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DOI: 10.17516/1997-1397-2021-14-4-445-451 УДК 517.9
Inverse Problem for Source Function in Parabolic Equation at Neumann Boundary Conditions
Victor K. Andreev* Irina V. Stepanova^
Institute of Computational Modelling SB RAS Krasnoyarsk, Russian Federation Siberian Federal University Krasnoyarsk, Russian Federation
Received 08.02.2021, received in revised form 10.03.2021, accepted 20.05.2021 Abstract. The second initial-boundary value problem for a parabolic equation is under study. The term in the source function, depending only on time, is to be unknown. It is shown that in contrast to the standard Neumann problem, for the inverse problem with integral overdetermination condition the convergence of it nonstationary solution to the corresponding stationary one is possible for natural restrictions on the input problem data.
Keywords: parabolic equation, inverse problem, source function, a priori estimate, nonlocal overdetermination condition.
Citation: V. K. Andreev, I. V. Stepanova, Inverse Problem for Source Function in Parabolic Equation at Neumann Boundary Conditions, J. Sib. Fed. Univ. Math. Phys., 2021, 14(4), 445-451. DOI: 10.17516/1997-1397-2021-14-4-445-451.
1. Introduction and preliminaries
We consider the parabolic equation with second-type boundary conditions
ut = vuxx + f (t) + g(x,t) in Qt = (0,0 X [0, T]; (1)
u(x, 0) = uo(x), x £ (0,l); (2)
-ux(0,t) = qi (t), ux(l,t) = q2(t), (3)
I u(x,t)dx = q3(t), t £ [0, T]. (4)
J 0
In (1)-(4) the functions g(x,t), u0(x), qi(t),i = 1, 2,3, and positive constants v, T, l are assumed to be given. The problem of finding a pair u(x,t) and f (t) is called inverse one.
Definition. The pair f (t) and u(x,t) from the class C[0,T] x C2'1(Qt) n C 1'0(Qt), for which equation (1) and conditions (2)-(4) are satisfied, is called a classical solution of the posed inverse problem.
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It is clear that for existence of smooth solution the consistency conditions should be fulfilled. They are the following
-uox(0) = qi(0), uox(l) = q2(0), / u0(x)dx = qs(0).
J 0
From the physical point of view the posed problem (1)-(4) allows to describe a motion of viscous fluid in the flat layer with two free boundaries. The function u(x,t) is the velocity in this case, f (t) is unknown pressure gradient, condition (4) means flow rate through the section of layer. The solution of the inverse problem in this case gives an answer on a question: what is pressure gradient needed for providing the given flow rate?
It is necessary to mention that there are many known results concerning to the inverse problems close to the posed problem. Among of them it can be distinguish the coefficient inverse problems (see, e.g. [1-3]), problems with unknown source function [4-6] and problems, where unknown function in the boundary condition occurs [7]. The authors, dealing with finding the source function, usually assume, that this function is included into the equation by multiplicative way (see, for example, [4]).
The overdetermination conditions can be nonlocal integral ones [1,8,9]. One-point and two-point overdetermination conditions are considered in [5,10]. As a rule, in the cited papers and books the existence and uniqueness of solution are proved, some asymptotic methods of solution construction are described. It is common situation when existence and uniqueness of solution are proved in the Sobolev's spaces. Usually, the same questions are considered in the uniform metric for 1-dimensional problems only. Concerning to different kinds of inverse problems and qualitative properties of their solutions we should also mention the monographs authored by Prilepko et al [11], Alifanov [12] and Belov [13].
1.1. Some remarks on corresponding direct problem
If the function f = 0 in equation (1), and condition (4) is not taken into account, then we deal with standard Neumann problem for the function u(x,t). It is well known that the direct initial boundary problem
ut = vAu + g(x,t), x £ tt C R", t £ [0,T]; (5)
du
u(x, 0) = u0(x), x £ tt; — = y(x), x £ dtt, t £ [0,T] (6)
on
has unique solution if the functions g(x,t), u0(x) and y(x) are smooth ones. The corresponding stationary problem
du
vAus = -gs(x), x £ tt C R", — = ys(x), x £ dtt has a countable number of solutions us(x) + const if and only if the following condition
1 i gsdtt+ i fsdr = 0 vJQ JSQ
is fulfilled. For the separation of unique solution it is necessary to give additional functional of us(x). For example, it could be us(x0), where x0 £ dtt.
