With the use of the method of summator–integral equations, an axisymmetric problem has been investigated
that deals with the development of spatial temperature fields appearing in a finite cylinder
with an arbitrary distribution of initial temperature when the cylinder comes in contact with a semiinfinite
body that has a constant initial temperature. The essential feature of the considered thermophysical
model of heat exchange is that mixed boundary conditions of the second and fourth kind are
assigned in the plane of contact of the finite body with the semispace. The thermophysical properties
of the bodies considered are different

Mandrik, P. A.

English

Электронная библиотека БГУ

SOLUTION OF A HEAT-CONDUCTION PROBLEM
FOR A FINITE CYLINDER AND SEMISPACE
UNDER MIXED LOCAL BOUNDARY CONDITIONS
IN THE PLANE OF THEIR CONTACT
P. A. Mandrik UDC 517.968,536.24
With the use of the method of summator–integral equations, an axisymmetric problem has been inves-
tigated that deals with the development of spatial temperature fields appearing in a finite cylinder
with an arbitrary distribution of initial temperature when the cylinder comes in contact with a semi-
infinite body that has a constant initial temperature. The essential feature of the considered thermo-
physical model of heat exchange is that mixed boundary conditions of the second and fourth kind are
assigned in the plane of contact of the finite body with the semispace. The thermophysical properties
of the bodies considered are different.
The formulation of the problem consists of the determination of the laws governing the development
of spatial nonstationary temperature fields in a semispace and a finite cylinder of radius R and height l, when
one of the end faces of the finite cylinder touches the semispace surface. The thermophysical characteristics
of the bodies considered and their initial temperatures are different and the distribution of the initial tempera-
ture of the cylinder is arbitrary. Outside the circular region of contact on the surface of the semispace and on
the lateral and noncontacting end-face surface of the cylinder there is ideal thermal insulation. Thereafter,
when writing down mathematical formulas, the subscript 1 is used for the semispace and 2 for the cylinder.
Let r and z denote cylindrical coordinates, τ time, T1(r, z, τ) the temperature of the semi-infinite body
(r > 0, z < 0. τ > 0), T2(r, z, τ) the temperature of the cylinder (0 < r < R, 0 < z < l, τ > 0), and λ1 > 0 and
a1 > 0 and λ2 > 0 and a2 > 0 the thermal conductivity and thermal diffusivity of the semi-infinite body and
cylinder, respectively.
Thus, it is necessary to solve a system of two differential equations of nonstationary heat conduction
(in the corresponding ranges of coordinates)
1
r
 
∂
∂r
 
r 
∂T1 (r, z, τ)
∂r
 + 
∂2T1 (r, z, τ)
∂z2
 = 
1
a1
 
∂T1 (r, z, τ)
∂τ
 ,   r > 0 ,   z < 0 ,   τ > 0 , (1)
1
r
 
∂
∂r
 
r 
∂T2 (r, z, τ)
∂r
 + 
∂2T2 (r, z, τ)
∂z2
 = 
1
a2
 
∂T2 (r, z, τ)
∂τ
 ,   0 < r < R ,   0 < z < l ,   τ > 0 , (2)
with the initial
T1 (r, z, 0) = T10 = const ,   T2 (r, z, 0) = f (r, z) ≠ T10 (3)
Journal of Engineering Physics and Thermophysics, Vol. 74, No. 5, 2001
Belarusian State University, Minsk, Belarus; email: mandrik@fpm.bsu.unibel.by. Translated from In-
zhenerno-Fizicheskii Zhurnal, Vol. 74, No. 5, pp. 153–159, September–October, 2001. Original article submit-
ted March 14, 2001.
1062-0125/01/7405-1262$25.00 2001 Plenum Publishing Corporation1262
and boundary-value conditions
∂T1 (r, − ∞, τ)
∂z  = 
∂T1 (0, z, τ)
∂r  = 
∂T1 (∞, z, τ)
∂r  = 
∂T2 (0, z, τ)
∂r  = 
∂T2 (R, z, τ)
∂r  = 
∂T2 (r, l, τ)
∂z  = 0 ,
(4)
T1 (r, 0, τ) = T2 (r, 0, τ) ,   0 < r < R , (5)
Kλ 
∂T1 (r, 0, τ)
∂z  = 
∂T2 (r, 0, τ)
∂z  ,   0 < r < R ,
(6)
∂T1 (r, 0, τ)
∂z  = 0 ,   R < r < ∞ , (7)
where Kλ = λ1 ⁄ λ2.
We note that, according to [1], expressions (5) and (6) determine the boundary condition of the fourth
kind in the region z = 0, 0 < r < R, and the system of expressions (5)–(7) determines the mixed boundary
conditions on the surface z = 0 in the corresponding regions of change of the variable r.
To solve the problem formulated, we first apply the Laplace integral transformation (L-transforma-
tion) [2], as a result of which, with account for the initial conditions, the problem at hand in the region of
transforms will be written as
1
r
 
