CC BY
8
DOI: 10.14529/mmph220203
ON UNIQUENESS IN THE PROBLEMS OF DETERMINING POINT SOURCES IN MATHEMATICAL MODELS OF HEAT AND MASS TRANSFER
L.V. Neustroeva
Yugra State University, Khanty-Mansiysk, Russian Federation E-mail: [email protected]
Abstract. We consider the problem of determining point sources for mathematical models of heat and mass transfer. The values of a solution (concentrations) at some points lying inside the domain are taken as overdetermination conditions. A second-order parabolic equation is considered, on the right side of which there is a linear combination of the Dirac delta functions S(x-xt) with coefficients that depend on time and characterize the intensities of sources. Several different problems are considered, including the problem of determining the intensities of sources if their locations are given. In this case, we present the theorem of uniqueness of solutions, the proof of which is based on the Phragmen-Lindelof theorem. Next, in the model case, we consider the problem of simultaneous determining the intensities of sources and their locations. The conditions on the number of measurements (the ovedetermination conditions) are described which ensure that a solution is uniquely determined. Examples are given to show the accuracy of the results. This problem arises when solving environmental problems, first of all, the problems of determining the sources of pollution in a water basin or atmosphere. The results are important when developing numerical algorithms for solving the problem. In the literature, such problems are solved numerically by reducing the problem to an optimal control problem and minimizing the corresponding objective functional. The examples show that this method is not always correct since the objective functional can have a significant number of minima.
Keywords: heat and mass transfer; parabolic equation; uniqueness; inverse problem; point source.
Introduction
Under consideration is the inverse problem of recovering the point sources in the model
m n
ut + Lu = ^N (t)S(x-Xj) + /0(t,x),Lu = -Au + ^at (x)ux + a0(x)u, (1)
7=1 7=1 '
where (x,i)e<2 = (0,r)xG, G is a domain in 1" (и = 2,3) with boundary ГеС2 . The unknowns are the functions N (t). The equation (1) is furnished with the initial and boundary conditions
Bus = g,ut=o = uo (x),S = (0,T)хГ, (2)
öu
where either Bu = — + au, or Bu = u (v is the outward unit normal to Г), and the overdetermination
öv
conditions
u(y,t) = W] (t), j = 1,2,...,s. (3)
These problems arise in mathematical modelling of heat and mass transfer processes, diffusion, filtration, and in many other fields (see [1-3]). In the theory of heat and mass transfer, the function u is the concentration of a transferred substance and the right part characterizes sources (sinks) [1]. In the most general formulation of the problem (1)-(3), the intensities Nt (t) of point sources, their locations xt and the number m are quantities to be determined. Some descriptions of models of this type can be found, for example, in [1]. A lot of articles are devoted to solving these inverse problems. The main results are connected with numerical methods of solving the problem and many of them are far from justified (see [4-16]). The problem is ill-posed and examples when the problem is not solvable or has many
solutions are easily constructed. Very often the methods rely on reducing the problem to an optimal control problem and minimization of the corresponding objective functional [2, 4, 5, 9, 16]. However, it is possible that the corresponding functionals can have many local minima. Some theoretical results devoted to the problem (1)-(3) are available in [17-21]. The stationary case is treated in [20], where the Dirichlet data are complemented with the Neumann data and these data allow to solve the problem on recovering the number of sources, their locations, and intensities using test functions and a Prony-type algorithm. The model problem (1)-(3) (G = Rn ) is considered in [21], where the explicit representation of solutions to the direct problem (the Poisson formula) and auxiliary variational problem are employed to determine numerically the quantities ^ Nrj (here Ni (t) = const for all i and rij = \ xi - yj |). The quantities found allow to determine the points {xi} and intensities Ni (see Theorem 2 and the corresponding algorithm in [21]). So the results of [21], for instance, say that the problem (1)-(3) and more general problem of simultaneous recovering points {xi} and intensities {Ni} in some model situations is uniquely solvable. In the one-dimensional case uniqueness theorem for solutions to the problem (1)-(3) with n = 1, m = 1 is stated in [17]. Similar results are presented also in [22].
