Mathematical analysis of transport systems modeled by the stationary Kolmogorov-Feller equation with a nonlinear drift coefficient

Firsov, Andrei; Zhilenkov, Anton

doi:10.21595/vp.2019.20795

Vibroengineering Procedia

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Published: 25 June 2019

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Mathematical analysis of transport systems modeled by the stationary Kolmogorov-Feller equation with a nonlinear drift coefficient

Andrei Firsov¹

Anton Zhilenkov²

^{1, 2}Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia

Corresponding Author:

Anton Zhilenkov

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Abstract

The paper proposes the formulation of problems modeled with the stationary Kolmogorov-Feller equations with a nonlinear drift coefficient. Mathematical analysis of the model is given. The basis of the proposed method is the application of the Fourier transform to obtain analytical solutions to the problems under consideration.

1. Introduction

Kolmogorov-Feller type equations, which are very popular in solving problems from various fields of natural science (Markov’s process theory, control theory, astronomy, physical and chemical processes), are integro-differential equations with (in general) non-linear coefficients. The theory of such equations (from a mathematical point of view) is sufficiently well developed only for the case of coefficients that depend linearly on the independent variable. In the general case, as far as we know, there are few accurate analytical results, and there are practically no strictly grounded constructive methods suitable for applications. In this paper, we propose an analytically justified constructive method for constructing solutions of the corresponding integro-differential equation of Kolmogorov-Feller type for the case of a quadratic dependence of the drift coefficient on the independent variable. The proposed method is very convenient for numerical implementation and, as far as we know, is not found in the literature.

2. Mathematical model of the problem

Consider a variant of the Kolmogorov-Feller Eq. (1), which occurs in control theory, communication theory, stellar dynamics. In the literature devoted to analytic constructions of solutions of equations of type Eq. (1), the case of a linear dependence ( $β =$ 0) of the drift coefficient on the coordinate is usually considered; see, for example, [1-7]. In this paper, we will consider $β \neq$ 0:

1

$\frac{d}{d x} [(α x + β x^{2}) W (x)] + ν \int_{- \infty}^{+ \infty} p (A) W (x - A) d A - ν W (x) = 0, - \infty < x < + \infty,$

under natural conditions:

2

$W (x) \underset{x \to \pm \infty}{\to 0}, \int_{- \infty}^{+ \infty} W (x) d x = 1,$

3

$p (A) \underset{|A| \to \infty}{\to} 0, \int_{- \infty}^{+ \infty} p (A) d A = 1 .$

Additionally, suppose $p (A)$ – analytical function with $|A| < R$ for $R$ big enough or at least its Fourier transform $\hat{p} (k) = \int_{- \infty}^{+ \infty} p (x) e^{i x k} d x$ exists and is an analytical function in a sufficiently large interval:

4

$\hat{p} (k) = {\hat{p}}_{0} + {\hat{p}}_{1} k + {\hat{p}}_{2} k + . . ., |k| < k_{0}, k_{0} ≫ 1 .$

Note that:

5

${\hat{p}}_{s} = \frac{{\hat{p}}^{(s)} (0)}{s!} = \frac{1}{s!} {(i)}^{s} \int_{- \infty}^{+ \infty} x^{s} p (x) d x .$

In particular, by virtue of Eqs. (2, 3):

6

${\hat{p}}_{0} = \hat{p} (0) = 1 .$

Also, if $p (x)$ – even function, then ${\hat{p}}_{2 s - 1} = 0$ , $s =$ 1, 2,... , and $\hat{p} (k)$ is real.

We now turn in Eq. (1) to the Fourier transform of the function $W (x)$ :

$\hat{W} (k) = \int_{- \infty}^{+ \infty} W (x) e^{i x k} d x,$

$k [i β \hat{W''} - α \hat{W'} + ν (\frac{\hat{p} - 1}{k}) \hat{W}] = 0 .$

Conditions Eq. (2) at the same time go to:

7

$\{\begin{array}{l} \int_{- \infty}^{+ \infty} |\hat{W} (k)| d k < \infty, \\ \hat{W} (0) = 1 . \end{array}$

Thus, the problem of solving an Eq. (1) with conditions Eq. (2) can be replaced by a solution of an equation:

8

$i β \hat{W''} (k) - α \hat{W'} (k) + ν ρ (k) \hat{W} (k) = 0,$

satisfying conditions Eq. (7). Denoted here:

9

$ρ (k) = \frac{\hat{p} (k) - 1}{k} .$

By virtue of Eq. (6), and the conditions imposed on $\hat{p} (k)$ , we have:

10

$ρ (0) = {\hat{p}}_{1} = \int_{- \infty}^{+ \infty} x p (x) d x .$

11

$ρ (k) = {\hat{p}}_{1} + {\hat{p}}_{2} k + {\hat{p}}_{3} k^{2} + . . ., |k| < k_{0} .$

Moreover, since $\hat{p} (k) \to 0$ , $|k| \to \infty$ (because $\hat{p}$ is a Fourier transform), then:

12

$ρ (k) ~ \frac{- 1}{k}, (|k| \to \infty) .$

3. Mathematical model analysis

Assume:

13

$\hat{W} (k) = φ (k) e^{- \int_{0}^{k} ψ (k) d k} .$

Notice, that:

14

$φ (0) = \hat{W} (0) = 1,$

15

$φ (- k) = \bar{φ (k)} .$

(The last equality is a consequence of the choice of $ψ (k),$ which will be done below – see Eq. (16)). Substituting Eq. (12) into Eq. (8), we get:

$φ'' + (- 2 ψ + i \frac{α}{β}) φ' + (ψ^{2} - ψ' - i \frac{α}{β} ψ - i \frac{ν}{β} ρ) φ = 0 .$

Assuming:

16

$ψ = i \frac{α}{2 β},$

yields the following equation for $φ (k)$ :

17

$φ'' - q (k) φ = 0,$

where:

18

$φ (k) = \hat{W} (k) e^{i \frac{α}{2 β} k},$

19

$q (k) = - \frac{α^{2}}{2 β^{2}} + i \frac{ν}{β} ρ (k) .$

Let us note some elementary properties of $q (k)$ .

1) By virtue of Eq. (12):

20

$q (k) + \frac{α^{2}}{2 β^{2}} ~ - i \frac{ν}{β k}, (|k| \to \infty),$

and in particular:

21

$q (k) \to - \frac{α^{2}}{2 β^{2}}, (|k| \to \infty) .$

2) By virtue of Eq. (11):

22

$q (k) = - \frac{α^{2}}{2 β^{2}} + i \frac{ν}{β} {\hat{p}}_{1} + i \frac{ν}{β} ({\hat{p}}_{2} k + {\hat{p}}_{3} k^{2} + . . .), |k| < k_{0},$

or

22a

$q (k) = q_{0} + q_{1} k + q_{2} k^{2} + . . ., |k| < k_{0},$

where:

22b

$\{\begin{array}{l} q_{0} = - \frac{α^{2}}{2 β^{2}} + i \frac{ν}{β} {\hat{p}}_{1}, \\ q_{n} = i \frac{ν}{β} {\hat{p}}_{n + 1} . \end{array}$

3) Let:

23

$δ = δ (k) + \frac{α^{2}}{2 β^{2}} + \frac{ν}{β} I m ρ (k) .$

Then:

24

$q (k) = - δ (k) + i \frac{ν}{β} R e ρ (k) .$

At the same time:

– if $ρ (k)$ is real (which will take place, if, for example, $p (x)$ – is an even function), then:

25

$δ = c o n s t = \frac{α^{2}}{2 β^{2}} > 0 .$

– if $I m ρ (k) \neq 0$ , then by Eq. (12), for sufficiently large $|k|$ :

26

$δ (k) = \frac{α^{2}}{4 β^{2}} + \frac{ν}{β} I m ρ (k) > 0 .$

Lemma 1. The branch of $\sqrt{q (k)} :$

27

${(\sqrt{q (k)})}_{1} = \sqrt{|q (k)|} {\{\frac{1}{2} (1 + {(1 + \frac{{[R e ρ (k)]}^{2} ν^{2}}{δ^{2} β^{2}})}^{- \frac{1}{2}})\}}^{1 / 2}$

$- i \sqrt{|q (k)|} {\{\frac{1}{2} (1 - {(1 + \frac{{[R e ρ (k)]}^{2} ν^{2}}{δ^{2} β^{2}})}^{- \frac{1}{2}})\}}^{1 / 2},$

is twice continuously differentiable by $k \in (0, + \infty)$ and $R e {(\sqrt{q (k)})}_{1} > 0$ for $k$ , which are large enough.

