Green’s Function For Radial Heat Conduction in Two-Region Composite Cylinders
With Imperfect Boundary Contact
by
Donald E. Amos
Albuquerque, NM
Abstract This paper presents the derivation of the Green’s function for composite cylinders 0<r<a and
r>a with contact resistance on the surface r=a. This imperfect contact is expressed by taking the flux
proportional to the temperature difference across the boundary where the constant of proportionality is
called the heat transfer coefficient h. Because the source function can be in either region, there are two
pairs of functions which define the Green’s function. Each pair is the solution to a two-region
conduction problem with zero initial temperatures and continuity of flux on the cylinder r=a. These pairs
are used in conjunction with a general formula to get the solution to other problems where the cylinders
have contact resistance, but may have non-zero initial conditions and/or possibly a distribution of
internal heat sources. The case of perfect contact is obtained by taking h to infinity and the case of
perfect insulation on r=a for both regions r<a and r>a is obtained by taking h to zero.
Keywords Two Regions Heat Conduction Unsteady State Laplace Transform
1.0 Introduction
The main thrust of this paper is to derive the radial Green’s function G(r,r',t) for adjacent cylinders with
contact resistance on the common surface r=a. The inner cylinder, 0  r  a , is denoted as Region 1 and
the outer cylinder, a  r   , is denoted as Region 2. Material properties are taken to be constant in
space and time.
In the notation of [7, Chapter 2], we are solving the heat conduction problem labeled (R0C30). In order
to apply the concept of a Green’s function to a problem, one has to use these functions in conjunction
with the general formula [6].
t

 t
Tj (P', t)  j    G jj (P, P', t   )q j (P, )dVd    G 3j  j (P, P', t   )q 3 j(P, )dVd 
k j  0 Vj

0 V3 j

  G jj (P, P', t  0)f j (P)dV 
k 3- j j
G
(P, P', t  0)f 3 j (P)dV
j  1,2
k j 3 j V3 j
t

G jj (P, P', t   )
T (P, ) 
  j   Tj (P, )
 G jj (P, P', t   ) j
dS jd

n

n
i
i
0 S j  SC 


k 3- j j t
G 3j  j (P, P', t   )
T (P, ) 

 G 3j  j (P, P', t   ) 3 j
T3 j (P, )
dS3 jd


k j 0 S3 j  SC 
n i
n i

The indices j and 3-j with j=1,2 can only have the values of 1 or 2. Here V1 and V2 are the volumes of
Regions 1 and 2 with surfaces S1 and S 2 ; Sc is the contact boundary and S j  Sc , j  1,2 are external
(1.1)
j
3 j
Vj
boundaries. q1, q2 and f1, f 2 are heat source and initial temperature distributions respectively. The
operators  / ni are internal normal derivatives. Since the Green’s function source can be in either
region, the construction requires the solution of two problems each of which satisfy the conditions on
the common boundary and each yielding two functions. The complete Green’s function therefore
consists of two pairs (G11( r, r ', t ), G21 ( r, r ', t )) and (G12 ( r, r ', t ), G22 ( r, r ', t )) where the superscript
1
designates the region where the source occurs and the subscript designates the region where the function
applies.
For this application, Region 1 has no external boundary and the external boundary for Region 2 is at
infinity where we select zero values for all temperatures. Consequently, (1.1) reduces to only the terms
involving the source distributions q1, q2 and the initial temperatures f1, f 2 .
Laplace transforms of the Green’s functions are used in most of the manipulations because the
transforms contain well documented handbook functions.
2. Conduction Equations for the Green’s Function
We solve the two-region problem of heat conduction in concentric cylinders with a heat source and
perfect boundary contact. The inner cylinder we designate as Region 1, 0<r<a, and the outer cylinder as
Region 2, a  r   . The conduction equations for the Green’s functions are
1 G1
1 G2
 2G1 
, 0<r<a ,  2G2 
, a<r< , G2 ( r, t )  0 for r  
(2.1)
1 t
 2 t
with boundary conditions, expressing continuity flux on r=a, as
G
G
 k1 1 ( a, t )   k2 2 ( a, t )  h(G1( a, t )  G2 (a, t )) .
(2.2)
r
r
The initial conditions are
(2.3)
G j ( r,0)  0 , j  1,2
except possibly at the source point where r  r ' . The energy comes from the point source solution
which can appear in either Region 1 or Region 2.
3. Green’s Function for the Source in Region 1 (G11( r, r ', t ), G21 ( r, r ', t ))
For the sake of simplicity, we suppress the superscripts for the analysis of this section, and set
G1( r, r ', t )  G11( r, r ', t ) and G2 ( r, r ', t )  G21 ( r, r ', t ) . With the source in Region 1 along the cylindrical
surface r  r ' , the source solution of the heat equation is
2
2
B
rr '
1
S1( r, r ', t )  1 e ( r  r ' ) /(41t ) I 0 (
) , B1 
(3.1)
.
2t
21t
21
We use the Laplace transform and write the solution in Region 1 as the sum of the source solution and
regular solutions of the conduction equation (2.1) which are finite for r=0. In Region 2 we use only
regular solutions of the conduction equation which vanish for r   . Then, the (Laplace) transformed
equations from Section 2 lead to modified Bessel functions and we select the appropriate solutions to
form the equations
G1( r, r ', p )  S1( r, r ', p )  AI 0 (q1r ) , 0  r  a
q j  p / j
(3.2)
G2 ( r, r ', p )  BK 0 (q2 r ) ,
ar
where the transform of (3.1) is[9, Chapter XIII]
 I (q r ')K0 (q1r ) r  r ' 
(3.3)
S1( r, r ', p )  B1  0 1

