LEGENDRE FUNCTIONS AS SOLUTIONS TO
THE INHOMOGENEOUS HEAT EQUATION
by
GLEN A. BEAR, B.A.
A THESIS
IN
MATHEMATICS
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
MASTER OF SCIENCE
Approved
August, 1995
Are • ^^V'7
•V-:'
; i'^^
ACKNOWLEDGEMENTS
C •^
I woukj like to express my sincere appreciation to my advisor Professor
Wayne T. Ford for his guidance and encouragement throughout the preparation
of this thesis.
I also wish to thank the State of Texas for support under the Texas Higher
Education Coordinating Board - Advanced Technical Program (Project No.
003644-162) and for additional support under the Minority Supplement Support
Program (Project No. 003644-162).
II
CONTENTS
ACKNOWLEDGEMENTS.
ABSTRACT
iv
I. INTRODUCTION.
II. THE WEIGHTED HEAT EQUATION
4
III.REDUCTIONOFORDER
10
IV. ADDITIONAL SOLUTIONS
17
V. CONCLUSION
20
REFERENCES
21
III
ABSTRACT
This thesis is a study of the classical and weighted heat equations, with a
detailed examination of solutions for a particular case of the weighted equation.
This particular case Involves the substitution of a weighting factor into the
equation which reduces it to Legendre's differential equation. Several methods
of finding solutions for this particular case are given, along with solutions
computed using the given techniques.
IV
CHAPTER I
INTRODUCTION
Consider the classical heat equation in the form
d (duix,t)\ _ du
dx\ dx J dt
X69t
(1.1)
/>0
and the weighted heat equation in the form
in
dx
(1.2)
Bt
where the w in w(x) stands for a weight. Although it might seem that (1.2) is quite
similar to (1.1), "Little attention has been paid to diffusion in nonhomogeneous
media in which the diffusion coefficients vary with distance measured in the
direction of diffusion" [6].
The outer partial in (1.2) with respect tox is indicated as an iterated partial to
emphasize that a major purpose of this paper is to insert a variable conductivity
between the outer and inner partials. This paper treats conductivity variable in
position in (1.2) rather than the nonlinear case, noted in Ford [10], as in
du
du
(1.3)
It
dx
which has been studied by Aronson [2] who gives a particular closed form
solution for the particular nonlinear version of the heat equation in
_i
m-\
dx
mu
du
dx_
= —(u"')
dx'
=~
dt
m>\
(1.4)
Consider the heat equation itself in (1.1). So that the computations in later
discussions may be fully understood, let us review the method of solution of this
particular equation through separation of variables. To begin, assume the
solution is of the form
uix,t) = vix)zit) .
1
(1.5)
Then
du
dvix)
dx
dx
d~u
dx^
d'vix)
z(t)
dx~
U)
,
(1 -6)
,
(1.7)
.
(1.8)
and
-
=v(.)—
Upon substituting (1.7), and (1.8) into (1.1), we find
d\ix)
^dzit)
zit) = v(x)——
dx^
dt
and dividing by v(x)z(/) gives
,
d\ix)
dzit)
dx"- _ dt
v(x)
zit)
, Q.
(1.9)
(1-10)
As can be seen, the left side of (1.10) is dependent on x only and the right side
is dependent on t only, hence, this equation must be equal to a constant.
Set
^'v(x)
dzit)
• ^ ^ = - ^ = -A^
vix)
.
(1.11)
zit)
which implies
d~vix)
and
- ^ = -A^
vix)
(1.12)
dzit)
-^-X-
.
Upon multiplying (1.12) by v(x) and (1.13) by c(r)and rearranging both
(1.13)
equations, we have
^'v(x)
and
dx'
^
+ A'v(^) = 0
+ A^z(r) = 0
(1.14)
.
(1.15)
dt
By (1.14), we find the solution to v(x) is
v(x)= AsinAx+BcosAt
,
(1-16)
where A and B are constants. Integrate (1.15) with respect to r to find
or
lnz(0 = -A^/ + C
zit)=De-'''
,
(1.17)
where again, C and D are constants. Recalling uix,t) = vix)zit) and substituting
the above results gives
-kh
M(x,r) = (AsinAx+BcosAx)D&"^'
This is a well-known solution to the classical heat equation [20].
