See discussions, stats, and author profiles for this publication at: https://www.researchgate.

net/publication/257845685

On a class of problems of determining the temperature and density of heat

sources given initial and final temperature

Article  in  Siberian Mathematical Journal · January 2012

DOI: 10.1134/S0037446612010120

CITATIONS READS

64 44

2 authors:

I. Orazov Makhmud A. Sadybekov

11 PUBLICATIONS   135 CITATIONS   
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
105 PUBLICATIONS   812 CITATIONS   
SEE PROFILE
SEE PROFILE

Some of the authors of this publication are also working on these related projects:

ДЛЯ СТАТЬИ ПО ОБРАТНЫМ ЗАДАЧАМ ОПЕРАТОРА ШТУРМА ЛИУВИЛЛЯ View project

All content following this page was uploaded by Makhmud A. Sadybekov on 15 March 2016.

The user has requested enhancement of the downloaded file.

Siberian Mathematical Journal, Vol. 53, No. 1, pp. 146–151, 2012
Original Russian Text Copyright c 2012 Orazov I. and Sadybekov M. A.

ON A CLASS OF PROBLEMS OF DETERMINING THE

TEMPERATURE AND DENSITY OF HEAT SOURCES
GIVEN INITIAL AND FINAL TEMPERATURE

c I. Orazov and M. A. Sadybekov UDC 517.956.4

Abstract: We consider a class of problems modeling the process of determining the temperature and
density of heat sources given initial and finite temperature. Their mathematical statements involve
inverse problems for the heat equation in which, solving the equation, we have to find the unknown
right-hand side depending only on the space variable. We prove the existence and uniqueness of classical
solutions to the problem, solving the problem independently of whether the corresponding spectral
problem (for the operator of multiple differentiation with not strongly regular boundary conditions) has
a basis of generalized eigenfunctions.
Keywords: inverse problem, heat equation, initial temperature, final temperature, not strongly regular
boundary conditions, Sturm-type boundary conditions, Fourier series, orthogonal basis

§ 1. Statement of the Problem

The problems of determining the coefficients or the right-hand side of a differential equation simulta-
neously with its solution are called the inverse problems of mathematical physics. These problems quite
often arise in various areas of human activity, which places them among the current problems of modern
mathematics. In this article we consider a class of problems which model the process of determining
the temperature and density of heat sources given initial and finite temperature. Their mathematical
statements involve inverse problems for the heat equation in which, solving the equation, we have to
find the unknown right-hand side depending only on the space variable. The questions of solvability of
various inverse problems for parabolic equations were studied in many articles (see [1–8] for instance). In
contrast to the previous articles, we study inverse problems for the heat equation with general boundary
conditions on the space variable which are regular but not strongly regular.
In the domain Ω = {(x, t), 0 < x < 1, 0 < t < T } consider the problem of finding the right-hand
side f (x) of the heat equation
ut (x, t) − uxx (x, t) = f (x) (1)
and its solutions u(x, t) satisfying the initial and final conditions
u(x, 0) = ϕ(x), u(x, T ) = ψ(x), 0 ≤ x ≤ 1, (2)
and the boundary conditions

a1 ux (0, t) + b1 ux (1, t) + a0 u(0, t) + b0 u(1, t) = 0,
(3)
c1 ux (0, t) + d1 ux (1, t) + c0 u(0, t) + d0 u(1, t) = 0.
The coefficients ak , bk , ck , and dk , with k = 0, 1, in (3) are real numbers, while ϕ(x) and ψ(x) are given
functions.
The Fourier method for solving (1)–(3) leads to the spectral problem for the operator l determined
by the differential expression l(y) = −y  (x), 0 < x < 1, and the boundary conditions

a1 y  (0) + b1 y  (1) + a0 y(0) + b0 y(1) = 0,
(4)
c1 y  (0) + d1 y  (1) + c0 y(0) + d0 y(1) = 0.
Shymkent; Almaty. Translated from Sibirskiı̆ Matematicheskiı̆ Zhurnal, Vol. 53, No. 1, pp. 180–186,
January–February, 2012. Original article submitted February 2, 2011.
146 0037-4466/12/5301–0146
These boundary conditions are called regular [9] if one of the following three conditions

a1 d1 − b1 c1 = 0;
a1 d1 − b1 c1 = 0, |a1 | + |b1 | > 0, a1 d0 + b1 c0 = 0; (5)
a1 = b1 = c1 = d1 = 0, a0 d0 − b0 c0 = 0

is satisfied. Regular boundary conditions are strongly regular in the first and third cases, while in the
second case this requires the additional condition

a1 c0 + b1 d0 = ±[a1 d0 + b1 c0 ]. (6)