 I 0 (q1r )K0 (q1r ') r  r ' 
and I 0 and K0 are modified Bessel functions of zero order. The next step is to apply the boundary
conditions for continuity of flux,
2
G1
G
( a, p )  k2 2 ( a, p )  h(G2 ( a, p)  G1( a, p))
r
r
and solve for A and B. Since a> r and a  r ' in Region 1, we use the first form of S1 for r  r ' with
(3.2) in (3.4),
 q2k2 BK1(q2a )  h  BK0 (q2a )  B1I 0 (q1r ')K0 (q1a )  AI 0 (q1a ) 
(3.5)
 q2k2 BK1(q2a )  q1k1B1I 0 (q1r ')K1(q1a )  q1k1AI1(q1a )
or
AhI 0 ( q1a )  ( hK 0 ( q2a )  q2k2 K1(q2a ))B   hB1I 0 (q1r ')K 0 (q1a )
(3.6)
.
q1k1 AI1(q1a )  q2k2 BK1(q2a )  q1k1B1I 0 (q1r ')K1(q1a )
Then
 hK0 (q1a ) ( hK0 ( q2a )  q2k2 K1(q2a ))
hI 0 (q1a )
hK 0 (q1a )
(3.4)
(3.7)
k1
q k K (q a )
A
 11 1 1
B1I 0 (q1r ')
or
q2k2 K1(q2a )
D
,
q k I (q a ) q1k1K1(q1a )
B
 111 1
B1I 0 (q1r ')
D
A  B1I 0 ( q1r ')   q2k2hK1(q2a )K 0 (q1a )  q1k1K1(q1a )(hK 0 (q2a )  q2k2 K1(q2a ))  / D
B  B1I 0 ( q1r ')  q1k1hK1(q1a ) I 0 (q1a )  q1k1hI1(q1a )K 0 (q1a )  / D 
(3.8)
k1h
B1I 0 (q1r ') / D
a
D  q2k2hK1(q2a ) I 0 ( q1a )  q1k1I1(q1a )(hK 0 (q2a )  q2k2 K1(q2a ))
where we have used the Wronskian of the I and K functions
I1( z )K0 ( z )  I 0 ( z )K1( z )  1/ z
(3.9)
in the computation of B. The solution (3.2) becomes
 I ( q r ') K 0 ( q1r ) 
G1( r, r ', p )  B1  0 1
  B1I 0 ( q1r ) I 0 ( q1r ')C ,
 I 0 ( q1r ) K 0 ( q1r ') 
G2 ( r, r ', p ) 
(3.10)
k1h
B1K 0 ( q2 r ) I 0 ( q1r ') / D ,
a
r'  r  a
r  r'  a
qj 
ar
C    q2k 2hK1( q2a ) K 0 ( q1a )  q1k1K1( q1a )( hK 0 ( q2a )  q2k 2 K1( q2a ))  / D
D  q2 k2 hK1( q2a ) I 0 ( q1a )  q1k1I1( q1a )( hK 0 ( q2a )  q2k 2 K1( q2a ))
or
(3.11)
 F ( r, r ', p )
G1( r, r ', p )   1
 F1( r ', r, p )
G2 ( r, r ', p ) 
where q j 
p / j
r'  r  a
r  r'  a
k1h
B1I 0 ( q1r ')K 0 (q2r ) / D ,
a
p /  j , j  1,2 and
3
ar
(3.12)
D  h  q2k2 K1( q2a ) I 0 (q1a )  q1k1I1 (q1a )K 0 (q2a )   q1k1q2k 2 I1 (q1a )K1 (q2a )
 hq2k2 K1( q2a )  I 0 ( q1a ) K 0 ( q1r )  I 0 ( q1r ) K 0 ( q1a )  


F1( r, r ', p )  B1I 0 ( q1r ')   hq1k1K 0 ( q2a )  I1( q1a ) K 0 ( q1r )  I 0 ( q1r ) K1 ( q1a )   / D


  q1k1q2k2 K1( q2a )  I1 (q1a )K 0 (q1r )  I 0 ( q1r ) K1( q1a ) 
.
Now we follow the procedure of [10, Chapter XII, Section 128] and invert the Laplace transform by the
complex inversion integral. Then with G as G1 or G2 ,
  i
1
G( r, r ' t ) 
G( r, r ', p )e pt dp ,
2 i  i
(3.13)
  0.
The indication from [10] is that the integrand has no poles, but it is clear from (3.11) that G1 and G2
have branch points at p=0. Therefore we follow [10] and select the branch cut along the negative real
axis and use Cauchy’s theorem to close the contour about the cut with a classical keyhole contour. Then
with G as G1 or G2 we have
0
G ( r, r ' t ) 
(3.14)

2
G ( r, r ', p )e pt

2 i 
pw e
2 i
1
  0 2 i
wei dw  lim


2

G( r, r ', p )e pt
2 i 0

 G( r, r ', p)e
pt

pw e
2  i
we
 i
 p  e
 iei d
i
.
dw  0
Now, the integrand is real on the positive real axis. Therefore, by the reflection principle, the values on
the top and bottom of the cut where p  w2ei and p  w2ei , 0  w  , are complex conjugates of
one another and the difference is 2i times the imaginary part on p  w2ei . Then, (3.14) becomes
(3.15) G( r, r ' t )  

Im G( r, r ', p )
pw e

2
2 i
we
 w2 t
0
1
dw  lim
  0 2

 G( r, r ', p)e

pt
 p  e
i
 ei d .
The task now is to explicitly compute the real and imaginary parts of G1 and G2 on p  w2ei and
evaluate the limit   0 . The details of the algebra are relegated to APPENDIX A and we jump to the
final result.
The Solution for G11( r, r ', t ) and G21 ( r, r ', t ) with the Source in Region 1
With the essentials presented in APPENDIX A, the final solution, with the proper superscript attached,
is
G11( r, r ', t )  B1
4h 2k1k 2
 2a 2

