THEORY OF HEAT CONDUCTION FOR TWO-REGION PROBLEMS USING
GREEN’S FUNCTIONS
Donald E. Amos
Albuquerque, NM 87110
Abstract This paper derives equations which describe transient temperature distributions in adjacent
regions which share a common boundary. These regions consist of materials with distinct, constant
physical properties. The theory is developed for two types of boundary contact. The first formula is
developed for perfect contact where there is continuity of both temperature and flux. The second
formula allows for a thermal resistance at the boundary which retains continuity of flux, but causes a
temperature drop across the boundary. As an example, the two-region theory is applied to quadrants 1
and 2 which are separated by an infinitely thin resistive layer on the y-axis and are heated along the xaxis. This problem was solved by a direct approach in a previous paper and the two-region Green’s
function approach gives the same results.
Keywords Two Regions
Heat Conduction
Unsteady State
Laplace Transform
1. Intoduction
The general theory of heat conduction centers mainly around one-region problems and the applicability
of the theory depends on the construction of the corresponding Green’s function. This limits the range of
problems to mostly constant material properties and simple geometries. However, two-region problems
on simple geometries occur frequently in engineering and the choices for solutions are mostly restricted
to direct assaults on the equations. It is the purpose here to expand the Green’s function theory to tworegion problems whose elements reflect both perfect and imperfect contact on a common boundary with
distinct, constant material properties in each region.
Section 2 reviews the one-region theory and generalizes the approach to a two-region problem with
perfect boundary contact where the temperature and flux are continuous. Section 3 continues the
development to bodies with imperfect contact where there is a thermal resistance which produces a
temperature difference across the boundary. Section 4 presents some Green’s functions for rectangular
bodies in contact which were developed in reference [4]. Section 5 applies the two-region theory of
Section 3 and the Green’s functions of Section 4 to quadrants 1 and 2 to obtain the temperatures
induced by heated surfaces along the x-axis. The results match those found in reference [2] where a
direct attack was used to solve the problem. Section 6 is an extension of Section 5 where the physical
interpretation of the Green’s function as a solution to a conduction problem with a heat source is used to
re-derive the results of Section 5.
2. Generalizing the One-Region Solution
In [8], [6] and elsewhere, the general theory for one-region, linear problems with constant parameters is
presented. The problem for a volume V with a surface S and heat generation q units per unit volume per
unit time is stated as
T

  2T  q,
q  0, t  0
t
k
(2.1)
T  f (x, y,z)
initially (t=0) inside the surface S
(a) T   (x, y,z, t) on S for t>0 or (b)
1
T
  (x, y,z, t) on S for t>0
n i
for two of the most common boundary conditions and the solution is presented as
T(P', t) 
(2.2)

t
k 0 V
G(P,P', t   )q(P, )dVd   G(P,P', t  0)f (P)dV
V

G(P,P', t   )
T(P, t   ) 
+   T(P, )
 G(P,P', )
dSd
n i
n i

0 S
where G(P, P', t   ) is the Green’s function associated with Case (a) or (b). The symbol  / ni is the
inward normal derivative at the surface S at P where P and P ' are the points (x,y,z) and (x',y',z')
respectively. If Case (a) prevails, we take G  0 on S and we get the classical result
T(P', t) 
(2.3a)
t

t
k 0 V
G(P,P', t   )q(P, )dVd   G(P,P', t  0)f (P)dV
V
G(P,P', t   )
     (P, )
dSd
n i
0 S
if Case (b) applies, we take G / ni  0 on S and we get
T(P', t) 
(2.3b)

