CC BY
6
УДК 517.9
The Solution of Adjoint Heat Problem in Spherical Area by Laplace Transform Method
Ilona A. Reznikova*
Institute of Mathematics and Computer Science, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 07.01.2013, received in revised form 14.03.2013, accepted 20.05.2013 The spherically symmetric adjoint initial-boundary value problem of heat propagation in closed bounded spherical regions has been researched. The exact analytical solution of the direct and inverse nonstationary problem has been obtained using Laplace transform method. The stationary state has been found and it is shown that the nonstationary solution converges to stationary one when time tends to infinity, if such are the heat sources in media.
Keywords: initial-boundary value problem, interface, Laplace transform method, inverse problem.
1. Problem statement
Assume that the functions u1(r,t), u2(r, t) are defined in areas Qi = {r|0 < r < R1}, Q2 = {r|R1 < r < R2}, respectively, and satisfy the equations
uit = Xi ^uirr + 2uir^ + fi(r,t), t> 0, r e Qi, (1.1)
U2t = X2 ^u2rr + 2u2r^ + f2 (r,t), t> 0, r e Q2 . (1.2)
The functions u, j = 1,2 are temperature fields, fj are defined internal heat sources; Xj are positive constants known as the thermal diffusivities. In addition, we have initial and boundary conditions
ui|t=0 = u2 |t=o = 0; (1.3)
|ui(0,t)| < to, (1.4)
ui|r = Ri = u2 |r=Ri, (1.5) du2
k dui
1 dr
= k2^ r=R1 dr
R ' (1-6)
r=R1
u2|r = R2 =0, (1.7)
where kj are heat conductivity coefficients. Condition (1.5) is equality of temperatures, and (1.6) is equality of heat fluxes on boundary surface r = R1. It is known [1] that Xj = kj/cjpj, where cj are specific heats, pj are densities of media.
It is necessary to find functions ui e C2(Qi) nC 1(ri), u2 e C2(Q2) nC1 (ri) nC(r2), which satisfy the equations (1.1), (1.2) and conditions (1.3)—(1.7), ri = {r|r = Ri} , r2 = {r|r = R2} .
* ilona_reznikova@mail.ru © Siberian Federal University. All rights reserved
2. The solution of nonstationary problem by Laplace transform method
We shall find the solution of the problem (1.1)-(1.7) by Laplace transform (its using is validated in [2]). We obtain
U' + 2U' - ^Uj = -Fj (2.1)
J r 0 Xj Xj
for images Uj (r,p) of functions uj (r,t) by use the relation feature of original function's differentiation [2] and initial conditions (1.3). Here, Fj = Fj(r,p) are images of functions fj(r,t). Boundary conditions become such as
Ui|r=fll = U2 |r=Ri, (2.2)
(2.3)
, dUi kl ~7T~ or
, dU2 = k2 r = Ri Or
r=R1
U2I r=R2 =0, (2.4)
|Ui(0,p)| < rc. (2.5)
Let us introduce change of variables Uj = vj/r, then equation (2.1) take on form
vj - PVj = -—Fj. (2.6)
Xj Xj
The solution of homogeneous differential equation (2.6) has the form [3]
Vj0 = Cl exp ( r. /— ) + Cl exp ( -r* — ) .