Remarks:

1) $\sqrt{| q (k) |} = {(δ^{2} + \frac{ν^{2}}{β^{2}} {[R e ρ (k)]}^{2})}^{1 / 4}$ .

2) If $ρ (k)$ - is real, then $R e {(\sqrt{q (k)})}_{1} > 0$ for all $k > 0$ .

3) By virtue of property Eq. (15), it suffices to construct a solution to Eq. (17) only for $k \geq 0$ [6-11].

4. Results and conclusions

An algorithm for the analytical solution of the Kolmogorov-Feller Eq. (1) is constructed and substantiated. This algorithm is as follows.

1) Enter the function $φ (k) = \hat{W} (k) e^{i \frac{α}{2 β} k}$ , where $\hat{W} (k)$ is a Fourier transform of the desired function $W (x)$ .

2) For function $φ (k)$ the equation holds:

$φ'' (k) - q (k) φ (k) = 0, k > 0,$

under conditions:

$\{\begin{array}{l} φ (0) = 1, \\ φ (k) \to 0, k \to + \infty, \end{array}$

$q (k) = q_{0} + q_{1} k + q_{2} k^{2} + . . ., |k| < k_{0} .$

3) The coefficients $q_{j}$ are from the relations:

$\{\begin{array}{l} q_{0} = - \frac{α^{2}}{2 β^{2}} + i \frac{ν}{β} {\hat{p}}_{1}, \\ q_{n} = i \frac{ν}{β} {\hat{p}}_{n + 1}, \end{array}$

where ${\hat{p}}_{j}$ are from decomposition coefficients of the Fourier transform $\hat{p} (k) = \int_{- \infty}^{+ \infty} p (x) e^{i x k} d x .$

4) The sought solution $φ (k)$ can be determined from a system of linear algebraic equations, which will be given in further work.

References

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About this article

Received

12 May 2019

Accepted

18 June 2019

Published

25 June 2019

SUBJECTS

Mathematical models in engineering

DOI

https://doi.org/10.21595/vp.2019.20795

Keywords

Kolmogorov-Feller equation

nonlinear drift coefficient

constructive method for solving

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A. Firsov and A. Zhilenkov, “Mathematical analysis of transport systems modeled by the stationary Kolmogorov-Feller equation with a nonlinear drift coefficient,” Vibroengineering PROCEDIA, Vol. 25, pp. 166–170, Jun. 2019, https://doi.org/10.21595/vp.2019.20795

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TY  - JOUR
DO  - 10.21595/vp.2019.20795
UR  - https://doi.org/10.21595/vp.2019.20795
TI  - Mathematical analysis of transport systems modeled by the stationary Kolmogorov-Feller equation with a nonlinear drift coefficient
T2  - Vibroengineering PROCEDIA
AU  - Zhilenkov, Anton
AU  - Firsov, Andrei
PY  - 2019
DA  - 2019/06/25
PB  - JVE International Ltd.
SP  - 166-170
VL  - 25
SN  - 2345-0533
SN  - 2538-8479
ER  - 

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@article{Zhilenkov_2019,
	doi = {10.21595/vp.2019.20795},
	url = {https://doi.org/10.21595/vp.2019.20795},
	year = 2019,
	month = {jun},
	publisher = {{JVE} International Ltd.},
	volume = {25},
	pages = {166--170},
	author = {Anton Zhilenkov and Andrei Firsov},
	title = {Mathematical analysis of transport systems modeled by the stationary Kolmogorov-Feller equation with a nonlinear drift coefficient},
	journal = {Vibroengineering {PROCEDIA}}
}

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[1]A. Zhilenkov and A. Firsov, “Mathematical analysis of transport systems modeled by the stationary Kolmogorov-Feller equation with a nonlinear drift coefficient,” Vibroengineering PROCEDIA, vol. 25, pp. 166–170, Jun. 2019, doi: 10.21595/vp.2019.20795.

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Zhilenkov, Anton, and Andrei Firsov. “Mathematical Analysis of Transport Systems Modeled by the Stationary Kolmogorov-Feller Equation with a Nonlinear Drift Coefficient.” Vibroengineering PROCEDIA 25 (June 25, 2019): 166–70. https://doi.org/10.21595/vp.2019.20795.

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Mathematical analysis of transport systems modeled by the stationary Kolmogorov-Feller equation with a nonlinear drift coefficient

Abstract

1. Introduction

2. Mathematical model of the problem

3. Mathematical model analysis

4. Results and conclusions

References

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