Solution of the boundary-value problem of heat conduction with periodic boundary conditions
UDC 517.9 We investigate the solution of the inverse problem for a linear two-dimensional parabolic equation with periodic boundary and integral overdetermination conditions.Under certain natural regularity and consistency conditions imposed on the input data, we establish the existence, uniqueness...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508266156523520 |
|---|---|
| author | Kanca, F. Baglan, I. Kanca, F. Baglan, I. Kanca, F. Baglan, I. |
| author_facet | Kanca, F. Baglan, I. Kanca, F. Baglan, I. Kanca, F. Baglan, I. |
| author_sort | Kanca, F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-04-07T12:13:22Z |
| description | UDC 517.9
We investigate the solution of the inverse problem for a linear two-dimensional parabolic equation with periodic boundary and integral overdetermination conditions.Under certain natural regularity and consistency conditions imposed on the input data, we establish the existence, uniqueness of the solution and its continuous dependence on the data by using the generalized Fourier method. In addition, an iterative algorithm is constructed for the numerical solution of this problem. |
| first_indexed | 2026-03-24T02:22:28Z |
| format | Article |
| fulltext |
UDC 517.9
F. Kanca (Dept. Comput. Eng., Fenerbahce Univ., Istanbul, Turkey),
I. Baglan (Kocaeli Univ., Turkey)
SOLUTION OF THE BOUNDARY-VALUE PROBLEM OF HEAT CONDUCTION
WITH PERIODIC BOUNDARY CONDITIONS
РОЗВ’ЯЗОК ГРАНИЧНОЇ ЗАДАЧI ТЕПЛОПРОВIДНОСТI
З ПЕРIОДИЧНИМИ ГРАНИЧНИМИ УМОВАМИ
We investigate the solution of the inverse problem for a linear two-dimensional parabolic equation with periodic boundary
and integral overdetermination conditions. Under certain natural regularity and consistency conditions imposed on the
input data, we establish the existence, uniqueness of the solution and its continuous dependence on the data by using the
generalized Fourier method. In addition, an iterative algorithm is constructed for the numerical solution of this problem.
Вивчається розв’язок оберненої задачi для лiнiйного двовимiрного параболiчного рiвняння з перiодичними гранич-
ними умовами та iнтегральними умовами перевизначення. За деяких природних умов регулярностi й узгодженостi,
що накладенi на початковi данi, встановлено iснування, єдинiсть розв’язку та його неперервну залежнiсть вiд даних
за допомогою узагальненого методу Фур’є. Крiм того, побудовано iтеративний алгоритм для побудови чисельного
розв’язку цiєї проблеми.
1. Introduction. The study of mathematical models for many important applications such as chemi-
cal diffusion, applications in heat conduction processes [5, 8], population dynamics, thermoelasticity,
medical science, electrochemistry, engineering, wide scope, chemical engineering [9] and control
theory give rise in the two-dimensional parabolic partial differential equation with nonlocal boundary
conditions [13, 14, 17].
Inverse problems are the problems that consist of finding an unknown property of an object,
or a medium, from the observation of a response of this object, or medium, to a probing signal.
Thus, the theory of inverse problems yields a theoretical basis for remote sensing and nondestructive
evaluation. For example, if an acoustic plane wave is scattered by an obstacle, and one observes the
scattered field far from the obstacle, or in some exterior region, then the inverse problem is to find the
shape and material properties of the obstacle. Such problems are important in identification of flying
objects (airplanes missiles, etc.), objects immersed inwater (submarines, paces of fish, etc.) and in
many other situations. In geophysics one sends an acoustic wave from the surface of the earth and
collects the scattered field on the surface for various positions of the source of the field for a fixed
frequency, or for several frequencies. The inverse problem is to find the subsurface inhomogeneities.
In technology one measures the eigenfrequencies of a piece of a material, and the inverse problem
is to find a defect in this material, for example, a hole in a metal. In geophysics the inhomogeneity
can be an oil deposit, a cave, a mine. In medicine it may be a tumor or some abnormality in a
human body. If one is able to find inhomogeneities in a medium by processing the scattered field
on the surface, then one does not have to drill a hole in a medium. This, in turn, avoids expensive
and destructive evaluation. The practical advantages of remote sensing are what make the inverse
problems important in [20].
There are several methods for the numerical approximation of two-dimensional parabolic inverse
problem. In [13], three different implicit finite difference schemes for solving the two-dimensional
c\bigcirc F. KANCA, I. BAGLAN, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2 209
210 F. KANCA, I. BAGLAN
parabolic inverse problem with temperature overspecification are considered. These schemes are
developed for identifying the control parameter which produces, at any given time, a desired tempe-
rature distribution at a given point in the spatial domain. The numerical methods discussed, are
based on the second-order Backward Time Centered Space (BTCS) implicit formula, and the second-
order Crank – Nicolson implicit finite difference formula and the fourth-order implicit scheme. These
finite difference schemes are unconditionally stable. The implicit formula takes a huge amount of
central processor (CPU) time, but its fourth-order accuracy is significant. The results of a numerical
experiment are presented, and the accuracy and CPU times needed for each of the methods are
discussed and compared. The implicit finite difference schemes use more central processor times
than the explicit finite difference techniques, but they are stable for every diffusion number.
