NUMERICAL SOLUTIONS TO THE CAUCHY PROBLEM FOR THE GENERALIZED NON-ISOTROPIC DIFFUSION EQUATION Текст научной статьи по специальности «Химические науки»

ARTICLE

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NUMERICAL SOLUTIONS TO THE CAUCHY PROBLEM FOR THE GENERALIZED NON-ISOTROPIC DIFFUSION

EQUATION Tozhiev Tokhirjon Khalimovich

Fergana State University, Khamraqulov Khamidullo Turgunboevich

Senior teacher at the National center for training pedagogues for new methods of the Fergana region https://doi.org/10.5281/zenodo.11505761

ABSTRACT

Received: 28th May 2024 Accepted: 05th June 2024 Online: 06th June 2024 KEYWORDS

Diffusion, method, solutions, estimates,

Monte Carlo approximate unbiased algorithm,

This article discusses one of the modern methods (Monte Carlo method) for solving boundary value problems for an ultraparabolic equation of mathematical physics. Based on the obtained results, some numerical estimates of the solution to the Cauchy type problem were carried out.

algorithm efficiency.

When studying dynamic systems on a computer, the statistical testing method (Monte Carlo method) is often used. The application of this method to study systems defined by stochastic differential equations requires their replacement with Euler and Runge-Kutta difference schemes. Such replacements are considered in the works [1, 2]. However, known estimates of the error of solutions of deterministic equations using difference methods cannot be used in the digital modeling of stochastic equations due to the fact that their solutions are not differentiable almost everywhere.

If we keep in mind that the application of the Monte Carlo method, which has demonstrated its effectiveness in multidimensional problems, then the development of methods of numerical integration in a system with many noises is very relevant.

Let us consider the Cauchy problem in the classical formulation in the (n+1)-

Q = Rn * [O, T]_

dimensional space R in the layer

(T \t A du (x,t ) k J d2u (x, t )

( Lu )(x,t ) =—^—L-y aJ-t—— +

V 1 A J * ^ dxl dxJ

lU )( X

k l

et

j=i

j

j=1 m=1

u ( x,0) = (( x )

mxJ = f ( x, t ),

dx

(1)

x G Rn,

é

Ws,

a = a I s l =

( s )

:es

mim.2

V m2 m3 y

q = q (s) = a 1 (s),

' h

- sßi Ii y mi =

a block

= exp 1

0 0

-sß 0

, d (p)

f „1/2 T

P Ik

0

n „3/2 t 0 P Ii y

(

T m2

size

4s i x k

a"1 + 3ß2aß

3

m2 =—2

size k x k, a block 2s

3

m3 =— ®

s~ size 1 x 1, matrices Ik ß'1

L

Here and below r is an identity matrix of size r x r, -t

(

s

a I

(s1 and q(s1

, where s>0, are positively defined. The ie equation (1) with a singularity at a point (has the form

r 1 \

y,r') = n 2 ||a||2 (t -t) 2 expy - Cx) d

V t-t y

f 1 \

ad

v t-t y

liai

= det ( a) y = k + 3i

n

1

é

Ws,

Br ( x, t )

T

(y,z) : (y - Cx) d

t-t

f 1 \

ad

Vt-tJ

( y - Cx )<-ln-^-, t > t v ' 2 t-t

(4)

From (4) it is clear that T satisfies the following conditions:

T< t,T> t - r

2

. Each

section of the spheroid by the horizontal plane t = const, t r < consl: <t, js an n-

dimensional ellipsoid centered at the point

At-T)ß

T. IfP< r, then Bp( xt )(= Br (t). At r ^ 0 Br (x' t), and ^Br (x' t) monotonically converge towards the center (x' t). Therefore, there is such a thing r > 0 that when (x't) eQ. B (x,t) C Q. Let r > 0 be that B (x,t) C Q

. Then to solve problem (1)-(2) the following relation is valid: u ( x, t ) = ( Er' u )( x, t) + f ( x, t)

,

where

(5)

,u )( x, t )

^ 1 12 f A--1 / n

ln Í1 ] 2

1 J VaJ

f ( x, t )= J J

Br (x,t)

J J u (y (A,û),t(A))ht (û) 4babTH (0)dsdA si (0)

Z(x,t;y,T)-n 2 ¡ai2

f ( y, t) dydT

Here (0) -is (n 1) - dimensional unit sphere with usual orthogonal coordinates:

e = (e2ieii..,en), 0<ej for 2< j<n-1, 0<en <2*. h(e)eS1 (0)

is a unit n-dimensional vector,

y (A,0)

t(A) = t - r2A2

e-r 2?}ßx +

- In Í1 11 2 d ( r2A2 ) b-iH (0)

V VAJJ

(6)

From (6) it follows that if

r (x, t)< t2

,, , ( x, t )eQ , Br ( x, t )cQ

, then for v ' we have r v ' / . . {( • '' )}

Let's proceed to constructing a Markov chain vs 0, on which we will construct

u ( x, t)

an unbiased estimate of the solution u ( x, t ) = 1

For

to problem (1)-(2). , applying formula (5) we obtain

r

1

1

é

Ws,

1 \

ln

V V^yy

d Ä

_p ¿ = e r

eplacing the variables ¿ = e , we get

r

t^- f e pp2dp

1+n f

. i + ^ V 2 y

re

r s r

r ( « )

