49
C H A P T E R 3
Differential Substitutions
One of the simplest NLPDEs is Burgers’ equation
u
t
C uu
x
D u
xx
: (3.1)
e PDE was introduced by Burgers in 1934 as a rough model for turbulence. is equation
has the steepening effect of the NPDE
u
t
C uu
x
D 0; (3.2)
and the diffusive effect of the heat equation
u
t
D u
xx
: (3.3)
Here, we first consider a related PDE–potential Burgers’ equation. If we let
u D v
x
; (3.4)
then (3.1) can be integrated once, giving
v
t
C
1
2
v
2
x
D v
xx
: (3.5)
Note that the function of integration can be set to zero with outloss of generality. A remarkable
fact is that this nonlinear PDE (3.5) can be linearized. If we let
v D f .w/; (3.6)
then (3.5) becomes
f
0
.w/w
t
C
1
2
f
02
.w/w
2
x
D f
0
.w/w
xx
C f
00
.w/w
2
x
: (3.7)
Choosing
f
00
D
1
2
f
02
(3.8)
gives (3.7) as the heat equation
w
t
D w
xx
Š (3.9)
50 3. DIFFERENTIAL SUBSTITUTIONS
Solving (3.8) gives
f D 2 ln jw C c
1
j C c
2
; (3.10)
where c
1
and c
2
are constants of integration. Setting c
1
D c
2
D 0 then f D 2 ln jwj and from
(3.6)
v D 2 ln jwj; (3.11)
transforming the potential Burgers’ equation (3.5) to (3.9), the heat equation. Now we return
to the actual Burgers’ equation (3.1). Combining (3.4) and (3.11) tells us that Burgers’ equation
and the heat equation are related via
u D 2
w
x
w
: (3.12)
is transformation is known as the Hopf–Cole transformation, introduced independently by
Hopf [53] and Cole [54], but appeared earlier in Forsythe [55]. If we start with Burgers’ equation
(3.1), we can see that it is transformed to the heat equation through the Hope–Cole transfor-
mation (3.12). e following illustrates:
u
t
D
u
x
1
2
u
2
x
2
w
x
w
t
D
2
w
xx
w
C
2
w
2
x
w
2
1
2
2
w
x
w
2
!
x
w
x
w
t
D
w
xx
w
x
(3.13)
w
t
w
x
D
w
xx
w
w
t
D w
xx
C c
0
.t/w;
where c
0
.t/ is arbitrary. However, we can set c
0
.t/ D 0 as introducing the new variable
Qw D e
c.t/
w, gives the last of (3.13) with c
0
.t/ D 0 and further leaves (3.12) unchanged. We
now consider two examples.
Example 3.1 Solve the initial value problem
u
t
C uu
x
D u
xx
; 0 < x < ;
u.0; t / D u.; t/ D 0; u.x; 0/ D sin x:
(3.14)
Passing the PDE, the boundary conditions, and the initial condition through the Hopf–Cole
transformation (3.12) gives
w
t
D w
xx
w
x
.0; t / D w
x
.; t/ D 0 (3.15)
w.t; 0/ D exp
1
2
cos x
:
3. DIFFERENTIAL SUBSTITUTIONS 51
With the usual separation of variables, we find the solution of (3.15)
w D
1
2
a
0
C
1
X
nD1
a
n
e
n
2
t
cos nx; (3.16)
where
a
n
D
2
Z
0
exp
1
2
cos x
cos nx dx D 2I
n
1
2
(3.17)
and I
n
.x/ is the modified Bessel function of the first kind. Passing (3.16) through the Hopf–
Cole transformation (3.12) gives
u
D
4
1
X
nD1
nI
n
1
2
e
n
2
t
sin nx
I
0
.
1
2
/ C 2
1
X
nD1
I
n
1
2
e
n
2
t
cos nx
: (3.18)
If we solve the heat equation with the same boundary and initial conditions as given in (3.14),
we obtain the simple solution
u D e
t
sin x: (3.19)
Figure 3.1 shows a comparison of the solutions (3.18) and (3.19) at a variety of times.
Example 3.2 Solve the initial value problem
u
t
C uu
x
D u
xx
; 1 < x < 1; u.t; 0/ D f .x/ (3.20)
Passing the initial condition through the Hopf–Cole transformation gives
w.t; 0/ D W .x/ D exp
1
2
Z
x
0
f ./ d
: (3.21)
e solution of the heat equation (3.9) with this initial condition is
w.x; t/ D
1
p
4t
Z
1
1
W ./ e
.x/
2
=4t
d (3.22)
from which we obtain
w
x
.x; t/ D
1
p
4t
Z
1
1
.x /
2t
W ./e
.x/
2
=4t
d : (3.23)
From (3.12) we find the following:
u.x; t/ D
Z
1
1
.x /
2t
W ./e
.x/
2
=4=t
d
Z
1
1
W ./e
.x/
2
=4=t
d
: (3.24)
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