In this article the main attention is paid to uniqueness questions of solutions to the problem in some model cases and the general case as well. Examples showing the accuracy of the results obtained are displayed. The constructions can be used when developing numerical algorithms. The results are based on asymptotic representations of the Green functions of the corresponding elliptic problems (see [23]).
Preliminaries
First, we describe our conditions on the data and some corollaries of the results in [23]. Let G be a domain in R" . The symbols Lp(G) and IV^ (G) ( \ < p< x ) stand for the Lebesgue and Sobolev
spaces [24]. We also use the spaces Ck (g) of k times differentiable functions (see the definitions in [24]). If r,S are some sets then the symbol p (r,S) stands for the distance between these sets. The symbol D(L) stands for the domain of an operator L. Denote by Br (x0) the ball of radius r centered at x0 . Let a = (aua2) for n = 2 and a = (a,,a2,a-, j for /7 = 3. The brackets (•,•) denote the inner product in R" . Let
1 f
Kx) = tJ(«(x0+ t(x - xo)X (x - Xo))dT-20
The coefficients in (1) are assumed to be real-valued and
a e W2(G)(i = 1,...,n),Vr,Ay,ao e(G),aeCl(T), (4)
Consider the problem
n
-Au + y^aiuY +a0u + Au = c>(x-x0),x e G cz M", (5)
i=1 '
Bu\T = 0. (6)
For the reader's convenience, here we present some results the article [23] (see Theorem 3.5, 3.9, 3.11, 3.12). We consider compact K c G, containing x0, with properties: if Bu = u and G is a domain with compact boundary then the convex hull of K is contained in G ; if G is a domain with compact boundary and Bu ^ u then K c /i^ p^ (x0); if G = M" or G = M" , then K is an arbitrary compact.
Theorem 1. [23]. Assume that the conditions (4) hold, K is a compact with the above properties, and if G = R" , then, in case Bu^u, cr = 0 (i.e., Bu=ux ) and at =0 for /=1,2,...,«. Then there
exists \ > 0 such that for all a solution un (x) (n = 2,3) to the problem (5), (6) in every domain
{y e K: 0 <s< | y - xi |, i = 1,2,., m} admits the representation
u2 (x) = 1 e<^(x)-^|x-x0 (1 + O{-U) ; (7)
^¡2n\x - x0\11/4 ^
u3 (x) =-1-x)-41 x-x0 (1 + O(-^)) . (8)
n ' 4n\x-x0\ ( ()
Next theorem deals with solvability of the direct problem (1), (2). Let
u0 ( x ) e W (G) ,u0 (x ) = g ( x, 0) if Bu = u. (9)
We also suppose that
f0 e L2 (Q), g ( x,t ) e W23/4,3/2 ( S ) if Bu = u, g ( x, t ) e W21/4,1/2 ( S ) if Bu * u. (10)
Consider auxiliary problems
ut + Lu = f0 (t, x),BuS = g,ut=0 = u0 (x), (11)
m
wt + Lw = £N (t)S(x - xt ),BwS = 0,wt=0 = 0. (12)
i=1
Let Wlp B (G) be a space of functions u eWlp (G) satisfying the homogeneous Dirichlet condition whenever Bu = u and W,B (G) = Wlp (G) if Bu ^ u . Denote by W~B (G) the dual space to Wr^B (G)
(the duality is defined by the inner product in L2 (G), see [25]).
The following theorem follows from [26], theorem 2 and [27], theorem 8.2.
Theorem 2. Let T< <x> and let p e(l, n/(n -1)). Assume that the conditions (9), (10) hold, a eLM(G) (i = 0,1,...,n), and N eL2(0,T) (i = 1,2,_,m) . Then there exists a unique solution to the problem (1), (2) such that u = w0 +w, where w0 eW21,2 (Q) is a solution to the problem (11), w is a solution to the problem (12), w eL2 (o,<x>;Wp B (G)), wt eL2 (o,^;Wp-B (G)) and w eW21,2 (Qe) with Qs = {(x,t )eQ :|x - xt \>sVi < m} for all £> 0 .