t
k 0 V
t
G(P,P', t   )q(P, )dVd   G(P,P', t  0)f (P)dV
V
t
+    G(P,P', t   ) (P, )dSd
;
.
0 S
In order to express the solution in this way, G(P, P', v) is defined by the equations
G
  2G
v0
v
(2.4)
G(P, P',0)  0
(v  0, P  P')
G
(a) G  0 on S for v  0 or (b)
 0 on S for v  0
n i
with the special property that G have the (asymptotic) form
2
1
(2.5)
GS (P,P', v) 
e R /[4 v] , R 2  (x  x ')2  (y  y') 2  (z  z') 2
3/ 2
8 v)
for v  0 in the neighborhood of the point P= P ' . Notice that G and G S satisfy the differential equation
and for v  0 both G and G S go to zero, except at the point P  P' where they assume a delta function
behavior:
1
 R 2 /  4 v 
(2.6)
lim
e
T(P)dV = T(P')
3/ 2 
v 0
8 v  VA
where V A is any neighborhood containing the point P ' . G S is the Green’s function for a solid which is
infinite in all directions. Reference [8] is replete with constructions which produce G from G S . One can
easily see the delta function behavior in the volume integral in (2.3) where the integral picks out the
initial value of T (  f ( P ') ) as t goes to zero. The Green’s function so constructed with v  t   for
(2.2) and (2.3) is interpreted as the temperature at a point P at time t due to a unit source of heat at a
point P ' at a previous time  or, because of symmetry in P and P ' , the temperature at P ' at time t due
to a unit source at P at time  . Because of this interpretation, G S is often called the source term.
In order to derive the temperatures for the two-region problem, we need the background on the
derivation for the one-region problem; that is, we need the equations leading to (2.2). These can be
found in [8, Chapter XIII] and [6, Chapter 3] and follow from a consideration of the product
G(P, P', t   )T(P, ) for time   t ,
2

T(P, )
G(P,P', v) v
G(P,P', t   )T(P, )  G(P,P', t   )
 T(P, )


v
 .
(2.7)

2
2


  G T  T G   Gq
k
Integration on V and  with   0 gives
t 
t 
 




 t  
2
2
(2.8)
0  V  GTdV d  k 0  V GqdV  d   0  V G T  T G  dV d .
Then, exchanging orders of integration on the left and applying Green’s theorem on the right, we get
t 

  T
 t  
G  
GT
dV

GT
dV

GqdV
d



T
(2.9)






  G
 dSd
V   t 
V   0


k 0 V
n
n  

0 S 
where  / n is an external normal derivative. Now, the first integral has the delta behavior of (2.6) in the
neighborhood of P ' as   0 and picks out the value of the integrand, T ( P ', t ) , at P ' ; the value of
T(P,0)  f (P) in the second integral is stated in the problem definition, and we get (2.2) when we
change from external to internal normal derivatives. It is conventional to express the solution in terms of
(x,y,z), but, since P ' is any point in V, (2.2) expresses the solution just as well.
To go forward, we define the Green’s function for a two-region problem with perfect contact along the
common boundary in the following way. We solve the heat conduction problem with the unit heat
source G S only in volume 1; the case for the source in volume 2 is obtained by exchanging subscripts 1
and 2. Continuity of temperature and flux along the common boundary SC reflects perfect contact with
the usual boundary conditions applied on the part of the boundary not common to each region, S j  SC ,
j=1,2. The mathematical formulation is given by the relations:
G j
  j 2G j , v  0 , j=1,2
v
(2.10)
lim G j (P, P', v)  0 except near P=P' in Region 1 where the behavior is like G S
v0
G j
(a) G j  0 or (b)
 0 on S j  SC , the part of S j not common to each region for v>0
n i
G
G 2
G1  G 2 , k 1 1   k 2
on the common boundary SC
n i
n i
G S (P, P', v) applies only for P and P' in volume 1
Examples of this construction are found in [4]. Opposite signs are required for continuity of flux because
the internal normal derivatives point in opposite directions on the boundary SC . The equations that have
to be satisfied by T1 and T2 are the same as (2.1(a)) and (2.1(b)) on S j  SC , j=1,2 with the continuity of
temperature and flux on the common boundary SC ,
Tj
j
  j 2Tj  q j ,
q j  0, t  0, j=1,2
t
kj
Tj  f j (x, y, z) initially (t=0) inside the surface S j
Tj
(2.11)
(a) Tj   j (x, y, z, t) on S j  SC for t>0 or (b)
  j (x, y, z, t) on S j  SC for t>0
n i
T
T
T1  T2 , k1 1   k 2 2 on the common boundary SC
n i
n i
Now for Region 1, we apply (2.9) and write the solution for T1 (P', t) as
3
T1 (P', t)   G1 (P, P', t  0)f1(P)dV=
V1
1
t
k1 0 V1
G1(P, P', t   )q1(P, )dVd
t