1+2

is the gamma function. Now, we can repr

d

/ f — 1 f

) = f P1 (p)dp f p (H)u y e r ,e e r

0 51 ( 0 ) V V y V y y

P (p)

re 1 v^/ is the density of a gamma-distributed ranc I p2 (H) = HT4BaB~ U

e future we will simulate a random vector with a dis

1 2

H\=P„(Hi H \ =--V K .M.M

0

é

Ws,

3. If

n

E Ktjataj

V L=1_:

r

> E

, then ® is accepted, otherwise point (1) is repeated.

},=1

Let J= be a sequence of independent gamma distributed random variables with

parameter P2 ( H )

1 +

71 {®J}"

2 y, J 1 be a sequence of independent vectors with distribution density

. Now, we define a Markov chain with the following recurrence relations:

x0 = x t0 = t

tj = tj 1 - exp

V r y

xt = xt 1 + tj-1exP

' 1 n

J_

v ry

E b COL

S j E m t '

m=1

(8)

/ 1 £ \

x

j-1

xLo- tj - iexP

k+p k+p j

2

V r y

c=1

\ 1

EßpcxC + tj2-1 exp —- fr E bk

V r y

where * 1,2,—,k , p obtained from (6). Now, we can write (5) in the form

1

1,2,...,l j = 1,2,..., r(xJ-1,tJ-1 ) = (tJ-1 )

and relation (8) is

u

(x-1, tJ-1 ) = E(xJ-1,tJ-1 )U (^ , t' ) + f (xj-1, f -1 ).

(9)

Let us define the sequence of random variables ^ ^=0 by the following equality,

l-1

Vl =2 h (xJ, fJ) f (yJ J) + «(xl, tl),

(10)

j=1

(y^). B (xt) (xj,tj)

\ ' ic a ranrlnm nnint nf thp cnVipriral r\ ' / for fixed

where is a random point of the spherical

distribution density in it

, having a

'( x, tj ; y,r)-

-n 1 ■n 2 IUI 2 r~r

h ( xj, tj )

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h ( x, t )= JJ

where Br( x,t )

Z(x,t;y,T)-n 2 h 2

Assuming that u (x't) t and applying formula (9) for J 1 from (8) we obtain

dydT.

j = 1

h(x,t) = r2 (x,t)(j/(r + 2})l1+ 2J. (n)

Let ^=0 be a sequence of a - algebras generated by random variables

0 „1 (yV0),(yV),...,(y-V-1 )

and a sequence of vectors a ,a and random points

f,A as uf Ax,t).

Let us denote the solutions to problem (1)-(2) corresponding to the given f ,V as "f ,(p Theorem 1

a) the sequence 1 l 'l=0 forms a martingale relative to the sequence of a -algebras

ft £0 ■

b) if

Uf ,0 ( x, t )<+œ , u\f\,0 ( x,t )<+œ

, then it is n1 square integrable.

in}

i •/ i 3

Proof: First we prove that ^ 'l=0 forms a martingale. From the definition of l it is

clear that •l is 3l - measurable, then using the property of conditional mathematical expectation and formulas (9, 11) we get

E( x, t) •+1 / & ] = xJ, tJ) f( yJ T) + u( xl, tl) = •

• H |3 H

Hence, it follows that 1 l il=0 is martingale with respect to 1 l il=0. We will prove that

E

(x,tn <

< œ

i-1

1 = Ex,t)(Z h(xj,tJ)f (yj,tJ))2 <+«

j=0

r2 ( x t ) < t

Dividing I into two terms, from the final condition V ' / we obtain that

h ( x, t )<

/ \1+n/ 7

v7 + 2 y

of these conditions we obtain the proof of the theorem.

Now we will show one of the ways to estimate the value from one random node

f ( ^ t )= JJ

Br (x,t)

-n 1

z (x, t;y,T)-n 2 H2

f ( y, t) dydT.

œ

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has the following relation:

Lemma 1. The function f ( ^ ' )

' 7 V 22Ef(j,t(Ï,Ç)),

f ( x, t ) = where

vr + 2 y

^1+n

-r2 exp

r+2

l^r f \ f

ßx +

r

r + 2

a

r2 exp

r + 2

y (€,C,®) =e

r(£^) =f - exP

Here ^- is a gamma distributed random variable with parameter

C/rb~C

(12)

r + 2

ç2t r.

2 ) Ç

beta

( /r,2 )

distributed random variable with parameter ^' ', a is a random unit vector.

Proof: Let's introduce the domain

,r): yTa(1/ r) ad(1/ r

Br =i( y,z) : yTa (1/z) ad (1/z) y <r/2ln r2/z, z> o},

r , r

The resulting mirror image of spheroids Br (0,0) relative to the plane r = 0. These domains will also be called spheroids (radius ). Then we have

II ||1/2 -

f (x, t) = ■^TV JJ [r J r 72 exp (-yTd (1 / r) ad (1 / r) y ) -

n/2 r n r'

Br

1

X

xf (e zßx + y,t-t)dydz

Let us make a change of variables and some integral transformation and obtain the proof of Lemma 1.

