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Pirniayzova Periuza Mambetniyazovna, Linguestic colleg by Kazakh University of the world Language E-mail: aydanushka@list.ru
ANALYTICAL SOLUTION OF DIFFUSION PROBLEMS IN THE SIMULATION OF IMPURITY DIFFUSION AND OBTAINING AN EXACT SOLUTION
Abstract: In this article given in demonstrative way, it is considered decision of a one-dimensional and two-dimensional problem of diffusion, by a recurrently operational method, which describes process of distribution of harmful impurity along a watercourse.
The received numerical results on the computer, where it is possible to define what time there is a distribution and river clarification. The received results are illustrated in drawings.
Keywords: distribution process, harmful impurity, the differential equation, the diffusion equation, problem Kashi.
The process of transporting pollutants to river water is described by diffusion equations. In many monographs, the solution of these models is given only with the use of numerical methods of difference schemes, and a number of questions arise related to the choice of methods for solving a number of various problems on which the practical realizability, accuracy and duration of the solution on the computer depend. In particular: a) an analytic description of a plane or spatial domain for which diffusion equations and boundary conditions are studied, i.e. analytical description of coastlines and river bottom; b) an analytical description of the dependence of the coefficients of the equation on the spatial coordinates; c) an analytical description of the dependence of the spatial coordinates and time of inho-mogeneous parts of the solved diffusion equation, i.e. from sources of pollution; d) the correct choice in the difference scheme of the relationship between the spatial steps of the gird, and also between them and the step of discretization of time.
In the work considered by us, the solution of these equations is given by analytical, where practical realization gives an exact solution. We consider the solution of a one-dimensional, two-dimensional,
three-dimensional differential equation of parabolic type diffusion. One-dimensional production, as a special case of the three-dimensional diffusion equation, the coefficients ofwhich determine the physical processes of interest to us, in particular the spread of pollution along the width of the watercourse and the infinite speed of propagation, and a number of other physical processes are determined by analogous equations, among them the potential flow, diffusion transfer of mass, flow through a porous medium and some fully developed currents in channels.
Integro-differential operators are introduced in the form of a special numbers with constant coefficients determined from the recurrence parity associated with the differential equation under consideration. The particular solutions obtained in this case have a simpler from than in other papers, since they are constructed on the basis of one recurrence relation, which leads to simpler differentiation formulas. With this approach, the general solutions are expressed in terms of arbitrary functions and are not related to the solution of the other equation. The resulting form of general solutions makes it possible to apply the method of initial functions for solving boundary value problems, since arbitrary analytic
functions foe functions entering into general solutions can be expressed in terms of the initial functions given by the condition. From these formulas of general solutions, it is easy to find all particular solutions, it is easy to find all particular solutions in various classes of analytic functions.
Such a recurrent-operator method is very effective in constructing the solution of the equation of the theory of heat conduction. The solution of the equation of the two-dimensional diffusion problem is the same as in the solution of the one-dimensional diffusion problem, that is, the solution is also sought with the introduction of an integro-differential operator in the form of a special series with constant coefficients, determined from the recurrence relation [4]
(dt + «02Ô2 + a01dx + a00 >t) = f(X>t) , C1)
a02 = kx ' a01 = Vx ' a00 = a,
Where q(x ,t ) is the concentration of emissions; pollutant NO3 - nitarate ion; vx - is the time - averaged velocity component along the x - axis; kx - is the diffusion coefficient, a - is the rate of destruction of the substance; f (x ,t ) is the source function.
The solution of equation(l) is obtained in the form
œ œ
q=HQ / j (x ) (2)
i=0 j=0
Substituting solution (2) into (1), we obtain the following recurrent equation:
Qi,j = ao,2Qi-i,j-2 - aoQ-i,j-i - aooQ,-1,j (3)
Under the initial conditions Q0>0 = 1> Qi,j = 0, at i < 0 or j < 0
Q0,0 = 1 Q0,1 = 0; Q1,0 = —a00;
Q2,0 = a020; Qi,i = -a0i; Q0,2 =0; (4)
Q3,0 = —a00' Q2,1 = 2a0ia00' Q12 = —a02' Q0,3 = 0
Writing out the first terms of the numbers (2), we have
q(t, g (x )) = g + [^tg (x )] + [a2wt vg (x ) - a01tg'(x )] +
+ [^t 3>!g (x ) + 2a22a2lt 2,!g' (x) - a3itg" (x )] +
+ [a4wt 4,!g (x ) - 3a022a2lt 3>!g' (x ) + (2aззaзo + a2m )t 2Jg" (x )] +
-a050t '"'g(x ) - 4a232a2lt 4,'g'(x ) - 1 + a3la33)t3>!g (x) + 2a30a31t2'g (x).
A function satisfying the initial conditions for 10 = 0.1, the next:
+... .
g (x ) =-^
2wtJ na20ft Substituting (6) in(5), we have
(6)
g (t, g (x )) = Ce
(x a10t0
a00 2 g(x ) - ai0t
Ce
-a00t
(x-ai0t 0) 4a20t 0
Ce
(x-aipt0)2 ^
This series is collapsed into a function
( (x-ai0(t0 +t))■'
g(x) =-, N e[ 4a20(t0+t)
a00 (t0 +t ) I
2wyl na 20(t 0 +1 )
4 2 0 2 Ax
Figure 1. Process of distribution of emission of harmful during the initial moment of time
■7 0.000S - ' A
0,0008 - / \
0,0007 - / 1 h =50
0,0006 - f V^
0,0005 - f 1 U =100
0,0004 - j \ /
0,0003 - / X i5=150
0,0002 - Y k =
o.oooy —,—;—,-' 1—,—,—»
Figure 2. Process of distribution of harmful impurity in time
Given the function g (t, g(x)) to the values t =11 , t =12 , t =13... We construct a combined graph of the function g*, g 2, g *,...
