We consider one of the models of information (or other substance) propagation in a spatially distributed multiphase environment, when the exchange rate between phases is much higher than the transfer rate, and the number 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 obtained formulas, it follows that in the first approximation, the initial perturbation propagates at a certain average speed with simultaneous diffusion spreading. 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.
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 variable 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 characteristics of the environment, "along" which the transfer substance.
In some cases, the exchange processes between phases have significantly higher
speeds than the transfer processes (which, for example, may correspond to the
processes of information transfer in a social environment, where the exchange
"horizontally" (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(
The presence of a small parameter at higher derivative makes the problem singularly 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 solution 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 continuous parameter p, p1≤p≤ p2 . In this case the system of equations becomes one differential 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
Consider the initial problem for a singularly perturbed differential operator equation 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 (
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
AE of solutions is constructed as the sum of the functions of the surge , concentrated in the neighborhood of a line – "Pseudocharacteristic" of equations and boundary layer functions concentrated in the neighborhood of the boundary and the remainder term : here surg ;
is a variable, which describes the function of the the stretched variables, which describe the boundary layer function . The algorithm for constructing a AE similar to the algorithm described in [3]. 3.1
Function S must satisfy the original equation (
(6)
is:
where
equations for
- as yet unknown function. Write conditions for the solvability of the
and [4]:
…
,
,
,
,
(
The condition (9) is true due to the choice of the variable ζ, therefore, written as: can be where – pseudo-inverse to operator nating φ1 derived equation to determine φ0: . Substituting (8), (11) in (10), and elimi
The equation for finding the following approximations ilarly. Omitting the calculations, we give only the result.
The function is: (12) are obtained simA function
defined by the equation: 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
A function under any initial conditions for equations (12), (13) does not
satisfy initial conditions (
Substituting (17) into (15) [4], we get the equations for determining pi , Substituting the series (8), (17) in condition (14), given that , we get the resulting equations for determining the initial conditions:
Imposing some additional conditions on the eigenvalues of the operator (IVVI,[3]) and omitting intermediate calculations, we give the result. The function has the form
Substituting (8), (21) in (19) we obtain the equation for determining the initial conditions for the equations (13) and functions : (21) (22) From (22) in the IV-VI [3] we get:
Thus, the obtained initial condition for equation (13) from which , as well as the function itself .
The construction of the subsequent functions is similar [4].
Thus, all members of the far solution (
If the condition III (M<0) is met, for any and for a any fixed N and are valid estimates all exist, are unique
When the conditions are met I-VI all pi exist, are unique and satisfy the estimate: 3.4
Write the solution of the original problem (
Then the solution of problem (
(233)
A full proof is given in [3]. 4
1. Built AE solutions of singularly perturbed differential-operator equation (
Functions is the solution of the initial problem for a parabolic equation:
(244) , , (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. Averaging 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 different .
3. For technical systems with well-defined inputs you can calculate an approximate 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 spatial 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 (
6. The obtained asymptotic formulas make it possible to significantly (up to several orders of magnitude) reduce the computational resources required for numerical calculation of the solution, since the solution of a singularly perturbed differential operator equation reduces to the solution of a parabolic equation (25) without a small parameter. 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,
despite 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
irreversibility.
6. Larkin, N.A.: Korteweg–de Vries and Kuramoto–Sivashinsky equations in bounded
domains. Journal of Mathematical Analysis and Applications. 297(
17(
Publishing house MSU (1978). (in Russian)
17. Butuzov, V.F.: Asymptotics of the solution of a system of singularly perturbed equations
in the case of a multiple root of the degenerate equation. Differential equations. 50(
Space. American Mathematical Society (1974). 23. Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations. MAIA, vol. 46. Springer Science & Business Media B.V. (1999). https://doi.org/10.1007/978-94-017-1965-0