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

2Corresponding author

Vibroengineering PROCEDIA, Vol. 26, 2019, p. 94-99. https://doi.org/10.21595/vp.2019.20799
Received 12 May 2019; accepted 18 June 2019; published 26 September 2019

Copyright © 2019 Andrei Firsov, et al. 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.
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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.

Keywords: mathematical model, analytical solution, Kolmogorov-Feller equation, nonlinear drift coefficient, constructive method for solving.

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)
d d x α x + β x 2 W x + ν - + p A W x - A d A - ν W x = 0 ,           - < x < + .

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

(2)
W x 0 x ± ,                 - + W x d x = 1 ,
(3)
p A A 0 ,         - + p A d A = 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 ^ 1 k + p ^ 2 k + . . . ,         k < k 0 ,         k 0 1 .

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 ^ k d k < , W ^ 0 = 1 .

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

(7)
i β W ' ' ^ k - α W ' ^ k + ν ρ k W ^ k = 0 .

From Eq. (6) we have:

(8)
ρ 0 = p ^ 1 = - + x p x d x ,
(9)
ρ k = p ^ 1 + p ^ 2 k + p ^ 3 k 2 + . . . ,         k < k 0 .

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

(10)
ρ k ~ - 1 k ,         k .

3. Mathematical model analysis

For:

(11)
W ^ k = φ k e - 0 k ψ k d k ,

we get:

φ ' ' + - 2 ψ + i α β φ ' + ψ 2 - ψ ' - i α β ψ - i ν β ρ φ = 0 .

Putting:

(12)
ψ = i α 2 β ,

we’ll get for φk following equation:

(13)
φ ' ' - q k φ = 0 ,

where:

(14)
φ k = W ^ k e i α 2 β k ,
(15)
q k = - α 2 2 β 2 + i ν β ρ k .

For qk we can highlight some properties.

1) From Eq. (10) it follows:

(16)
q k - α 2 2 β 2 ,         k .

2) From Eq. (19) we have:

(17)
q k = - α 2 2 β 2 + i ν β p ^ 1 + i ν β p ^ 2 k + 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)
q 0 = - α 2 2 β 2 + i ν β p ^ 1 , q n = i ν β p ^ n + 1 .

3) Also:

(19)
q k = - δ k + i ν β R e ρ k .

Lemma 1. For qk:

(20)
q k 1 = q k 1 2 1 + 1 + R e ρ k 2 ν 2 δ 2 β 2 - 1 / 2 1 / 2
          - i q k 1 2 1 - 1 + R e ρ k 2 ν 2 δ 2 β 2 - 1 2 1 / 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 - q x u x = 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:

u x = q - 1 4 x e -   x q t d t 1 + ε 2 x ,         ε 2 x 0 ,       x .

Moreover, for x>0:

u x u ~ x - 1 2 e 2 x α 1 t d t - 1 ,
u ' x q x u ~ x + 1 1 4 q ' x q 3 2 x + 4 1 + 1 4 q ' x q 3 2 x × e 2 x α 1 t d t - 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 - q k φ k = 0 ,         k > 0 ,
(23)
φ 0 = 1 , φ k 0 ,         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 + 1 n + 2 a n + 2 - s = 0 n a s q n - s = 0 ,         n = 0,1 , 2 , . . . a 0 = 1 .

For a2 we immediately get at n=0:

(25)
a 2 = 1 2 q 0 = - α 2 4 β 2 + i ν 2 β p ^ 1 .

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

(26)
N = d e t A N = ( 2 3 ) ( 3 4 ) . . . ( N + 1 ) ( N + 2 )
            = 1 2 N + 1 ! N + 2 ! = N + 2 2 N + 1 ! 2 > 0 .

In these designations for φk 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
            a 1 g 1 k + h 1 k ,

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)
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 .

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

(29)
a ~ 1 g 1 k 1 + h 1 k 1 = C ~ q - 1 / 4 k 1 , a ~ 1 g ' 1 k 1 + h ' 1 k 1 = - C ~ q 1 / 4 k 1 ,

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

(30)
a ~ 1 = - h 1 q 1 / 2 + h ' 1 g 1 q 1 / 2 + g ' 1 , C ~ = q 1 / 4 g ' 1 h 1 - g 1 h ' 1 g 1 q 1 / 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 ~ 1 g 1 k   + h 1 k   ,         0 k k 1 , g ' 1 h 1 - g 1 h ' 1 g 1 q 1 2 + g ' 1 k = k 1 q 1 4 k 1 q - 1 4 k e - k 1 k q t d t ,           k k 1 , k 1 1 ,           k 1 < k 0 , φ ~ - k = φ ~ k , ¯             k 0 .

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:

φ 0 = 1 , φ k 0 ,         k + ,
q k = q 0 + q 1 k + q 2 k 2 + . . . ,         k < k 0 .

3) We can get qj from:

q 0 = - α 2 2 β 2 + i ν β p ^ 1 , q n = i ν β p ^ 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 + 1 n + 2 a n + 2 - s = 0 n a s q n - s = 0 n = 0,1 , 2 , . . . ,         a 0 = 1 ,

and:

a 1 = - l i m k + h 1 k q 1 2 k + h ' 1 k g 1 k q 1 2 k + g ' 1 k ,

where h1(k), g1(k) are determined from Eqs. (30), (31).

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