(1.18)
CHAPTER II
THE WEIGHTED HEAT EQUATION
Let us now return to the weighted heat equation
d_
dx
du
du
(2.1)
and apply methods similar to those used to find a solution to the unweighted
heat equation to attempt to find a solution to (2.1). If we again assume the
solution M(X,/) can be expressed in the form
(2.2)
M(X,/) = V(X)Z(0
then, as before,
du
dx
dvix)
Zit)
dx
(2.3)
d^u _ d^vix)
Zit)
dx'
dx^
and
du
dt
(2.4)
dzit)
dt
(2.5)
Upon substituting (2.3) and (2.5) into (2.1), we find
d[
^ ^dvix) ^ 1
^ -^ ( 0
dt
(2.6)
or, equivalently
dwix)dvix)^^^
. , .,fi? v(x) ^^,
..^.dzit)
zit) + wix)lit) = vix)dx
dx
dx'
dt
(2.7)
As before, divide by v(x)z(/) to get
dwix) dvix)
dx
dx
+
vix)
^^)/v(x)
dx'
vix)
dzit)
dt
zit)
(2.8)
which again can be seen to be equal to a constant, due to the fact that the left
side is dependent on x only and the right side is dependent on r only.
Hence, set
^(x)^x)
dx
dx
v(x)
. .d'vix)
dzit)
w(x)
^—
j
dx' _ dt _
v(x)
zit)
^
(2 9)
which implies
^(x) ^ x )
and
, .d~vix)
w(x)
=—
= -?:
dx
dx +.
dx
v(x)
v(x)
(2.10)
dzit)
dt = -A=
zit)
(2.11)
.
Upon multiplying (2.10) by v(x) and (2.11) by z(r) and rearranging both
equations, we have
^w(x)^x)
, ^d^x)
dx
dx ^ - ( x ) ^ ^ + A-v<x) = 0
(2.12)
and
^ ^ + A'::(/) = 0 .
dt
(2.13)
By (2.13), we arrive at the same solution for zHt) as before, namely
zit)=Ae-'''
.
(2.14)
As it is most likely that "The term wix) in (2.7) makes the medium
inhomogeneous, and the solution in closed form then ranges from difficult (for
special cases) to impossible" [21], the rest of this paper will be limited to
considering one special case of the weighted heat equation, namely when
wix) = 1 - x^ . Therefore, consider (2.12) again, but this time make the
substitution wix) = l-x'. Then we have
d /,
2\dvix) I
^sd\ix)
.-, , , ^
— l - x ' ) - - 3 - ^ + ( l - x - ) — - ^ + A-v(^) = 0
dx
dx
dx
2.15)
or, carrying through the derivation and rearranging gives
/,
2\d'vix) ^ dvix) <,.
^
(1-x )-—^- 2x—^—^ + X-v{x) = 0
dx^
dx
Compare (2.16) with Legendre's differential equation
{l.,^-)£^.2x^^
+ viy + mx)^0
^
^ dx^
dx
,/s-.,,v
(2.16)
.
(2.17)
It is obvious these are the same equation with the identification
v(v + l) = A'
.
(2.18)
Carrying through the multiplication and rearranging we have
v' + v - A ' = 0
(2.19)
and upon solving for v using the quadratic formula, we find
-l±Vl+4A'
(2.20)
V =
Make the following designations for clarity in future calculations:
-1+V1+4A'
V, =
z
,„^^,
(2.21)
-1-V1+4A'
(2.22)
and
^2
=
Solutions to Legendre's equation (2.17) are
r
v(v +1) . . V(V+1XV-2XV-H3) 4
S =fln1
X +
X
1
2!
4!
v(v-HXv-2Xv + 3Xv-4Xv + 5) ,
^
and
T^a.
iv-Wv
+ l) 3 (V-1XV-H2XV-3XV+4) 5
—
X
X +
X
3!
5
(V-1XV + 2 X V - 3 X V + 4 X V - 5 X V - H 6 )
7
7!
1
J
(2.24)
where a^ and a, are arbitrary constants [16]. For comparison with later
calculations, call the terms inside the brackets above 5;^ and T^, respectively.
As 5;^ and 7^ are linearly independent, and each series is a solution of
Legendre's equation, any linear combination of S^^ and T;^ Is a solution. Hence,
the general solution is given by
(2.25)
v(x) = ao5;i+a,ri
Given the computation of v, and v^ in (2.21) and (2.22), we are now in a
position to substitute Into (2.23) and (2.24) to determine two linearly
independent solutions to (2.16). Upon substituting v, into (2.23), we have
1 1
1
_i^iViT3F
-^^-ivrr^
2 2
2
1 ( 1
S,ix) = l - 2!LV 2
l_
6!