∂
∂r
 
r 
∂T
_
1 (r, z, s)
∂r
 + 
∂2T
_
1 (r, z, s)
∂z2
 − 
s
a1
 T
_
1 (r, z, s) + 
T10
a1
 = 0 ,   r > 0 ,   z < 0 , (8)
1
r
 
∂
∂r
 
r 
∂T
_
2 (r, z, s)
∂r
 + 
∂2T
_
2 (r, z, s)
∂z2
 − 
s
a2
 T
_
2 (r, z, s) + 
f (r, z)
a2
 = 0 ,   0 < r < R ,   0 < z < l , (9)
∂T
_
1 (r, − ∞, s)
∂z  = 
∂T
_
1 (0, z, s)
∂r  = 
∂T
_
1 (∞, z, s)
∂r  =  
∂T
_
2 (0, z, s)
∂r  = 
∂T
_
2 (R, z, s)
∂r  = 
∂T
_
2 (r, l, s)
∂z  = 0 , (10)
T
_
1 (r, 0, s) = T
_
2 (r, 0, s) ,   0 < r < R , (11)
Kλ 
∂T
_
1 (r, 0, s)
∂z  = 
∂T
_
2 (r, 0, s)
∂z  ,   0 < r < R , (12)
∂T
_
1 (r, 0, s)
∂z  = 0 ,   R < r < ∞ , (13)
in which
1263
T
_
i (r, z, s) = L [Ti (r, z, τ)] = ∫ 
0
∞
Ti (r, z, τ) exp (− sτ) dτ ,   i = 1, 2 , (14)
and the restriction Re s > 0 on the parameter of Laplace transformation (L-parameter) is omitted for brevity
here and hereafter.
Solution of Eq. (8) under the corresponding conditions from (10) is known [3]:
T
_
1 (r, z, s) = 
T10
s
 + ∫ 
0
∞
C
_
 (ρ, s) exp 
− z √ρ2 + sa1   J0 (ρr) ρdρ ,   r > 0 ,   z < 0 , (15)
Here J0(ρr) is the Bessel function of the first kind and zero order and C
__
(ρ, s) is the unknown analytical
function.
To solve Eq. (9), we will apply to it the Hankel finite integral transformation [2]
T
_
2H (p, z, s) = H [T
_
2 (r, z, s)] = ∫ 
0
R
rT
_
2 (r, z, s) J0 (pr) dr . (16)
Then Eq. (9), with account for the corresponding conditions from (10), is transformed into the equation
∂2T
_
2H (p, z, s)
∂z2
 − ψ2 (p, s) T
_
2H (p, z, s) + F (p, z) = 0 ,   0 < z < l , (17)
in which
F (p, z) = 1
a2
 ∫ 
0
R
rf (r, z) J0 (pr) dr , (18)
ψ(p, s) = √p2 + s ⁄
 