Over the last couple of years, considerable efforts have been put in to develop either approximate
analytical solution or purely numerical solution to nonlocal boundary-value problems [5, 10 – 12,
15], implemented finite difference scheme to obtain the numerical solution of the one dimensional
nonlocal boundary-value problem [12, 13, 16].
The periodic boundary conditions arise from many important applications in heat transfer, life
sciences [1 – 4].
In this paper, we prove the existence, the uniqueness and the continuous dependence on the data
of the solution and we will develop the numerical solution of two-dimensional diffusion problem
with periodic boundary conditions. We will use Fourier method and the finite difference method for
two-dimensional inverse parabolic equation [1 – 3].
The paper is organized as follows. In Section 2, the existence and uniqueness of the solution of
the problem are proved by using the Fourier method. In Section 3, stability of the solution is shown.
In Section 4, the numerical procedure for the solution of the problem is given.
Let T > 0 be fixed number and denote by \Omega :=
\bigl\{
(x, y, t) : 0 < x < \pi , 0 < y < \pi , 0 < t < T
\bigr\}
.
Consider the problem of finding a pair of functions
\bigl\{
r(t), u(x, y, t)
\bigr\}
satisfying the following
equations:
\partial u
\partial t
=
\partial 2u
\partial x2
+
\partial 2u
\partial y2
+ r(t)f(x, y, t), (x, y, t) \in \Omega , (1)
u(0, y, t) = u(\pi , y, t), y \in [0, \pi ], t \in [0, T ],
(2)
u(x, 0, t) = u(x, \pi , t), x \in [0, \pi ], t \in [0, T ],
ux(0, y, t) = ux(\pi , y, t), y \in [0, \pi ], t \in [0, T ],
(3)
uy(x, 0, t) = uy(x, \pi , t), x \in [0, \pi ], t \in [0, T ],
u(x, y, 0) = \varphi (x, y), x \in [0, \pi ], y \in [0, \pi ], (4)
E(t) =
\pi \int
0
\pi \int
0
xyu(x, y, t) dx dy, t \in [0, T ], (5)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
SOLUTION OF THE BOUNDARY-VALUE PROBLEM OF HEAT CONDUCTION WITH PERIODIC . . . 211
for a two-dimensional parabolic equation with the periodic boundary conditions. The functions
\varphi (x, y) and f(x, y, t) are given functions on [0, \pi ]\times [0, \pi ] and \Omega , respectively. In heat propagation
in a thin rod in which the law of variation E(t) of the total quantity of heat in the rod is given in
[18]. This integral condition in parabolic problems is also called heat moments which are analyzed
in [19].
Condition (4) is initial condition, conditions (2) and (3) are periodic Dirichlet and Neumann
conditions, respectively.
Problem (1) – (5) will be called an inverse problem, the pair
\bigl\{
r(t), u(x, y, t)
\bigr\}
from the class
C[0, T ] \times
\bigl(
C2,2,1(\Omega ) \cap C1,1,0(\Omega )
\bigr)
for which conditions (1) – (5) are satisfied, is called a classical
solution of the inverse problem (1) – (5).
The inverse problem of finding the heat source in a parabolic equation has been investigated in
many studies for the cases when the unknown heat source is space-dependent in [6, 7] and time-
dependent in [5].
Nomenclature:
\varphi (x, y) — initial temprature,
r(t) — unknown coefficient,
E(t) — energy,
u(x, y, t) — temperature distribution,
f(x, y, t) — source function,
u0(t), ucmn(t), ucsmn(t), uscmn(t), usmn(t) — Fourier coefficients,
M — arbitrary constant,
M1,M2,M3,M4,M5,M6 — dimensionless constants,
F (t) — continuous function, K(t, \tau ) — kernel function,
\Omega :=
\bigl\{
(x, y, t) : 0 < x < \pi , 0 < y < \pi , 0 < t < T
\bigr\}
— domain of x, y, t.