G21 ( r, r ', t )  B1
(3.16)
2k1h
 a 0


J 0 ( 1wr '/ a ) J 0 ( 1wr / a )
e  w t dw ,
2
0  r, r '  a
2
2
w
(
D

D
)
1
2
0
r'
J 0 ( 1w )
r'  a
a  D J ( w r )  D Y ( w r )  e  w2 t dw ,
1
0
2
2
0
2
ar
a
a 
( D12  D22 ) 
D1( w)  h   2 J 0 ( 1w)Y1( 2 w)  1J1( 1w)Y0 ( 2 w)   w1 2 J1( 1w)Y1( 2 w)
D2 ( w)  h   2 J 0 ( 1w) J1( 2 w)  1J 0 ( 2 w) J1( 1w)   w1 2 J1( 1w) J1( 2 w)
B1  1/(21 )
,  j  kj /  j ,  j  a/  j
4
, j  1,2
It should be noted that for h   the transform (3.11) and the solution (3.16) agree with the results of
[8].
4. Green’s Function for the Source in Region 2 (G12 ( r, r ', t ), G22 ( r, r ', t ))
We take the source in Region 2, a  r '   , on the cylindrical surface r  r ' ,
2
2
B
rr '
1
S2 ( r, r ', t )  2 e ( r  r ' ) /(4 2 t ) I 0 (
) , B2 
(4.1)
2t
2 2t
2 2
and solve the heat conduction problem in the manner of Section 3. We use the Laplace transform and
write the solution in Region 2 as the sum of the source solution and regular solutions of the conduction
equation which vanish for r   . In Region 1 we use only regular solutions of the conduction equation
which are finite at r=0. Then, proceeding as before with the superscript suppressed, we have
G1( r, r ', p )  AI 0 ( q1r ) ,
0ra ,
(4.2)
G2 ( r, r ', p )  S2 ( r, r ', p )  BK 0 ( q2r ) ,
a  r ', r  
qj 
p /  j , j  1,2
where [10, Chapter XIII ]
 I (q r ')K0 (q2r ) r  r ' 
S2 ( r, r ', p )  B2  0 2

 I 0 (q2r )K0 (q2r ') r  r ' 
and I 0 and K0 are modified Bessel functions of zero order. The next step is to apply the boundary
conditions
G
G
k1 1 ( a, p )  k2 2 ( a, p )  h(G2 ( a, p)  G1( a, p))
(4.4)
r
r
and solve for A and B. Since a  r and a  r ' in Region 2, we use the second form of S 2 with (4.2),
(4.3)
q1k1 AI1( q1a )  h  B2 I 0 ( q2a ) K 0 ( q2 r ')  BK 0 ( q2a )  AI 0 ( q1a ) 
(4.5)
q1k1 AI1( q1a )  k2  q2 B2 I1( q2a ) K 0 ( q2 r ')  q2 BK1(q2a ) 
or
 q1k1I1(q1a)  hI 0 (q1a) A  BhK0 (q2a)  hB2 I 0 (q2a )K0 (q2r ')
(4.6)
q1k1 AI1(q1a )
 q2k2 BK1(q2a )  q2k2 B2 I1(q2a )K0 (q2r ')
and
(4.7)
A

B2 K 0 ( q2 r ')
hI 0 ( q2a )
 hK 0 ( q2a )
q2k2 I1( q2a ) q2k2 K1(q2a )
D
,
B

B2 K 0 ( q2 r ')
k1q1I1(q1a )  hI 0 (q1a )
hI 0 (q2a )
q1k1I1(q1a )
q2k2 I1(q2a )
D
A  B2 K 0 (q2 r ')q2k2h  K1(q2a ) I 0 (q2a )  I1(q2a )K 0 (q2a ) / D  B2 K 0 (q2r ')(k 2h / a ) / D
(4.8)
B  B2 K 0 (q2 r ')  q2k2 I1(q2a )  k1q1I1(q1a )  hI 0 (q1a )   q1k1hI1(q1a )I 0 (q2a )  / D
D  h  k2q2 K1(q2a ) I 0 (q1a )  q1k1I1(q1a )K 0 (q2a )  q1k1q2k 2 I1(q1a ) K1( q2a )
where we have used the Wronskian of the I and K functions
5
I1( z )K0 ( z )  I 0 ( z )K1( z )  1/ z
(4.9)
in the computation of A. The solution (4.2) becomes
G1( r, r ', p ) 
k2h
B2 K 0 ( q2 r ') I 0 ( q1r ) / D ,
a
0  r  a  r'
a  r'  r
a  r  r'
 I ( q r ') K 0 ( q2 r ) 
(4.10) G2 ( r, r ', p )  B2  0 2
  B2 K 0 (q2r )K 0 (q2r ')C ,
 I 0 ( q2 r )K 0 (q2 r ') 
qj 
p / j
C   h   q1k1I1( q1a ) I 0 ( q2a )  q2k 2 I 0 (q1a ) I1(q2a )   q1k1q2k 2 I1(q1a )I1(q2a )  / D
or
G1( r, r ', p ) 
(4.11)
where
(4.12)
k2h
B2 I 0 ( q1r )K 0 (q2r ') / D ,
a
 F ( r, r ', p )
G2 ( r, r ', p )   2
 F2 ( r ', r, p )
0  r  a  r'
a  r'  r
a  r  r'
D  h  q2k2 K1( q2a ) I 0 (q1a )  q1k1I1 (q1a )K 0 (q2a )  q1k1q2k 2 I1 (q1a )K1 (q2a )
 q2k2hI 0 ( q1a )  K1( q2a ) I 0 (q2 r ')  I1(q2a )K 0 (q2 r ') 



F2 ( r, r ', p )  B2 K 0 ( q2 r )   q1k1hI1(q1a )  K 0 (q2a ) I 0 (q2 r ')  I 0 (q2a )K 0 (q2 r ')   / D


  q1k1q2k2 I1(q1a )  K1(q2a ) I 0 (q2 r ')  I1( q2a ) K 0 ( q2 r ') 
qj 
p /  j , j  1,2
Again, we follow Section 3 and invert the Laplace transform by the complex inversion integral using the
keyhole contour. The solution with the proper designation (G12 ( r, r ', t ), G22 ( r, r ', t )) is
(4.13) G 2j ( r, r ' t )

2


0

 pw e
Im G j ( r, r ', p )
2 i
we
 w2 t
1
dw  lim
  0 2

 G j (r, r ', p )e
pt
 p  e
i

 ei d
for j=1 or 2. The details for F2 ( r, r ', p ) on p  w2ei , 0  w   and p   ei , -      are
contained in APPENDIX B.
The Solution for G12 ( r, r ', t ) and G22 ( r, r ', t ) with the Source in Region 2
G12 ( r, r ' t )
(4.14)

2k 2 h
 D J ( w 2r '/ a )  D2Y0 ( w 2r '/ a ) e w2t dw, 0  r  a  r '