Let us consider the question of the computational feasibility of estimate (10). Let's take s

(an)e= (Rn *[o,£])

small enough and consider

NS = min il :(xl, tl ) g (öQ)s}

Let 1 ^ be the moment of the first hit of process (x ,^ )

within (, i.e. Ns. moment of stopping the process (Markov moment).

Lemma 2. The inequality holds:

E(x,t) Ns <

r = 2

r

! n A1+T

V / y

2 t S

Proof. Taking u(x f) f and applying formulas (10) and (11), we get

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Ne-1

t = U^xt) > E(x,t) Z h (x, t

j=1

i \

j -

-

V- + 2 J

Ne-1

From the definition of r (x' t) it follows that

t >

\x,t) Z r2 (x , tj

j=1

r 2( xj, tt ) = {tj }>£

/ \1+n/2

-

From here we get that

V- + 2 J

sE( x,t ) Ns

Hence

E(x,t)Ns <

/ , W+n/2 - + 2

V - J

. The lemma has been proved.

Theorem 2. Let the conditions of Theorem 1 be satisfied. Then £ is an unbiased

estimate for u( x't). Its variance is finite.

Proof. From Theorem 1 it follows that it is r1 quadratically integrable and hence r1 is

uniformly integrable and N . Further, the moment the process stops is a Markov moment. Therefore, ac cording to Doob's theorem "On the transformation of free choice" and

ErN = Er = u (x, t) rN u( • t)

formula (9) £ i.e. £ is an unbiased estimate for From the

rN r DrN < Dr^ rN

definition of random variables £ and it is clear that £ . From £ a mixed

*

rN

one is built using the standard method, but practically realizable estimate s . Let ^ (x' 0) = • e R and (t ) be the point closest to . Estimated

n£-1

/ = Z h(x, tJ )f (y t ) + u(xN, tN )

j=0

replace

u(x, N ) and u(x, t " )

and

Ns-1

and get

*ne .

r/N = Z h(xj,tj)f(yj,tj) + ^(x,t )

j=0

Theorem 3. Let

u( x, t )

satisfy the Lipschitz condition and the modulus of

u( x, t )

continuity u(x1) . Then the random variable is e is a biased estimate for limited parameter function s .

E(x,t)rNE = u(x,t) u(x, t)

............. >J, x^n

*

u(x, t) D/ne

Proof. Since ^^) N u(x1), then

| u(x't) - E(xt)rfN£ 1=1 E(x,t)VNe - E(x,t)VNs 1=1 E(x,t)u(.xtNe ) -

-E{x,t)Vl(x t* ^ E( x,t) 1 u(x'tNs ) - u(x'^ £)l= A(s)

The theorem 3 has been proved.

é

Ws,

a =

at

1 0

ß =

1

v^

Exact solution u(x, y,z,f) = e sin(x + y + z)

The results of the numerical experiment are given in the table.

Number Point exact Selective

of tracks x,y,z,t s solution assessment 3-sigma

100 0.7, 0.7, 0.7, 0.7 0.005 1.738 1.732 0.017

500 0.6, 0.6, 0.6, 0.6 0.005 1.774 1.647 0.005

100 0.6, 0.6, 0.6, 0.3 0.005 1.315 1.317 0.005

100 0.8, 0.8, 0.8, 0.8 0.005 1.503 1.513 0.020

100 0.5, 0.5, 0.5, 0.3 0.005 1.349 1.358 0.0058

References:

1. S.M. Ermakov, A.S. Rasulov, M.T. Bakoev, A.Z. Vaselovskaya. Selected algorithms of the Monte Carlo method / Tashkent: University, 1992. 132 p.

2. M.Bakoev Solution of a mixed problem for the Kolmogorov equation. // Problems of computational and applied mathematics, No. 2 2019, pp. 60-70. (01.00.00., No. 9)

3. Doob J.L. Classical Potential Theory and its. Probabilistic Counterpart. Springer-Varlag. 1984. 846 p.

4. Kolmogorov A.N. Is ber die analitichen Metho der in der Wahrscheinlichkeizsrechnung. Mathemat. Annalen. 1931. 104 p. 415-458.

5. T.T Halimovich, I.S Mamirovich. Monte Carlo method for constructing an unbelised assessment of diffusion problems. European science review, 2020

6. Tozhiev T.Kh., Ibragimov Sh.M. Stochastic approximation methods for solving diffusion problems. "Fundamental and applied scientific research: current issues, achievements and innovations" collection of articles of the XVI International Scientific and Practical Conference. - Penza: ICNS "Science and Enlightenment". - 2018, 13-15 p.

7. Tozhiev T, Abdullaev SH, Creation of new numerical simulation algorithms for solving initial-boundary-value problems for diffusion equations - AIP Conference Proceedings, 2023

8. Тожиев.Т-Применение методов монте-карло для аппроксимации диффузионных задач.international journal of scientific researchers (IJSR) 2024