Particular cases of (1)–(3) were considered in [10] with boundary conditions (3) which are not strongly
regular: the case of periodic boundary conditions and the case of conditions of Samarskiı̆–Ionkin type.
But the method of proof of [10] does not automatically carry over to problems with arbitrary not strongly
regular boundary conditions (3). This has essentially to do with the use in [10] of a basis of eigenfunctions
and generalized eigenfunctions of the corresponding problem (4) for the operator of multiple differenti-
ation. But [11] implies that not all problems of this type enjoy the basis property. We propose a new
method for studying these problems independently of the basis property

§ 2. The Case of Sturm-Type Boundary Conditions

A particular case of strongly regular boundary conditions are Sturm-type conditions: b0 = b1 = c0 =
c1 = 0. Denote by λk the eigenvalues of the operator l enumerated in the increasing order of their absolute
values, and by yk (x), for k = 1, 2, . . . , the associated normalized eigenfunctions. It is known [9] that the
eigenvalues of these problems are real and simple, while the system of their eigenfunctions constitutes
an orthonormal basis for the L2 (0, 1) space. Thus, we can represent the solution u(x, t), f (x) to (1)–(3)
as the series:
∞ ∞

u(x, t) = uk (t)yk (x), f (x) = fk yk (x). (7)
k=1 k=1

Inserting (7) into (1) and (2), we obtain the problems

uk (t) + λk uk (t) = fk , uk (0) = ϕk , uk (T ) = ψk (8)

for finding the unknown functions uk (t) and coefficients fk , where ϕk and ψk are the Fourier coefficients
of ϕk and ψk with respect to the system {yk (x)}; thus, ϕk = (ϕ(x), yk (x)) and ψk = (ψ(x), yk (x)).
A solution to (8) exists, is unique, and can be written explicitly as

1 − e−λk t
uk (t) = e−λk t ϕk + (ψk − e−λk T ϕk ), (9)
1 − e−λk T

λk
fk = (ψk − e−λk T ϕk ). (10)
1 − e−λk T
We should note that (9) and (10) remain valid in the case λ0 = 0. Then, passing to the limit as
λ0 → 0, from (9) and (10) we obtain u0 (t) = ϕ0 + ψ0 −ϕT
0
t and f0 = ψ0 −ϕ
T
0
. Thus, we will not treat this
particular case separately.
Inserting (9) and (10) into (7), we arrive at a formal solution to the problem. In order to complete
our study, it is necessary, as in the Fourier method, to justify the smoothness of the resulting formal
solutions and the convergence of all appearing series. Let us state the main result of this section.

147
 Lemma 1. Suppose that b0 = b1 = c0 = c1 = 0, i.e., that the boundary conditions (4) are Sturm-
type conditions. If ϕ(x), ψ(x) ∈ C 4 [0, 1] and the functions ϕ(x), ψ(x), ϕ (x), and ψ  (x) satisfy (4) then
2,1
there exists a unique classical solution u(x, t) ∈ Cx,t (Ω), f (x) ∈ C[0, 1] to the problem (1)–(3).
Proof. Since the functions ϕ (x), ψ  (x) ∈ C 2 [0, 1] satisfy (4), by Steklov’s theorem [12, p. 41] they
admit expansions into absolutely and uniformly converging Fourier series in the eigenfunctions {yk (x)}.
Thus, the series
∞ ∞
 
ϕ (x) = − λk ϕk yk (x), ψ (x) = − λk ψk yk (x) (11)
k=1 k=1

converge absolutely and uniformly.