Xj Xj
We define fundamental system of solutions = exp (r^p/xj^ , fj2 = exp (^-r^/p/xj^ for
finding of particular solution. Then Wronsky's determinant W (r) = flf^' - = - 2\Jp/Xj and solution of equations (2.1) can be represented by formulae (subject to boundary conditions (2.2)-(2.5))
sh (rJïp) exp ((R-2 - Ri)J]^ rR2 ( [V)
Ui = v Vxiy-\-xj ÏF2 sh (r - 4) J X- d4+
r^pX2 sh {Ri^H) JRi \ \X2)
1
+
^/pxi
¡hS^ f sh ((Ri- VXD «-I'№ sh ((r - 4) VXD A +
\
C sh (r, A-
+^tr^ lexp №-exp ((2R- Ri) ] (2J)
U2 = ? ("P (VXD -exK(2R2-r)/X9) -rr/mi ^sh ((r -4)/X9 d4+
1 N IT ) ^2
+
ex^(R2 - r')\[]2) jR 2 4F2 shf^R2 - 4)^X2) d£> (2.8)
r
where
Cx eXp((R2-Ri)J
C =
fR1 eF sh (№ - e) Д) de- xi MkR, ^, fRi eF sh de
VPXÏ JRi ^ 2 Vv 2 sy V*2/ xi sh( Ri^J _£_) JO
C2 exp (Ri + Cl exP ((-R - Ri ) yfOD '
(2.9)
Ci = # + ^ + ^ cth fRi^/j^^ , C2 = - - k cth f R./^
V X2 Ri V xi V Vxi/ V X2 Ri V xi V V xi
(2.10)
3. The solution of stationary problem
Assume that fj(r,t) ^ /¿(r), when t ^ то. Then the question arise whether solution of nonstationary problem will converge to solution of stationary one. The steady-state condition of heat conduction satisfies the equations
2 1
«0» + - u0'(r) =--f0(r) (3.1)
with boundary conditions
xj
«i(Ri) = u2(Ri), (3.2)
ki«0'(Ri) = k2 «2' (Ri), (3.3)
u
2(R2) = 0, (3.4)
|u0(0)| < TO. (3.5)
It is easy to see that the problem (3.1)-(3.5) has the following solution:
- - iR1 e2f0(e) de - - /R7^ - ^ f0(e) de-
R i R2/ X wo X wo \R
- ^ Г (R - 0 /20(e) de + - /7- - e) /0(e) de, (3.6)
X2 J Ri \ R2 y X i Л Vr /
1
^ - i) - Г e2/0(e) de - - Г (^ - 0 /0(e) de+
,r R2/ X i Jo X2 JRi VR
if ^e2
r
+ XL Г - A/2o(e) de. (3.7)
X2 J rA r /
We use the well-known limiting relations lim pF(p) = f (to) [2], sh x ~ x, ch x ~ 1, exp x 1 + x, when x ^ 0, and formulae (2.7)-(2.10) for finding of limits
o
«
2
limpu(r,p) = — lim 1 fRl e(R i - e)/0 de + - f e(e - / de) +
p^0 x i p^0 y R 17o w o J
1+(R2-Rl^" (/xi+Ri) /rRi2 e(R - e)/0 de - xiR hRl e2/0 de
, 2(R i - R2) r VX2 +----lim
R 1 p^o
X2 - (1 + R n/X2) + U/X2 + Rr) (- + № - ri)JX2
x2 Rl
+
1 + (R2 - Ri V X2 /• R2 0 0
+ lim-s— e(R2 - e)/? de = uftr),
P^0 R iX2 J Ri
limpU2(r,p) = — lim ( /e(e - r)/0de + fi + № - r),/^ f 2 e№ - e/ de i +
rx2 P^0 \./ V VX2/ JRi /
+ Ri) JrR2 e(R2 - e)/0 de - xiR hRl e2/0 de
2(r-R2) Hf r V X2
+ lim ■
p^0
p^0 i+(R2-RiU X2
X2 - 1 + R iVX2 +VX2 + Ri) (1 + (2R2 -R iK/X2
= u2(r)
with functions u0, u!° from (3.6), (3.7). In other words, we have proved that the nonstationary solution converges to stationary one when time tends to infinity, if such are the heat sources in media.
Let us now put fj = fj(t). In this case formulae (2.7)-(2.9) are simplified up to
Ui
Sh fc^ exp ((R2 - R i) - R iF ) +
rp sh (Ri Jxl^ V VX2,
+
C Sh )
r sM R i VXi
exp ( R i^XQ - ex^(2R2 - Ri) JX^)
Fi
+ —, (3.8) p
U2 = C4 (-(VS - ex^<2R2 " ^ ) + f + ^ exp (№ - r^ -
where
FT
rp
it sK,r - Ri >/XD+Ri ch((r - Ri '/XT
C3 = JXsh((R - R i)yXXr) + R i *((R - R i) JX^) - R2,
C4 =
fcF1
ypxi
Xi (^ - R i cth (R i VXD) + ^ exp ((R2 - R i^
C2 exp (R i /Xi) + Ci exp ((2R - R i ^^Xi) and constants Ci; C2 are determined in (2.10).
