Internat. J. Math. & Math. Sci.
(1987) 163-172
Vol. i0 No.
163
ON THE SOLUTION OF REACTION-DIFFUSION
EOUATIONS WITH DOUBLE DIFFUSlVITY
B.D. AGGARWALA and C. NASIM
Department of Mathematics and Statistics
The University of Calgary
Calgary, Alberta TIN 1N4 Canada
(Received September 9, 1985 and in revised form March 28, 1986)
In this paper, solution of a pair of Coupled Partial Differential equations
is derived. These equations arise in the solution of problems of flow of homogeneous
liquids in fissured rocks and heat conduction involving two temperatures. These
equations have been considered by Hill and Aifantis, but the technique we use appears to
ABSTRACT.
be simpler and more direct, and some new results are derived.
the
Also,
about
discussion
propagation of initial discontinuities is given and illustrated with graphs of some
special cases.
KEY WORDS AND PHRASES.
Reaction Diffusion equations,
Fluid
Flow
in
rocks,
Fissured
Laplace and Fourier Transforms, Propagation of Initial Discontinuities.
35K57
Subject Classification:
i.
INTRODUCTION.
In
this
paper we solve a pair of coupled Partial Differential Equations, which
pair may arise in a number of physical situations, including, e.g., flow of
homogeneous
in fissured rocks [i] and heat conduction involving two temperatures [2].
liquids
equations have been considered by Hill [3], by Hill and Aifantis [4,5] and
by
Lee
Such
and
Hill [6], wherein they have arrived at the same solutions as those in this paper, albeit
by
a
different technique.
Our method, in which inversion depends upon the idea behind
equation (3.5), appears to be simpler and more direct than that of these
Also,
we
have
deduced
some
results
from
these
investigators.
solutions which are not derived in
See also Gopalsamy and Aggarwala [7] where a slightly different
coupled partial differential equations is considered in a different setting.
literature.
2.
pair
of
THE PROBLE
We wish to find solutions u(x,t) and v(x,t) of the equations
Ou
Ot
v
O’(where a l,
bl,a2
and b
2
DlV2
D2v2
u
v +
alu
a2u
+
bl v
b2v
are constants), with the initial conditions
u
f(x) and v
0 at t
homogeneous boundary conditions on u and v.
0, and with (identical)
(2 la)
(2.1b)
B.D. AGGARWALA and C. NASIM
164
3.
SOLUTION
(x,s)) of (2.1),
Taklng the Laplace Transform (u(x,t)
iv2
D2v2
D
su
sT
We
now
v25 _2.
where
()
that
assume
blV
a2 b2
/
alu
+
we get
(3.]a
f(x)
(3.1b
((x,s)
there is a Fourier Transform
(,s)) such that
Taking this Fourier Transform, we get
s
s
-D12- al bl$
-D22 a2- b2
+
+
()
(3.2a)
(3.25)
+
is the Fourier Transform of f(x).
These equations give
a2f(
v
(3.3a)
D1D2(2+21) (2’+ 22)
whe re
J (s+al)D2-(s+b2)D1]2+4D1D2bla2
2D1D2
(s+al)D2+(s+b2)D 1]
C2+ 2,
(3.3b)
A
say.
To invert v, consider the problem Oh
v2h with
f(x) at t
h
0 and
the
same
boundary conditions as on u and v.