G (P, P', t   )
T (P, ) 
 1   T1 (P, ) 1
 G1 (P, P', t   ) 1
dS1d
n i
n i 
0 SC 
(2.12)

G1 (P, P', t   )
T (P, ) 
 G1 (P, P', t   ) 1
T1 (P, )
dS1d
n i
n i 
0 S1  SC 
t
 1 

and for Region 2, (2.9) gives
0   G 2 (P, P', t  0)f 2 (P)dV 
2
t
G 2 (P, P', t   )q 2 (P, )dVd
k 2 0 V2
t

G (P, P', t   )
T (P, ) 
+ 2   T2 (P, ) 2
 G 2 (P, P', t   ) 2
dS2d
n i
n i 
0 SC 
V2
(2.13)

G 2 (P, P', t   )
T (P, ) 
 G 2 (P, P', t   ) 2
T2 (P, )
dS2d
n i
n i 
0 S2  SC 
t
 2 

In Region 1 the delta behavior of G1 pulled out the value of T1 (P', t) from the volume integral for
  t , but G 2 does not have the delta behavior and is continuous everywhere. Therefore in (2.9), the
initial condition on G 2 gives limG 2 (P,P',  )  G 2 (P,P',0)  0 . Consequently, in Region 2 there is no
 0
contribution from the volume integral over G 2T2 as   t . Now we multiply (2.12) by k1 / 1 and (2.13)
by k 2 /  2 and apply the continuity equations at the SC boundary. The integrals over the common
boundary SC become negatives of one another. We therefore add the two equations and solve for
T1 (P', t) :
T1 (P', t) 
(2.14)
1  t

   G1 (P, P', t   )q1 (P, )dVd +   G 2 (P, P', t   )q 2 (P, )dVd 
k1  0 V1
0 V2

t
k 21
G 2 (P, P', t  0)f 2 (P)dV
k1 2 V2
V1
t

G (P, P', t   )
T (P, ) 
+1   T1 (P, ) 1
 G1 (P, P', t   ) 1
dS1d

n

n

i
i
0 S1  SC 
t

k 21
G 2 (P, P', t   )
T2 (P, ) 

