Construction of an algorithm for the analytical solution of the Kolmogorov-Feller equation with a nonlinear drift coefficient

Firsov, Andrei; Zhilenkov, Anton

doi:10.21595/vp.2019.20799

Vibroengineering Procedia

Browse Procedia

Published: 26 September 2019

Check for updates

Construction of an algorithm for the analytical solution of the Kolmogorov-Feller equation with a nonlinear drift coefficient

Andrei Firsov¹

Anton Zhilenkov²

¹Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia

²Saint Petersburg State Marine Technical University, Saint Petersburg, Russia

Corresponding Author:

Anton Zhilenkov

Cite the article Download PDF

Downloads 1335

Abstract

The paper proposes a constructive method for solving the stationary Kolmogorov-Feller equation with a nonlinear drift coefficient. The corresponding algorithms are constructed and their convergence is justified. The basis of the proposed method is the application of the Fourier transform.

1. Introduction

This paper proposes an approach to constructing solutions of differential equations of fractional order of the Kolmogorov-Feller type. We consider equations with nonlinear coefficients, namely the case of a quadratic dependence of the drift coefficient on the independent variable. As far as we know, this method of construction is not presented in the literature. The advantage of the method is its effectiveness in numerical implementation.

2. Mathematical model of the problem

Let’s consider the form of the Kolmogorov-Feller Eq. (1) with the drift coefficient $β \neq$ 0, which depends nonlinearly on the coordinate:

1

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

In the literature, it is customary to consider the simplified case $β =$ 0. In our case for normal form we have:

2

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

3

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

We assume $p (A)$ – analytical function and $\hat{p} (k) = \int_{- \infty}^{+ \infty} p (x) e^{i x k} d x$ – it’s Fourier transform, where $|A| < R$ or:

4

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

Insofar as Eqs. (2-3), we have:

5

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

In case $p (x)$ – even function, we have ${\hat{p}}_{2 s - 1} = 0$ , $s =$ 1, 2,..., and $\hat{p} (k)$ – is real analytical function. From Eq. (2) we have:

6

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

Obviously, we can go from solving the Eq. (1) with Eq. (2), to the equation:

7

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

From Eq. (6) we have:

8

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

9

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

Again, since $\hat{p} (k) \to 0$ , $|k| \to \infty$ , we get:

10

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

3. Mathematical model analysis

For:

11

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

we get:

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

Putting:

12

$ψ = i \frac{α}{2 β},$

we’ll get for $φ (k)$ following equation:

13

$φ'' - q (k) φ = 0,$

where:

14

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

15

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

For $q (k)$ we can highlight some properties.

1) From Eq. (10) it follows:

16

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

2) From Eq. (19) we have:

17

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

or

18a

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

where:

18b

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

3) Also:

19

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

Lemma 1. For $\sqrt{q (k)}$ :

20

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

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

is $C^{2}$ by $k \in (0, + \infty)$ and $R e {(\sqrt{q (k)})}_{1} > 0$ for $k$ , which are large enough.

4. Construction of the solution of the transfer theory problem

We will use the well-known asymptotic theorem for solving the equation:

21

$u'' (x) - q (x) u (x) = 0,$

when $x \to + \infty$ .

Theorem 1. Let in the Eq. (21) $q (x) \in C^{2} (0, \infty)$ , $q (x) \neq 0$ for sufficiently large $x$ and let there exist a branch $\sqrt{q (x)}$ of class $C^{2} (b, \infty)$ such that $R e \sqrt{q (x)} > 0$ , $x > b \geq 0 .$ Let further $α_{1} (x) = \frac{1}{8} \frac{q''}{q^{3 / 2}} - \frac{s}{32} \frac{{[q']}^{2}}{q^{5 / 2}}$ and $\overset{\infty}{\int ‍} |α_{1} (x)| d x < \infty$ . Then Eq. (21) has a solution:

$u (x) = q^{- \frac{1}{4}} (x) e^{- \int_{}^{x} \sqrt{q (t)} d t} [1 + ε_{2} (x)], ε_{2} (x) \to 0, (x \to \infty) .$