It should be noted that if g(x,t) ^ gs(x) and y(x,t) ^ ys(x) in the uniform metric at t ^ x> for all x £ tt, then it is not difficult to prove that nonstationary solution u(x,t) does not tend
to us (x) at t ^ to. We confirm this fact using an example. For the problem in the space R1 we consider equation (5) with conditions u0(x, 0) = x, ux(0,t) = ux(l,t) = 1 and the right hand side in the form
cos ln M v
g(x,t)= M , m = 1 + 1.
It has the solution
l2
u(x,t) = x +--sin ln M,
which has no a limit at t ^ to while the corresponding stationary solution is us(x) = x + const at = ^2 = 1 and gs(x) = 0.
We should also mention that for the Dirichlet and Robin problems for multidimensional linear parabolic equation (5) the sufficient convergence conditions of solution of nonstationary problem to corresponding stationary one are described in [14]. According to the example above, for the Neumann problem there is no such convergence. However, it turn out well to show the convergence of nonstationary solution to the corresponding stationary one for the posed inverse problem (1)-(4).
Below, using the specific features of the considered problem and their 1-dimensionality, we derive sufficient conditions for the initial data for which the nonstationary solution tends to stationary one at t ^ to in the uniform metric.
2. Analysis of the inverse problem (1)—(4)
Integrating equation (1) by x from 0 till l and using condition (4), the function f (t) can be found in the form
f (t) = 1
dq fi
— v (rJi(t) + q2(t)) — J g{x,t)do.
(7)
Substitution expression (7) into equation (1) leads to direct problem for the function u(x,t) with conditions (2) and (3). The solution of the obtained problem can be constructed as follows [15]
(x,t)= uo(y)G(x,y,t)dy + / F(y,T)G(x,y,t - t) dydr+ J 0 J 0 J 0
+ v / qi(T)G(x, 0,t — t) dT + v q2 (t)G(x,l,t — t) dT,
00
0
where G is the Green's function:
12 (nnx\ (nny\ ( vn2n21\ G(x,y,t) = J + cos ^ —— J cos ^—J exp I--^— I , F(x,t) = f (t) + g(x,t).
n=l ^ ^
The examples of construction of a priori estimate for the functions presented as series with the Green's functions are given in [16,17]. As it can be observed in those works, the deriving the estimate of the function \u(x,t)\, x G [0, l], t G [0, T] from this expression is a cumbersome task. We suggest another way and reduce problem (1)-(4) to the axillary problem with the first-type boundary conditions.
u
Differentiating equation (1) with respect to variable x, we obtain the initial boundary value problem for the function w(x,t) = ux(x,t):
wt = vwxx + 9x(x,t), x G (0,l), t G [0,T];
w(x, 0) = wo(x) = uox(x), x G [0, l]; w(0,t) = -qi(t), w(l,t) = q2(t), t G [0,T].
(8)
The corresponding stationary problem has the solution
S(x) = -ql + j
ql + ql + - I gs(y)dy - - ( v Jo v jo
gs(y)dy,
(9)
where qf, q%, gs(x) are given constants and function respectively.
Let the functions q1j2(t) be known for all t > 0 and the following inequalities be fulfilled
q(t)| < Nj(1 + T)-a, j = 1, 2, \g(x,t)\ < N3(1 + T)
\gx(x,t)\ < N4(1 + t)
(10)
with some positive constants Nl7... ,N4 and a for all x G [0, /], where t = vt/l2 is the dimen-sionless time here and below. Then with respect to the results from [14] it can be concluded that the function w(x,t) can be restricted as
\w(x,t\ < N5(1+ t)-a
with constant N5 > 0 at x G [0, /].
In order to find the stationary solution us(x), expression (9) should be integrated
us(x) = -qfx - 1 fs Y + J (x - y)gs(y)d^j + C.