a2 , and pR = µ are zeros of the Bessel function J1(µ).
The solution of differential equation (17) can be easily obtained by the classical method of varying
an arbitrary constant (see, e.g., [14]):
T
_
2H (p, z, s) = A
_
1 (p, s) cosh [zψ (p, s)] + A
_
2 (p, s) sinh [zψ (p, s)] −
− 
1
ψ (p, s) ∫ 
0
z
F (p, ξ) sinh [(z – ξ) ψ (p, s)] dξ , (19)
Here the unknown functions-transforms A
_
i(p, s) are determined by the boundary condition
∂T
_
2H (p, l, s)
∂z  = 0 .
(20)
Substituting Eq. (19) into Eq. (20), we obtain the value
A
_
1 (p, s) = 
1
ψ (p, s) sinh [lψ (p, s)] ∫ 
0
l
F (p, ξ) cosh [(l − ξ) ψ (p, s)] dξ − A
_
2 (p, s) cot [lψ (p, s)] ,
1264
the substitution of which into Eq. (19) allows us to write the following expression:
T
_
2H (p, z, s) = 
cosh [zψ (p, s)]
ψ (p, s) sinh [lψ (p, s)] ∫ 
0
l
F (p, ξ) cosh [(l − ξ) ψ (p, s)] dξ −
− 
1
ψ (p, s) ∫ 
0
z
F (p, ξ) sinh [(z − ξ) ψ (p, s)] dξ − A
_
2 (p, s) 
cosh [(l − z) ψ (p, s)]
sinh [lψ (p, s)]  ,   0 < z < l . (21)
The formula of the reversal of the Hankel transform for expression (21) can be written in the form
T
_
2 (r, z, s) = 
2
R2
 T
_
2H (0, z, s) + 
2
R2
  ∑ 
n=1
∞
 
J0 (pnr)
J0
2
 (pnR)
 T
_
2H (pn, z, s) ,   0 < r < R ,   0 < z < l , (22)
in which pnR = µn are the zeros of the Bessel function J1(µn).
Expression (21) explicitly yields the following value needed to calculate the Hankel inverse transform
(22):
T
_
2H (0, z, s) = 
cosh 
z √ sa2  

√ sa2  sinh l √ sa2  
  ∫ 
0
l
F (0, ξ) cosh 
(l − ξ) √ sa2  
 dξ −
− √ a2s   ∫ 
0
z
F (0, ξ) sinh 
(z − ξ) √ sa2  
 dξ − A
_
2 (0, s) 
cosh 
(l − z) √ sa2  
sinh 
l √ sa2  
 ,   0 < z < l ,
which makes it possible, by using notation (21), to write the solution in the region of Laplace transforms for
the finite cylinder as
T
_
2 (r, z, s) = 
2 cosh 
z √ sa2  
R2 √ sa2  sinh l √ sa2  
  ∫ 
0
l
F (0, ξ) cosh 
(l − ξ) √ sa2   dξ −
− 
2
R2
 √ a2s   ∫ 0
z
F (0, ξ) sinh 
(z − ξ) √ sa2  
 dξ − 2A
_
2 (0, s) 
cosh 
(l − z) √ sa2  

R2 sinh 
l √ sa2  

 +
1265
+ 
2
R2
  ∑ 
n=1
∞
 
J0 (pnr)
J0
2
 (pnR)
 T
_
2H (pn, z, s) ,   0 < r < R ,   0 < z < l , (23)
in which it is necessary to determine the unknown functions A
_
2(pn, s) to satisfy the mixed boundary condi-
tions (11)–(13).
Formulas (15) and (23) yield the corresponding expressions
T
_
1 (r, 0, s) = 
T10
s
 + ∫ 
0
∞
C
_
 (ρ, s) J0 (ρr) ρdρ ,   r > 0 ;
∂T
_
1 (r, 0, s)
∂z  = − ∫ 
0
∞
C
_
 (ρ, s) √ρ2 + sa1  J0 (ρr) ρdρ ,   r > 0 ;
T
_
2 (r, 0, s) = 2
R2 √ sa2  sinh 
l √ sa2  
  ∫ 
0
l
F (0, ξ) cosh 
(l − ξ) √ sa2   dξ − 2R2 A
_
2 (0, s) cot 
l √ sa2   −
− 
2
R2
  ∑ 
n=1
∞
 
J0 (pnr)
J0
2
 (pnR)
 A
_
2 (pn, s) cot [lψ (pn, s)] +
+ 
2
R2
  ∑ 
n=1
∞
 
J0 (pnr)
J0
2
 (pnR) ψ (pn, s) sinh [lψ (pn, s)]
  ∫ 
0
l
F (pn, ξ) cosh [(l − ξ) ψ (pn, s)] dξ ,   0 < r < R ,
∂T
_
2 (r, 0, s)
∂z
 = 
2
R2
 √ s
a2
 A
_
2 (0, s) cot 
l √ sa2  
 + 
2
R2
  ∑ 
n=1
∞
 
