Published: 26 September 2019

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
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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 0, which depends nonlinearly on the coordinate:

1
ddxαx+βx2Wx+ν-+pAWx-AdA-νWx=0, -<x<+.

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

2
Wx0x±, -+Wxdx=1,
3
pAA0, -+pAdA=1.

We assume pA – analytical function and p^k=-+px eixkdx – it’s Fourier transform, where A<R or:

4
p^k=p^0+p^1k+p^2k+..., k<k0, k01.

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

5
p^0=p^0=1.

In case px – even function, we have p^2s-1=0, s= 1, 2,..., and p^k – is real analytical function. From Eq. (2) we have:

6
-+W^kdk<,W^0=1.

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

7
iβW''^k-αW'^k+νρkW^k=0.

From Eq. (6) we have:

8
ρ0=p^1=-+xpxdx,
9
ρk=p^1+p^2k+p^3k2+..., k<k0.

Again, since p^k0, k, we get:

10
ρk~-1k, k.

3. Mathematical model analysis

For:

11
W^k=φke-0kψkdk,

we get:

φ''+-2ψ+iαβφ'+ψ2-ψ'-iαβψ-iνβρφ=0.

Putting:

12
ψ=iα2β,

we’ll get for φk following equation:

13
φ''-qkφ=0,

where:

14
φk=W^keiα2βk,
15
qk=-α22β2+iνβρk.

For qk we can highlight some properties.

1) From Eq. (10) it follows:

16
qk-α22β2, k.

2) From Eq. (19) we have:

17
qk=-α22β2+iνβp^1+iνβp^2k+p^3k2+..., k<k0,

or

18a
qk=q0+q1k+q2k2+..., k<k0,

where:

18b
q0=-α22β2+iνβp^1,qn=iνβp^n+1.

3) Also:

19
qk=-δk+iνβReρk.

Lemma 1. For qk:

20
qk1=qk121+1+Reρk2ν2δ2β2-1/21/2
-iqk121-1+Reρk2ν2δ2β2-121/2,

is C2 by k0,+ and Reqk1>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-qxux=0,

when x+.

Theorem 1. Let in the Eq. (21) qxC20,, qx0 for sufficiently large x and let there exist a branch qx of class C2b, such that Reqx>0, x>b0. Let further α1x=18q''q3/2-s32q'2q5/2 and α1xdx<. Then Eq. (21) has a solution:

ux=q-14xe- xqtdt1+ε2x, ε2x0, x.

Moreover, for x>0:

uxu~x-12e2xα1tdt-1,
u'xqxu~x+114q'xq32x+41+14q'xq32x×e2xα1tdt-1.

If q'xq32x0, x, then u'x=q1/4xe -xqtdt1+ε1x, ε1x0, x+.

Lemma 2. If p'^kO1k and p''^kO1k, then for Eq. (12) the previous theorem is valid.

Thus, further we solve the following problem:

22
φ''k-qkφk=0, k>0,
23
φ0=1,φk0, k+.

Here qk is given by Eq. (15). Further, we assume that the assumptions of Theorem 1 are fulfilled. In particular, the function qk is analytic when k<k0, k01 (see Eq. (18a)).

From the theory of differential equations, we obtain for the coefficients an following infinite system of equations:

24
n+1n+2an+2-s=0nasqn-s=0, n=0,1,2,...a0=1.

For a2 we immediately get at n=0:

25
a2=12q0=-α24β2+iν2βp^1.

In case of even p(x): p^1=0, a2=-α24β2. The determinant of the matrix AN of this system is:

26
N=detAN=(23)(34)...(N+1)(N+2)
=12N+1!N+2!=N+22N+1!2>0.

In these designations for φk we have the expression:

27
φk=1+a1k+a2k2+hk+a1gk=a1k+gk+1+a2k2+hk
a1g1k+h1k,

where k+gk=g1k, 1+a2k2+hk=h1k.

To find the coefficient a1, we use the asymptotic solution φk (k+), given by Theorem 1. Let k1<k0. Then by Theorem 1 we get:

28
a1g1k1+h1k1=Cq-1/4k11+ε2k1,a1g'1k1+h'1k1=-Cq1/4k11+ε1k1.

If k11, then ε1k11, ε2k11 [7, 8]. Therefore, Eq. (28) can be approximately replaced by the system:

29
a~1g1k1+h1k1=C~q-1/4k1,a~1g'1k1+h'1k1=-C~q1/4k1,

where a~1 and C~ are approximate values for a1 and C. From Eq. (29) we find:

30
a~1=-h1q1/2+h'1g1q1/2+g'1,C~=q1/4g'1h1-g1h'1g1q1/2+g'1,

where all functions are calculated when k=k1. For an approximate value φ~k of φk we therefore have:

31
φ~k=a~1g1k +h1k , 0kk1,g'1h1-g1h'1g1q12+g'1k=k1q14k1q-14ke-k1kqtdt, kk1,k11, k1<k0,φ~-k=φ~k,¯ k0.

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 φk=W^keiα2βk.

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

φ0=1,φk0, k+,
qk=q0+q1k+q2k2+..., k<k0.

3) We can get qj from:

q0=-α22β2+iνβp^1,qn=iνβp^n+1,

where p^j – are from p^s=p^s0s!=1s!is-+xspxdx, or from p^k=-+pxeixkdx with p^k=p^0+p^1k+p^2k+..., k<k0, k01.

4) Then we have solution in form φk=1+a1k+a2k2+..., 0k<k0, where aj, j2 are determined from equations:

n+1n+2an+2-s=0nasqn-s=0, n=0,1,2,..., a0=1,

and:

a1=-limk+h1kq12k+h'1kg1kq12k+g'1k,

where h1(k), g1(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.
  • Blackledge J., Lamphiere M., Panahi A. Simulation and analysis of stochastic signals using the Kolmogorov-Feller equation. IET Irish Signals and Systems Conference, 2012.
  • 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.
  • 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.
  • 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.

About this article

Received
12 May 2019
Accepted
18 June 2019
Published
26 September 2019
SUBJECTS
Mathematical models in engineering
Keywords
mathematical model
analytical solution
Kolmogorov-Feller equation
nonlinear drift coefficient
constructive method for solving