In the recurrent-operatror method, the solution is obtained in the form (5), while and the authors of K, I, Kachiashvili, D. G. Gordeziyani obtained
(x-V (to+t ))2
g (x ,t ) = ■
N
4Kx (t 0 +t )
-a(t 0 +t )
2wyJ nKx (t0 +1 ) By the same algorithm for solving the problem of a simulation one-dimensional diffusion equation, a solution is obtained for two-dimensional diffusion equation
I= kx 0+"> 0 - S -aq+f (x 'y't > (7)
At the initial and boundary conditions
q(x, 7 ,t )| t=o =■
N
(x-ux (to +t ))2
7
4kx(to+t) 4ky(to +t )
a (to+t)
2WyJ nkxky (t 0 +1 )
Boundary conditions
dq(x, y ,t ), _dq(x, y ,t )i
dx
dx
x=\
= 4>(y ,t ) ;
0<7 <l2, 0<t<T
dq(x, 7 ,t )i = dq(x, 7 ,t ) dy '7=0 87
7=l2
= V(x ,t )
0 < x < Zp 0 < t < T; For convenience, finding the coefficients of the series that rewrote equation (7) in the form
(d2 + a1d y - a 2 dx - a3dt - a 4)q(x, y ,t) = f (x, y ,t) (8) Where a1 = -L; a2 = ^; a3 = a4 = a;
kx kx kx kx
We seek the solution of this equation in the form of a series
m M M
qr(x, y,gr) = ZZZQij X+,+k+r)ldddk g (y,t )); r = 0,1(9)
i=0 j=0 k=0
Substituting equation (9) into equation (8) and making the appropriate mixing of the indices and talking the common factor in parenthesis and equating the expressions in parenthesis to zero, we obtain the following recurrence equation
Q, j ,k =-aQij-2,k +a2Qi-i, j ,k +aA-i,j ,k-i + a4Qi-2,j ,k (10)
Under the initial Q0>0>0 = 1, ,j,k = 0 by i < 0 or j < 0, or k < 0 (11)
From the recurrence relation (10) we find coefficients of the series (9). Then changing the order of summation in (9), we obtain a solution of equation (8) in the from
qr =1 x'+^+k!(XXQ',
- j, j-k
d'-j dj-
;gr (7 ,t ), r = 0,1. (12)
=0 j=0 k=0 dy' j dtj -
We next write out the first few terms of the series
(3) substituting the coefficients
.....x2
q0 = g 0 + a 3xg 0 + [(a22 + a 4 )g 0- aig 0 + a 3 g 0]—+
+ [a2(a22 + a 4g + 2a2a3go + (-2aia2 )go]~ + ...
qi = a3go + [(a22 + a4 )go - aigo + a3go ]x +
I x2
+ [a 2 (a^ + a 4)g 0 + 2a 2a3 g o + (-2a^)g o]y +...
The particular solution of the inhomogeneous equation (8) in accordance with the recursion-operator method [5] has the form
?o u = o.i
œ œ œ
€ *( / ) =
Qi. j.
d
I (i+j+k+2) psj
dj dk
=0 j=0 k=0 i'j,k dx-i+j+k+2) dyj dtk
Thus, the general solution of the cranes (8) will be q(x, 7, t ) = q0 (x, 7, g 3 ) + qi (x, 7, g 1 ) + q * ( f ). ( 13) Solving the Cauchy problem foe equation (l) with f (x, 7 ,t ) = 0, for an instantaneous point source of a unit mass of a pollutant with an initial condition
( (x-Ux (tp +t))2 72 _a(t +t)! , M N I 4kx(t3 +t) 4k7(t3 +t) " 3 I
q(x> 7A =0 = 1 , , , , 7 7
2^ ^ (t 0 +1 ) ,
N - the capacity source of pollution for the river (mg/s); w - is the area if the rivers live section (m2). Substituting this function in a numbers
f (x, 7 ,t ),
(x > 7 > gr ) = E
YZQ j j
(i+j+k+r )
d 7 dk (g (7 ,t ));
j=0 k=0
(i + j + k + r )!
t—u „
r = 0,1
by t = 0, we find that q0 = g0, we obtain a power numbers that depends on the variable to. This numbers converges into an elementary function
N
q(x, 7 ,t ) = -
(x-uxt )2 72 4kxt 4k7t
, which coin-
cides with the results of the authors of K. I. Kachi-ashvili, D. G. Gordesiyani, D. I. Melikzhanyan, obtained by a finite-difference method.
According to (Figure 5), with the passage of time the intensity of emission of harmful impurities decreases, and according to the graph it is possible to determine the values x = l, y = m meaning that the concentration reaches the maximum allowable emission rate. The results obtained coincide with the results of other authors.
0,08
o,oe
0,04
0,02
t. = 20
Figure 3. Process of distribution of harmful impurity
Figure 4. Process of distribution of emission of harmful impurity on an axis x impurity to axes x, y
Figure 5. The process of distribution of harmful impurities over time
3
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