J0 (pnr)
J0
2
 (pnR)
 A
_
2 (pn, s) ψ (pn, s) ,   0 < r < R ,
making it possible to write down the system of equations resulting from the mixed boundary conditions (11)–
(13):
T10
s
 + ∫ 
0
∞
C
__
 (ρ, s) J0 (ρr) ρdρ = 
2
R2 √ sa2  sinh 
l √ sa2  

  ∫ 
0
l
F (0, ξ) cosh 
(l − ξ) √ sa2   dξ −
− 
2
R2
 A
_
2 (0, s) cot 
l √ sa2  
 − 
2
R2
  ∑ 
n=1
∞
 
J0 (pnr)
J0
2
 (pnR)
 A
_
2 (pn, s) cot [lψ (pn, s)] +
+ 
2
R2
  ∑ 
n=1
∞
 
J0 (pnr)
J0
2
 (pnR) ψ (pn, s) sinh [lψ (pn, s)]
  ∫ 
0
l
F (pn, ξ) cosh [(l − ξ) ψ (pn, s)] dξ ,   0 < r < R ; (24)
1266
− Kλ ∫ 
0
∞
C
_
 (ρ, s) √ρ2 + sa1  J0 (ρr) ρdρ = 2R2 √ sa2  A
_
2 (0, s) cot 
l √ sa2   +
+ 
2
R2
  ∑ 
n=1
∞
 
J0 (pnr)
J0
2
 (pnR)
 A
_
2 (pn, s) ψ (pn, s) ,   0 < r < R ; (25)
 ∫ 
0
∞
C
__
 (ρ, s) √ρ2 + sa1  J0 (ρr) ρdρ = 0 ,   R < r < ∞ , (26)
and, moreover, according to [1], when z = 0, the following expression is valid:
A
_
2 (0, s) = ∫ 
0
R
f (r, 0) rdr , (27)
in which f(r, 0) is the value of the initial temperature of the contacting end-face surface of the finite cylinder.
Moreover, it is known (see, e.g., [5, p. 634]) that if a certain function Φ
__
(r, s) is representable, in the
interval 0 < r < R, by the Fourier–Dini series
Φ
__
 (r, s) =  ∑ 
n=1
∞
 BnJ0 
µn 
r
R
 , (28)
in which µn are the positive zeros of the Bessel function of the first kind and first order, J1(µ), then the
coefficients of this series are calculated from the formula
Bn = 
2
R2J0
2
 (µn)
 ∫ 
0
R
Φ
__
 (r, s) J0 µn 
r
R

 rdr . (29)
With the use of formulas (27)–(29), Eq. (25) yields
Kλ
−1
 ψ 

µn
R , s
 A
_
2 

µn
R , s
 =
= ∫ 
0
R
J0 

µn
R
 r
 





2
R2Kλ
 √ s
a2
  ∫ 
0
R
f (ρ, 0) ρdρ + ∫ 
0
∞
C
__
 (ρ, s) √ρ2 + sa1  J0 (ρr) ρdρ





 rdr . (30)
Since in our case
 ∫ 
0
R
J0 

µn
R  r
 rdr = 
R2
µn
 J1 (µn) = 0 ,   ∫ 
0
R
J0 (pr) J0 

µn
R  r
 rdr = 





0   for   p ≠ 
µn
R  ,
R2
2  J0
2
 (µn)   for   p = 
µn
R
1267
by virtue of the orthogonality of the corresponding functions (see, e.g., [5]), from formula (30) the following
expression can be written for the unknown function:
A
_
2 

µn
R , s
 = 
KλR
2
2ψ 

µn
R , s

 J0
2
 (µn) ∫ 
0
∞
C
_
 (ρ, s) √ρ2 + sa1  ρdρ , (31)
the substitution of which into Eq. (24) leads to the formula
 ∫ 
0
∞
C
__
 (ρ, s) J0 (ρr) ρdρ = − 
T10
s
 − 
2
R2
 cot 
l √ sa2  
 ∫ 
0
R
f (ρ, 0) ρdρ +
+ 
2
R2 √ sa2  sinh l √ sa2  
 ∫ 
0
l
F (0, ξ) cosh 
(l − ξ) √ sa2  
 dξ +
+ 
2
R2
  ∑ 
n=1
∞
 