 G 2 (P, P', t   )
T2 (P, )
dS2d
k1 0 S2  SC 
n i
n i 
  G1 (P, P', t  0)f1(P)dV +
Now, all of the quantities in the integrals are known from the boundary or initial conditions on Tj and
G j and we have an expression which looks very much like (2.2) for the single region case (i.e. time
integrated boundary and heat source integrals added to volume integrals over the initial values). This
formula agrees with (2.2) when 1   2 , k1  k2 and the respective volume and surface integrals are
combined.
The formulas for the temperature in Region 2 are obtained by exchanging the subscripts 1 and 2.
Remember that the pair for G1 and G 2 in (2.14) is a solution of (2.10) and this exchange puts the source
term in volume 2. Consequently (2.10), modified for the source term in volume 2, defines a new pair for
G1 and G 2 since Regions 1 and 2 may not be symmetric about the boundary SC . Thus the complete
4
Green’s function consists of two pairs (G1k , G2k ) , k  1,2 where k defines the region where the delta
behavior is active. In terms of this notation, (G1, G2 ) used in (2.14) is (G11 , G21 ) and (2.14), with the
subscripts 1 and 2 exchanged, uses (G12 , G22 ) .
For boundary condition (a) where we specify the temperature on a boundary, it is common practice to
take the Green’s function to be zero on the boundary and for (b), where we specify a derivative on the
boundary, the practice is to take the Green’s function derivative to be zero there. This is how the oneregion cases in (2.3) were derived. Equation (2.14) allows for the possibility of mixed boundary
conditions if one can solve the mixed problem for the Green’s function.
In two dimensions the source term G S takes the form
(2.15)
GS (P,P', t   ) 
2
1
e R /[4 (t  )] , R 2  (x  x')2  (y  y') 2
4 (t   )
Volume integrals in (2.14) are replaced by (x,y) plane surface areas and the surface integrals are
replaced by boundary contours.
The one-dimensional source term is
(2.16)
GS (P,P', t   ) 
2
1
e R /[4 (t  )] , R 2  (x  x') 2
[2  (t   )]
and boundary integrals are replaced by end point values on the adjacent intervals. Both (2.5) and (2.15),
called product solutions, are obtained by applying (2.16) in each coordinate and forming the product.
3. Two-region Solution with Imperfect Heat Transfer Between Regions
The equations, taken from (2.10),
T
T
T1  T2 , k1 1   k 2 2 on the common boundary SC
n i
n i
express the continuity of temperature and flux across the physical boundary and represent surfaces in
perfect contact. When these media are not in perfect contact, or have been soldered together by a
bonding material with a finite conductivity, there is a thermal resistance and the amount of heat (flux)
crossing this boundary is assumed to be proportional to the temperature difference,
(3.1)
q  h T1(P,P', t)  T2 (P,P', t) , P on SC .
The constant of proportionality h (=1/thermal resistance) is called the heat transfer coefficient. When we
combine this with the equations for the propagation of heat normal to the boundary ( n is the exterior
normal vector)
T
q j   k j j   k jTj n
(3.2)
n
we get the boundary conditions
T
T
 k1 1 (P, P', t)  k 2 2 (P, P', t)  h T1 (P, P', t)  T2 (P, P', t)  , P on SC
(3.3a)
n
n
5
where  / n denotes the external normal derivative and  / n   / ni where ni is the interior normal.
The fluxes have opposite signs because heat flow is from one material (exterior normal) into the other
(interior normal). Then, in terms of the interior normal,
T
T
k1 1 (P, P', t)   k 2 2 (P, P', t)  h T1 (P, P', t)  T2 (P, P', t)  , P on SC .
(3.3b)
n i
n i
To proceed, we re-examine (2.12) and (2.13) where we select G j to have the same boundary condition
as T j ,
(3.4)
k1
G1
G 2
(P, P', t   )   k 2
(P, P', t   )  h G1 (P, P', t   )  G 2 (P, P', t   )  , P on SC .
n i
n i
The plan is to use (2.12) and (2.13) again and show that with (3.3) and (3.4) the integrals on the
boundary SC cancel one another and we obtain (2.14) again. Now we multiply (2.12) by k1 / 1 and
(2.13) by k 2 /  2 so that when we add these equations, we can examine the sum of the integrals on SC :

 T (P, )k
1
(3.5)
SC
1
G1 (P,P', t   )
T (P, ) 
 G1 (P,P', t   )k1 1
dS1
n i
n i 
G 2 (P,P', t   )
T (P, ) 
 G 2 (P,P', t   )k 2 2
dS2
n i
n i 
SC
Now we use (3.3b) and (3.4) to write the fluxes of second integral in terms of the fluxes in the first
integral,

G (P, P', t   )
T (P, ) 
 G 2 (P, P', t   )k 2 2
dS2
S T2 (P, )k 2 2 ni

n

i
C
(3.6)
.

G1 (P, P', t   )
T1 (P, ) 
   T2 (P, )k1
 G 2 (P, P', t   )k1
dS2
n i
n i 
SC 


 T (P, )k
2
2
Then substituting (3.6) into (3.5) and applying (3.3b) and (3.4) again, we have

G (P,P', t   )
T (P, ) 
 G1 (P,P', t   )  G 2 (P,P', t   )  k1 1
dS1
S T1(P, )  T2 (P, ) k1 1 ni

n

i
C
(3.7)
.
1  T1 (P, ) G1 (P,P', t   )
G1 (P,P', t   ) T1 (P, ) 
   k1
k1
 k1
k1
dS1  0
h SC 
n i
n i
n i
n i 
These manipulations show that the contact boundary terms cancel one another, as they did in Section 2.
Then, multiplying (2.12) by k1 / 1 , (2.13) by k 2 /  2 , adding and solving for T1 (P', t) , we get (2.14).
While the derivations give the same formula, the Green’s functions are entirely different.
4. Green’s Functions for Some Two-region Rectangular Geometries
The Laplace transforms of Green’s functions appear to be more useful than the direct inversions [4],
especially when one has to apply the general formulas of Sections 2 and 3. It is with this in mind that we
present a few examples of two-region Green’s functions which are directly applicable to the formulas
derived above. An example is provided in Section 5.
The Green’s function for the rectangular system defined by the half planes x>0 (Region 1) and x<0
(Region 2) with imperfect contact along x=0 was developed in [4] and given as
6