. 2 2
A2
2
)]
SM) = • - x\^'V + ^[(^'XA^ -6)]x^ -1[(A^XA^ -6XA' -20)];c' +...
2!
4!
6!
120A' - 26A^ + t
5.(^) = 1 - ^ ^ '
^-^
Upon substituting Vj into (2.24), we have
6!
x^+...
(2.26)
5!LV 2
2
A2
2
A 2
2
A2
2
J.
7!LV 2
2
A2
2
A 2
2
A2
2
J
2
2
2
T,(x) = x-^[X--2]x'
X +...
2
+1[(A' _2XA' -12)]r
-:|;[(A' -2X A' -12X A' -30)]A:' +...
.-. 7i(x) = x^-
2 - A ' 3 24-14A'+A^ 5 720-444A' + 44A^-A' 7
•X' +
x'+...
3!
5!
7!
.(2.27)
Hence, we have two solutions of (2.16). A calculation similar to that which led to
(2.26) and (2.27) will give additional solutions by substituting v. into (2.23) and
(2.24). However, these solutions will be the same as (2.26) and (2.27), as is
apparent by comparing the form of v,and v^.
Let us conclude this section by taking a brief look at a substitution which will
be considered in more detail in the next section, namely, let wix) = 14- x'. Upon
substitution into (2.12), and rearranging the equation, we see
dr
dvix)-]
= -iMx)
dx
(2.28)
Then, make a change of variables, letting x = irf, hence dx = idr\. After
substituting into (2.28) we see
\d_ 1-ry dvjir]) = -A-v(/r])
i dt]
i
dr] _
(2.29)
Letting V(T]) = viij]) and simplifying, we are left with
_d_
dt]
1-7]'
dvir])
dr]
2
'^y
= A^ ^T))
8
(2.30)
As in (2.16) and (2.17), this is again simply Legendre's differential equation with
the identification
i7(v+i)=-A-
,
(2.31)
where the "bar" designation on v distinguishes (2.31) from (2.18). Using the
quadratic formula as before, and solving for v, we find
- -l±Vl-4A'
V=
^
.
,^^^,
(2.32)
Make the following designations for clarity in future calculations:
_
-1+V1-4A'
^. =
2
(2.33)
and
'
2
Hence, by using the given solutions of Legendre's equation in (2.23) and (2.24),
we are now in a position to find additional solutions of (2.30). Namely, upon
substituting the above values of v, and v. into (2.23) and (2.24) and following
similar calculations which led us to (2.26) and (2.27) we find
- ^ ^ , A' 2 6A'+A' 4 120A'+26A'' + A' ,
X +
S^ix) = 1 + —X• ++
—X
+
X +...
2!
4!
6!
,^ ^..
(2.35)
and
2 + A' 3 24 + 14A'+A'' s
ri(x) = x-i- 3! X +
5
+•
720-444A- + 44A'+A'
x'+...
7!
(2.36)
where again the "bar" designation on the S and T distinguishes (2.35) and (2.36)
from (2.26) and (2.27).
CHAPTER ill
REDUCTION OF ORDER
We have been considering
dx
wix)^\
= -?Cv{x)
dxj
(3.1)
in the special case
_d^
i.dvi^)
dx
(i+r)
= -v(^
(3.2)
Let ir] = ^ to obtain
1
_
\d_
1-77^ dvjr))
i dr)
i
dr)
(3.3)
-v(n)
or
dr)
(3.4)
= V(77)
il-r)')''^''^
dr)
If we imagine vir)) to be transposed to the left side of (3.4), we obtain
Legendre's differential equation with the identification
(3.5)
v(v + l) = - l
_
which implies
(3.6)
V =—±/ —
2
2
If V, is defined using the plus sign, and v^ is defined using the minus sign, it is
interesting to observe
s
I .^^ . 1
v , + l = - - + 1+ 1 = - + / — = _v,
2
2 J
2
2
and
V2 + 1 =
1 .^^
2
^2
,
1
.Ji
+1 = - - / - = - = - V ,
10
(3.7)
As (3.4) has a regular point at x = 0 , the power series method provides a
solution of (3.4) in a neighborhood of this point [22]. Hence, let
v(n) = X^;t^*
(3.8)
it=0
Then, suppress subscripts on v to obtain
^k+2 ~
m-^D-vi v-n)
for
ik + l)lik + 2)
(3.9)
0<k<<^
It should be noted that If v is an integer, (3.8) reduces to a polynomial.