J0 

µn
R  r

J0
2
 (µn) ψ 

µn
R , s
 sinh 
lψ 

µn
R , s


  ∫ 
0
l
 F 

µn
R , ξ
 cosh 
(l − ξ) ψ 

µn
R , s

 dξ −
− ∫ 
0
∞
C
__
 (ρ, s) √p2 + sa1  pdp  ∑ 
n=1
∞
 
Kλ cot 
lψ 

µn
R , s


ψ 

µn
R , s

 J0 

µn
R  r
 ,   0 < r < R . (32)
Finally, using, on the right-hand side of Eq. (32), the equality proven in [3]
   ∑ 
n=1
∞
 
Kλ cot 
lψ 

µn
R , s


ψ 

µn
R , s

 J0 

µn
R  r
 = 
Kλ cot [lψ (p, s)]
ψ (p,s)  J0 (pr) ,
we arrive at the first integral equation with the L-parameter to find the unknown analytical function C
_
(p, s)
in the region 0 < r < R, z = 0:
 ∫ 
0
∞
C
__
 (p, s) 





1 + Kλ 
√p2 + sa1
√p2 + sa2
 cot 
l √p2 + sa2  






 
J0 (pr) pdp = − 
T10
s
 + Φ
__
 (r, s) ,   0 < r < R , (33)
1268
Φ
__
 (r, s) = 2
R2
  ∑ 
n=1
∞
 
J0 

µn
R  r

J0
2
 (µn) ψ 

µn
R , s
 sinh 
lψ 

µn
R , s


  ∫ 
0
l
 F 

µn
R , ξ
 cosh 
(l − ξ) ψ 

µn
R , s

 dξ −
− 
2
R2
 cot 
l √ sa2  
 ∫ 
0
R
f (ρ, 0) ρdρ + 2
R2√ sa2  sinh 
l √ sa2  

  ∫ 
0
l
 F (0, ξ) cosh 
(l − ξ) √ sa2  
 dξ .
In the region R < r < ∞, z = 0 the second integral equation with the L-parameter for finding the un-
known analytical function C
_
(p, s) has the form of Eq. (26):
 ∫ 
0
∞
C
_
 (p, s) √p2 + sa1  J0 (pr) pdp = 0 ,   R < r < ∞ . (34)
To solve the paired integral equations (33), (34) the substitution
C
__
 (p, s) 1
√p2 + sa1
  ∫ 
0
R
ϕ
_
 (t, s) cos 
t √p2 + sa1   dt , (35)
is used, which ensures automatic fulfillment of the second paired equation (34) at any choice of the new
unknown analytical function ϕ
_
(t, s) = ϕ
_
(−t, s) owing to the value of the corresponding discontinuous integral
(see [3, p. 489]) and from the first paired integral equation (33) provides the integral equation with the L-pa-
rameter
 ∫ 
0
R
ϕ
_
 (t, s) ∫ 
0
∞
 
cos 
t √p2 + sa1  

√p2 + sa1
 J0 (pr) pdp dt +
(36)
+ Kλ ∫ 
0
R
ϕ
_
 (t, s) ∫ 
0
∞
 
cot 
l √p2 + sa2  
√p2 + sa2
 cos 
t √p2 + sa1   J0 (pr) pdp dt = − T10s  + Φ
__
 (r, s) ,   0 < r < R .
We note that the improper internal integrals on the left-hand side of Eq. (36) converge at any
0 < r < R, Re s > 0. In particular,
1269
  ∫ 
0
∞
 
cos 
t √p2 + sa1  
√p2 + sa1
 J0 (pr) pdp = 









exp 
− √sa1 (r2 − t2)

√r2 − t2  ,
sin 
− √sa1 (t2 − r2) 
√t2 − r2  ,
     
0 < t < r ,
0 < r < t ,
 ∫ 
0
∞
 
cot 
l √p2 + sa2  
√p2 + sa2
 cos 
t √p2 + sa1   J0 (pr) pdp =
= 
1
l
 ∫ 
s
∗
∞
cot [x] cos 




t √x2l2  + s a2 − a1a1a2   




 J0 




r √x2l2  − sa2  




 dx ,   s∗ = l √ s
a2
 ,
which allows one, just as in [3], to transform the integral equation (36) as
  ∫ 
0
r
ϕ
_
 (t, s)
√r2 − t2  exp 
− √sa1 (r2 − t2)  
 dt − ∫ 
r
R
ϕ
_
 
