4.1. CONTACT TRANSFORMATIONS 75
or
U
2
Y
U
XX
2U
X
U
Y
U
XY
C U
2
X
U
Y Y
C U
2
Y
D 0:
So one may ask—what was the point? We clearly made the problem more difficult. However,
suppose we started with the NLPDE
u
2
y
u
xx
2u
x
u
y
u
xy
C u
2
x
u
yy
C u
2
y
D 0; (4.13)
a PDE given in (3.67) (with c.v/ D 1), then under (4.5), (4.13) becomes
U
XX
D U
Y
;
the linear heat equation!
4.1 CONTACT TRANSFORMATIONS
We now extend transformations to include first derivatives
x D F .X; Y; U; P; Q/; y D G.X; Y; U; P; Q/; u D H.X; Y; U; P; Q/; (4.14)
where U D U.X; Y /, P D U
X
and Q D U
Y
. We also need the relation that
u
x
D M.X; Y; U; P; Q/; u
y
D N.X; Y; U; P; Q/; (4.15)
and impose the condition that (4.15) cannot have higher order derivatives. Before talking about
condition which will ensure this, we consider three very famous contact transformations:
1. the Hodograph transformation,
2. the Legendre transformation, and
3. the Ampere transformation.
4.1.1 HODOGRAPH TRANSFORMATION
ese transformations are of the form
x D X; y D U; u D Y; (4.16)
or
x D U; y D Y; u D X: (4.17)
e first transformation (4.16) was previously given in (4.5). For the second transformation
(4.5), first-order derivatives transform as
u
x
D
1
U
X
; u
y
D
U
Y
U
X
; (4.18)
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.