to determine φ0:
(12)
The equation for finding the following approximations are obtained sim-
ilarly. Omitting the calculations, we give only the result.
The function is:
.
A function defined by the equation:
(13)
where is a linear combination of the functions , and their
derivatives.
Thus, the obtained expression for finding and equations to determine the members
in these expressions for all functions .
Apply the condition of parabolicity on equations (12), (13):
III.
3.2 The construction of boundary layer functions
A function under any initial conditions for equations (12), (13) does not sat-
isfy initial conditions (2). To meet these conditions is constructed, the boundary layer
function [4]. The boundary layer function P needs together
with the function to satisfy the initial conditions (2):
(14)
the original equation (1):
(15)
and the condition:
(16)
The function P is constructed as:
(17)
Substituting (17) into (15) [4], we get the equations for determining pi
, (18)
.
Substituting the series (8), (17) in condition (14), given that , we get the re-
sulting equations for determining the initial conditions:
(19)
(20)
Imposing some additional conditions on the eigenvalues of the operator (IV-
VI,[3]) and omitting intermediate calculations, we give the result. The function
has the form
(21)
Substituting (8), (21) in (19) we obtain the equation for determining the initial con-
ditions for the equations (13) and functions :
(22)
From (22) in the IV-VI [3] we get:
Thus, the obtained initial condition for equation (13) from which is determined
, as well as the function itself .
The construction of the subsequent functions is similar [4].
Thus, all members of the far solution (3) - functions and , clearly
defined.
3.3 The function evaluation of splash and border functions
If the condition III (M<0) is met, for any all exist, are unique
and for a any fixed N and are valid estimates
.
When the conditions are met I-VI all pi exist, are unique and satisfy the estimate:
.
3.4 Evaluation of the residual term
Write the solution of the original problem (1)-(2) in the form:
(233)
where built above the AE of the solution, R - the residual term.
Fair
Theorem. Let the conditions I-IX [3].
Then the solution of problem (1)-(2)can be represented in the form
where built above the AE of the solution, the residual term satisfies the as-
ymptotic bound on the discrepancy:
A full proof is given in [3].
4 Discussion
1. Built AE solutions of singularly perturbed differential-operator equation (1) for
t>t0 where t0 >0 is any positive, independent of the ε, taking into account the estimate
(23), has the form
(244)
Functions is the solution of the initial problem for a parabolic equation:
, (25)
, (25)
which is neither a small parameter nor a parameter p.
2. The above result can be interpreted in terms of a qualitative description of the
evolution of the solution. The main term in AE has the form ,
there is a solution to the parabolic equation (26) where
. This suggests that the initial perturbation is transferred from the effective
(average) speed, and the transfer is accompanied by a "pseudodiffusion" blur. Aver-
aging the rate and "pseudodiffusion" the blur is due to the fact that there is a "rapid
mixing solution" for the variable , and the speed of migration is different for differ-
ent .
3. For technical systems with well-defined inputs you can calculate an approxi-
mate solution for t>t0 using the problem (26)-(27). For processes with poorly defined
input data (social, economic, informational) in equation (26) can give a qualitative
description of the process of moving a heterogeneous interactive information ( the
dependence on a parameter p) along social strata ( variable x) with some "effective
speed" while blur, that describes the slow the spread and diffusion of information
thanks to the intensive exchange.
4. The results obtained can be generalized to equations with a large number of spa-
tial variables, to equations with variable coefficients.
5. The most interesting results are obtained if a weak nonlinearity is added to the
right side of the equation
(26)
The AE of the solution of the equation (26) with the initial condition (2) has the
same form (3), but the equation for the determining becomes nonlinear
, (267)
,
where is determined through and the problem data. For a
different form of weak nonlinearity on the right side, the equation (27) can take the
form of a generalized Burgers equation.
6. The obtained asymptotic formulas make it possible to significantly (up to several
orders of magnitude) reduce the computational resources required for numerical cal-
culation of the solution, since the solution of a singularly perturbed differential opera-
tor equation reduces to the solution of a parabolic equation (25) without a small pa-
rameter.
5 Conclusion
1. An asymptotic expansion of the solution of the initial problem for a singularly
perturbed differential operator transfer equation is obtained. Under the conditions
imposed on the problem, the main term of the asymptotics is described by a parabolic
equation, linear or nonlinear, depending on the presence of a small nonlinearity in the
original problem. The resulting formulas can be used to calculate the solution and for
qualitative analysis of the solution behavior.
2. We can figuratively say that "strong mixing generates irreversibility", since, de-
spite the reversibility of time in the original problem, the solution quickly begins to
evolve as a solution of the parabolic equation, which is characterized by irreversibil-
ity.
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