Considering that limk→∞ λk = +∞, it is not difficult to obtain from (9) and (10) the estimates

|uk (t)| ≤ C(|ϕk | + |ψk |), |uk (t)| ≤ C(|ϕk | + |ψk |)|λk |, |fk | ≤ C(|ϕk | + |ψk |)|λk |,

which are uniform in k. Hence, the uniform and absolute convergence of (11) implies that (7) converges
2,1
and the solution to (1)–(3) is of the following class: u(x, t) ∈ Cx,t (Ω), f (x) ∈ C[0, 1]. Since the system
{yk (x)} constitutes an orthonormal basis for L2 (0, 1), we can express every solution to (1)–(3) in this
class as a series (7). The uniqueness of the construction of solutions (9) and (10) to (8) implies the
uniqueness of solutions to (1)–(3).

§ 3. Regular but Not Strongly Regular Boundary Conditions

To begin with, we describe a class of regular but not strongly regular boundary conditions in a con-
venient form.
Lemma 2. If the boundary conditions (4) are regular but not strongly regular then the boundary
conditions (3) reduce to

a1 ux (0, t) + b1 ux (1, t) + a0 u(0, t) + b0 u(1, t) = 0,
|a1 | + |b1 | > 0, (12)
c0 u(0, t) + d0 u(1, t) = 0,

of one of the following four types:

a1 + b1 = 0, c 0 − d0 = 0;
a1 − b1 = 0, c 0 + d0 = 0;
(13)
c 0 − d0 = 0, a1 + b1 = 0;
c 0 + d0 = 0, a1 − b1 = 0.

Proof. According to [9, p. 73], if (4) are regular but not strongly regular boundary conditions, then
c1 = d1 = 0 and
b1 c0 + a1 d0 = 0, (14)
a1 c0 + b1 d0 = ±[a1 d0 + b1 c0 ]. (15)
In turn, we can express (15) as (a1 ± b1 )(c0 ± d0 ) = 0, which amounts to the fulfillment of at least one
equality in (13). If one of these equalities is fulfilled then (14) yields the corresponding inequality in (13).
The proof of Lemma 2 is complete.
Henceforth we consider only the boundary conditions of the form (12).
Introduce the even and odd parts C(x, t) and S(x, t) of the function u(x, t) = C(x, t) + S(x, t) with
respect to the variable x, where

2C(x, t) = u(x, t) + u(1 − x, t), 2S(x, t) = u(x, t) − u(1 − x, t). (16)

