CC BY
4
MSC 65M32; 80A23
DOI: 10.14529/ mmp210208
THE PROBLEM OF IDENTIFYING THE TRAJECTORY OF A MOBILE POINT SOURCE IN THE CONVECTIVE TRANSPORT EQUATION
Kh.M. Gamzaev, Azerbaijan State Oil and Industry University, Baku, Azerbaijan, [email protected]
We consider the problem of identifying the trajectory of a mobile point source described by the Delta function in a one-dimensional linear convective transport equation under a given additional boundary condition. To solve this problem, the Delta function is approximated by a continuous function and a discrete analog of the problem is constructed using finite-difference approximations in the form of an implicit difference scheme. To solve the resulting difference problem, we propose a special representation that allows to split the problem into two mutually independent linear first-order difference problems at each discrete value of a time variable. The result is an explicit formula for determining the position of a mobile point source for each discrete value of a time variable. Based on the proposed computational algorithm, numerical experiments were performed for model problems.
Keywords: convective transport equation; mobile point source; identification problem; source motion law; delta function approximation.
Introduction
It is known that the process of one-dimensional transfer of a substance or any physical quantity (mass, momentum, energy, etc.) by a medium moving at a speed v(x,t), with disregard of diffusion and in the presence of sources (drains) in the medium, is described by the convective transfer
du du
——I- u(x, t) ——I- A(x, t)u = g(x, t),
d t dX
where the term A(x, t)u(x, t) describes the absorption or release of a substance and the term g(x,t) describes the action of an external source. This equation is used to describe a wide class of processes in ecology, heat transfer, hydrodynamics, acoustics, plasma physics, etc. [1-3]. Numerous papers are devoted to analytical and numerical research of direct initial boundary value problems for the convective transport equation [1-5].
It should be noted that in many practical cases, external sources are represented as a mobile point source and the Dirac Delta function is used to describe such sources, i.e.
g(x,t) = q(t)8(x — r(t)),
where 8(x — r(t)) is the Dirac Delta function, r(t) is the law of motion of a point source, and q(t) is the power of the source. One of the main problems that arise in the study of processes with mobile point sources is to determine the law of motion of the source, i.e. the trajectory of the source. However, at present, the problems associated with identifying the trajectory of mobile point sources are not sufficiently studied. In this paper, the problem of identifying the trajectory of a mobile point source is presented as an inverse problem of mathematical physics.
1. Problem Statement and Solution Method
Let us consider the convective transport equation in the presence of a mobile point source
du du
— + u(x, t)— + A(x, t)u = q(t)5(x - r(t)), 0<x<l,0<t<T, (1)
dt dx
under the following conditions
u(x, 0) = p(x), 0 < x < /, (2)
u(0, t) = 0(t), 0 < t < T. (3)
It is known that the direct problem for equation (1) consists of defining a function from equation (1) with the given coefficients v(x,t), A(x,t), the right-hand side q(t)i(x — r(t)), and additional conditions (2), (3). Let us assume that in addition to the unknown function u(x,t), the trajectory of a mobile point source r(t) is also unknown, and we need to construct this function using the following additional condition
u(1,t) = f (t), 0 < t < T, (4)
where f (t) is the specified function.
Thus, the identification problem is to determine the functions u(x,t) and r(t) that satisfy equation (1) and conditions (2) - (4). This problem belongs to the class of inverse problems related to the recovery of the right parts of partial differential equations [6,7]. To eliminate the singularity in equation (1), we approximate the Delta function with a continuous function [8]. For this purpose, we use the following ratio
8(x — r(t)) = v^Ae-£(:c-r(i))2,
where e is a positive number. Having previously de-dimensioned the spatial variable x by introducing a scale l-1, we take into account the approximation of the Delta function and conditions (2) - (4) in order to represent equation (1) as follows
du v(x,t)du w . ,\/eh s(x-m)2 „ , „ m
-7T- + , 7T- + A x, t)u = q(t)^-r—e , 0 < x < 1, 0 < t < T, 5
dt l dx l
u(x, 0) = p(x), 0 < x < 1, (6)
u(0, t) = 0(t), 0 < t < T, (7)
u(1,t) = f (t), 0 < t < T, (8)
where £(t) = r(t)1-1.
First, we construct a discrete analog of problem (5) - (8). To this end, we introduce a uniform difference grid in a rectangular domain {0 < x < 1, 0 < t < T}
u = {(tj,x,): x, = ¿Ax, tj = jAt, i = 0, 1, 2,...,n, j = 0, 1, 2, ...,m} with the step Ax = 1/n for the variable x and the step At = T/m for the time t.