2. Existence and uniqueness of the solution of inverse problem. Let us look for the solution
of (1) – (5) in the form:
u(x, y, t) =
u0(t)
4
+
\infty \sum
m,n=1
ucmn(t) \mathrm{c}\mathrm{o}\mathrm{s} 2mx \mathrm{c}\mathrm{o}\mathrm{s} 2ny =
+
\infty \sum
m,n=1
ucsmn(t) \mathrm{c}\mathrm{o}\mathrm{s} 2mx \mathrm{s}\mathrm{i}\mathrm{n} 2ny +
\infty \sum
m,n=1
uscmn(t) \mathrm{s}\mathrm{i}\mathrm{n} 2mx \mathrm{c}\mathrm{o}\mathrm{s} 2ny+
+
\infty \sum
m,n=1
usmn(t) \mathrm{s}\mathrm{i}\mathrm{n} 2mx \mathrm{s}\mathrm{i}\mathrm{n} 2ny.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
212 F. KANCA, I. BAGLAN
By applying the standard procedure of Fourier method, we obtain Fourier coefficients:
u0(t) = \varphi 0 +
4
\pi 2
t\int
0
\pi \int
0
\pi \int
0
r(\tau )f (\xi , \eta , \tau ) d\xi d\eta d\tau ,
ucmn(t) = \varphi cmne
- ((2m)2+(2n)2)t +
4
\pi 2
t\int
0
\pi \int
0
\pi \int
0
r(\tau )e - ((2m)2+(2n)2)(t - \tau )\times
\times f(\xi , \eta , \tau ) \mathrm{c}\mathrm{o}\mathrm{s} 2m\xi \mathrm{c}\mathrm{o}\mathrm{s} 2n\eta d\xi d\eta d\tau ,
ucsmn(t) = \varphi csmne
- ((2m)2+(2n)2)t +
4
\pi 2
t\int
0
\pi \int
0
\pi \int
0
r(\tau )e - ((2m)2+(2n)2)(t - \tau )\times
\times f(\xi , \eta , \tau ) \mathrm{c}\mathrm{o}\mathrm{s} 2m\xi \mathrm{s}\mathrm{i}\mathrm{n} 2n\eta d\xi d\eta d\tau ,
uscmn(t) = \varphi scmne
- ((2m)2+(2n)2)t +
4
\pi 2
t\int
0
\pi \int
0
\pi \int
0
r(\tau )e - ((2m)2+(2n)2)(t - \tau )\times
\times f(\xi , \eta , \tau ) \mathrm{s}\mathrm{i}\mathrm{n} 2m\xi \mathrm{c}\mathrm{o}\mathrm{s} 2n\eta d\xi d\eta d\tau ,
usmn(t) = \varphi smne
- ((2m)2+(2n)2)t +
4
\pi 2
t\int
0
\pi \int
0
\pi \int
0
r(\tau )e - ((2m)2+(2n)2)(t - \tau )\times
\times f(\xi , \eta , \tau ) \mathrm{s}\mathrm{i}\mathrm{n} 2m\xi \mathrm{s}\mathrm{i}\mathrm{n} 2n\eta d\xi d\eta d\tau ,
where
\varphi 0 =
4
\pi 2
\pi \int
0
\pi \int
0
\varphi (x, y) dx dy, \varphi cmn =
4
\pi 2
\pi \int
0
\pi \int
0
\varphi (x, y) \mathrm{c}\mathrm{o}\mathrm{s} 2mx \mathrm{c}\mathrm{o}\mathrm{s} 2ny dx dy,
\varphi csmn =
4
\pi 2
\pi \int
0
\pi \int
0
\varphi (x, y) \mathrm{c}\mathrm{o}\mathrm{s} 2mx \mathrm{s}\mathrm{i}\mathrm{n} 2ny dx dy,
\varphi scmn =
4
\pi 2
\pi \int
0
\pi \int
0
\varphi (x, y) \mathrm{s}\mathrm{i}\mathrm{n} 2mx \mathrm{c}\mathrm{o}\mathrm{s} 2ny dx dy,
\varphi smn =
4
\pi 2
\pi \int
0
\pi \int
0
\varphi (x, y) \mathrm{s}\mathrm{i}\mathrm{n} 2mx \mathrm{s}\mathrm{i}\mathrm{n} 2ny dx dy.