148
Furthermore, for all (x, t) ∈ Ω we have
C(x, t) = C(1 − x, t), S(x, t) = −S(1 − x, t),
(17)
Cx (x, t) = −Cx (1 − x, t), Sx (x, t) = Sx (1 − x, t).
It is obvious that in order to construct the solutions u(x, t) it suffices to determine the functions C(x, t)
and S(x, t) on the “half” of the domain Ω, the subdomain Ω0 = {(x, t), 0 < 2x < 1, 0 < t < T }.
It is not difficult to verify that C(x, t) and S(x, t) on Ω0 are solutions to the heat equation:
Ct (x, t) − Cxx (x, t) = f0 (x), (18)
St (x, t) − Sxx (x, t) = f1 (x), (19)
and satisfy the initial and final conditions
C(x, 0) = ϕ0 (x), C(x, T ) = ψ0 (x), 0 ≤ 2x ≤ 1, (20)
S(x, 0) = ϕ1 (x), S(x, T ) = ψ1 (x), 0 ≤ 2x ≤ 1, (21)
where
2f0 (x) = f (x) + f (1 − x), 2f1 (x) = f (x) − f (1 − x);
2ϕ0 (x) = ϕ(x) + ϕ(1 − x), 2ϕ1 (x) = ϕ(x) − ϕ(1 − x);
2ψ0 (x) = ψ(x) + ψ(1 − x), 2ψ1 (x) = ψ(x) − ψ(1 − x).
Find the boundary conditions satisfied by C(x, t) and S(x, t) on the boundary of Ω0 . Imposing (3)
on u(x, t) = C(x, t) + S(x, t) and taking (17) into account, we obtain
(a1 − b1 )Cx (0, t) + (a1 + b1 )Sx (0, t) + (a0 + b0 )C(0, t) + (a0 − b0 )S(0, t) = 0,
(22)
(c0 + d0 )C(0, t) + (c0 − d0 )S(0, t) = 0.
If each of the conditions (13) of regularity, but not strong regularity, of boundary conditions is fulfilled
then one of the “principal” coefficients in (22) always vanishes. Using this property, separately for each
of the types (13) we arrive at the following boundary conditions for the functions C(x, t) and S(x, t) on
the left boundary of Ω0 .
(I) For a1 + b1 = 0 and c0 − d0 = 0, we have
(a1 − b1 )(c0 − d0 )Cx (0, t) − (a0 d0 − b0 c0 )C(0, t) = 0, (23)
(c0 + d0 )
S(0, t) = C(0, t). (24)
(c0 − d0 )
(II) For a1 − b1 = 0 and c0 + d0 = 0, we have
(a1 + b1 )(c0 + d0 )Sx (0, t) + (a0 d0 − b0 c0 )S(0, t) = 0, (25)
(c0 − d0 )
C(0, t) = S(0, t). (26)
(c0 + d0 )
(III) For c0 − d0 = 0 and a1 + b1 = 0, we have
C(0, t) = 0, (27)
(a1 + b1 )Sx (0, t) + (a0 − b0 )S(0, t) = −(a1 − b1 )Cx (0, t). (28)
(IV) For c0 + d0 = 0 and a1 − b1 = 0, we have
S(0, t) = 0, (29)
(a1 − b1 )Cx (0, t) + (a0 + b0 )C(0, t) = −(a1 + b1 )Sx (0, t). (30)
Also from (17) we deduce on the right boundary of Ω0 the boundary conditions
Cx (1/2, t) = 0, (31)
S(1/2, t) = 0. (32)
Consequently, each of the types in (13) of not strongly regular boundary conditions reduces to solving
two boundary value problems successively.

149
 Problem I. In the domain Ω0 find a solution C(x, t) to (18) satisfying the initial and final condi-
tions (20) and boundary conditions (23), (31). Using this C(x, t), in the domain Ω0 find a solution S(x, t)
to (19) satisfying the initial and final conditions (21) and the boundary conditions (24), (32).
Problem II. In the domain Ω0 find a solution S(x, t) to (19) satisfying the initial and final condi-
tions (21) and boundary conditions (25), (32). Using this S(x, t), in the domain Ω0 find a solution C(x, t)
to (18) satisfying the initial and final conditions (20) and the boundary conditions (26), (31).
Problem III. In the domain Ω0 find a solution C(x, t) to (18) satisfying the initial and final con-
ditions (20) and boundary conditions (27), (31). Using this C(x, t), in the domain Ω0 find a solution
S(x, t) to (19) satisfying the initial and final conditions (21) and the boundary conditions (28), (32).
Problem IV. In the domain Ω0 find a solution S(x, t) to (19) satisfying the initial and final con-
ditions (21) and boundary conditions (29), (32). Using this S(x, t), in the domain Ω0 find a solution
C(x, t) to (18) satisfying the initial and final conditions (20) and the boundary conditions (30), (31).
It is not difficult to see that all new boundary conditions for C(x, t) and S(x, t) obtained on the
boundary of Ω0 are separated and, consequently, strongly regular.
Therefore, we have established
Lemma 3. We can always equivalently reduce the solution of the problem (1)–(3) in the case of
regular but not strongly regular conditions to solving successively two problems with strongly regular
Sturm boundary conditions.
Using the solutions to boundary value problems in the domain Ω0 , we construct a solution to (1)–(3):

C(x, t) + S(x, t), 2x ≤ 1,
u(x, t) =
C(1 − x, t) − S(1 − x, t), 2x ≥ 1,

f0 (x) + f1 (x), 2x ≤ 1,
f (x) =
f0 (1 − x) − f1 (1 − x), 2x ≥ 1.
Furthermore, (17) ensures the smoothness of the resulting solution on the entire domain Ω.