Вестник ^ЭУрГУ. Серия «Математическое моделирование и программирование» (Вестник ЮУрГУ ММП). 2021. Т. 14, № 2. С. 78-84
Using implicit time approximation, the discrete analog of problem (5) - (8) on the difference grid To is represented as
ul~ul 1 , vl ul ~ ul-i ,
At + I Ax + % %
1, 2,
,n - 1, (9)
j u0
oj,
(10)
ui
fj ,j = 1, 2, ...,m,
(11)
u° = (p(xi ), i = 0,1, 2,... ,n,
(12)
where uj « u(xi,tj), £j « £(tj), Aj = A(xi,tj), vj = v(xi,tj), = q(tj), ^ = 9(tj), = f (tj).
The constructed difference problem (9) - (12) is a system of linear algebraic equations in which the approximate values of the desired functions u(x,t) and £(t) in the nodes of the difference grid To are unknown, i.e. u\, , i = 0,1, 2,..., n,, j = 1,2, ..., m.
In order to divide difference problem (9) - (12) into mutually independent subproblems, each of which can be solved independently, we take
e-e(xi-)2 _ e-e(xi-tj)2
and the solution to problem (9)-(12) for each fixed value j, j as [9-11]
u
wi + Pi e
j es(xi-tj )2
0, 1, 2,
1, 2,
n,
, m, we represent
(13)
where wj, pj are variables, which are not yet known.
Substituting the expression uj in each equation of system (9), (10), we get
j j—i w — u
At
j j j vj wi — wi—i 3 3
I -JL—L-— -f
I Ax 11
+
+e—s(xi—ij )2
3 j j
Pl_ _
At I Ax
pI-1 . w- j + K'Pi - -
i
wj + pj e—£ (x—3 )2 = 0j.
From the last relations we obtain the following first order difference problems for determining auxiliary variables wj, pj
j j— i wi — u
At
vij wij — wij i j j
+ -r A + W = 0, i = 1, 2,
i
Ax
w
0j,
(14)
(15)
j j j
PL + ^n_
At I Ax
j
pi—i
+ % Pj
qty/epK
1, 2,
n,
(16)
0
.. , n
= 0, j = 1, 2, 3,..., m. (17)
It is obvious that the solutions to obtained difference problems (14), (15) and (16), (17) for each fixed value j = 1, 2, . . . , m, regardless of £j, can be determined by the formulas
j /Дж j—i
wJ, =---щ +
г /Дж + vjAt + Aj/ДжД^ г
vj At
+-—-=-.,i=l,n,wi = e3, (18)
lAx + uAt + XlAxAt ° V ^
v]At ,• qj \/e / 7Г Ax At - , , N
=-—-=-p ,+-UVJ.-:-,i=l,n,d = 0. 19
г /Дж + vj At + Aj/ДжДГ i-1 /Дж + vjAt + Aj /ДжД^
And substituting representation (13) in (11), we have
< + -?J )2 = fj. From here, we can determine the approximate value of the desired function £(t) for t = tj, i.e. £j
£j
--In
£
f j - wn
- A --m --r— . (20)
Thus, the computational algorithm for solving difference problem (9) - (12) by determining u], i = 0,n, and for each fixed value j = 1, 2, . . . ,m consists of the following steps.
Step 1. Solve two independent first-order difference problems (14), (15) and (16), (17) with respect to auxiliary variables wf, pi, i = 0,n, using formulas (18) and (19).
Step 2. Determine the approximate value of the desired function £(t) for t = tj, i.e. £j, by formula (20).
Step 3. Calculate the values of variables uj using formula (13).
2. Results of Numerical Calculations
To find out the effectiveness of the proposed computational algorithm, a numerical experiment was conducted for model problems with dimensionless variables. The numerical experiment was carried out according to the following scheme.
1. For a given function £(t), 0 < t < T, the solution to problem (5) - (7) is defined, i.e. the function u(x,t), 0 < x < 1, 0 < t < T.
2. The found dependency f (t) = u(1,t) is accepted as accurate data for solving the inverse recovery problem £ (t).
Table presents the results of the numerical experiment conducted for the case l =10000, T =200, e =12,57, A(x,t) =0,002, v(x,t) = 2, 6(t) = 2, (p(x) = 0. Here t is the time, C and £ are the exact and calculated values of the function £(i), respectively. As recoverable functions, we use £(t) = 0, 6 + 0, 3sin4t, £(t) = t/T, £(t) = -4t2/t2 + 4t/T.