B e
T 1 ( x, y , p )  1 
2 
[ x  x '  2  12 i  ( y  y ')] p
 2  12

T 2 ( x, y , p ) 
hB1

k 2 

1
p

e
 (  ) p

    e 1
d 
 1  p     2   2 d  , x  0
2  


1
 2 (  ) p
 2  12  2   22
(4.1)  1 (  )  ( x  x ' )  2  12  i ( y  y ')  ,
B1
d

1
 k 2  2   22
x0
,
j  1/  j , j  1,2
 2 (  )  x  2   22  x '  2  12  i ( y  y ')  ,
  h

B1  1/(21 )



1
1
2 h

,   h 
, - 
2
2
2
2
2
2
 k1   1 k 2    2 
  1 
k1  2  12
1

k1
for the source at P ' in the right half plane where h is the heat transfer coefficient between Regions 1 and
2. Because of the symmetry of the geometry about the y-axis, the Green’s function for a source in the
left half plane is obtained by exchanging 1 and 2, replacing x with –x and x ' with  x ' . The amount of
heat (flux) being transferred between regions is taken to be proportional to the temperature difference on
each side of the thermal resistance and (4.1) satisfies
T
k1 1 (0, y, p)  h T1 (0, y, p)  T 2 (0, y, p) 
x
T
k 2 2 (0, y, p)  h T1 (0, y, p)  T 2 (0, y, p)  .
x
There are two important special cases which reflect extremes in the rate at which heat is transferred from
one region to the other. The first, with h   , reflects perfect contact between materials with maximum
heat transfer,

[ x  x '  2   2  i  ( y  y ')] p

 2   22 
 k1  k 2
d

 2  12 
1
B e
T 1 ( x, y , x ', y ', p )  1 
2  k1  2  12  k 2  2   22
(4.2)

[( x  x ' )  2   2  i  ( y  y ')] p
1
B e
+ 1
2  k1  2  12  k 2  2   22

T 2 ( x, y , x ', y ', p )  B1k1 

e
, x>0

 2   22 
 k1  k 2
d

 2  12 
[( x  2  22  x '  2  12  i  ( y  y ')] p
k1  2  12  k2  2   22
d ,
x<0
The second special case for h  0 represents a perfect barrier or insulated boundary along x=0 and
we have the Green’s function for the right half plane, x  0 , x '  0

(4.3)
B e
T 1 ( x, y, x ', y ' p )  1 
2 
T 2 ( x, y, x ', y ' p )  0 ,
[ x  x '  2  12 i ( y  y ')  ] p
 2  12
 1 (  )  ( x  x ')  2  12  i ( y  y ')  , x '  0
7
d 
B1
2



e
 1 (  ) p
 2  12
d , x  0
x0
The second integral is the image of the source (the first integral) in the y-axis [4].
Standard Green’s functions for quarter planes 1 and 2 can be obtained by the method of images for any
of the functions above. In particular, if we take y '  0 and place a sink (negative source) at an image
point  y ' in the lower half plane, we obtain the Green’s function with a zero boundary condition on the
x-axis,

 T1 (x, y, x ', y', p)  T1(x, y, x ',  y', p)
(4.4a)
G( P,P', p)= 

T 2 (x, y, x ', y', p)  T 2 (x, y, x ',  y', p)
and direct computation with any of the above gives
(4.4b)
G( P(x,0),P', p)=0 .
To get the Green’s function for a zero derivative boundary condition on y=0, we add these two images
to obtain