However, in the case wherev is not an integer, whether (3.8) is an acceptable
solution, and therefore of interest, depends upon its convergence properties.
The ratio test for convergence says that if in a series of positive numbers the
ratio of the (n+1)th term to the nth term approaches a limit L as n increases
without limit, and if L is less than one, the series converges. That is,
convergence requires
where
L< 1
(3.10)
It is evident that in equations (2.23) and (2.24) the ratio is
(3.11)
N
but from equation (3.9) it is readily seen ttiat
A:(;t + l ) - v ( v + 1)
limlT^=lim
ik+m+2)
= lim
k
k
V(V+1)
k+2
ik+lXk+2)
=1
(3.12)
Hence, the condition that equations (2.23) and (2.24) converge is that x^ be less
than 1, and this is true only If |x| < 1 • Hence, for values of x in the range
-1 < X < 1, the series solutions are valid.
Alternatively, consider (3.1) In the special case
11
dx
\^-^'')^V-X'yix)
.
(3.13)
dx J
where the normalization replacing x by ^ using ^ = Ax has been dropped, and
wix) has a minus sign leading directly to Legendre's equation in the form
d__
( l - ; c ^ ) ^ l + A^(x) = 0 ,
dx
dx J
(3.14)
Where we identify
v(v + l) = v ' + v = A'
.
(3.15)
Solve (3.15) for V in the form
v=
^
(3.16)
Overcome the tendency to imagine some restrictions on choices of A , and
consider a sequence of values of A defined by
l+4A'=(2v + l)^
4A'=4v-+4v ,
A^ = v^ + V
and
A = ±Vv' + V
(3.17)
where v takes the successive values in 5i = {o, l, 2, ... }corresponding to the
positive sign in (3.16). If the values in (3.14) were used with the negative signs
in (3.16) the successive values of v would lie in ^^ = {-1,-2,-3 ... }. The sum of
each element in S^ and the corresponding element in ^2 is - 1 . Then (3.14)
becomes
- ^ [ ( l - x ' ) ^ ^ 1 + «(/i + l)v(x) = 0
where n = v E:S,
(3.18)
dx J
dxL
which has the Legendre polynomial given by Rodrigue's formula in
PnM-—
\ „
^
as one of Its solutions.
12
wheren = v e 5 , .
(3.19)
If we let Vj^ =P„ ,a second solution of (3.18) can be obtained by reduction of
order to obtain Vj „ = q„P^ using
-nin + l)q„ix)P^ix)
dq„ix)
=-7-i (1 - xOI P , ( J : ) - ^ 5 ^ + q',(x)
dx
dx J.
dx
-'{
= ^[(i-.^)/',(.)^]
dx
dx
dxL
^
(3.20)
J
where « = v 6 5*,. Set
(3.21)
^^n(^)
dx
and use /J in (3.18) to work with (3.20) as follows:
0 = sn(n + l)P„(x) + ~ [ ( l - x ' ) ^ ^ 'q.ix)
^£r(j_,.)^(,^^L(i_,.)^^
dxL
dx A
dx
dx
^±[i\^x')P„ix)z„ix)]+il-x'')^^^z„ix)
dx
dx'
= ( 1 - x ' ) P , ( x ) ^ ^ + ^[(1 -x')P^ix)]z^ix)
dx
dx
, dPix)
+il-x')^^^zM)
.
dx
(3.22)
Thus,
_(l_;,^)P,(;,)*5(i>={i-[(l_;,^)/>,(;c)]+(l-;c^)^k(;c) ,
d[r
l<xc
(3.23)
dx )
and
1 dzix)
H^
''
zjix) dx
_
1
-i[(l_^^)P^(;,)]^-i«^^W
(l-x')P„(x)d!r^
" •• P„(x) dx
13
(3.24)
This may be integrated to obtain
ln[(l-x')P„(x)] + lnP,(x) + lnc„(x)
= ln[(l-x')P/(x)z„(x)] = 0
.
(3.25)
Hence, by (3.25) we see
.2\n2,
l=(l-x^)P;(x):;(x)
or
1
,ix) =
(3.26)
il-x')P^ix)
Thus, we know by (3.21) that
^"<^^ = l(l-i=(x)
and
- P{
xf
(3.27)
dx
(3.28)
•'(l-x')P;(x
where w = v € 5,.