We obtain the solution of the problem (1) – (4) for arbitrary r(t) \in C[0, T ] by
u(x, y, t) =
1
4
\left( \varphi 0 +
4
\pi 2
t\int
0
r(\tau )f0(\tau )d\xi d\eta d\tau
\right) +
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
SOLUTION OF THE BOUNDARY-VALUE PROBLEM OF HEAT CONDUCTION WITH PERIODIC . . . 213
+
\infty \sum
m,n=1
\left( \varphi cmne
- ((2m)2+(2n)2)t +
4
\pi 2
t\int
0
r(\tau )e - ((2m)2+(2n)2)(t - \tau )fcmn(\tau )d\tau
\right) \mathrm{c}\mathrm{o}\mathrm{s} 2mx \mathrm{c}\mathrm{o}\mathrm{s} 2ny+
+
\infty \sum
m,n=1
\left( \varphi csmne
- ((2m)2+(2n)2)t +
4
\pi 2
t\int
0
r(\tau )e - ((2m)2+(2n)2)(t - \tau )fcsmn(\tau ) d\tau
\right) \mathrm{c}\mathrm{o}\mathrm{s} 2mx \mathrm{s}\mathrm{i}\mathrm{n} 2ny+
+
\infty \sum
m,n=1
\left( \varphi scmne
- ((2m)2+(2n)2)t +
4
\pi 2
t\int
0
r(\tau )e - ((2m)2+(2n)2)(t - \tau )fscmn(\tau ) d\tau
\right) \mathrm{s}\mathrm{i}\mathrm{n} 2mx \mathrm{c}\mathrm{o}\mathrm{s} 2ny+
+
\infty \sum
m,n=1
\left( \varphi smne
- ((2m)2+(2n)2)t +
4
\pi 2
t\int
0
r(\tau )e - ((2m)2+(2n)2)(t - \tau )fsmn(\tau ) d\tau
\right) \mathrm{s}\mathrm{i}\mathrm{n} 2mx \mathrm{s}\mathrm{i}\mathrm{n} 2ny,
(6)
where
f0(t) =
4
\pi 2
\pi \int
0
\pi \int
0
f(x, y, t) dx dy, fcmn(t) =
4
\pi 2
\pi \int
0
\pi \int
0
f(x, y, t) \mathrm{c}\mathrm{o}\mathrm{s} 2mx \mathrm{c}\mathrm{o}\mathrm{s} 2ny dx dy,
fcsmn(t) =
4
\pi 2
\pi \int
0
\pi \int
0
f(x, y, t) \mathrm{c}\mathrm{o}\mathrm{s} 2mx \mathrm{s}\mathrm{i}\mathrm{n} 2ny dx dy,
fscmn(t) =
4
\pi 2
\pi \int
0
\pi \int
0
f(x, y, t) \mathrm{s}\mathrm{i}\mathrm{n} 2mx \mathrm{c}\mathrm{o}\mathrm{s} 2ny dx dy,
fsmn(t) =
4
\pi 2
\pi \int
0
\pi \int
0
f(x, y, t) \mathrm{s}\mathrm{i}\mathrm{n} 2mx \mathrm{s}\mathrm{i}\mathrm{n} 2ny dx dy.
Theorem 1. Suppose that the following conditions hold:
(A1) E(t) \in C1[0, T ],
(A2) \varphi (x, y) \in C2,2
\bigl(
[0, \pi ]\times [0, \pi ]
\bigr)
, \varphi (0, y) = \varphi (\pi , y), \varphi x(0, y) = \varphi x(\pi , y), \varphi (x, 0) = \varphi (x, \pi ),
\varphi y(x, 0) = \varphi y(x, \pi ) and
\pi \int
0
\pi \int
0
xy\varphi (x, y) dx dy = E(0),
(A3) f(x, y, t) \in C2,2,0(\Omega ), f(0, y, t) = f(\pi , y, t), fx(0, y, t) = fx(\pi , y, t), f(x, 0, t) =
= f(x, \pi , t), fy(x, 0, t) = fy(x, \pi , t) and
\int \pi
0
\int \pi
0
xyf(x, y, t) dx dy \not = 0,
then solution of the system (1) – (5) has a unique solution.
Proof. The assumptions \varphi (0, y) = \varphi (\pi , y), \varphi (x, 0) = \varphi (x, \pi ), f(0, y, t) = f(\pi , y, t),
f(x, 0, t) = f(x, \pi , t) are consistency conditions which are necessary for solution u(x, y, t) to be in
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
214 F. KANCA, I. BAGLAN
C2,2,1 (\Omega ) \cap C1,1,0
\bigl(
\Omega
\bigr)
. Further, under the smoothness assumptions \varphi (x, y) \in C2,2
\bigl(
[0, \pi ] \times [0, \pi ]
\bigr)
and f(x, y, t) \in C2,2
\bigl(
[0, \pi ] \times [0, \pi ]
\bigr)
\forall t \in [0, T ], the series (6) and its x, y-partial derivative con-
verge uniformly in \Omega since their majorizing sums are absolutely convergent. Therefore, their
sums u(x, y, t), ux(x, y, t) and uy(x, y, t) are continuous in \Omega . In addition, the t-partial deriva-
tive and xx, yy-second order partial derivative series are uniformly convergent for t > 0. Thus,
u(x, y, t) \in C2,2,1 (\Omega ) \cap C1,1,0(\Omega ) and satisfies conditions (1) – (5). In addition, ut(x, y, t) is
continuous in \Omega because the majorizing sum of t-partial derivative series is absolutely conver-
gent under the conditions \varphi x(0, y) = \varphi x(\pi , y), \varphi y(x, 0) = \varphi y(x, \pi ), fx(0, y, t) = fx(\pi , y, t) and
fy(x, 0, t) = fy(x, \pi , t) in \Omega .