B2  J 0 ( w 1r / a ) 1 0
a
( D12  D22 )
0
 
 B2 

2
G2 ( r, r ' t )   0
B
 2
 0
 A( r ', w)  B( r ', w)

r
'

r


D12  D22
 A( r, w)  B( r, w) D J ( wr '/ a )  D Y ( wr '/ a ) we w2t dw, a  r  r ' 
 1 0 2

2 0 2

D12  D22

D1J 0 ( 2wr / a )  D2Y0 ( 2wr / a ) we w t dw, a 
2
6
A( r ', w)  h  2 J 0 ( 1w) ( r ', w,  2 )  1J1( 2 w) ( r ', w,  2 ) 
B( r ', w)  1 2 wJ1( 1w) ( r ', w,  2 )
 ( r ', w,  2 )  J1( w 2 )Y0 ( w 2 r '/ a )  J 0 ( w 2 r '/ a )Y1( w 2 )
(4.15)
 ( r ', w,  2 )  J 0 ( w 2 r '/ a)Y0 ( w 2 )  J 0 ( w 2 )Y0 ( w 2 r '/ a )
D1  h   2 J 0 ( 1w)Y1( 2 w)  1J1( 1w)Y0 ( 2 w)   w1 2 J1( 1w)Y1( 2 w)
D2  h   2 J 0 ( 1w) J1( 2 w)  1J 0 ( 2 w) J1( 1w)   w1 2 J1( 1w) J1( 2 w)
B2  1/(2 2 )
,  j  kj / j ,  j  a/ j
, j  1,2
5. Special Cases h   and h  0
If we divide the numerator and denominators of equations (4.14) and (4.15) by h 2 , and take h   , we
get the formulas for perfect boundary contact (R0C0). This problem was solved in [8] and agrees with
the result we get here. Several special cases were developed in [8] which gives some confidence in the
formulas. If we go to the other extreme and take h  0 , the terms which were dominant in the perfect
contact case are now zero and the terms which were eliminated are now dominant. This gives a partial
check on the algebra if we can apply the result to a problem with a known solution. Further, the case for
h  0 gives perfect boundary insulation on r=a and we have G12 ( r, r ', t )h  0  0 and G21 ( r, r ', t )h  0  0 .
Since the temperatures in Regions 1 and 2 are not affected by what happens in the other region, the
temperatures are uncoupled and one can treat the temperatures independently as one region problems
with boundary conditions G12 (a, r ', t) / r  0 (R02) and G 22 (a, r ', t) / r  0 (R20).
Green’s functions with zero boundary conditions are used with the general formula when one wants to
compute the solution where boundary fluxes are specified. We therefore apply (1.1) to a problem on
Region 2 where we have zero initial temperature with a constant heat flux of Q units per unit area per
T
unit time on the external boundary r=a. Applying (1.1) for j=2, with 2 ( a, t )  Q / k 2 and
r
 2
G 2 (a, r ', t   ) h  0  0 we get
r
t 2

G 22 (a,r ', t   )
T (a, ) 
(5.1)
T2 (P', t)   2   Tj (P, )
 G 22 (a,r ', t   ) 2
ad d
r
r 
0 0 
and (4.15) with h=0 gives

(5.2)
G 22 (r,r',t)h=0 =B2

 ( r, w,  2 ) ( r ', w,  2 )
we w t dw, a  r  r ' .
2
Y12 ( 2 w)  J12 ( 2 w) 


With a change of variable    2 w , this agrees with the R20 case of [9, p. 546]. Notice
0

(5.3)
G 22 (a,r',t)h=0
 B2 
r
0
  ( r, w,  2 ) / r r  a  ( r ', w,  2 )
Y12 ( 2 w)  J12 ( 2 w) 


Then, with the Wronskian  (a, w,  2 )  2 /( w 2 ) and B2  1/(2 2 )
7
we w t dw  0 .
2
T2 ( r ', t ) 
 2Q
k2
t


t
G22 ( a, r ', t
0
Q 2
 ( a, w,  2 ) ( r ', w,  2 )  w2 ( t  )
  )2 ad 
B2  2 a 
we
dwd
2
k2
 2

0
0 Y1 ( 2 w)  J1 ( 2 w ) 
t
2
Qa
 ( r ', w,  2 )
2

we w ( t  )dwd


2
2
k2 0 0 Y1 ( 2 w)  J1 ( 2 w)   2 w


2
2Q  2 
 ( r ', w,  2 )

(1  e w t )
dw

2
2
 k2 0
w Y1 ( 2 w)  J12 ( 2 w) 


and this agrees with C&S,[10, Chapter XII, Section 127] with a change of variables, w  u  2 . The
asymptotic behavior at r=a for large and small t is also given in [10].
(5.4)
There is another way to arrive at the solution for the case for h  0 . If we put a cylindrical source
T
q2  Q r  (a   ) just to the right of r=a and take 2 ( a, t )  0 with (1.1), the boundary terms drop
r
out and we get the solution as the body source term with heat generation at the point r  a  
T2 (r ', t) 
(5.5)

t
2
k 2 0 a
 2Q
k2
t

G 22 (r, P', t
  )q 2 (r, )2 rdrd 
 2Q
k2
t 
 
0
G 22 (r, P', t   ) r  (a   )2 rdrd
a
G 22 (a   , r ', t   )2 (a   )d
0
Except for a small amount of heat capacity in (a,a   ) all of the heat flows into the region r  a  
because the boundary r=a is insulated making the flux to the right of r  a   close to Q. Therefore in
T
the limit for   0 , we get (5.4) and the boundary condition 2 ( a, t )  0 is changed into the condition
r
T2
( a  0, t )  Q / k 2 because the heat source is flush against the insulated boundary.
r
The case for h  0 in Region 1 is more difficult to get. Equation (3.16) results in an indeterminant form
since the integral appears to diverge as h  0 . This can be resolved if we take the limit in the Laplace
transform (3.12)
(5.6) F1( r, r ', p)  B1I 0 (q1r ') (q1a) I1(q1a)K0 (q1r )  I 0 (q1r )(q1a)K1(q1a)  / (q1a) I1(q1a) , q1 
p / 1 .
At first glance, it looks like this expression has a branch point at p=0. However, the power series,
( q1a ) K 0 ( q1r )  ( q1a )ln( q1r / 2) I 0 ( q1r )  power series in p
(5.7)
( q1a ) K1( q1a )  1  ( q1a )ln( q1a / 2) I1( q1a )  power series in p
I 0 ( q1a )  1  power series in p
( q1a ) I1( q1a )  ( pa 2 ) 1  power series in p  /(21 ),
having infinite radii of convergence, show that the logarithmic terms in ln(q1 ) cancel and the numerator
and denominator are analytic functions of p. The poles on the negative real axis at the zeros of
(q1a) I1(q1a) are the sole contributors to the inversion integral (3.13)
8
  i
1
G( r, r ' t ) 
F1( r, r ', p )e pt dp 