(t, s)
√t2 − r2  sin 
√sa1 (t2 − r2)  
 dt +
+ 
Kλ
l
 ∫ 
0
R
ϕ
_
 
(t, s)  ∫ 
s
∗
∞
cot [x] cos 




t √x2l2  + s a2 − a1a1a2   




 J0 




r √x2l2  − sa2  




 dxdt = − 
T10
s
 + Φ
__
 (r, s) ,   0 < r < R .
(37)
If the height of the cylinder l → ∞, then Eq. (37) is the basis for determining the unknown auxiliary
function in the case of the model of thermal contact of a semi-infinite cylinder and semispace.
If the radius of the cylinder R → ∞ at a constant height of the cylinder l, i.e., a model of thermal
contact of an infinitely long plate and semispace is considered, Eq. (37) does not appear, which is natural in
the absence of mixed boundary conditions.
If simultaneously R → ∞ and l → ∞, we have a model of thermal contact of two semi-infinite bodies
with different thermophysical properties and different initial temperatures (this case is considered in [1]).
Here, to illustrate the solution of the equations of the type (37) we will consider a simplified model
of heat exchange at f(r, z) = 0, i.e., when Φ(r, s) = 0 in Eqs. (33), (36), and (37).
To solve a corresponding equation of the form (37), we replace in it the variable r by µ, multiply
both parts of the resulting equation by the integrating factor 2µ cos (√(r2 − µ2)s ⁄
 
a1) /√r2 − µ2  and integrate
over µ from zero to r. Using the equality
d
dr
 ∫ 
0
r 2 cos 
√ sa1  √r2 − µ2

√r2 − µ2  J0 




µ √xl2 − sa2  




 µdµ = 2 cos 




r √x2l2  − (a2 − a1) sa1a2  




 ,
1270
we obtain an integral equation with the L-parameter written in a standard fashion [see [3]):
ϕ
_
 (r, s) − 1
π
 ∫ 
0
R
ϕ
_
 (t, s) K (r, t, s) dt = − 2T10
πs
 cos 
r √ sa1   ,   0 < r < R ,
in which
K (r, t, s) = 
sin 
(t − r) √ sa1  

t − r
 + 
sin 
(t + r) √ sa1  

t + r
 +
+ 
Kλ
l
  ∫ 
s
∗
∞
cot [x] 




cos 




(t − r) √x2l2  + s a2 − a1a1a2   




 +  cos 




(t + r) √x2l2  + s a2 − a1a1a2   









 dx ,   s∗ = l √ s
a2
 ,
The methods of solution of this equation are considered in [3].
Thus, after determining the function ϕ
_
(t, s), from formula (35) we find the unknown function C
__
(p, s),
the substitution of which into (15) makes it possible to determine the temperature field T
_
1(r, z, s) in the
region of L-transforms for the semispace. Applying the inverse Laplace transform, we find the inverted trans-
form T1(r, z, τ). The temperature field T
_
2(r, z, s) in a finite cylinder is determined by formula (23) with the
use of the value of A
_
2(µn ⁄ R, s) from (31).
In conclusion, it should be noted that when s → 0 (τ → ∞) the analytical results obtained describe
stationary models of contact heat exchange between a finite cylinder and a semi-infinite body.
REFERENCES
1. A. V. Luikov, Theory of Heat Conduction [in Russian], Moscow (1967).
2. V. P. Kozlov, Two-Dimensional Axisymmetric Nonstationary Problems of Heat Conduction [in Rus-
sian], Minsk (1986).
3. V. P. Kozlov and P. A. Mandrik, Systems of Integral and Differential Equations with the L-Parameter
in Problems of Mathematical Physics and Methods of Identification of Thermal Characteristics [in
Russian], Minsk (2000).
4. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers [Russian transla-
tion], Moscow (1978).
5. G. N. Watson, A Treatise on the Theory of Bessel Functions [Russian translation], Pt. 1, Moscow
(1949).
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Solution of a heat-conduction problem for a finite cylinder and semispace under mixed local boundary conditions in the plane of their contact

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Solution of a heat-conduction problem for a finite cylinder and semispace under mixed local boundary conditions in the plane of their contact

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