 T1 (x, y, x ', y', p)  T1(x, y, x ',  y', p)
(4.5a)
G( P,P', p)= 

T 2 (x, y, x ', y', p)  T 2 (x, y, x ',  y', p)
which gives
 G( P(x,0),P', p)
=0 .
(4.5b)
y
Details of the inversions are contained in [3], [2], and [4].
5. Heating of Quadrants 1 and 2 Along the x-axis- Part 1.
Reference [2] deals with temperature distributions in adjacent quadrants (1 and 2) which are initially at
zero temperature and heated on the x-axis. In problem I, constant heat fluxes for x>0 and x<0 on the xaxis boundary are used to heat the system and, in Problem II, constant temperatures on these boundaries
are used. The general problem which contains a thermal resistance with a heat transfer coefficient h
between quadrants is solved for both problems I and II by a direct attack on the equations. It is the
purpose of this section to apply the two-region theory of Sections 2 and 3 to either get a new
representation or verify the solution in [2]. We will work with Laplace transforms.
We start with the zero flux Green’s function (4.5a) for y=0 with the functions of (4.1) and note that by
direct observation T1 (x, y, x ', y', p) and T1 (x, y, x ',  y ', p) are equal for y=0,

 G1 (P(x,0),P', p) 
 T1 (x,0, x ', y', p)
G( P(x,0),P', p)= 
=2 
G 2 (P(x,0),P', p) 
T 2 (x,0, x ', y', p)

where the normal derivative on the x-axis (y=0) is zero. The general formula (2.14) for the first quadrant
temperatures for this two-region problem reduces to
t
t


T1 (P, ) 
k 21
T2 (P, ) 
(5.2) T1 (P', t)  1   G1 (P,P', t   )
dS1d 
G 2 (P,P', t   )
dS2d


n i 
k1 0 S2  SC 
n i 
0 S1  SC 
(5.1)
where SC  0  y   , S1  SC  0  x   and S2  SC    x  0 . The equations for constant
flux boundary conditions on y=0 are
(5.3)
T1 (P(x,0), )  F1

y
k1
and
8
T2 (P(x,0), )  F2

y
k2
and (5.2) gives
t 
t 0


F
k
F 
T1 (P', t)  1   G1 (P(x,0),P', t   ) 1  dxd  2 1   G 2 (P(x,0),P', t   ) 2  dxd .
k1 
k1 0  
k2 
0 0
The Laplace transform is
0
 
F
k 
F 
(5.5)
T1 (P', p)  1  G1 (P(x,0),P', p) 1  dx  2 1  G 2 (P(x,0),P', p) 2  dx .
p 0
k1 
k1p  
k2 
(5.4)
To proceed, we need to integrate G j (P(x,0), P', p)  2T j (x,0, x ', y ', p) , j=1,2 from (5.1). We take
T j (x,0, x ', y ', p) from (4.1) and integrate on x in Region 1. The integrals for the two terms are

e
(5.6)
0
=
 a1 x  x '
x'
dx   e
 a1  x '  x 
0

dx   e
 a1  x  x ' 
dx  e
x'
 a1 x '
e
x'
1
1 1
a x '
a x '
1  e 1     2  e 1 

a1 
 a1 a1 

a1x
dx  e
a1x '
0

and
0 e
 a 1x
dx 
1
a1
,

x ' e
 a 1x
dx 
a1  p  2  12
and for Region 2 we have,
0
(5.7)
e

a 2 x
dx 
1
,
a2
a 2  p  2  22 .
With B1  1/(21 ) , the integrals in (5.5) become

a1x '  i y' p

 B  1  2  e
 e

1
 
d
2
2
2
a





1

1

F
F 
(5.8)  G1 (P(x,0),P', p) 1  dx  2 1 
, a1  p  2  12
k1 
k1 
0
a1x ' i y' p 

e
   

 + B1 1 e
1


 a1  2   2  p    d
 2 


1

a x' i y' p
0


F2 
F2 B1h 1 e 1 e
1
d , a 2  p  2   22
(5.9)  G 2 (P(x,0),P', p)  dx  2

2
2
2
2
k
k
k
a
  1    2 p  
2
2
2  2
 
or
(5.10)
(5.11)

 F1 / k1  ei y' p
d
 3/ 2  2
2
1  
F1 
  p    1
G1 (P(x,0),P', p)  dx  
p 0 
k1 
2
2