2 i  i
(5.8)

(5.9)

2
,
 0.
m 0
The computation is
( ei / 2 wm ) I 0 ( ei / 2 wm r ')  (iwm ) J 0 ( wm r '),
K 0 ( wmaei / 2 ) 
 (residues at p  1wm2 )
 iJ 0 ( wma )  Y0 ( wma ) ,
I1( ei / 2 wma )  iJ1( wma )  0 ,
K1( wmaei / 2 ) 

2
q1 
p / 1
  J1(wma )  iY1(wma )
F1( r, r ', p )  B1I 0 ( q1r ')  ( q1a ) I1( q1a ) K 0 ( q1r )  I 0 ( q1r )( q1a ) K1( q1a ) /  ( q1a ) I1( q1a )  ,
d
 (q1a ) I1(q1a ) p  w21 =  a 2 J 0 ( wma )  /(21)
dp
Re sidue of F1( r, r ', p ) at p  1wm2 
r  r'

 J 0 ( wmr )Y1( wma ) ( wma )e
2
where J1( wma )  0 .The residue of (5.6) at p=0 is
 21B1J 0 ( wm r ')
(5.10)
1wm2 t
/ a J 0 ( wma ) 


2
Re sidue of F1( r, r ', p ) at p  0 is 21B1 / a 2
The solution is
 2 B  B  J ( w r ) J ( w r ')Y ( w a )  w a  e 1wm2 t 
m
(5.11) G ( r, r ' t )   12 1  12 1  0 m 0 m 1 m

J
(
w
a
)
a m 1
0 m
 a

where wm is defined by the positive zeros of J1( wma )  0 , m=1,2,3,…. But a reduction is possible from
the Wronskian,
2
2
J1( wma )Y0 ( wma )  J 0 ( wma )Y1( wma ) 
or J 0 ( wma )Y1( wma ) 
(5.12)
 wma
 wma
and with B1  1/(21 )
2

1 
J 0 ( wmr ) J 0 ( wmr ')e1wmt

(5.13)
 2 1  
 a  m1
J 02 ( wma )
With a change of variable  m  wma , this agrees with R02 in [9, p.541]
G11( r, r ' t )h  0


.


In summary, we have used special cases associated with h   and h  0 to check essential terms of
the numerator and denominator of each solution.
References
[1] Abramowitz S, Stegun IA (1965) Handbook of Mathematical Functions, AMS 55, Dover
Publications Inc., New York, 1046pp
[2] Amos DE (2006) Handbook of Integrals Related to Heat Conduction and Diffusion,
http://nanohub.org/resources/13874
[3] Amos DE, Beck JV, de Monte F (2011) Transient Heat Conduction in Adjacent Quadrants
Separated by a Thermal Resistance, http://nanohub.org/resources/12465
9
[4] Amos DE (2011) Transient Heat Conduction in Adjacent Materials Heated on Part of the
Common Boundary, http://nanohub.org/resources/12390
[5] Amos, DE (2012), Green’s Functions For Heat Conduction in Adjacent Materials,
http://nanohub.org/resources/12856
[6] Amos, DE (2012), Theory of Heat Conduction for Two-region Problems Using Green’s Functions,
http://nanohub.org/resources/13671
[7] Amos, DE (2012), 1-D Green’s Functions For Heat Conduction Between Semi-infinite Slabs
With Perfect and Imperfect Boundary Contact, http://nanohub.org/resources/15237
[8] Amos, DE (2013), Green’s Function For Radial Heat Conduction in Two-Region Composite
Cylinders With Perfect Boundary Contact, http://nanohub.org/resources/16482
[9] Cole DC, Beck JV, Haji-SheikhA, Litkouhi B (2010) Heat Conduction Using Green’s Functions,
2nd Ed., CRC Press, 643p.
[10] Carslaw HS, Jaeger JC (1948) Conduction of Heat in Solids, Oxford Univ Press, London, 386pp
APPENDIX A
Calculation of the integrands for (3.15)
Evaluation of F1( r, r ', p ) and G11( r, r ', p ) on the contour p  w2ei , 0  w  
Because of the symmetry in the cases for r  r ' and r  r ' we consider only the case for r  r ' . We take
F1( r, r ', p ) ,
D  h  q2k2 K1( q2a ) I 0 (q1a )  q1k1I1 (q1a )K 0 (q2a )   q1k1q2k 2 I1 (q1a )K1 (q2a )
(A.1)
 hq2k2 K1( q2a )  I 0 ( q1a ) K 0 ( q1r )  I 0 ( q1r ) K 0 ( q1a )  


F1( r, r ', p )  B1I 0 ( q1r ')   hq1k1K 0 ( q2a )  I1( q1a ) K 0 ( q1r )  I 0 ( q1r ) K1 ( q1a )   / D ,


  q1k1q2k2 K1( q2a )  I1 (q1a )K 0 (q1r )  I 0 ( q1r ) K1( q1a )  
qj 
p /  j , j  1,2
and start by applying the analytic continuation relations
(A.2)
K0 ( zei / 2 ) 
K1( zei / 2 ) 
and the Wronskian
(A.3)

2

2
i   J 0 ( z )  iY0 ( z ) =
  J1( z )  iY1( z )
,

 iJ 0 ( z )  Y0 ( z )
2
I 0 ( zei / 2 )  J 0 ( z ) ,
J1( z )Y0 ( z )  J 0 ( z )Y1( z )  2 /( z )
to F1( r, r ', p ) . Then with the definitions,
(A.4)
 j  kj / j ,
 j  a/  j
we have
10
, j  1,2
I1( zei / 2 )  iJ1( z )
D  h  q2k 2 K1( q2a ) I 0 ( q1a )  q1k1I1(q1a )K 0 (q2a )   q1k1q2k 2 I1(q1a )K1(q2a )