Moreover, for $x > 0$ :

$|\frac{u (x)}{\tilde{u} (x)} - 1| \leq 2 (e^{2 \int_{x}^{\infty} |α_{1} (t)| d t} - 1),$

$|\frac{u' (x)}{\sqrt{q (x)} \tilde{u} (x)} + 1| \leq \frac{1}{4} |\frac{q' (x)}{q^{\frac{3}{2}} (x)}| + 4 (1 + \frac{1}{4} |\frac{q' (x)}{q^{\frac{3}{2}} (x)}|) \times (e^{2 \int_{x}^{\infty} |α_{1} (t)| d t} - 1) .$

If $\frac{q' (x)}{q^{\frac{3}{2}} (x)} \to 0$ , $(x \to \infty),$ then $u' (x) = q^{1 / 4} (x) e^{- \int_{}^{x} \sqrt{q (t)} d t} (1 + ε_{1} (x))$ , $ε_{1} (x) \to 0$ , $x \to + \infty$ .

Lemma 2. If $|\hat{p'} (k)| \leq O (\frac{1}{k})$ and $|\hat{p''} (k)| \leq O (\frac{1}{k}),$ then for Eq. (12) the previous theorem is valid.

Thus, further we solve the following problem:

22

$φ^{'' (k)} - q (k) φ (k) = 0, k > 0,$

23

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

Here $q (k)$ is given by Eq. (15). Further, we assume that the assumptions of Theorem 1 are fulfilled. In particular, the function $q (k)$ is analytic when $|k| < k_{0}$ , $k_{0} ≫ 1$ (see Eq. (18a)).

From the theory of differential equations, we obtain for the coefficients $a_{n}$ following infinite system of equations:

24

$\{\begin{array}{l} (n + 1) (n + 2) a_{n + 2} - \sum_{s = 0}^{n} a_{s} q_{n - s} = 0, n = 0,1, 2, . . . \\ a_{0} = 1 . \end{array}$

For $a_{2}$ we immediately get at $n = 0$ :

25

$a_{2} = \frac{1}{2} q_{0} = - \frac{α^{2}}{4 β^{2}} + i \frac{ν}{2 β} {\hat{p}}_{1} .$

In case of even $p (x)$ : ${\hat{p}}_{1} = 0$ , $a_{2} = - \frac{α^{2}}{4 β^{2}}$ . The determinant of the matrix $A_{N}$ of this system is:

26

$∆_{N} = d e t A_{N} = (2 \cdot 3) (3 \cdot 4) . . . (N + 1) (N + 2)$

$= \frac{1}{2} (N + 1)! (N + 2)! = \frac{N + 2}{2} {[(N + 1)!]}^{2} > 0 .$

In these designations for $φ (k)$ we have the expression:

27

$φ (k) = 1 + a_{1} k + a_{2} k^{2} + h (k) + a_{1} g (k) = a_{1} (k + g (k)) + 1 + a_{2} k^{2} + h (k)$

$\equiv a_{1} g_{1} (k) + h_{1} (k),$

where $k + g (k) = g_{1} (k)$ , $1 + a_{2} k^{2} + h (k) = h_{1} (k)$ .

To find the coefficient $a_{1}$ , we use the asymptotic solution $φ (k)$ ( $k \to + \infty$ ), given by Theorem 1. Let $k_{1} < k_{0}$ . Then by Theorem 1 we get:

28

$\{\begin{array}{l} a_{1} g_{1} (k_{1}) + h_{1} (k_{1}) = C q^{- 1 / 4} (k_{1}) (1 + ε_{2} (k_{1})), \\ a_{1} {g'}_{1} (k_{1}) + {h'}_{1} (k_{1}) = - C q^{1 / 4} (k_{1}) (1 + ε_{1} (k_{1})) . \end{array}$

If $k_{1} ≫ 1$ , then $|ε_{1} (k_{1})| ≪ 1$ , $|ε_{2} (k_{1})| ≪ 1$ [7, 8]. Therefore, Eq. (28) can be approximately replaced by the system:

29

$\{\begin{array}{l} {\tilde{a}}_{1} g_{1} (k_{1}) + h_{1} (k_{1}) = \tilde{C} q^{- 1 / 4} (k_{1}), \\ {\tilde{a}}_{1} {g'}_{1} (k_{1}) + {h'}_{1} (k_{1}) = - \tilde{C} q^{1 / 4} (k_{1}), \end{array}$

where ${\tilde{a}}_{1}$ and $\tilde{C}$ are approximate values for $a_{1}$ and $C$ . From Eq. (29) we find:

30

$\{\begin{array}{l} {\tilde{a}}_{1} = - \frac{h_{1} q^{1 / 2} + {h'}_{1}}{g_{1} q^{1 / 2} + {g'}_{1}}, \\ \tilde{C} = q^{1 / 4} \frac{{g'}_{1} h_{1} - g_{1} {h'}_{1}}{g_{1} q^{1 / 2} + {g'}_{1}}, \end{array}$

where all functions are calculated when $k = k_{1} .$ For an approximate value $\tilde{φ} (k)$ of $φ (k)$ we therefore have:

31

$\{\begin{array}{l} \tilde{φ} (k) = \{\begin{array}{l} {\tilde{a}}_{1} g_{1} (k_{}) + h_{1} (k_{}), 0 \leq k \leq k_{1}, \\ {(\frac{{g'}_{1} h_{1} - g_{1} {h'}_{1}}{g_{1} q^{\frac{1}{2}} + {g'}_{1}})|}_{k = k_{1}} q^{\frac{1}{4}} (k_{1}) q^{- \frac{1}{4}} (k) e^{- \int_{k_{1}}^{k} \sqrt{q (t)} d t}, k \geq k_{1}, \\ k_{1} ≫ 1, k_{1} < k_{0}, \end{array} \\ \tilde{φ} (- k) = \bar{\tilde{φ} (k),} k \geq 0 . \end{array}$

5. Results and conclusions

For construction of the analytical solution of the Kolmogorov-Feller Eq. (1) one can use the following algorithm.

1) Take the desired function $φ (k) = \hat{W} (k) e^{i \frac{α}{2 β} k}$ .

2) For $φ (k)$ we have $φ'' (k) - q (k) φ (k) = 0$ , $k > 0$ under:

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

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

3) We can get $q_{j}$ from:

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

where ${\hat{p}}_{j}$ – are from ${\hat{p}}_{s} = \frac{{\hat{p}}^{(s)} (0)}{s!} = \frac{1}{s!} {(i)}^{s} \int_{- \infty}^{+ \infty} x^{s} p (x) d x$ , or from $\hat{p} (k) = \int_{- \infty}^{+ \infty} p (x) e^{i x k} d x$ with $\hat{p} (k) = {\hat{p}}_{0} + {\hat{p}}_{1} k + {\hat{p}}_{2} k + . . .$ , $|k| < k_{0}$ , $k_{0} ≫ 1 .$

4) Then we have solution in form $φ (k) = 1 + a_{1} k + a_{2} k^{2} + . . .$ , $0 \leq k < k_{0}$ , where $a_{j}$ , $j \geq 2$ are determined from equations:

$(n + 1) (n + 2) a_{n + 2} - \sum_{s = 0}^{n} a_{s} q_{n - s} = 0, n = 0,1, 2, . . ., a_{0} = 1,$

and:

$a_{1} = - \underset{k \to + \infty}{l i m} \frac{h_{1} (k) q^{\frac{1}{2}} (k) + {h^{'}}_{1} (k)}{g_{1} (k) q^{\frac{1}{2}} (k) + {g^{'}}_{1} (k)},$

where $h_{1} (k)$ , $g_{1} (k)$ are determined from Eqs. (30), (31).

References

Kim Ju Gyong, Choe Il Su A solution to Kolmogorov-Feller equation and pricing of option. International Symposium in Commemoration of the 65th Anniversary of the Foundation of Kim Il Sung University (Mathematics), 2011.

Search CrossRef
Blackledge J., Lamphiere M., Panahi A. Simulation and analysis of stochastic signals using the Kolmogorov-Feller equation. IET Irish Signals and Systems Conference, 2012.