Perhaps the complexities in the above computations suggest checking the
solution in (3.28) against Legendre's equation. Compute
'^-'^'-'^'^.ixHP.M^'-
dx
dx
dx
1
dPnix)
dx '^''^''^•^(l-x^)P,(x)
and
dxl
i\-x
)•
dx _
= --^(l-x^)
dx
2jdP„ix)^
d_jn
< ( 1 - X ),
1
qnix)^
1~^
dx
dq„ix)
^_
dx
(3.29)
4P„(X)^„(X)]^
dx
-\
Pnix)j
2,dPJ^
(l-x')^^^-^?.(x)j + —j^d-acO dx
^ ^"^""'
., J
dx
dxl
-iK-'-^V'-^-'-'^^^^L^J
14
/ ^i^D/. / .
1 dp^ix) ^ r 1 1
= -nin + l)PM)^ix) + i
T
+ —I
I
[P„ix)] dx
dxUix)\
= -n(n + l)P,(x)^(x) = -n(/i + 1)V2 „(x) ,
(3.30)
and V2^(x) satisfies Legendre's equation.
To continue, use ^, from (3.27) to compute
. ._ f
ax
f ax
•'(l-x=)/^^(x)"J(l-x>^
1^
(3.31)
X
which has singularities for x € { - l , 0, 1} . However, use v,, from (3.28) to
compute
V2.,(jc) = /f(x)(?,(x)
= x^,(x) = xlnJ-;
-1
(3.32)
I 1-x
which has singularities for x e {-1, 1} .
As an example, note that Legendre's polynomial for n = 1 compares with
(2.24) as shown in
/> = 7;(x) = x .
(3.33)
To develop v^^ from (3.32) for comparison with (3.33) recall
I
I
1 2 1 3 1 4
I n l + x =J<^—X + - X — X .+ ...
'
2
3
(3.34)
4
Although unnecessary, write
I
i
1
2
1
3
1
ln|l-x| = - x - - x - - X - - X
for Clarity in using (3.32) to compute
15
4
(3.35)
v.M+i = x l n J i i ^ =-ln — = i[ln|l+x|-ln|l-x|l
Vl-x
2 1-x
2^ '
'
(3.36)
J
X
2
3
4
J
^r _ i 2 _ i 3 _ 1 4_ 1
2
= X
1
3
3
4
1
2
J
1 4
X + - X +... = x +—X +
3
J
(3.37)
3
Thus,
V2 i(x) = - 1 +x^ +-x'^ +... = -Sj^ix)
16
(3.38)
CHAPTER IV
ADDITIONAL SOLUTIONS
Recall, from the variation of parameters method, a second solution to
Legendre's equation may be found as
^2.n = P.Mq.ix)
(4.1)
where/i(x) are the associated Legendre polynomials and
(4.2)
"^^^'^-^IT^)
Let us find a few more solutions to (3.15), given the following Legendre
polynomials [22]:
1
P,ix) = 1
P,ix)^x
(4.3)
P2(x) = 1(3x^-1)
2
P3(x)=i(5x^-3x)
2
P4(x) = -i(35x'^-30x'+3)
o
From (4.2), we see
r dx
'?o(^) = J^IT7^
or
(4.4)
1+X
^o(^)=jl°
(4.5)
1-X
Hence, by (4.1)
Mm
l+x
V2.o=Po(^)^o(^) = ( l ( j l n 1-x
J
2
17
1-fx
1-x
-"H
(4.6)
We have already calculated a solution associated with /f(x). For comparative
purposes it will be restated here, that is
V2 ,(x) = /f (x)^,(x) = x^,(x) = X
Vl - x
(4.7)
-1 •
To find V,. we have
f
%M = }
dx
ax
7^
(1-JC')J-(3J:--1)J
(4.8)
or, by partial fractions
r i
1
i
6
3 ,,
f1 2
2
+
: 'dtc
^^' ~ -I I (1 - x) "^ (1 + x) '^JSx^ -1)' • (3X--1)
(4.9)
Hence,
^2(-t) = ln
ll + x
3x
(3x- -1)
Vl-x
and
v., =PJx)q2ix) =
1
r
-i3x'--l)
(4.10)
\\-x
3x 1
(3x^-1)
(4.11)
3x
(4.12)
Hence,
V2.2 = P2(^)^2(^) = ^(3-^' - i^^^ylTlf
Y
Similar calculations will show that
%ix) = J
dx
(4.13)
(l-x^-)[ix(5x^-3)]
I2
or
qiix)
J
l+x
—
4
25x
9x ~ 9(5x- - 3 )
18
(4.14)
Again by (4.1)
V2.3 = P3(^)^3(^) = ^(5^' -3x)| li^ J 7 7 7
4
25r
1
9x 9(5x--3)J
(4.15)
Hence,
Vo 3 = -(Sx" - 3 x ) l n J
•
2
Vll-^
5 o 2
—X" + -
2
(4.16)
3
Continued calculations in a manner similar to those above would generate an
infinite number of solutions to (3.14). It is interesting to note that each solution is
dependent on a combination of a logarithmic term times the nth degree
Legendre polynomial plus linear combinations of lower order polynomials.