 [x '   1 i y'] p
hF
/
k
e
 1 1
  p3 / 2  k  2   2 (  2   2 )

1
1
1

0

k 21 
F2 
1 h / k1 e
 G2 (P(x,0),P', p) k 2  dx   p3 / 2 
k1p 
[x'  2  12 i y'] p
 
2
2
1
The first integral on the right in (5.10) is a Fourier transform with the value
9
F2 / k 2
 2  22
1
p
d
1
p
d .
i y' p
e
 y ' p / 1
(5.12)
.
  2  12 d   1 e
The final transform from (5.5) is
 y' p / 1
  (  ) p
F1 1 e
1 e 1
T1 (P', p) 

 p3 / 2 ( p   ) H1(  )d
k1
p3 / 2
 

(5.13)
H1 (  ) 
 F2 / k 2
F /k 
 21 1 2 

2
2
 2  12     2   1 
h
k1
 1 (  )  x '  2  12  i y '
and this agrees with the formula on page 4 of [2]. Since the Regions 1 and 2 are mirror images of one
another, exchanging the subscripts and replacing x ' with  x ' gets the formula for the second quadrant
temperatures T 2 (P', p) . In this case, x '  0 and  2 (  )  x'  2  22  i y' .
The solution for Problem II of [2], where each temperature is defined on the x-axis boundary, is
developed in the same way with (2.14), (4.4a) and the Green’s function of (4.1). The main content of [2]
is devoted to the inversion and convergence analysis of these two problems.
6. Heating of Quadrants 1 and 2 Along the x-axis-Part 2
In the previous section, the Green’s function is employed as the kernel of the inverse operator to obtain
the solution of the problem. In this section we take advantage of its physical interpretation as the
solution of a conduction problem with a point source of heat in its interior. The goal is to re-derive the
solution for the adjacent quarter planes and compare it to the formula on page 4 of [2] and the result in
(5.13). Remember that in (5.13) x ' and y ' are x and y in these comparisons. We take the Green’s
function (4.5a) associated with a zero normal derivative on the x-axis where T 1 and T 2 are define in
(4.1). For the moment, we consider a line of sources in the first quadrant parallel to the x-axis at a small
distance,   0 , from the x-axis. Then



x>0
 T1 (x, y, x ',  , p)  T1 (x, y, x ',  , p)
(6.1)
G(
P,P'(x',

),
p)dx'=
0
0 T2 (x, y, x ',  , p)  T 2 (x, y, x ',  , p)dx' , x<0

represents the temperatures in quadrants 1 and 2 due to heat flow from an instantaneous, uniform line
source in quadrant 1 at time zero with insulated boundaries on y=0. To obtain a continuous line source
in time, we divide the transform by p. Except for some minor storage of energy associated with a rise in
temperature in the mass contained between y=0 and y=  , physical reasoning suggests that all of the
heat generated from the line source will flow from the source into quadrant 1 and, in the limit for
  0 , the interior normal derivative will not be zero, but will be proportional to the heat generated on
y=0. The derivative on the negative x-axis remains unchanged with a value of zero. The uniform source
will guarantee a constant flux and we have to relate the source strength, which is contained as a
multiplier in the parameter B1 , to the flux. To simplify the algebra, we take the limit   0 and work
with the forms

x>0
2 
 T1 (x, y, x ',0, p)
T(x,y,p)=  
dx'
(6.2)
.
x<0
p 0
T
(x,
y,
x
',0,
p)
2

Now, the integrations on x ' in the equations of (4.1) are
10

e
 a1 x  x '
0
x
dx '   e
 a1  x  x ' 

dx '   e
0
x
 a1  x '  x 
1
a x
dx '  2  e 1  ,
a1 


e
0
1
 a1 x '
dx ' 
a1
a1  p  2  12
Then, (4.1) becomes

 
2 B
1
T1 (x, y, p)   1 
p  2  a1

a1 x  i y p

 a x i y p 

2  e
e
B
1e 1 e
   



d + 1 
1

 d , x  0
2  a1
p   
 2  12
 2  12 

(6.3)
Bh 1
T2 (x, y, p)  2 1 
k 2 p  a1
e
 a 2 x i y p
e
 
2
 
2
1
2
1
p
2
2
a1  p  2  12 , a 2  p  2   22 ,  -ˆ 
d ,
x0
2h
k1  2  12
Reduction of (6.3) with (5.12) gives
T1 (x, y, p)  2B1 1
e
 y p / 1
p3 / 2

e
p




(6.4)
k1
3/ 2

p
2h

H(  )d
x0

H(  )d ,
x0
 B1 
 B1 
, H(  ) 