We differentiate equation (5) under the condition (A1) to obtain
E
\prime
(t) =
\pi \int
0
\pi \int
0
xyut(x, y, t) dx dy. (7)
Further, under the consistency assumption
\int \pi
0
\int \pi
0
xy\varphi (x, y) dx dy = E(0), formulas (6), (7)
yield the following Volterra integral equation of the second kind:
r(t) = F (t) +
t\int
0
K(t, \tau )r(\tau ) d\tau , t \in [0, T ],
where
F (t) =
E
\prime
(t) +
\pi 2
4
\sum \infty
m,n=1
(2m)2 + (2n)2
mn
\varphi smne
- ((2m)2+(2n)2)t
f0(t) +
\pi 2
4
\sum \infty
m,n=1
(2m)2 + (2n)2
mn
fsmn(t)
, (8)
K(t, \tau ) =
\pi 2
4
\sum \infty
m,n=1
(2m)2 + (2n)2
mn
fsmn(\tau )e
- ((2m)2+(2n)2)(t - \tau )
f0(t) +
\pi 2
4
\sum \infty
m,n=1
(2m)2 + (2n)2
mn
fsmn(t)
. (9)
Under the assumption (A1) – (A3) the function F (t) and the kernel function K(t, \tau ) are continuous
functions in [0, T ] and [0, T ]\times [0, T ], respectively. We obtain a unique function r(t) continuous on
[0, T ] which, together with the solution of the problem (1) – (4) given by the Fourier series u(x, y, t),
form the unique solution of the inverse problem (1) – (5).
Theorem 1 is proved.
3. Continuous dependence of (\bfitr , \bfitu ) upon the data. The following result on continuous
dependence on the data of the solution of the inverse problem (1) – (5) holds.
Theorem 2. If \Phi = \{ \varphi ,E, f\} satisfies the assumptions (A1) – (A3) of Theorem 1, then the
solution (r, u) of problem (1) – (5) depends continuously upon the data \varphi , E, f.
Proof. Let \Phi = \{ \varphi ,E, f\} and \Phi = \{ \varphi ,E, f\} be two sets of the data, which satisfy the
assumptions (A1) – (A3). Suppose that there exist positive constants Mi, i = 1, 2, 3, 4, 5, such that
\| f\| C2,2,0(\Omega ) \leq M, \| f\| C2,2,0(\Omega ) \leq M,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
SOLUTION OF THE BOUNDARY-VALUE PROBLEM OF HEAT CONDUCTION WITH PERIODIC . . . 215
\| \varphi \| C2,2([0,\pi ]\times [0,\pi ]) \leq M1, \| \varphi \| C2,2([0,\pi ]\times [0,\pi ]) \leq M1,
\| E\| C1[0,T ] \leq M2, \| E\| C1[0,T ] \leq M2, \| F\|
C[0,T ]
\leq M3, \| F\|
C[0,T ]
\leq M3,
\| K\| C([0,T ]\times [0,T ]) \leq M4, 0 < M5 \leq \mathrm{m}\mathrm{i}\mathrm{n}
(x,y,t)\in \Omega
| f(x, y, t)| , 0 < M5 \leq \mathrm{m}\mathrm{i}\mathrm{n}
(x,y,t)\in \Omega
\bigm| \bigm| f(x, y, t)\bigm| \bigm| .
Let us denote \| \Phi \| = (\| E\| C1[0,T ] + \| \varphi \|
C2,2
\bigl(
[0,\pi ]\times [0,\pi ]
\bigr) + \| f\| C2,2,0(\Omega )). Let (r, u) and (r, u) be
solutions of inverse problems (1) – (5) corresponding to the data \Phi = \{ \varphi ,E, f\} and \Phi =
\bigl\{
\varphi ,E, f
\bigr\}
,
respectively, where
\| \varphi - \varphi \|
C2,2
\bigl(
[0,\pi ]\times [0,\pi ]
\bigr) \leq
\| \varphi 0 - \varphi 0\| C2,2
\bigl(
[0,\pi ]\times [0,\pi ]
\bigr)
4
+
+
1
6
\infty \sum
m,n=1
\bigm\| \bigm\| \bigm\| (\varphi xy)cmn - (\varphi xy)cmn
\bigm\| \bigm\| \bigm\|
C2,2
\bigl(
[0,\pi ]\times [0,\pi ]
\bigr) + 1
6
\bigm\| \bigm\| \bigm\| (\varphi xy)csmn - (\varphi xy)csmn
\bigm\| \bigm\| \bigm\|
C2,2
\bigl(
[0,\pi ]\times [0,\pi ]
\bigr) +
+
1
6
\bigm\| \bigm\| \bigm\| (\varphi xy)scmn - (\varphi xy)scmn
\bigm\| \bigm\| \bigm\|
C2,2
\bigl(
[0,\pi ]\times [0,\pi ]
\bigr) + 1
6
\bigm\| \bigm\| \bigm\| (\varphi xy)smn - (\varphi xy)smn
\bigm\| \bigm\| \bigm\|
C2,2
\bigl(
[0,\pi ]\times [0,\pi ]
\bigr) ,
(\varphi xy)cmn =
4
\pi 2mn
\pi \int
0
\pi \int
0
\varphi xy(x, y) \mathrm{s}\mathrm{i}\mathrm{n} 2mx \mathrm{s}\mathrm{i}\mathrm{n} 2ny dx dy,
(\varphi xy)csmn =
4
\pi 2mn
\pi \int
0
\pi \int
0
\varphi xy(x, y) \mathrm{s}\mathrm{i}\mathrm{n} 2mx \mathrm{c}\mathrm{o}\mathrm{s} 2ny dx dy,
(\varphi xy)scmn =
4
\pi 2mn
\pi \int
0
\pi \int
0
\varphi xy(x, y) \mathrm{c}\mathrm{o}\mathrm{s} 2mx \mathrm{s}\mathrm{i}\mathrm{n} 2ny dx dy,
(\varphi xy)smn =
4
\pi 2mn
\pi \int
0
\pi \int
0
\varphi xy(x, y) \mathrm{c}\mathrm{o}\mathrm{s} 2mx \mathrm{c}\mathrm{o}\mathrm{s} 2ny dx dy.