Main results
Here we present our uniqueness theorem for solutions to the problem (1)-(3). We introduce the functions
1 J. _
<Pj (x) = "tJ (aiyj+Tix-y^Xx-y^yiT.
20
*Pi ( Xi)
Let Sj = min;. r^, j = 1,2,...,s, where r^ =\ xi -yj |. Let A0 be the matrix with entries a^ = e J if | xi -yj | = Sj and aJi = 0 otherwise. We assume, that:
det A0 * 0 (13)
Condition (4) is rewritten as follows: the coefficients of L are real-valued and
ai e Wj (G)(i = 1,...,n), V<pj, Aq>j,a0 e LX(G)(j < s),a e C1 (r). (14)
Firstly, we justify uniqueness in the inverse problem (1)-(3) of recovering a solutions u and intensities Nj (i = 1,...,/»). Points {x(} and their number are assumed to be known.
Theorem 3. Assume that T < oo, m = s , and the conditions (13), (14) hold. Then a solution (u. N) to the problem (1)-(3) such that u belongs to the class described in Theorem 2 and Ni e L2 (0,T) is unique.
Proof. It suffices to demonstrate that a solution to the problem (1)-(3) with homogeneous data is zero. In this case the auxiliary function w0 = 0. Let a function u such that u e W22 (Qs) for any s > 0,
u e L2 ( 0,T ; Wp B (G )), ut e L2 ( 0,T ; W p B (G )) be a solution to the problem
ut + Lu = £Nt (t)S(x - xt ), (15)
t=1
Bus = 0, ut=0 = 0, (16)
u(y ,t) = 0, j = 1,2,...,s. (17)
We integrate the equation (15) with respect to t and make the change of variables w = j^u(r)dr. This function is a solution to the problem
m t
wt + Lw = £ st (t )?( x - xt), st = j N, (r) dr e ( 0,T), s, ( 0) = 0, (18) ,=1 0
Bws = 0, wt=0 = 0, (19)
w( yj, t ) = 0, j = 1,2,..., s. (20)
Put w = eAt v , where let. This function satisfies the equation
vt+Lv + Xv = £ s(t)e~Xtó(x-x¡). (21)
i=l
Let Vj (x, X) be a solution to the problem
Lv*j + Avj =s(x - yj), B* v* |r= 0, (22)
where £ - formally adjoint operator to operator L, B v = v, if Bu = u and B v (a, v))v
otherwise. The problem (22) is the adjoint problem to the problem
Lv + Av = s(x -yj ),Bv|r = 0. (23)
Multiplying the equation (21) by Vj , integrating the result over Q , and using (20), we obtain the equalities
m T
(v (T, x), vj (T, x)) = j v (T, x) vj (T, x) dx = s, (t) e~Atdtvj (x,). (24)
G i=1 0
The equality (24) can be rewritten as
A(X)S = F, (25)
where the vectors S, F have the coordinates St■= js,(t}e~Xtdt and I<] = 4/rtr [v(7-.x). vj (x)je 7
0
for n = 3 and Fi = 2yj2Sj^A1'4 (v(T,x), vj (T,x))ej for n = 2 . Transform the representation f =( v (T, x ), vj (T, x )) = e"AT (w(T, x),(A + L*)~ XS( x - yj )) = e-Xr ((A + L)-1 w(T, x), £(x - y )) = e"A (A + L)-1 w(T, x)|x=y¡ , Note that the last expression makes sense and these formal transformations are justified. Indicate that w,wt e L2 (0,T; Wp B (G)). In particular, we infer w e C([0,T]; Wlp (G)) after a possible change on a set
of zero measure. By embedding theorems, w e C([0,T];Lq (G)) c q < 3p/(3 - p) for n = 2,3 . In this
_i o
case the expression (A + L) w(T, x) e Wq (G) is well-defined if the parameter X is sufficiently large, say A > A0 > 0 for some A0 . However, Wq2 (G) c C(g) when n = 2,3 and q > 3/2 . Thus, we can consider the value (A + L)-1 w(T, x)| _ . There is the estimate
x=yj
m
|(X + L)-1 w(T, yj )|<| |(X + L)-1 w(T, x)| | c (G) < C, | |(X + L)-1 w(T, x)| | ^ (G)< q | | w(T, x)| | Lq {G), (26)
where the constants do not depend on the parameter X>X0 and we use resolvent estimates for the elliptic operators (see. [27, Ch.2]). As a consequence, we obtain the estimate
| Fj |< c2Xre~T1eV2>A0, where is the constant c2 does not depend on X and y = 0 when n = 3 and y = 1/4 when n = 2 . Fix an arbitrary s e (0,T). The above estimate implies that there exists a constant C0 (s) > 0 such that
C0(e)e
-(T-e)X
VX>X0.