Taking the Laplace and Fourier Transforms, we get
f()
s +
(3.4)
2
If we now consider the fact that the inverse Fourier Transform of
-st
e
is given by
h(x,t)dt, then since
0
a
v
()
2
2-
2 +
D1D2( -1
2
2
i 2+2
it follows that the inverse Fourier Transform of
is given by
2
Ie
D1D2(2- 1)
a2
D1D2
DID2
0 -0
e
(3.5)
h(x,t)dt
0
-At
0
sinh(tJ 2
is2 +
+ C2
(3.6)
h(x,t)dt
c2
e-Ath(x,t)Io(Bt sin 8)cosh(Ct cos 8)t sin 8
dSdt
(3.7)
(see [8], p.743(2))
a2
2D1D2
f f/2 [e-SklU
0
"e
-k3u + e-sk2u
e
-k4u h(x,u)I
0
sin 8)u sin 8 dSdu
0 (Bu
(3.8a)
where
D
+ D
D
D
1
2
cos
kl,2 1 2 2D
1D2
D lb +
2
D2a1 D lb2 D2a1 cos
k3,4
2D1D 2
2D1D 2
(3.8b)
8
(3.8c)
SOLUTION OF REACTION-DIFFUSION EQUATIONS WITH DOUBLE DIFFUSIVITY
The inverse Laplaqe Transform must now be performed w.r.t, s (s
165
t) and noting
that, with the usual notation,
e-St6(t-)dt
e
-s
(3.9)
0
(where 6 is the Dirac Delta function), it is given by
v
[/2
2DID r [e-k3
a2
+ e
6(t-Ukl)
2 -0-0
Using
-k4u6(t-uk 2)
u
0 f(u)6(t-klU)du iii f[l)’
a
v
2
2D1D2
a
+
2
2biD2
I
1
+
IO
fv/2
|-0
kll
t
k2
e
e
we get
-k3t/kl
-k4t/k2
Io(t
B sin
8/kl)h(x,t/kl)Sin
8 d8
Io(t
B sin
8/k2)h(x,t/k2)sin
8 d8
(3.11)
12
1
Putting u
/2 t
I 0 (Bu sin 8)h(x,u)u sin 8 dOdu. (3.10)
]
1
in I 1, u
]
D1
a2te-6t
v
e
D1 D2
and combining the two integrals, we get
12,
in
-Tut
(3.12)
Io(qt)h(x,ut)du
2
where
b
a
1
Y= D
1
2
D2
n
Now,
(2.1b)
using
6:
and
the
fact
-6t
D1
D1
and
D2
(3.13)
](Dl-U)(u-D2)].
J
D2.
[D 1
Dlb 2 -alD 2
that
v2h
@h/@t,
get,
we
after
some
simplification
u
e
-alt h(x,D
;D2
1Jl-le
lt) D1 D2
+
t
e_TUt I l(t)h(x,ut)
D2
u
jD
u
1
du.
(3.14)
Equations (3.12), (3.13) and (3.14) give the solution to our problem.
It
be
may
that
noted
equations (3.5) and (3.1b) may be used to invert these
relations [3].
4.
SPECIAL CASE
The most important case for applications is when a I
a
2
b
I
b
2
a.
In this
case the solution becomes
v
ae
-at
DI
t
Dl D2
(4.1a)
h(x,ut)Io(/t)du
and
u
e-ath(x,D1 t) + D1_e-atD2
a
D1
Ut h(x,ut)Ii(t)
D1
D2
J
D2
u
(4. Ib)
where
2a
[D 1
V2[
J’ ’[ (u-D2) (Dl-U)
(4 lc)
B.D. AGGARWALA and C. NASlM
166
v dx
1 for all t, we readily obtain
h(x,t)dx
-t
e
sinh at
(4.2a)
u dx
e
If, in this case, we assume that
(4.2b)
cosh at
which shows that if initially the "u-sys%em is supplied with a "unit" of
heat
and
no
heat is escaping from the boundaries (the boundary conditions on h are the sme as those
"u-system" and the other half goes into
It is interesting to note that
this result is independent of the relative magnitudes of D and D
2.
1
on
or v) then half of the heat goes into the
u
the "v-system" at the rate implied by equations (4.2).
In many cases
h(x,t)
is
multiplied by a function of t.
given
a
s
(or integral) of a function of x
sum
If we assume
F (x) e -t
h(x,t)
(4.3a)
integration gives
u
e
-t
and
h x
Dlt)
v
e
+ e-t Z
-t
Z
F(x)G(t)
(4.3b)
(4.3c)
F$(x)H(t)
where
G(t)
e
-lit (DI+D2)/2
AI
cosh(tJA+A)
2
2
sinh(tJ A+;)-e
-Alt]
(4.3d)
(4.3e)
where
(DI-D2),
A1
A
(4.3f)
a
2
To obtain equations (4.3), we need
Dl
;D2
I1
e-$utll (nt) ]
u
DI
_, Du2 du
(4.4a)
and
D1
where
n
e-Xut I0(nt)du
(4.4b)
is given by (4.1c).