Вестник !Ю"УрГ"У. Серия «Математическое моделирование и программирование» (Вестник ЮУрГУ ММП). 2021. Т. 14, № 2. С. 78-84
Table
Results of numerical calculations
tj m = o,6 + 0,3 sin At № = t/T № -At2/T2 + At/T
e £ e £ ? £
10 0,824 0,824 0,050 0,050 0,190 0,190
20 0,302 0,302 0,100 0,100 0,360 0,360
30 0,774 0,774 0,150 0,150 0,510 0,510
40 0,666 0,666 0,200 0,200 0,640 0,640
50 0,338 0,338 0,250 0,250 0,750 0,750
60 0,884 0,884 0,300 0,300 0,840 0,840
70 0,484 0,484 0,350 0,350 0,910 0,910
80 0,472 0,472 0,400 0,400 0,960 0,960
90 0,888 0,888 0,450 0,450 0,990 0,990
100 0,345 0,345 0,500 0,500 1,000 0,997
110 0,653 0,653 0,550 0,550 0,990 0,989
120 0,785 0,785 0,600 0,600 0,960 0,960
130 0,301 0,301 0,650 0,650 0,910 0,910
140 0,814 0,815 0,700 0,700 0,840 0,840
150 0,613 0,613 0,750 0,750 0,750 0,750
160 0,368 0,368 0,800 0,800 0,640 0,640
170 0,896 0,897 0,850 0,850 0,510 0,510
180 0,437 0,437 0,900 0,900 0,360 0,358
190 0,521 0,522 0,950 0,950 0,190 0,191
200 0,868 0,868 1,00 0,998 0,000 0,490
Table 1 shows that, in all three cases, the values of the desired function are restored with high accuracy. In this case, the relative error of restoring the function values does not exceed 0,07%, 0,22% and 0,3%, except at the point x = 0, for the first, second and third case, respectively. The results of numerical calculations show that when a point source approaches the observation point x =1, the accuracy of identifying source coordinate increases.
Analysis of the results of the numerical experiment shows that the proposed computational algorithm can be used in the study of a wide class of processes with a mobile point source.
Conclusion
The identification problem for a one-dimensional linear convective transport equation related to the restoration of the trajectory of a mobile point source is considered. The computational algorithm for solving this problem is based on the approximation of the Delta function, the discretization of the problem, and the use of a special representation for solving the difference problem. The proposed method allows to consistently determine the coordinates of a mobile point source and the distribution of the substance in the considered area in each time layer.
References
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Received December 18, 2020
УДК 519.63 БЭТ: 10.14529/mmp210208
ЗАДАЧА ИДЕНТИФИКАЦИИ ТРАЕКТОРИИ ПОДВИЖНОГО ТОЧЕЧНОГО ИСТОЧНИКА В УРАВНЕНИИ КОНВЕКТИВНОГО ПЕРЕНОСА
Х.М. Гамзаев, Азербайджанский государственный университет нефти и промышленности, г. Баку, Азербайджан
Рассматривается задача идентификации траектории подвижного точечного источника, описываемого дельта функцией, в одномерном линейном уравнении конвективного переноса по заданному дополнительному граничному условию. Для решения рассматриваемой задачи сначала дельта функция аппроксимируется непрерывной функцией и строится дискретный аналог задачи с помощью конечно-разностных аппроксимаций в виде неявной разностной схемы. Для решения полученной разностной задачи предлагается специальное представление, позволяющее на каждом дискретном значении временной переменной расщепить задачу на две взаимно независимые линейные разностные задачи первого порядка. В результате получена явная формула для определения положения подвижного точечного источника при каждом дискретном значении
Вестник !Ю"УрГ"У. Серия «Математическое моделирование и программирование» (Вестник ЮУрГУ ММП). 2021. Т. 14, № 2. С. 78-84
временной переменной. На основе предложенного вычислительного алгоритма были проведены численные эксперименты для модельных задач.
Ключевые слова: уравнение конвективного переноса; подвижный точечный источник; задача идентификации; закон движения источника; аппроксимация дельта функции.
Ханлар Мехвали оглу Гамзаев, доктор технических наук, профессор, кафедра <Общая и прикладная математика:», Азербайджанский государственный университет нефти и промышленности (г. Баку, Азербайджан), [email protected].
Поступила в редакцию 18 декабря 2020 г.