J' iXi
By Theorem 1, the entries b, (X) of A(X) are representable as
(27)
4x5,-
f
bjt (X) = 4KÔjVj (xi )e 1 = a
1 + O
лД
for n = 3 and
bß (X) = 2j2S^v* (x, )X1/4e^ = a i ^ 1 + O¡J=
for n = 2 . Under the condition (13), we can assume that the matrix A(X) is invertible for X > A0 and the elements of the inverse matrix A_1 = {j are bounded by a constant independent of X ; otherwise, we increase the parameter A0 . Therefore, we have
m
S ( A )=! j A ) F ( A )
j=i
and estimate (27) ensures that
a (*)(*)e-(T-£)A
(28)
, ч С (e)(e)e-(T-s)X i St ( X) i < u ,,,-VX > X0.
|X|
Consider the functions Si (Aq + z), where z is a complex parameter, Re z > 0 . The function S[ (X() +z) = s[ (i)e ^e~ztdt is the Laplace transform of the function si(t) = si(t)e for t<T
and Sj (/) = 0 for i '/'. Introduce an additional function W(z) = ze:{T~':) St (Xq + z). It is analytic in the
right half-plane and is bounded by a constant ( \ on the real semi-axis R+ . Estimate this function on the on the imaginary axis. Integrating by parts, we have
S, ( I0+ z ) = -For z = iy we have the estimate
1
Xo + z
f T T
st (T)e"V e-zT + jst, (t)e~^e~ztdt v 0 y
iW (z)i < is, (T ) i + i is,
i iL1( 0, T )
= C3 Vz = iy, y e R.
In each of the sectors 0 < arg z < n /2, -n/2 < arg z < 0 the function W (z) admits the estimate
iW ( z ) i< e1
zi(T-e)
(t )i-
iiL1(0, T ).
V Re z > 0.
(29)
(30)
Applying the Fragment-Lindelef Theorem (see theorem 5.6.1 in [28]) we obtain that in each of the sectors 0 < argz < n/2, -n/2 < argz < 0 the function W(z) admits the estimate
(31)
W(z)< max (C1, C3 ) = C4 VRe z > 0.
Therefore,
IS (2 + z) | = | L(S, (t))(z) | < C4 4e"(rRez /1 z | VRez > 0. (32)
We have equality (a>2, p = c +
C+iro ro
S (t) = J eptL(S,)(p)dp = ±. J e^e^L(s,)(c + i#)d#.
c—iro —ro
and, thereby,
ro
s,i)e"c^-(r-£)) = — f ei^ec(r-£)L(s,)(c + i£)d£. 2k j
—ro
The Parseval identity yields
(t)->) ii^, = ^ J -*' (',df:S^ J ■Jdf S^
'M-vx^f ^ 2k f c2 2c
—ro —ro ^
Since this inequality is true for all a> 0, (/) = 0 for t<T-s . Since the parameter e is arbitrary, .v, (i) 0 for i < I'. We infer Nt (/) = 0 for t<T and every / and, therefore, the right-hand side
of (15) vanishes which implies that u = 0 . □
We note that the following condition is actually a necessary condition for the uniqueness of solutions to the problem (1)-(3). If it fails then any number of the points {y,} does not ensure uniqueness of solutions (see examples below).