 2
2
2
2
 2  12    1 
k 2  2   22    1 
2h
H(  ) 

p
 2 (  ) p
3/ 2
e
p
T2 (x, y, p) 
 1 (  ) p
 j (  )  x  2   j2  i y, j  1,2
The boundary condition
(6.5)
k1
T1
F
( x,0, p )   1 , x>0
y
p
with
(6.6)
 T1 (x, y, p)
e
 2 B1
y
 y p / 1
p



e
 1 (  ) p
(i )
p( p   )

H(  )d
gives
B1  F1 /(2 k1 ) .
(6.7)
The integral in (6.6) vanishes for y=0 because the integrand (with the  multiplier) is an odd function of
.
(6.4) is a special case of the results in [2] with boundary values F1 >0 for x  0 and F2 =0 for x  0 . To
obtain the solution corresponding to the fluxes F1 =0 for x  0 and F2 >0 for x  0 , we take advantage of
the symmetric geometry by exchanging the subscripts 1 and 2 and replace x by –x in (6.4). Then
superposition, by adding the two T1 solutions and the two T2 solutions, gets the solution for the first and
second quadrant temperatures with non-zero flux values on the x-axis. This manipulation for T1 is
11
implemented by adding the second equation in (6.4), with the exchanges, to the first equation in (6.4) to
obtain
 y p / 1
  (  ) p
S
 B1 
e
2h e 1
1
T1 (x, y, p)  2B1 1


d
3/ 2

3/ 2
2
2
p
k1  p ( p   )   1   2  12 
(6.8)
 (  ) p

 B2 
2h
e 1
1


d

k1  p3 / 2 p    2  12   2   22 


or
(6.9)
S
1
T (x, y, p)  (F1 / k1 ) 1
e
 y p / 1
p
3/ 2

1


p

 1 (  ) p
 F2 / k 2
F /k 
 21 1 2 d

2
2
3/ 2
( p   )  2  12     2   1 
(h / k1 )e
This superposition agrees with page 4 of [2] and the result in (5.13). Exchanging subscripts and
S
replacing x by –x in (6.9) gives the solution for T2 (x, y, p) or, alternatively, in the first equation of (6.4)
exchange the subscripts, replace x by –x and add the result to the second equation of (6.4).
References
[1] Abramowitz S, Stegun IA (1965) Handbook of Mathematical Functions, AMS 55, Dover
Publications Inc., New York, 1046 pp
[2] Amos DE, Beck JV, de Monte F (2011) Transient Heat Conduction in Adjacent Quadrants
Separated by a Thermal Resistance, http://nanohub.org/resources/12465,
http://thermalhub.org/resources/470.
[3] Amos, DE (2011), Transient Heat Conduction in Adjacent Materials Heated on Part of the Common
Boundary, http://nanohub.org/resources/12390.
[4] Amos, DE (2012), Green’s Functions For Heat Conduction in Adjacent Materials,
http://nanohub.org/resources/12856.
[5] Amos DE (2006) Handbook of Integrals Related to Heat Conduction and Diffusion,
http://thermalhub.org/resources/459.
[6] Beck JV, Cole KD, Haji-Sheikh A, Litkouhi B (2010) Heat Conduction Using Green’s Functions,
2nd Ed., CRC Press, Boca Raton, 643 pp
[7] Morse PM, Feshbach H (1953) Methods of Theoretical Physics, Part I, McGraw-Hill, New York,
997 pp
[8] Carslaw HS, Jaeger JC, (1948) Conduction of Heat in Solids, Oxford Univ Press, London, 386pp
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