(A.5)
 h iw 2 J 0 ( 1w)   J1( 2 w)  iY1( 2 w)  

2

  w 1 2iJ1( 1w)   J1( 2 w)  iY1( 2 w) 
2   w1J1( 1w)  iJ 0 ( 2 w)  Y0 ( 2 w) 
2


w
2
 D1  iD2 
,
1
2 D1  iD2

D  w D12  D22
.
D1  h   2 J 0 ( 1w)Y1( 2 w)  1J1( 1w)Y0 ( 2 w)   w1 2 J1( 1w)Y1( 2 w)
D2  h   2 J 0 ( 1w) J1( 2 w)  1J 0 ( 2 w) J1( 1w)   w1 2 J1( 1w) J1( 2 w)
The evaluations for (A.1) are:
 I 0 (q1a)K0 (q1r )  I 0 (q1r )K0 (q1a) p  w e
2 i


r
r 
 J 0 ( 1w )Y0 ( 1w)  J 0 ( 1w)Y0 ( 1w ) 
2
a
a 


2
 ( r, w,  1 ) (note  is real and  ( a, w,  1 )  0)
q2k2 K1(q2a)  I 0 (q1a)K0 (q1r )  I 0 (q1r )K0 (q1a) p  w e
2 i
 I1(q1a )K0 (q1r )  I 0 (q1r )K1(q1a ) p  w e
2 i
  2 w
2
Y1( 2 w)  i 2w
2
4
4

r
r

 i  J1( 1w)Y0 ( 1w )  J 0 ( 1w )Y1( 1w) 
2
a
a


=  i  ( r, w,  1 )
2
J1( 2w)
(note  is real and  ( a, w,  1 )  2 /( 1w)
q1k1K0 (q2a)  I1(q1a)K0 (q1r )  I 0 (q1r )K1(q1a ) p  w e
2 i
 1w
2
4
Y0 ( 2 w)  i1w
2
4
J 0 ( 2 w)
Then,

(A.6)

F1( r, r ', w2ei )  B1J 0 ( 1wr '/ a )

h 2 w   2Y1( 2 w) ( r, w,  1 )  1Y0 ( 2 w) ( r, w,  1 )

 / D( w)
4  i   2 J1( 2 w) ( r, w,  1 )  1J 0 ( 2 w) ( r, w,  1 ) 


w2 2
 B1J 0 ( 1wr '/ a )
1 2  iJ1( 2 w)  Y1( 2 w)  ( r, w,  1 ) / D( w)
4
r
r
a
a
(A.7)
r
r
 ( r, w,  1 )  J1( 1w)Y0 ( 1w )  J 0 ( 1w )Y1( 1w)
a
a
Note that  and  are real and  (a, w,  1 )  0,  (a, w,  1 )  2 /( 1w).
 ( r, w,  1 )  J 0 ( 1w )Y0 ( 1w)  J 0 ( 1w)Y0 ( 1w )
 h   2Y1( 2 w) ( r, w,  1 )  1Y0 ( 2 w) ( r, w,  1 )  





wY
(

w
)

(
r
,
w
,

)

 D1  iD2
1
 1 2 1 2
(A.8) F1( r, r ', w 2ei )  B1J 0 ( 1wr '/ a ) 

2  ih   2 J1( 2 w) ( r, w,  1 )  1J 0 ( 2 w) ( r, w,  1 )   D12  D22


 i1 2 wJ1( 2 w) ( r, w,  1 )

and
11
   h  2 J1( 2 w) ( r, w,  1 )  1J 0 ( 2 w) ( r, w,  1 )  
 D1 

  1 2 wJ1( 2 w) ( r, w,  1 )
 
 J 0 ( 1wr '/ a ) 
(A.9) Im F1( r, r ', w 2ei )   B1


2 D12  D22 
 h  2Y1( 2 w) ( r, w,  1 )  1Y0 ( 2 w) ( r, w,  1 )  
  D2 


 1 2 wY1( 2 w) ( r, w,  1 )
 
 h  2 D1J1( 2 w)   2 D2Y1( 2 w) ( r, w,  1 ) 

 J 0 ( 1wr '/ a ) 

(A.10)
Im F1( r, r ', w2ei )   B1

h

D
J
(

w
)


D
Y
(

w
)

(
r
,
w
,

)



1 1 0 2
1 2 0 2
1 .
2
2
2 D1  D2 




w
D
J
(

w
)

D
Y
(

w
)

(
r
,
w
,

)




1
2
1
1
2
2
1
2
1


Using the definitions of D1 and D2 , we can further reduce (A.10)
(A.11)
 2h 21 2 J1( 1w)

 ( r, w,  1 )


 2 w




 J 0 ( 1wr '/ a )   2h1 2 J 0 ( 1w) 212 2 J1( 1w) 

2 i
Im F1( r, r ', w e )   B1

 ( r, w,  1 ) 
 h 
2
2
2 D1  D2  
 2 w
 2




 2h12 2 J1( 1w)

 ( r, w,  1 )


 2


=  B1

 J 0 ( 1wr '/ a )  2h 21 2
 J1( 1w) ( r, w,  1 )  J 0 ( 1w) ( r, w,  1)

2
2
2 D1  D2   2 w

Im F1( r, r ', w2ei )   B1
(A.12)
2h21 2 J 0 ( 1wr '/ a ) J 0 ( 1wr / a )
 1 2w2
D12  D22
For G2 ( r, r ', p ) we have
G2 ( r, r ', p ) 
(A.13)
k1h
B1I 0 ( q1r ') K 0 ( q2 r ) / D ,
a
ar
r'  
r
r  2 D1  iD2
 J 0 ( 1w )  iJ 0 ( 2 w )  Y0 ( 2 w ) 
a 2
a
a   w D12  D22 .
r'
J 0 ( 1w )
a   D J ( w r )  D Y ( w r ) 
Im I 0 ( q1r ') K 0 ( q2 r ) / D p  w2 ei 
1 0 2
2 0 2
2
a
a 
w( D1  D22 ) 
I 0 (q1r ')K0 (q2r ) / D p  w e 
2 i
Evaluation of F1( r, r ', p ) and G21 ( r, r ', p ) on the contour p   ei ,       for   0
We have completed the combination of the contours along the top and bottom of the branch, but we still
need to evaluate the line integral around the origin with the radius  and take  to zero,
(A.14)
1
  0 2
lim