Condition (A). For n = 2, any three points {y,} do not lie on the same straight line and, for n = 3 any four points {y,} do not lie on the same plane.
Next, we describe some model situation in which Lu = -Am + , ^ > 0. G = M" and functions N, on the right-hand side of (1) are real constants.
Theorem 4. Let uj,u2 be two solutions to the problem (1)-(3) from class described in the theorem
rJ
1 with the right-hand sides in (1) of the form Y N{5(x — xi) (Nj = const, j = 1,2), the condition (A)
i=1
holds, and s > 2r +1 in the case n = 2 and s > 3r +1 in the case n = 3, where r > max(r1, r2) (i. e., there is the upper bound for the number max (r1, r2)). Then u1 = u2 , r1 = r2 , and N1 = N2 for all i, i. e., a solution to the problem of recovering the number m , points xi , and constants Ni is unique.
Proof. Let the functions u1,u2 do not coincide and let w = u1 — u2 . The function w satisfies the homogeneous initial data and over determination conditions (3) and we have (after renumbering the constants Nij and points xi )
r3 r4
wt + Lw =
УNtS(x-x)-£C,S(x-x*), 2r >Г3 + r4,Nt,Ct = const, (33)
vt
i=1 i=1
where Ni, Cj > 0 for all i, j. Without loss of generality, we can assume that all the numbers N, , Ci are
not equal to zero and all points xi,xi are distinct. Let, for example, n = 3. For simplicity, take 20 = 0 . The proof is the same for other values of this parameter. Applying the Laplace transform, we infer
V Ni Ix— X, I _ V Ci r.—'x— x* I
w ( x ) = V-^-x—^ - V-x—x-1. (34)
W i 4k \ x — x, \2 i 4k| x — x*¡2
Using (3), we obtain
Ni_-xi| _ V Ci r-^y> - x
_ 4*| y} - x_ | Л ___ 4*1 yf - х,1Л
У , Ci *, /-xi 1,j = 1,2,...,J. (35)
r-
3
Neustroeva L. V. On Uniqueness in the Problems of Determining Point Sources
in Mathematical Models of Heat and Mass Transfer
For definiteness, we assume that r3 > r4 . Let us show that the sets of numbers {rj =| xi - yj |: i = 1,2.,r3}, {rij =| x* - yj |,i = 1,2.,r4} coincide for all j. Fix the parameter j . Let S1 j = minf rj , S1j. = minf rj . Demonstrate that S1 j = S1j . Assume the contrary. Let, for example,
S1 j < S1 j . Multiply the system (35) by 4nS1 jXe 1j and passing to the limit as X ^ +j we obtain the equality
Z N, = 0. (36)
i:|xi -yj\=S1j
■JXS ■
It is a contradiction, since Nf > 0. So, S1 j = S1j and multiplying the system (35) by 4nS1 jXe 3 and passing to the limit as X ^ + j we also derive that
Z N; = Z C.
i:|xi ~y1\=S1j i:|x* -y^Sj
So, we can reduce the following sums on the left and on the right in the equalities (35):
y^ Nf c-^lyj -x\ Ci_e-^lyj-xi*l
,:|x, Z=Sj 4n | yj - x | X ,^=Sj. 4n I yj - x* I ^ .
^ ry and S2j = minf^ >S
S2j = S2j and, thereby,
Denote S2j = min..^ ^ Tj and S2j = min..^ rtj . Repeating the arguments, we obtain that
Z ^ = Z
Again, abbreviated equal summands (35), we arrive at the system (35), where the sums on the left and on the right are taken over i: rfj > S2j u i: r^ > S2j. , respectively. It is now obvious by induction that
there are pairs of equal numbers Sj Sj k = 1,2,., r0j < min (r3, r4) and
Z N,= X = (37)
moreover, the left-hand and right-hand sides of these equalities are positive. So, the sets of numbers {ry =| Xj-yj \: i = 1,2.. ,,r3}, {fjj =\xt -y} \,i = 1,2.. ,,r4} coincide for all j. In particular, it follows that
for any point, for example, x1 and any j , there exists a point xi such that
| x1 -yj |=| -yj |, j = s.