§ 4. Solution to the Problem in the Case of

Not Strongly Regular Boundary Conditions
Using Lemma 3, we can obtain the existence of a solution to (1)–(3), as well as its uniqueness and
smoothness, from Lemma 1 for the corresponding problems with strongly regular Sturm-type boundary
conditions. The presence of inhomogeneous boundary conditions on the left boundary of the domain Ω0
is not a substantial obstacle and can be bypassed in a standard way.
The main result of this article is
Theorem. Consider regular but not strongly regular boundary conditions (3); therefore, their co-
efficients satisfy one of the conditions in (13). If ϕ(x), ψ(x) ∈ C 4 [0, 1] and the functions ϕ(x), ψ(x),
ϕ (x), and ψ  (x) satisfy the boundary conditions (4) then there exists a unique classical solution
2,1
u(x, t) ∈ Cx,t (Ω), f (x) ∈ C[0, 1] to the problem (1)–(3).
Proof. It is not difficult to verify by direct calculations that under the hypotheses of the theorem the
initial and finite functions ϕ0 (x), ϕ1 (x), ψ0 (x), and ψ1 (x) belong to C 4 [0, 12 ] and satisfy the corresponding
Sturm-type boundary conditions at x = 0 and x = 12 . Applying Lemma 1 for the corresponding problems
with Sturm-type boundary conditions on Ω0 , we complete the proof of the theorem.
In closing, note that we solve the problem (1)–(3) using this method independently of whether the
corresponding spectral problem for the operator of multiple differentiation with boundary conditions (4)
enjoys the basis property for generalized eigenfunctions.

150
 References
1. Anikonov Yu. E. and Belov A. Ya., “On unique solvability of an inverse problem for a parabolic equation,” Soviet Math.,
Dokl., 39, No. 3, 601–605 (1989).
2. Anikonov Yu. E. and Bubnov B. A., “Existence and uniqueness of a solution to an inverse problem for a parabolic
equation,” Soviet Math., Dokl., 37, No. 1, 130–132 (1988).
3. Bubnov B. A., “Existence and uniqueness of solution to inverse problems for parabolic and elliptic equations,” in:
Nonclassical Equations of Mathematical Physics [in Russian], Izdat. Inst. Mat., Novosibirsk, 1986, pp. 25–29.
4. Kostin A. B. and Prilepko A. I., “On certain inverse problems for parabolic equations with final and integral observation,”
Russian Acad. Sci., Sb., Math., 75, No. 2, 473–490 (1993).
5. Kozhanov A. I., “Nonlinear loaded equations and inverse problems,” Comput. Math. Math. Phys., 44, No. 4, 657–678
(2004).
6. Monakhov V. N., “Inverse problems for relaxation models of hydrodynamics,” Vestnik NGU Ser. Mat. Mekh. Informat.,
3, No. 3, 72–80 (2003).
7. Kaliev I. A. and Pervushina M. M., “Inverse problems for the heat equation,” in: Modern Problems of Physics and
Mathematics. Vol. 1 [in Russian], Gilem, Ufa, 2004, pp. 50–55.
8. Kaliev I. A. and Pervushina M. M., “Some inverse problems for the heat equation,” in: Proceedings of the Confer-
ence “Information Technology and Inverse Problems of Rational Use of Natural Resources” [in Russian], Poligrafist,
Khanty-Mansiı̆sk, 2005, pp. 39–43.
9. Naı̆mark M. A., Linear Differential Operators. Pts. 1 and 2 [in Russian], Nauka, Moscow (1969).
10. Kaliev I. A. and Sabitova M. M., “Problems of determining the temperature and density of heat sources from the initial
and final temperatures,” J. Industr. Appl. Math., 4, No. 3, 332–339 (2010).
11. Lang P. and Locker J., “Spectral theory of two-point differential operators determined by −D2 ,” J. Math. Anal. Appl.,
146, No. 1, 148–191 (1990).
12. Levitan B. M. and Sargsyan I. S., Sturm–Liouville and Dirac Operators [in Russian], Nauka, Moscow (1988).

I. Orazov
South Kazakhstan State University, Shymkent, Kazakhstan
M. A. Sadybekov
Institute of Mathematics, Informatics and Mechanics, Almaty, Kazakhstan
E-mail address: makhmud-s@mail.ru

151

View publication stats