 F1(r, r ', p)e
pt

This requires the evaluation of
12
 p  e
i
 ei d .
 hq2k2 K1( q2a )  I 0 ( q1a ) K 0 ( q1r )  I 0 ( q1r ) K 0 ( q1a )  


F1( r, r ', p )  B1I 0 ( q1r ')   hq1k1K 0 ( q2a )  I1( q1a ) K 0 ( q1r )  I 0 ( q1r ) K1 ( q1a )   / D


  q1k1q2k2 K1( q2a )  I1 (q1a )K 0 (q1r )  I 0 (q1r )K1 (q1a )  
D  h  q2k2 K1( q2a ) I 0 (q1a )  q1k1I1 (q1a )K 0 ( q2a )  q1k1q2k 2 I1(q1a )K1 (q2a )
(A.15)
qj 
p /  j , j  1,2
with p   ei for small  . The power series representations for the I and K functions are needed to get
the proper limits:
K0 ( z )   ln( z / 2)    O( z 2 ln z ) , I 0 ( z )  1  O( z 2 ) , I1( z )  z / 2  O( z 3 )
(A.16)
.
1
K1( z )   ( z / 2)ln( z / 2)  ( z / 2)(  1/ 2)  O( z 3 ln z)
z
The limit lim  ln b  0 in the form lim  ln b   0 helps in the evaluation. Then, the dominant
 0
 0
terms for   0 are
(A.22)


( hk2 / a )  K 0 ( q1r )  K 0 ( q1a )



F1( r, r ', p ) B1I 0 ( q1r ')  (hk1 / a ) K 0 ( q2a ) ( q1a ) 2 K 0 ( q1r ) / 2  ( q1a ) K1( q1a )   / D




  ( k k / a 2 ) ( q a )2 K ( q r ) / 2  ( q a ) K ( q a ) 

0 1
1
1 1 
 1 2

 1
D  (k2h / a)(q2a )K1(q2a )  (k1h / a )(q1a )I1(q1a )K0 (q2a )
 (k1k2 / a 2 )(q1a) I1(q1a)(q2a)K1(q2a ) hk2 / a
One can see clearly now that the powers of p dominate the logarithmic terms and with p   ei the limit
is
(A.23)
1
lim
  0 2

 F1( r, r ', p)e
pt

 p  e
i
 ei d  0 .
Similarly for G2 ( r, r ', p ) , it is clear that  in the integrand dominates the logarithmic term in K0 (q2r ) ,
kh
G2 ( r, r ', p )  1 B1I 0 ( q1r ') K 0 ( q2 r ) / D ,
ar
(A.24)
a
and
(A.25)
1
  0 2
lim

 G2 ( r, r ', p)e

pt
 p  e
i
 ei d  0 .
APPENDIX B
Calculation of the integrands for (4.13)
Evaluation of F2 ( r, r ', p ) and G22 ( r, r ', p ) on the contour p  w2ei , 0  w  
Because of the symmetry in the cases for r  r ' and r  r ' we consider only the case for r  r ' . We take
F2 ( r, r ', p ) ,
13
D  h  q2k2 K1( q2a ) I 0 (q1a )  q1k1I1 (q1a )K 0 (q2a )  q1k1q2k 2 I1 (q1a )K1 (q2a )
 q2k2hI 0 ( q1a )  K1( q2a ) I 0 (q2 r ')  I1(q2a )K 0 (q2 r ') 



F2 ( r, r ', p )  B2 K 0 ( q2 r )   q1k1hI1(q1a )  K 0 (q2a ) I 0 (q2 r ')  I 0 (q2a )K 0 (q2 r ')   / D


  q1k1q2k2 I1(q1a )  K1(q2a ) I 0 (q2 r ')  I1( q2a ) K 0 ( q2 r ') 
(B.1)
qj 
p /  j , j  1,2
and start by applying the analytic continuation relations
K0 ( zei / 2 ) 
(B.2)
K1( zei / 2 ) 

2

2
i   J 0 ( z )  iY0 ( z ) =
  J1( z )  iY1( z )
and the Wronskian
(B.3)

,
 iJ 0 ( z )  Y0 ( z )
2
I 0 ( zei / 2 )  J 0 ( z ) ,
I1( zei / 2 )  iJ1( z )
J1( z )Y0 ( z )  J 0 ( z )Y1( z )  2 /( z )
to F2 ( r, r ', p ) . Then with the definitions,
 j  kj / j ,
(B.4)
 j  a/  j
, j  1,2
the evaluations for (B.1) are:
D  h  q2k2 K1(q2a ) I 0 (q1a )  q1k1I1(q1a )K 0 (q2a )  q1k1q2k 2 I1(q1a )K1(q2a )

(B.5)
w
2
 D1  iD2 
,
1
2 D1  iD2

D  w D12  D22
D1  h   2 J 0 ( 1w)Y1( 2 w)  1J1( 1w)Y0 ( 2 w)   w1 2 J1( 1w)Y1( 2 w)
.
D2  h   2 J 0 ( 1w) J1( 2 w)  1J 0 ( 2 w) J1( 1w)  w1 2 J1( 1w) J1( 2 w)
The first term of (B.1) is

r'
r'

J1( 2 w)Y0 ( 2 w )  J 0 ( 2 w )Y1( 2 w) 

2
a
a


=  i  ( r ', w,  2 ) (note  is real and  ( a, w,  2 )  1/( 2 w))
2

q2k2 I 0 (q1a ) I 0 (q2r ')K1(q2a )  I1(q2a )K 0 (q2r ') p  w2 ei   2wJ 0 ( 1w) ( r ', w,  2 )
2
and the second term is

r'
r'
 I 0 (q2r ')K 0 (q2a )  I 0 (q2a )K 0 (q2r ') p  w2 ei   2  J 0 ( 2w a )Y0 ( 2w)  J 0 ( 2w)Y0 ( 2w a ) 