But we have s > 3r +1 and r4 < r is the number of points {x*}. Hence, among the points {x* }sf=1 there
j j
are four coinciding points. After renumbering if necessary we can assume that these points are
x*, x* , x*, x* . Then the equalities
'1 '2 '3 4 n
| x1 -yj |=| -yj |, j = l2,.s.
imply that the points yj with j = 1,2,3,4 lie in the same plane which is perpendicular to the segment
X1 5 Xi
, but this fact contradicts to the conditions (A). So, w = 0.
The proof in the case of n = 2 is almost the same but we use an asymptotic representation for a
fundamental solution _h01) ('"JX | x - x0 |) defined by the equality (7), where \y = 0 . As in the case of
n = 3, we arrive at contradiction with the condition (A). □
We display the corresponding examples showing the accuracy the results obtained. The following example shows that if the condition (A) fails then the problem of recovering the intensities of sources
(sinks) located at x1, x2 has a nonunique solution. At the same time, it is an example of the nonuniqueness in the problem of recovering the intensity of one source and its location. Note that the problem of determining the location of one source x0 and its intensity N (t) is simple enough and to
uniquely recover these parameters we need two measurements in the case of n = 1 [22], three measurements in the case of n = 2 [28] and four measurements (that is s = 4 in (3)) in the case of n = 4 [25]. The smaller number of points does not allow to define the parameters N (t), x0 uniquely. We should also require that the point x0 lie between two measurement points in the case of n = 1 and the condition (A) holds in the case of n = 2,3. The numerical solution of the problem of recovering one source is treated in the articles [6, 9-15, 19, 28].
Example 1. First we take /7 = 3. G = M" , Lu = -Am . Let u be a solution to the equation (1) satisfying the homogeneous initial conditions with the right-hand side in (1) of the form
N (t )(£( x — x ) — S( x — x2 )). The Laplace transform of this solution to the problem (1)-(2) is written as
u = N (l)(-\-e'2'-^----e~42x—
4k | x - x11 4k | x - x2 |
Let P be the plane perpendicular to the segment [ x1, x2 ] and passing through its center. We have
u (y,2) = 0 Vy e P.
So, u (y, t) = 0 for all y e P . Precisely the same example can be constructed in the case n = 2 . We take the perpendicular to the segment [ x1, x2 ] passing through its center rather than the plane P . Thus, if
condition (A) fails then any number of measurement points does not allow to determine the intensity and the location of the sources.
Example 2. Consider the case of G = M" , Lu = -Au. Let us show that the conditions (3) with s = 4 in the case of n = 2 and s = 6 in the case of n = 3 does not allow to determine location of two sources and their intensities even if the condition (A) holds. Let u1 , u2 be solutions to the equation (10) satisfying the homogeneous initial conditions in which the right-hand sides are of the form N (t )£( x — x1) + N (t )£( x — x2 ) ,N (t )<?( x — x* ) + N (t x — x2 ).
Let, for example, n = 3. Then the Laplace transforms of u1,u2 are as follows:
- ( — Y---e^1 x—x^, — — Y-N-x& (38)
1 ( , ) i 4k | x — x, | , 2( , ) i 4k| x — x* | ( )
Here we use explicit representations of the fundamental solution for the Helmgoltz equation (see, for example, in [30, §3.1] or [31, ch. 4, 8]). We take xl = (a, a,0), x** = (a, — a,0),
x2 =(—a, — a,0), x* =(—a,a,0) (a > 0). As is easily seen, the functions ux,u2 coincide at the points y1 = (M ,0,0), y2 =(—M ,0,0), y3 =(0, M ,0), y4 =(0, —M ,0), y5 =(0,0, M ), y6 =(0,0, —M ), where M > 0 and, thus, the problem of recovering the locations of 2 sources and their intensities admits several solutions in the case of s = 6 . It follows from the theorem 2 that in the case of s = 7 points x1, x2 and the intensities are determined uniquely (if the condition (A) holds and the intensities are constants).