If we put u
D
1
sin28
+
D2cos28
we get
D1
II
and
D
’’2
2 e -Xt(DI+D2)/2
(I3-I 4)
/2
2
12
cosh(Alt
"0
cos
x)I0(A2t
sin x)sin x dx
where
13
and
2
"0
cosh(Alt
’cos
x)I0(A2 t
sin x)dx
SOLUTION OF REACTION-DIFFUSION EQUATIONS WITH DOUBLE DIFFUSIVITY
;
167
/2
2
14
sinh(Alt
0
x)II(A 2
cos
t sin x)cos x dx.
These integrals may be evaluated with the help of known
pages 742(1) and 743(2)).
5. SEPARATION OF VARIABLES
It is to be noted
solution to be of the type u
v2F
H
E
+
EF
with
G
results
(see,
e.g.[B]
equations (4.3) may also be derived by assuming the
that
__FB(x)G(t)’
____FB(x)HB(t)
v
where
F(x)
are
solving the resultant pair of ordinary differential equations in G
0,
I, H
0 at t
E
0 and superposing.
of
solutions
The same process may also be
E
and
used
to
solve equations (2.1), in which case the results are
u
-alt
e
Z
v
(5. la)
FE(x)GE(t)
F(x)HE(t)
h(X,Dlt)
+ Z
(5.1b)
where
where now
k
AI
6.
(DIE + D2 + a I + b2)/2
(DIE + a I D2 b 2)/2
PROPAGATION OF INITIAL DISCONTINUITIES
It
is
term in u.
to
be
noticed that the last term in
We shall disregard this term in
GE(t)
simply cancels out the first
GE(t)
in this discussion.
If now
then the initial discontinuities of u and v are immediately smoothed out.
-b
D
2
O,
for
then
that if Z
FE(X)
large values of
,
in (5.1),
HE(t)
e
2t /E
and
G/(t)
#
O,
If D 1 # 0 and
e
-b2t/E2,
so
(= f(x)) is discontinuous at some point, then this initial discontinuity
of u is smoothed out for t
of
E
DID2
HB(t)
e
>
-alt /E
O.
If, however, D 1
and
GB(t)
e
-alt
0 and D
+
}
O(),
2
#
so
O, then for
that,
while
large
values
the
initial
discontinuities of u do not get propagated into v, they stay as discontinuities of u and
decay exponentially as e
-alt
The same statement is also true
for
discontinuities
in
normal derivatives across some surface.
If D
1
O, and we eliminate v from equations (2.1), we get
@2u
If
now
we
D2
prescribe
V2u
the
alD2V2
u +
(al+b 2)
@u
initial values of u and
+
@u
(alb2-a2b 1)u
O.
(6.1)
and the boundary values of u for
B.D. AGGARWALA and C. NASIM
168
(6.1) and if the initial values of u as we approach the boundary do not coincide with
0, then there is an inltia] discontinuity in
u near the boundary which does not die out immediately. The boundary condition on u in
this case should be modlfled from the consideration that for D # 0, there are no
2
the boundary values of u as we approach t
discontinuities in v and then, integration of (2.1a) gives
u(s,0)e
u(s,t)
where
a
is
s
point
the
on
-alt
-alt rt e alO
Jo
alP 1
out exponentially as e
die
-alt
-
term is
(6.3)
absent
PO
In (6.3),
Lim u(x,O).
(6.2), u(s,O)
equation
xs
0
(P0-PI
this
in
The result in equation (6.3) coincides with the
@v
one obtained in [i] for the case when
>
for t
-alt
This equation once again indicates that the initial discontinuities of u
case)
PI
u(s,t)
P0’
0), (6.2) gives
for t
P1 + (P0-PI)e
u(s,t)
(6.2)
blV(S,0)d0
If, e.g., u(x,0)
boundary.
blV(S,t)
(which, using (2.1a), gives
+ e
and
(2.1b)
in
and
are constants.