From (10), the following equality can be written:
F (t) - F (t) =
E
\prime
(t) + \pi 2
4
\sum \infty
m,n=1
(2m)2 + (2n)2
mn
\varphi smne
- ((2m)2+(2n)2)t
f0(t) +
\pi 2
4
\sum \infty
m,n=1
(2m)2 + (2n)2
mn
fsmn(t)
-
-
E
\prime
(t) +
\pi 2
4
\sum \infty
m,n=1
(2m)2 + (2n)2
mn
\varphi smne
- ((2m)2+(2n)2)t
f0(t) +
\pi 2
4
\sum \infty
m,n=1
(2m)2 + (2n)2
mn
fsmn(t)
.
Applying Hölder inequality and taking maximum of both sides of the last inequality, we obtain
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
216 F. KANCA, I. BAGLAN
\bigm\| \bigm\| F - F
\bigm\| \bigm\| \leq 2M
M2
5
\bigm\| \bigm\| \bigm\| E\prime (t) - E\prime (t)
\bigm\| \bigm\| \bigm\| +
2M\pi 3
\surd
6M2
5
\| \varphi - \varphi \| +M0
\bigm\| \bigm\| f - f
\bigm\| \bigm\| ,
and, similarly, we have \bigm\| \bigm\| K - K
\bigm\| \bigm\| \leq M6
\bigm\| \bigm\| f - f
\bigm\| \bigm\| ,
\| r - r\| \leq
\bigm\| \bigm\| F - F
\bigm\| \bigm\| +M4T \| r - r\| + MT
1 - TM4
\bigm\| \bigm\| K - K
\bigm\| \bigm\| ,
\| r - r\| \leq 2M
M2
5 (1 - TM4)
\bigm\| \bigm\| \bigm\| E\prime (t) - E\prime (t)
\bigm\| \bigm\| \bigm\| +
M0
1 - TM4
\bigm\| \bigm\| f - f
\bigm\| \bigm\| +
+
TM
(1 - TM4)
2
\bigm\| \bigm\| f - f
\bigm\| \bigm\| +
2M\pi 3
\surd
6M2
5 (1 - TM4)
\| \varphi - \varphi \| ,
\| u - u\| \leq M7
\bigm\| \bigm\| \bigm\| E\prime (t) - E\prime (t)
\bigm\| \bigm\| \bigm\| +M8 \| \varphi - \varphi \| +M9
\bigm\| \bigm\| f - f
\bigm\| \bigm\| ,
\| u - u\| \leq M10
\bigm\| \bigm\| \Phi - \Phi
\bigm\| \bigm\| ,
where
M0 = \mathrm{m}\mathrm{a}\mathrm{x}
\Biggl\{
\pi 4M2
6M2
5
,
\pi 3M2\surd
6M2
5
\Biggr\}
, M6 = \mathrm{m}\mathrm{a}\mathrm{x}
\Biggl\{
\pi 32M\surd
6M2
5
,
\pi 3
\surd
6M2
5
\Biggr\}
,
M7 =
2MT
3M2
5 (1 - TM4)
, M8 = \mathrm{m}\mathrm{a}\mathrm{x}
\Biggl\{
1,
4M2\pi 3T
3
\surd
6M2
5 (1 - TM4)
\Biggr\}
,
M9 = \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
2MT
3M2
5 (1 - TM4)
,
2M2M6T
2
3(1 - TM4)
\biggr\}
, M10 = \mathrm{m}\mathrm{a}\mathrm{x}\{ 1,M7,M8,M9\} .
If \Phi \rightarrow \Phi then r \rightarrow r and u \rightarrow u.
Theorem 2 is proved.