Perhaps further computations would yield a simple algorithm for computing a
particular order solution without the necessary integrations.
19
CHAPTER V
CONCLUSION
In this thesis, we have studied the classical and weighted heat equations,
with an emphasis on the weighted equation
d^
>v(x)-dx
ox.
du
By assuming the solution is of the form
w(x,0 = v(x)z(0
(5.2)
and applying the method of separation of variables, we were able to reduce the
problem to one that resembled Legendre's differential equation as shown in
Chapter II. Solutions were then computed by substitution into well-known
solutions for Legendre's equation.
Furthermore, it was shown in Chapter III, that an infinite number of solutions
could be generated for (5.1) when wix) = 1 - x^ by using the method of reduction
of order. In this particular case, the additional solutions were
given by the formula
y2.n = P.M^n(x)
(5.3)
where
'?»W = I(1_;,= )/.^(;C)
<^*)
and P (x) are the Legendre polynomials. Several solutions were calculated
using this formula, and It was noted that successive solutions were
combinations of preceding solutions. Perhaps further investigation would yield
a simple algorithm which would allow one to forego the prohibitive integrations
necessary to compute additional solutions. I feel the best step in this direction
could be made with careful considerations of inductive arguments.
20
REFERENCES
[1] M. Abramowltz and I. A. Stegun, Hanc^book of Mathematical Jmctions,
NBS, 1964, Dover reprint, Mineola, NY, 1972.
[2] D. G. Aronson, "Regularity Properties of Flows Through Porous Media,"
SD/m 'JoiArml of/^piied Mathematics 17, pp. 461-467, 1969.
[3] Richard M. Barrer, Diffi^sion in ani^l Through Solic^s, Cambridge
University Press, Cambridge, 1951.
[4] Jacob Bear, Dynamics ofjiuiids in Porous Media, Elsevier, 1972,
Dover, Mineola, NY, 1988.
[5] William H. Beyer, Ci^e Standard Mathematical Tables. O^C Press, Boca
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[6] J. Crank, The Mathematics of Diffusion. 1st edition corrected, Oxford,
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[7] J. T. Gushing, Applied Analytical Mathematics for physical Sciences.
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[8] Ross L. Finney and Donald R. Ostberg, Elementary Differential Equations
with Linear Algebra. Addison-Wesley, Reading, MA, 1976.
[9] Wayne T. Ford, Elements of Simulation ofjluid Jlow in Porous Media.
Texas Tech University Mathematical Series No. 8, 1983.
[10] Wayne T. Ford, Solutions of the Heat Equation in a inhomogeneous Rod.
Texas Tech University, 1995.
[11 ] Wayne T. Ford, Porous Media Jlow: An introduction. Texas Tech
University, 1994.
[12] Wayne T. Ford and Ronald M. Anderson, Mathematical Methodology
for Evaluating Simulations of Jlow in Porous Media. Technical
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Inc., Monticello. NY, 1983.
21
[14] Seize Ito, Diffusion Equations, Translations of Mathematical
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Providence, Rl, 1992.
[15] W. Jost, Diffusion in Solids. Liquids. Qases. Academic Press Inc., San
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[16] William Wayne Kitts, Legendre Junctions. Master's Thesis, Texas Tech
University, 1949.
[17] ZhuU, The Discrete Observability of the Heat Equation. Master's Thesis,
Texas Tech University, 1987.
[18] James A. Liggett and Philip L-F. Liu, The Boundary integral Equation
Method for Porous Media Jlow. George Allen and Unwin, London, 1983.
[19] Jin Liu, A Numerical Method for inverse Heat Conduction Problems.
Master's Thesis, Texas Tech University, 1989.
[20] Zaiman Rubenstein, A Course in Ordinary and Partial Differential
Equations. Academic Press, San Diego, CA, 1969.
[21] Paul G. Shewmon, Diffusion in Solids. McGraw-Hill Book Company, New
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[24] H. F. Weinberger, A Jirst Course in Partial Differential Equations. John
Wiley and Sons, New York, 1965.
22
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