   ( r ', w,  2 ) (note  is real and  ( a, w,  2 )  0) .
2

q1k1I1(q1a )  I 0 (q2r ')K0 (q2a )  I 0 (q2a )K 0 (q2r ') p  w2 ei  2 1wJ1( 1w) (r ', w,  2 )
Then,
 I 0 (q2 r ')K1(q2a )  I1(q2a )K 0 (q2r ') p  w e
2 i
 i
14
F2 (r, r ', p)
2 i
pw e
 B2
2
(B.6)
 B2


4
 2 whJ 0 ( 1w) ( r ', w,  2 ) 
 / D( w)
 1whJ1( 1w) ( r ', w,  2 ) 
 iJ 0 ( w 2r / a )  Y0 ( w 2r / a ) 
2
4
 2w
 iJ 0 ( w 2r / a )  Y0 ( w 2r / a )1 2w2 J1( 1w) (r ', w,  2 ) / D(w)
2 D1  iD2
4
 w D12  D22
2
 w
2 D1  iD2
 B2
i  A( r ', w) J 0 ( 2 wr / a)  B( r ', w ) J 0 ( 2 wr / a )
4
 w D12  D22
F2 ( r, r ', w 2 ei )   B2
 A( r ', w)Y0 ( 2wr / a )  B( r ', w)Y0 ( 2wr / a )
A( r ', w)  h  2 J 0 ( 1w) ( r ', w,  2 )  1J1( 2 w) ( r ', w,  2 ) 
(B.7)
B( r ', w)  1 2 wJ1( 1w) ( r ', w,  2 )
 ( r ', w,  2 )  J1( w 2 )Y0 ( w 2 r '/ a )  J 0 ( w 2 r '/ a )Y1( w 2 )
 ( r ', w,  2 )  J 0 ( w 2 r '/ a )Y0 ( w 2 )  J 0 ( w 2 )Y0 ( w 2 r '/ a )


Im F2 ( r, r ', w 2ei )   B2
  A( r ', w)  B( r ', w)
2( D12  D22 )
D1J 0 ( 2wr / a )  D2Y0 ( 2wr / a )
D1  h   2 J 0 ( 1w)Y1( 2 w)  1J1( 1w)Y0 ( 2 w)   w1 2 J1( 1w)Y1( 2 w)
(B.8)
D2  h   2 J 0 ( 1w) J1( 2 w)  1J 0 ( 2 w) J1( 1w)   w1 2 J1( 1w) J1( 2 w)
For r<a
G1( r, r ', p ) 
(B.9)
k2h
B2 I 0 ( q1r )K 0 (q2 r ') / D ,
a
0  r  a  r'
and
G12 ( r, r ', w2ei ) 
(B.10)

k2h

2 D1  iD2
B2 J 0 ( w 1r / a )  iJ 0 ( w 2 r '/ a )  Y0 ( w 2 r '/ a ) 
,
a
2
 w D12  D22

Im G12 ( r, r ', w2ei )  
k2h
 D J ( w 2r '/ a )  D2Y0 ( w 2r '/ a )
B2 J 0 ( w 1r / a ) 1 0
a
w( D12  D22 )
Evaluation of F2 ( r, r ', p ) and G2 ( r, r ', p ) on the contour p   ei ,       for   0
We have completed the combination of the contours along the top and bottom of the branch, but we still
need to evaluate the line integral around the origin with the radius  and take  to zero,
(B.11)
1
lim
  0 2

 F2 (r, r ', p)e
pt

This requires the evaluation of
15
 p  e
i
 ei d .
D  h  q2k2 K1( q2a ) I 0 (q1a )  q1k1I1 (q1a )K 0 (q2a )  q1k1q2k 2 I1 (q1a )K1 (q2a )
(B.12)
 q2k2hI 0 ( q1a )  K1( q2a ) I 0 (q2 r ')  I1(q2a )K 0 (q2 r ') 



F2 ( r, r ', p )  B2 K 0 ( q2 r )   q1k1hI1(q1a )  K 0 (q2a ) I 0 (q2 r ')  I 0 (q2a )K 0 (q2 r ')   / D


  q1k1q2k2 I1(q1a )  K1(q2a ) I 0 (q2 r ')  I1( q2a ) K 0 ( q2 r ') 
qj 
p /  j , j  1,2
with p   ei for small  . The power series representations for the I and K functions are needed to get
the proper limits:
K0 ( z )   ln( z / 2)    O( z 2 ln z ) , I 0 ( z )  1  O( z 2 ) , I1( z )  z / 2  O( z 3 )
(B.13)
.
1
K1( z )   ( z / 2)ln( z / 2)  ( z / 2)(  1/ 2)  O( z 3 ln z)
z
The limit lim  ln b  0 in the form lim  ln b   0 helps in the evaluation. Then, the dominant
 0
 0
terms for   0 are
D
(B.14)
q2k 2hK1( q2a ) I 0 ( q1a )  q1k1hI1( q1a ) K 0 ( q2a )
F2 ( r, r ', p )
hk 2 / a
( k h / a ) ( q a ) K ( q a )  ( q a ) 2 K ( q r ') / 2 

1 2
2
0 2
 2

 2



B2 K 0 ( q2 r )  ( k1h / a ) ( q1a ) 2 / 2   K 0 ( q2a )  K 0 ( q2r ') 
/D





  ( k k / a 2 ) ( q a )2 / 2  ( q a ) K ( q a )  ( q a ) 2 K ( q r ') / 2  
1 2
2
0 2
 1 2
 1
 2
 
p /  j , j  1,2
qj 
On can see that the powers of p dominate the logarithmic terms and with p   ei we get
(B.15)
1
  0 2
lim

 F2 (r, r ', p)e

pt
 p  e
i
 ei d  0 , a  r '  r .
For G1( r, r ', p ) in (4.11), it is clear that  in the integrand dominates the logarithmic term in K0 (q2r ) ,
k h
G1( r, r ', p )  2 B2 I 0 ( q1r )K 0 (q2 r ') / D ,
0  r  a  r'
(B.16)
a
and
(B.17)
1
lim
  0 2

 G1(r, r ', p)e

16
pt
 p  e
i
 ei d  0 .