4. Numerical method for the problem (1) – (4). In this section, we use implicit finite-difference
approximation for the discretizing problem (1) – (5):
1
\tau
\Bigl(
uk+1
i,j - uki,j
\Bigr)
=
1
h2
\Bigl(
uk+1
i - 1,j - 2uk+1
i,j + uk+1
i+1,j
\Bigr)
+
+
1
h2
\Bigl(
uk+1
i,j - 1 - 2uk+1
i,j + uk+1
i,j+1
\Bigr)
+ rk+1fk+1
i,j ,
u0i,j = \phi i,
(10)
uk0,j = ukM+1,j , ukM+1,j =
uk1,j - ukM,j
2
,
uki,0 = uki,M+1, uki,M+1 =
uki,1 - uki,M
2
,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
SOLUTION OF THE BOUNDARY-VALUE PROBLEM OF HEAT CONDUCTION WITH PERIODIC . . . 217
where we discretize the computing domain [0, \pi ] \times [0, \pi ] \times [0, T ] by xi = ih, i = 0, 1, . . . ,M,
yj = jh, j = 0, 1, . . . ,M, and tk = k\tau , k = 0, 1, . . . , N, where h = \pi /M and \tau = T/N are
the space and time steps, respectively, and M, N are two positive integers, uki,j = u(xi, yj , tk),
fk
i,j = f(xi, yj , tk), r
k = r(tk).
Let us integrate equation (1) with respect to x and y from 0 to \pi and use (2), (3) and (5), we
obtain
r(t) =
E\prime (t) -
\int \pi
0
yux(\pi , y, t) dy -
\int \pi
0
xuy(x, \pi , t) dx\int \pi
0
\int \pi
0
xyf(x, y, t) dx dy
. (11)
The finite difference approximation of (11) is
rk+1 =
(Ek+2 - Ek)/\tau -
\biggl( \int \pi
0
yux(\pi , y, t) dy
\biggr) k
-
\biggl( \int \pi
0
xuy(x, \pi , t) dx
\biggr) k
\biggl( \int \pi
0
\int \pi
0
xyf(x, y, t) dx dy
\biggr) k
,
where Ek = E(tk), k = 0, 1, . . . , N. We mention that the integrals are numerically calculated using
Simpson’s rule of integration and also the first derivatives are calculated using central difference
scheme.
rk(s), u
k(s)
i,j are the values of rk, uki,j at the sth iteration step, respectively. At each (s + 1)th
iteration step, rk+1(s+1) is as follows:
rk+1(s+1) =
(Ek+2 - Ek)/\tau -
\biggl( \int \pi
0
yux(\pi , y, t) dy
\biggr) k(s)
-
\biggl( \int \pi
0
xuy(x, \pi , t) dx
\biggr) k(s)
\biggl( \int \pi
0
\int \pi
0
xyf(x, y, t) dx dy
\biggr) k
.
The iteration of (10) is
1
\tau
\Bigl(
u
k+1(s+1)
i,j - u
k+1(s)
i,j
\Bigr)
=
1
h2
\Bigl(
u
k+1(s+1)
i - 1,j - 2u
k+1(s+1)
i,j + u
k+1(s+1)
i+1,j
\Bigr)
+
+
1
h2
\Bigl(
u
k+1(s+1)
i,j - 1 - 2u
k+1(s+1)
i,j + u
k+1(s+1)
i,j+1
\Bigr)
+ rk+1(s+1)fk+1
i,j ,
u0i,j = \phi i,
(12)
u
k+1(s)
0,j = u
k+1(s)
M+1,j , u
k+1(s)
M+1,j =
u
k+1(s)
1,j - u
k+1(s)
M,j
2
,
u
k+1(s)
i,0 = u
k+1(s)
i,M+1 , u
k+1(s)
i,M+1 =
u
k+1(s)
i,1 - u
k+1(s)
i,M
2
.
The system of the equations (12) is solved and u
k+1(s+1)
i,j is determined. If the difference of values
between two iterations reaches the prescribed tolerance, the iteration is stopped.
In order to illustrate the behavior of our numerical method, an example is considered.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
218 F. KANCA, I. BAGLAN
Example 1. This example investigates finding the exact solution\bigl\{
r(t), u(x, y, t)
\bigr\}
=
\bigl\{
2 \mathrm{e}\mathrm{x}\mathrm{p}(t2), (2 + \mathrm{c}\mathrm{o}\mathrm{s} 2x+ \mathrm{c}\mathrm{o}\mathrm{s} 2y) \mathrm{e}\mathrm{x}\mathrm{p}(t2)
\bigr\}
.
for the given functions
\varphi (x, y) = (2 + \mathrm{c}\mathrm{o}\mathrm{s} 2x+ \mathrm{c}\mathrm{o}\mathrm{s} 2y), E(t) =
\pi 4
2
\mathrm{e}\mathrm{x}\mathrm{p}(t2),
F (x, y, t) = 2t+ (t+ 2)(\mathrm{c}\mathrm{o}\mathrm{s} 2x+ \mathrm{c}\mathrm{o}\mathrm{s} 2y).