Consider the case of n = 2 . As before, we construct functions u1, u2 whose Laplace transform is of the form
ui = YE ) (iJ21 x—u 2 = YE 77 ) ix—x* ix
where HI is the Hankel functions [32]. Let us take x1 =(a,a),x* =(a, —a),x2 =(—a, —a),x** =(—a,a)
(a > 0). It is easy to check that
«! (yj ,a) = ù2 (yJ, a) Vj = 1,..4, A e K+- (39)
where y1 = (M,0),y2 = (-M,0),y3 = (0,M),y4 =(0,-M). It follows from the theorem 2 that the points x1, x2 and intensities are determined uniquely in case 5 = 5 (if condition (A) holds and the intensities are constants).
Remark 1. The examples show that the number of minima of the corresponding objective functionals introduced if we solve the problem (1)-(3) numerically reducing the problem to an optimal control problem can be large and even can be a manifold.
Remark 2. Relying on asymptotic representations and Theorem 1 in the case of constant values N , we can construct a numerical algorithm for finding sources [xt} employing the ideas from the article [19]. Some review of the results connected with numerical determining point sources can be found in the article [33] and some results in [34-37].
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Received March 9, 2022
Information about the author
Neustroeva Lyubov' Vladimirovna is Post-graduate Student, Digital Technologies Department, Yugra State University, Khanty-Mansiysk, Russian Federation, e-mail: [email protected]
Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2022, vol. 14, no. 2, pp. 31-43
УДК 517.95 DOI: 10.14529/mmph220203
О ЕДИНСТВЕННОСТИ В ЗАДАЧАХ ОПРЕДЕЛЕНИЯ ТОЧЕЧНЫХ ИСТОЧНИКОВ В МАТЕМАТИЧЕСКИХ МОДЕЛЯХ ТЕПЛОМАССОПЕРЕНОСА
Л.В. Неустроева
Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация E-mail: [email protected]
Аннотация. В работе рассмотрены задачи об определении точечных источников для математических моделей тепломассопереноса. В качестве условий переопределения берутся значения решения (концентраций) в некоторых точках лежащих внутри области. Рассматривается параболическое уравнение второго порядка, в правой части которого присутствует линейная комбинация дельта-функций Дирака S(x-xj) с коэффициентами, зависящими от времени и характеризующими мощность источников. Рассматриваются несколько различных задач, в том числе задача определения интенсивностей источников в случае, если их местоположение задано.
В этом случае мы приводим теорему единственности решений, доказательство которой основано на теореме Фрагмена-Линделефа. Далее в модельном случае мы рассматриваем задачу об одновременном определении мощностей источников и их местоположения. Описаны условия на числе замеров (условий переопределения), когда решение определяется единственным образом. Приведены примеры, показывающие точность полученных результатов. Проблема возникает при решении экологических задач, прежде всего задач определения источников загрязнения в водоеме или атмосфере. Результаты важны при построении численных алгоритмов решения задачи. В литературе такие задачи решаются численно с помощью сведения задачи к задаче оптимального управления и минимизации соответствующего целевого функционала. Примеры показывают, что такой способ решения не всегда корректен, поскольку целевой функционал может иметь значительное количество минимумов.
Ключевые слова: тепломассоперенос; параболическое уравнение; единственность; обратная задача; точечный источник.
Литература
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Поступила в редакцию 9 марта 2022 г.
Сведения об авторе
Неустроева Любовь Владимировна - аспирант, кафедра цифровых технологий, Югорский государственный университет, г. Ханты-Мансийск, Российская Федерация, e-mail: [email protected]