(11)
(12)
Here
fs = -^[ql + ql + - 0 gs(y)d^j, (13)
i i (x - y)gs(y)dydx - l2 f gs(y)dy\ .
o o o
C = ql + qîl_l_qi + —
l + 3 6 + vl
After that the stationary solution of the posed inverse problem is constructed, we can start obtaining a priori estimates of the corresponding nonstationary solution.
2.1. A priori estimates of the solution of problem (1)—(4) It should be noted that if the following conditions are fulfilled
^ 0, qj (t) ^ qj, g(x,t) ^ gs (x) at t ^ x> and x G [0, l], then it can be concluded that
f (t) ^ fs at t ^^
x
is valid. The function f (t) is from formula (7), and fs is from (13).
According to the integral mean-value theorem there is the point x0 £ (0, l) such that u(x0,t) = l-1q3(t) (see (4)). That is why for every x £ [0, l], t > 0 it follows that
fx fx
u(x,t) = u(x0,t)+ uy (y,t)dy = l-1qs(t)+ w(y,t)dy.
Jx 0 J'x 0
Using estimate (11), it can be obtained that
\u(x,t)\ < l-1\qs(t)\ +f \w(y,t\dy < l-1\qs(t)\ + N5l(1 + r)-a. (14)
■J 0
Let the following inequalities should be fulfilled
\qj (t) - qj \ < Dj(1 + t)-a, j = 1, 2, 3;
^ 0,t ^m, \g(x,t) - gs(x)\ < Da(1+ t)-a, (15)
\gx(x,t) - gx(x)\ < D5(1+ t)-a
dq3
dt
with constants Di > 0 (i = 1,..., 5), a > 0 for every x £ [0, l]. Then the following estimates can be provided
\u(x,t) - us(x)\ < D6(1+ t)-a,
\ux(x,t) - usx\ < D7(1+ t)-a, (16)
\f (t) - fx\ < Ds(1+ t)-a,
where D6, D7, D8 are positive constants.
For the deriving the estimates in (16) it needs to make a change u(x,t) = u(x,t) - us(x), w(x,t) = w(x,t) - ws(x) and f(t) = f - fs in equation (1) and condition (4). The boundary conditions should be rewritten as qj(t) = qj(t) - qx, j = 1,2, 3, and g(x,t) = g(x,t) - gs(x) in this case. Applying estimates (14) and (11), formulas (7) and (13), using assumptions (15) we derive estimates (16). It concludes that the solution of inverse problem (1)-(4) converges to the corresponding stationary solution (12), (13) in the class C[0, m] x C2'1(Qt) D Cl'0(Qt).
Conclusion
For the conclusion some remarks can be made. The first one is following. If the right hand sides of inequalities (15) are restricted by exponent function (exp(-aT), a > 0), then the solution of problem (1)-(4) tends to stationary regime us(x), fs (see (12), (13)) with respect to exponent law at t ^ m. The second remark is concerned to question of stabilization. The results obtained can be interpreted as stability of stationary solution (12), (13) if conditions (15) are fulfilled.
The authors were surprised at research of some aspects of solution stabilization in problems on binary mixtures motion that the question on solutions solvability and stability in the problems close to (1)-(4) was not described anywhere in literature. And we were glad to fill this gap in the investigation of such problems.
This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry
of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement
no. 075-02-2020-1631).
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Обратная задача определения функции источника для параболического уравнения c краевыми условиями Неймана
Виктор К. Андреев Ирина В. Степанова
Институт вычислительного моделирования СО РАН Красноярск, Российская Федерация
Аннотация. В работе изучается вторая начально-краевая задача для параболического уравнения, когда часть функции источника, зависящая только от времени, неизвестна. Показано, что в отличие от классической задачи Неймана для обратной задачи с интегральным условием переопределения возможна сходимость ее нестационарного решения к соответствующему стационарному при естественных ограничениях на входные данные.
Ключевые слова: параболическое уравнение, обратная задача, функция источника, априорная оценка, нелокальное условие переопределения.
