132 6. FUNCTIONAL SEPARABILITY
Requiring that (6.46) be compatible gives
F D
1
u
; F D
3
u
:
Each case will be considered separately.
Case 1: F D u
1
. Since F D f
00
=f
0
, this gives
f
00
f
0
D
1
u
;
and integrating twice gives
f .u/ D c
1
C c
2
ln u:
us, our equation admits a separable solution of the form
ln u D T .t / C X.x/:
Note that we set c
1
D 1 and c
2
D 0 without loss of generality. With a resetting of variables,
this is just u D T .t /X.x/.
Case 2: F D u
3
: Since F D f
00
=f
0
, this gives
f
00
f
0
D
3
u
;
and integrating twice gives
f .u/ D c
1
C
c
2
u
2
:
us, our equation admits a separable solution of the form
u D
1
p
T .t/ C X.x/
: (6.47)
Note that we set c
1
D 1 and c
2
D 0 without loss of generality. Substituting (6.47) into our orig-
inal PDE shows that
2XX
00
X
02
C 2TX
00
2T
0
D 0;
which eventually leads to the exact solution
u D
1
p
c
2
x
2
C c
1
x C c
0
C c
3
e
2c
2
t
:
6. FUNCTIONAL SEPARABILITY 133
GENERAL TRAVELING WAVE SOLUTIONS
We now wish to consider a slightly different form of solutions, namely
f .u/ D A.t/x C B.t/; (6.48)
which are known as general traveling wave solutions. As we did previously, we differenti-
ate (6.48), but this time with respect to x twice, giving
f
0
u
xx
C f
00
u
2
x
D 0; (6.49)
or
u
xx
D F .u/u
2
x
; (6.50)
where F D f
00
=f
0
. If our given equation is compatible with (6.50), then it will admit solutions
in the form (6.48). e following examples illustrate this.
Example 6.4 Consider the diffusion equation with a nonlinear source
u
t
D u
xx
C Q.u/: (6.51)
Using (6.50), (6.51) becomes
u
t
D F .u/u
2
x
C Q.u/: (6.52)
Imposing compatibility between (6.50) and (6.52) gives
Q
00
FQ
0
QF
00
C
F
00
C 4FF
0
C 2F
3
u
2
x
D 0; (6.53)
from which we obtain the determining equations
Q
00
FQ
0
QF ” D 0; (6.54a)
F
00
C 4FF
0
C 2F
3
D 0: (6.54b)
Since F D f
00
=f
0
, then (6.54a) integrates readily, giving
Q D
c
1
f C c
2
f
0
; (6.55)
where c
1
and c
2
are arbitrary constants. e remaining equation in (6.54) becomes
f
.4/
f
0
7f
00
f
000
f
02
C
8f
003
f
03
D 0: (6.56)
Fortunately, (6.56) admits several integrating factors
1
f
02
;
f
f
02
;
f
2
f
02
;
ln f
0
f
02
; (6.57)
134 6. FUNCTIONAL SEPARABILITY
and can be used to integrate (6.56). e first integrating factor leads to
f
000
f
03
2
f
002
f
04
D c
3
; (6.58)
the integrating factor f
0
leads to
f
00
f
02
D c
3
f C c
4
; (6.59)
and the integrating factor f
0
leads to
ln jf
0
j D
1
2
c
3
f
2
C c
4
f C c
5
; (6.60)
or
f
0
D e
c
3
f
2
Cc
4
f Cc
5
; (6.61)
where c
3
c
5
are further arbitrary constant. Note that we have absorbed the factor of 1=2 into
c
3
. For example, if we choose c
3
D 0; c
4
D 1; c
5
D 0 then (6.61) integrates to give
f D ln u; (6.62)
where we have suppressed the constant of integration. In this case, Q in (6.55) becomes
Q D .c
1
ln u C c
2
/u; (6.63)
and PDEs of the form
u
t
D u
xx
C .c
1
ln u C c
2
/u (6.64)
admits solutions of the form
u D e
A.t/xCB.t /
: (6.65)
e reader can verify that substituting (6.65) into (6.64) shows that A and B satisfy the ODE
A
0
D c
1
A; B
0
D 2A
2
C c
1
B C c
2
: (6.66)
Example 6.5 Consider the nonlinear diffusion equation
u
t
D
.