P1
D1
In
0
Also f(x) (or
@f/Sn in the case of possible discontinuities in the normal derivatives) in the above
respectively) is
(or
dlscussion is assumed to be such that Z
uniformly convergent.
1.
the following examples and graphs are given for n
Examples"
In
1.
0 and D
2.
2
this
1.
a
I
case
a
b
2
x
I.
b
I
Fig.
D1
I
.5 and f(x)
x
2
for .5
illustrates
O.
l, D 2
Fig. 2
x S 1.
.5.
the
illustrates
O.
1 for .5
.5 and f(x)
0 for 0 S x
3
<
Fig. 1 illustrates the decay of discontinuity in 8u/#x at x
the
decay
modification of boundary value at x
4.
2
f(x) is the same as in example 1.
f(x)
are
1.
x/2 for 0 S x
f(x)
smoothing of discontinuity for t
3.
Numerical values
We give some examples to illustrate the above results.
indicated in Figs. 1-6.
D1
Z(@F(x)/@n)/
F(x)/
The symbol x denotes a vector in n-dimensions throughout, however
D1
i.
Same as example 3 but with D
1
of
1.
x
discontinuity
in
u
u
v
0 at x
at
x
0 and
.5 and the
I.
O, D 2
O. Discontinuities are smoothed out
l, D
2
in Fig. 4.
5.
In this case f(x)
1
1/2, v
1/2 as t
illustrates u
6.
Same as example 5 with D
.
D
1
l, D2
cos wx, @u/@n
1
O, D 2
O.
0
at
1.
x
0
and
x
1.
Fig.
5
SOLUTION OF REACTION-DIFFUSION EQUATIONS WITH DOUBLE DIFFUSIVITY
0.28
0.24
0.20
0.16
0.12
0.08
0.04
0.0
0.2
0.4
d .0
0.8
0.6
FIG. I. DECAY OF DISCONTINUITY IN
au
at x
0.5.
0.28
0.24
0.20
0.16
0.12
0.08
0.04
0.2
0.4
0.6
0.8
SMOOTHING OF DISCONTINUITY FOR t
1.0
0.
169
170
B.S. AGGARWALA and C. NASIM
0.2
0.4
0.6
0.8
DECAY OF DISCONTINUITY IN u at x
0.8
0.6
0.4
"’’
0.0
FIG. 4.
0.2
0.4
0.6
0.8
DISCONTINUITIES ARE SMOOTHED OUT.
1.0
.0
0.5.
SOLUTION OF REACTION-DIFFUSION EQUATIONS WITH DOUBLE DIFFUSIVITY
1.6
1.2
(x.-s)
0.o 0
0.2
0.0
,
0.4
FIG. 5. u
AS t
v
1.0
0.6
=, D
0.8
0.6
0.8
0;
D2
=-[.
1.6
1.2
Q. 0
FIG.
.
0.2
u
,
-
0.4
v
AS t
oo,
DI
i,
1.0
D2
0.
171
B.D. AGGARWALA and C. NASIM
172
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Ill
Barenblatt, G.I., Zhetlov, Iu. P. and Kochina, I.N., Basic concepts in the theory
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1286-]303 (1960).
[2]
[3]
Chen, P.J.
temperatures,
and
M.E., On
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ZAM.. 1__9,
a
theory of heat conduction involving two
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I.M.A.
Journal
of
A__ppl. Math.
[4]
Aifantis, E.C. and Hill, J.M., On the theory of Diffusion in media with double
diffusivity I, Ouar. Jour. Mech.j_& App1. Math. 2__3, 1-21 (1980).
[5]
Hill, J.M. and Aifantis, E.C., On the theory of Diffusion in media
Jour. Mech. & Appl. Math. 2__3, 23-41 (1980).
diffusivity II,
[6]
Lee, A.I. and Hill, J.M., On the solution of Boundary Value Problems for Fourth
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[7]
Gopalsamy, K. and Aggarwala, B.D., On the non-existence of periodic solutions
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[8]
Gradshtem, I.S. and Ryzhik,
Academic Press (1965).
with
double
uar.
I.M.,
Table
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