The step sizes are h = 0.0393, \tau = 0.005.
Note that the convergence criterion for r(t), was
\bigm| \bigm| rk+1(s+1) - rk+1(s)
\bigm| \bigm| \leq h/200.
The comparisons between the exact solution and the numerical finite difference solution are
shown in Figs. 1 – 3 and Table 1 when T = 1.
0 0.2 0.4 0.6 0.8
2
2.5
3
3.5
4
4.5
5
(a)
t
r(t)
0 0.5 1 1 .5 2 2 .5 3 x
-1
0
1
2
3
4
5
u(1,y,1)
(a) (b)
Fig. 1. The exact and approximate solutions of r(t) (a) and of u(1, y, 1) (b). The exact solution is shown with dashes
line.
0
20
40
60
0
20
40
60
−1
0
1
2
3
4
x
y
u
0
20
40
60
0
20
40
60
0
1
2
3
4
x
y
u
(a) (b)
Fig. 2. The approximate (a) and the numerical (b) solutions of u(x, y, 1/10).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
SOLUTION OF THE BOUNDARY-VALUE PROBLEM OF HEAT CONDUCTION WITH PERIODIC . . . 219
0
20
40
60
0
20
40
60
0
0.02
0.04
0.06
x
y
Absolute
error
Fig. 3. The absolute error of u(x, y, 1/10).
Table 1. The some values of r(t)
Exact Approximate Error Relative error
2 2.0614 0.0614 0.0307
2.0201 2.0717 0.0516 0.0255
2.0816 2.1098 0.0282 0.0135
2.1883 2.2099 0.0216 0.0099
2.3470 2.3664 0.0194 0.0083
2.5681 2.5874 0.0193 0.0075
2.8667 2.8873 0.0206 0.0072
3.2646 3.2878 0.0232 0.0071
3.7930 3.8201 0.0272 0.0072
4.4958 4.5287 0.0410 0.0075
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Received 14.12.16,
after revision — 21.12.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
|
| id | umjimathkievua-article-2367 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:28Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9e/0a1404a3995f889592d3d79ec64ee29e.pdf |
| spelling | umjimathkievua-article-23672020-04-07T12:13:22Z Solution of the boundary-value problem of heat conduction with periodic boundary conditions Розв'язок граничної задачі теплопровідності з періодичними граничними умовами Kanca, F. Baglan, I. Kanca, F. Baglan, I. Kanca, F. Baglan, I. граничної задачі теплопровідності гранична задача теплопровідності boundary-value problem periodic boundary conditions UDC 517.9 We investigate the solution of the inverse problem for a linear two-dimensional parabolic equation with periodic boundary and integral overdetermination conditions.Under certain natural regularity and consistency conditions imposed on the input data, we establish the existence, uniqueness of the solution and its continuous dependence on the data by using the generalized Fourier method. In addition, an iterative algorithm is constructed for the numerical solution of this problem. УДК 517.9 Вивчається розв’язок оберненої задачi для лiнiйного двовимiрного параболiчного рiвняння з перiодичними граничними умовами та iнтегральними умовами перевизначення. За деяких природних умов регулярностi й узгодженостi, що накладенi на початковi данi, встановлено iснування, єдинiсть розв’язку та його неперервну залежнiсть вiд даних за допомогою узагальненого методу Фур’є. Крiм того, побудовано iтеративний алгоритм для побудови чисельного розв’язку цiєї проблеми. Institute of Mathematics, NAS of Ukraine 2020-02-11 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2367 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 2 (2020); 209-220 Український математичний журнал; Том 72 № 2 (2020); 209-220 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2367/1561 Copyright (c) 2020 F. Kanca,I. Baglan |
| spellingShingle | Kanca, F. Baglan, I. Kanca, F. Baglan, I. Kanca, F. Baglan, I. Solution of the boundary-value problem of heat conduction with periodic boundary conditions |
| title | Solution of the boundary-value problem of heat conduction with periodic boundary conditions |
| title_alt | Розв'язок граничної задачі теплопровідності з періодичними граничними умовами |
| title_full | Solution of the boundary-value problem of heat conduction with periodic boundary conditions |
| title_fullStr | Solution of the boundary-value problem of heat conduction with periodic boundary conditions |
| title_full_unstemmed | Solution of the boundary-value problem of heat conduction with periodic boundary conditions |
| title_short | Solution of the boundary-value problem of heat conduction with periodic boundary conditions |
| title_sort | solution of the boundary-value problem of heat conduction with periodic boundary conditions |
| topic_facet | граничної задачі теплопровідності гранична задача теплопровідності boundary-value problem periodic boundary conditions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2367 |
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