D.u/u
x
/
x
: (6.67)
Expanding (6.67) and using (6.50) gives
u
t
D .D
0
C DF /u
2
x
: (6.68)
If we set G D D
0
C DF , then (6.68) becomes
u
t
D Gu
2
x
: (6.69)
6. FUNCTIONAL SEPARABILITY 135
Requiring that (6.50) and (6.69) be compatible gives
G
00
C 3F G
0
C .F
0
C 2F
2
/G D 0; (6.70)
and since F D f
00
=f
0
, then (6.70) becomes
G
00
C 3
f
00
f
0
G
0
f
000
f
0
3
f
002
f
02
G D 0: (6.71)
Equation (6.71) can be explicitly solved, giving
G D
.
c
1
f C c
2
/
f
0
; (6.72)
where c
1
and c
2
are constants of integration. Further, since G D D
0
C FD, then
D
0
f
00
f
0
D D
.
c
1
f C c
2
/
f
0
: (6.73)
Since f .u/ D A.t/x CB.t/ we have the freedom to set the constants c
1
and c
2
without loss
of generality. If c
1
D 0, then we can set c
2
D 1 without loss of generality. If c
1
¤ 0, we can set
c
1
D ˙1 and c
2
D 0. In the first case, where c
1
D 0, we can solve (6.73) giving
f .u/ D
Z
D.u/
u C d
du D A.t/x CB.t/; (6.74)
(d is an arbitrary constant) and substitution of (6.74) into (6.67) gives
A
0
D 0; B
0
D A
2
: (6.75)
ese easily integrate, giving
Z
D.u/
u C d
du D c
1
x C c
2
1
t C c
2
; (6.76)
is an exact solution to the NPDE
u
t
D
.
D.u/u
x
/
x
: (6.77)
Of course, (6.76) is nothing more than the usual traveling solution.
In the second case we consider c
1
D 1, (6.73) becomes
D
0
f
00
f
0
D D ff
0
: (6.78)
Substitution of f .u/ D A.t/x C B.t/ into (6.67), with D and F satisfying (6.78), leads to
A
0
D A
3
; B
0
D A
2
B: (6.79)
136 6. FUNCTIONAL SEPARABILITY
ese easily integrate
A D
1
p
2.t C t
0
/
; B D
x
0
p
2.t C t
0
/
; (6.80)
and we set t
0
D x
0
D 0 without loss of generality. us, we obtain solutions of the form
f .u/ D
x
p
2t
: (6.81)
Now (6.78) would need to be integrated for a given D. However, this is a nonlinear ODE for F .
Instead, for a given F , we will deduce a form of D, where exact solutions (6.67) can be obtained.
Integrating (6.78) gives
D.u/ D
d
Z
f .u/ du
f
0
.u/; (6.82)
where
d
is a constant of integration. Table 6.1 contains several examples where exact solutions
of the nonlinear diffusion equation (6.67) are given for a variety of diffusivities.
Table 6.1: Exact solutions of the nonlinear diffusion equation (6.67)
f(u)
Solution
D(u)
u
u =
x
√2t
d –
1
u
2
2
u
2
u = ±
x
√2t
2u d –
1
u
3
3
1
u
u =
√2t
x
1n |u| – d
u
2
e
u
u = 1n
x
√2t
(d – e
u
)e
u
tan u
u = tan
–1
x
√2t
(
d + 1n | cos u|) sec
2
u
Example 6.6 Consider the nonlinear dispersion equation
u
t
C k.u/u
x
C u
xxx
D 0; (6.83)
where k.u/ is an arbitrary function of u. Our goal is to determine forms of k such that (6.83)
admits separable solutions. Expanding (6.83) and using (6.50) gives
u
t
D k.u/u
x
F
0
C 2F
2
u
3
x
: (6.84)
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