(3.9)
(3.10)
(3.11)
r
4. On the evaluation of internal heat sources
Consider the problem (1.1)-(1.7) when f2 =0 (there is no heat sources in second medium), add condition of overdetermination (we specify heat flux on surface) and regard temperature on sphere surface as non-nil constant. Then problem take on form
"it
"2t
Xi ( "irr + 2"ir ) + fi(r,t), t> 0, r G Oi,
2
X2 "2rr + - "2r r
t > 0, r G O2;
"i|t=0 = "2 |t=0 = 0; |"i(0,t)| < to,
"i|r=Ri = "2 |r=Ri, ki"ir |r = Ri = k2"2r |r=Ri,
"2 |r=R2 = T = const, "2r |r= R2 = Q = const.
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
Let us introduce change of variables uj = wj — T and apply Laplace transform to the problem (4.1)-(4.8). Thus we get following problem for images Uj of functions uj (F is image of function f1):
Ui' + 2Ui - ^Ui = rX1 - — (Fi - T), Xi (4.9)
U2' + 2 U2 - -— U2 = r X2 T ; X2 ' (4.10)
|Ui(0,p)| < to, (4.11)
Ui|r=Ri = U2 |r = Ri , (4.12)
kiUir |r=Ri = ^2 U2r |r =Ri , (4.13)
U2|r=R2 =0, (4.14)
U2r |r=R2 = Q. (4.15)
By using of the task (2.1)-(2.5) solution (see formulae (2.7)-(2.10)) and condition of overdetermination (4.15) we derive integral equation of the first kind for F1(r,p) :
jf iFi sh ^) d£ = (^^ — sh a. — ^^ | +
+
XiR2 sh a(Cieb + C^e-6) f T
2^v/pX2
R2
^^((Ri + 1)ch b - R2 - l) + (y + sh b
QX2A P )
(4.16)
where a = R1 -y/p/x1, b = (R2 — ROx/p/x2. Assume that f1 = f1(t), then F1(p) can be found from (3.9) and (4.15), or (4.16), as
Fi(p)= T + ( (QR2 + T)A/^ - T
Xp2 + RiJ exp^R - ri)^X2) - R2
P
X
C2 exp ((Ri - R)^) + Ci exp ((R - R) CiCsTexp ((R - Ri) ^J)
x +k C1 - cth (Ri/X1))'
(4.17)
The constants Ci; C2, C3 are specified by formulae (2.10), (3.10). So the heat source in sphere 0 < r < Ri (that is function fi(t)) can be obtained by using of inverse Laplace transform. Here
we evaluate only lim fi(t).
t—
Since ex « 1 + x + x2/2 when x ^ 0, we derive from (4.17)
Xi i- f ГУ 1 kpR\ f ГР p(Ri - Д2)
limpFi(p) = lim ,/f- - — - 1 + (Ri - R),/f- +
p^0 kR2 p^oVV X2 Ri 2xi У V V X2 2x2
2
x ( QR^/X2 - T
R2 - R2 [Г + pRi(R2 - Ri)2
2 V X2 2X2
kR2 P-4V X2 1 Ri 1 2xi JV R2)V X2 ' 2X2
2
Xi li^/? + fi - (Ri -*),/? + ) x
x (QR2^X2 -T
R2 - R2 pRi(R2 - Ri)2
X2 2X2
T 2xiQR2 , . . + pm Tp = - ЛМ.
Thus the boundedness of internal heat sources at infinity is proved.
Author expresses her gratitude to professor V. K. Andreev for problem definition and unceasing attention throughout fulfilment of work.
The work is implemented with financial support of SB RAS, integration project no. 44-
References
[1] A.N.Tihonov, A.A.Samarsky, Equations of mathematical physics, Nauka, Moscow, 1972 (in Russian).
[2] M.A.Lavrent'ev, B.V.Shabat, The function of complex variable theory methods, Nauka, Moscow, 1987 (in Russian).
[3] E.Kamke, The ordinary differential equations reference-book, Nauka, Moscow, 2003 (in Russian).
Решение сопряженной тепловой задачи в шаровой области методом преобразования Лапласа
Илона А. Резникова
Исследована сферически симметричная сопряжённая начально-краевая задача распространения тепла в замкнутых ограниченных шаровых областях. Точное аналитическое решение прямой и обратной нестационарной задачи получено методом преобразования Лапласа. Найдено стационарное состояние и показано, что оно является предельным при больших временах, если таковыми служат источники тепла в средах.
Ключевые слова: начально-краевая задача, поверхность раздела, преобразование Лапласа, обратная задача.
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