Publisher
Popov A. V., Seredenko N. A., Zhilenkov A. A. Control system of multi-level manipulator with rotational degrees of mobility. Proceedings of the IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering, 2019.

Publisher
Zhilenkov A. High productivity numerical computations for gas dynamics modelling based on DFT and approximation. IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering, 2019.

Publisher
Nyrkov A. P., Chernyi S. G., Sokolov S. S., Zhilenkov A. Optimization problem of thermal field on surface of revolving susceptor in vapor-phase epitaxy reactor. IOP Conference Series: Earth and Environmental Science, Vol. 87, Issue 8, 2017, p. 082060.

Publisher

About this article

Received

12 May 2019

Accepted

18 June 2019

Published

26 September 2019

SUBJECTS

Mathematical models in engineering

DOI

https://doi.org/10.21595/vp.2019.20799

Keywords

mathematical model

analytical solution

Kolmogorov-Feller equation

nonlinear drift coefficient

constructive method for solving

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Previous article in issue Previous Next article in issue Next

Research article

2019 06 25

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

Andrei Firsov, Anton Zhilenkov

Research article

2018 09 30

Analytical solution for modal analysis of Euler-Bernoulli and Timoshenko beam with an arbitrary varying cross-section

Fatemeh Sohani, H. R. Eipakchi

Research article

2017 08 15

The analytical solution of 2D electromagnetic wave equation for eddy currents in the cylindrical solid rotor structures

Lale T. Ergene, Yasemin D. Ertuğrul

A. Firsov and A. Zhilenkov, “Construction of an algorithm for the analytical solution of the Kolmogorov-Feller equation with a nonlinear drift coefficient,” Vibroengineering PROCEDIA, Vol. 26, pp. 94–99, Sep. 2019, https://doi.org/10.21595/vp.2019.20799

Copy Extrica

Copied to clipboard!

TY  - JOUR
DO  - 10.21595/vp.2019.20799
UR  - https://doi.org/10.21595/vp.2019.20799
TI  - Construction of an algorithm for the analytical solution of the Kolmogorov-Feller equation with a nonlinear drift coefficient
T2  - Vibroengineering PROCEDIA
AU  - Zhilenkov, Anton
AU  - Firsov, Andrei
PY  - 2019
DA  - 2019/09/26
PB  - JVE International Ltd.
SP  - 94-99
VL  - 26
SN  - 2345-0533
SN  - 2538-8479
ER  - 

Copy Ris

Copied to clipboard!

@article{Zhilenkov_2019,
	doi = {10.21595/vp.2019.20799},
	url = {https://doi.org/10.21595/vp.2019.20799},
	year = 2019,
	month = {sep},
	publisher = {{JVE} International Ltd.},
	volume = {26},
	pages = {94--99},
	author = {Anton Zhilenkov and Andrei Firsov},
	title = {Construction of an algorithm for the analytical solution of the Kolmogorov-Feller equation with a nonlinear drift coefficient},
	journal = {Vibroengineering {PROCEDIA}}
}

Copy Bibtex

Copied to clipboard!

[1]A. Zhilenkov and A. Firsov, “Construction of an algorithm for the analytical solution of the Kolmogorov-Feller equation with a nonlinear drift coefficient,” Vibroengineering PROCEDIA, vol. 26, pp. 94–99, Sep. 2019, doi: 10.21595/vp.2019.20799.

Copy IEEE

Copied to clipboard!

Zhilenkov, Anton, and Andrei Firsov. “Construction of an Algorithm for the Analytical Solution of the Kolmogorov-Feller Equation with a Nonlinear Drift Coefficient.” Vibroengineering PROCEDIA 26 (September 26, 2019): 94–99. https://doi.org/10.21595/vp.2019.20799.

Copy Chicago

Copied to clipboard!

Construction of an algorithm for the analytical solution of the Kolmogorov-Feller equation with a nonlinear drift coefficient

Abstract

1. Introduction

2. Mathematical model of the problem

3. Mathematical model analysis

4. Construction of the solution of the transfer theory problem

5. Results and conclusions

References

About this article

Related Articles