73
C H A P T E R 4
Point and Contact
Transformations
In an introductory course in PDEs we found that by introducing new variables
x D f .r; s/; y D g.r; s/;
@.x; y/
@.r; s/
¤ 0; (4.1)
we were able to transform PDEs of the form
Au
xx
C Bu
xy
C C u
yy
C l.o.t.s D 0; (4.2)
where A; B and C are functions of .x; y/ and l.o.t.s are lower-order terms to standard form, i.e.,
u
ss
C l.o.t.s D 0 parabolic;
u
rs
C l.o.t.s D 0 modified hyperbolic; (4.3)
u
rr
C u
ss
C l.o.t.s D 0 elliptic:
We now ask, can we generalize these transformation? For example, can we consider transfor-
mations of the form
x D F .X; Y; U /; y D G.X; Y; U /; u D H.X; Y; U /; (4.4)
where U D U.X; Y /? Transformations of this type are referred to as point transformations. Con-
sider, for example, the transformation
x D X; y D U; u D Y: (4.5)
In order to see the effect on a PDE, we will need to calculate how derivatives transform. e
easiest way is using Jacobians. For example, the derivative u
x
transforms as
u
x
D
@.u; y/
@.x; y/
D
@.u; y/
@.X; Y /
@.x; y/
@.X; Y /
D
@.Y; U /
@.X; Y /
@.X; U /
@.X; Y /
D
U
X
U
Y
; (4.6)
noting that we have used the transformation (4.5) in (4.6). Similar for u
y
,
u
y
D
@.x; u/
@.x; y/
D
@.x; u/
@.X; Y /
@.x; y/
@.X; Y /
D
@.X; Y /
@.X; Y /
@.X; U /
@.X; Y /
D
1
U
Y
: (4.7)
74 4. POINT AND CONTACT TRANSFORMATIONS
e process easily extends to second-order derivatives. For example, u
xx
transforms as
u
xx
D
@.u
x
; y/
@.x; y/
D
@.u
x
; y/
@.X; Y /
@.x; y/
@.X; Y /
D
@.
U
Y
U
X
; U /
@.X; Y /
@.X; U /
@.X; Y /
D
U
2
Y
U
XX
2U
X
U
Y
U
XY
C U
2
X
U
Y Y
U
3
Y
; (4.8)
while u
xy
and u
yy
transform as
u
xy
D
@.u
y
; y/
@.x; y/
D
@.u
y
; y/
@.X; Y /
@.x; y/
@.X; Y /
D
@.
1
U
Y
; U /
@.X; Y /
@.X; U /
@.X; Y /
D
U
X
U
Y Y
U
Y
U
XY
U
3
Y
; (4.9)
and
u
yy
D
@.x; u
y
/
@.x; y/
D
@.x; u
y
/
@.X; Y /
@.x; y/
@.X; Y /
D
@.X;
1
U
Y
/
@.X; Y /
@.X; U /
@.X; Y /
D
U
Y Y
U
3
Y
: (4.10)
us, under the transformation (4.5), the first- and second-order derivatives transform as
u
x
D
U
X
U
Y
; u
y
D
1
U
Y
; (4.11)
u
xx
D
U
2
Y
U
XX
2U
X
U
Y
U
XY
C U
2
X
U
Y Y
U
3
Y
; u
xy
D
U
X
U
Y Y
U
Y
U
XY
U
3
Y
; u
yy
D
U
Y Y
U
3
Y
:
We now ask—why would this be useful? Consider, for example, the heat equation
u
xx
D u
y
: (4.12)
Under (4.5) (using the derivatives obtained in (4.11)), (4.12) would become
U
2
Y
U
XX
2U
X
U
Y
U
XY
C U
2
X
U
Y Y
U
3
Y
D
1
U
Y
;
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