4.4. EXERCISES 87
Catalan’s Surface
By choosing
W D 4 sinh a sin b 2a cosh a sin b 2b sinh a cos b; (4.118)
as a solution of Laplace’s equation, (4.106) gives
x D 2b cosh 2a sin 2b;
y D 1 cosh 2a cos 2b; (4.119)
u D 4 sinh a sin b;
known as Catalan’s minimal surface [68].
Bour’s Surface
By choosing
W D
1
2
.
cosh 3a Csinh 3a
/
cos 3b; (4.120)
as a solution of Laplace’s equation gives, from (4.106)
x D e
2a
cos 2b
e
4a
2
cos 4b;
y D e
2a
sin 2b
e
4a
2
sin 4b; (4.121)
u D
4
3
e
3a
cos 3b:
Identifying that r D e
2a
and b D =2 gives rise to
x D r cos
r
2
2
cos 2;
y D r sin
r
2
2
sin 2; (4.122)
u D
4
3
r
4=3
cos
3
2
;
the minimal surface known as Bour’s surface [68].
4.4 EXERCISES
4.1. Show the following are contact transformations:
.i/ x D X C Y C U
X
; y D X C 3U
X
; u D 2XU
X
2U CX
2
C Y
2
;
.i i/ x D e
Y
; y D U; u D X
2
e
Y
;
.i i i/ x D Y C
2U
X
U
Y
; y D U; u D e
X
U
2
X
U
2
Y
;
.iv/ x D U
X
; y D
1
U
Y
; u D
U XU
X
Y U
Y
U
Y
:
88 4. POINT AND CONTACT TRANSFORMATIONS
4.2. Given X; Y; P and Q, show a U exists such that the contact conditions are satisfied;
further, find U
.i/ X D u; Y D y
xp
q
; P D
x
q
; Q D x;
.i i/ X D x 2p; Y D y; P D p; Q D q;
.i i i/ X D q; Y D y
u
q
; P D
u
pq
; Q D
q
p
;
.iv/ X D x C
1
p
; Y D y C
1
q
; P D A.p; q/; Q D B.p; q/:
In part (iv), you will need to determine the forms A and B.
4.3. Show that the following is a contact transformation:
t D 2T; x D
2U .T C X/u
X
2.T C X/
; u D
U .T C X/U
X
2.T C X/
2
: (4.123)
Further, show that the Hunter–Saxon equation [65]
u
tx
C uu
xx
C
1
2
u
2
x
D 0; (4.124)
is transformed to [66]
U
TX
D
2.U
T
C U
X
/
T C X
4U
.T C X/
2
: (4.125)
4.4. e boundary Layer equations from fluid mechanics are
u
x
C v
y
D 0; (4.126a)
uu
x
C vu
y
D u
yy
: (4.126b)
4 .i / Show that (4.126b) can be written as
u
2
x
C
uv u
y
y
D 0: (4.127)
4 .i i/ Show that with the introduction of the stream functions D .x; y/ and D
.x; y/ such that
u D
y
; v D
x
; u
2
D
y
; uv u
y
D
x
; (4.128)
4.4. EXERCISES 89
(4.126a) and (4.127) are automatically satisfied. Furthermore, from (4.128), and
satisfy
2
y
D
y
; (4.129a)
yy
C
x
y
D
x
: (4.129b)
4 .i ii/ Show that under the Hodograph transformation,
x D X; y D ‰; D Y; D ˆ; (4.130)
system (4.129) becomes
ˆ
Y
D
1
‰
Y
; ˆ
X
D
‰
Y Y
‰
3
Y
; (4.131)
which, on eliminating ˆ, gives
‰
Y
‰
XY
‰
X
‰
Y Y
2‰
2
Y Y
‰
3
Y
C
‰
Y Y
‰
2
Y
D 0; (4.132)
or
1
‰
Y
X
D
‰
Y Y
‰
3
Y
Y
: (4.133)
Under the Hodograph transformation (4.130), u becomes
u D
1
‰
Y
; (4.134)
giving (4.133) as
u
X
D
.
uu
Y
/
Y
(4.135)
a nonlinear diffusion equation! is has been noted in Ames [67].
4 .iv/ Show under the Hodograph transformation,
x D X; y D ˆ; D ‰; D Y; (4.136)
system (4.129) becomes
‰
2
Y
ˆ
2
Y
1
ˆ
Y
D 0;
‰
Y Y
ˆ
2
Y
‰
Y
ˆ
Y Y
ˆ
3
Y
C
‰
X
‰
Y
ˆ
Y
‰
2
Y
ˆ
X
ˆ
2
Y
C
ˆ
X
ˆ
Y
D 0; (4.137)
which on simplifying gives
‰
2
Y
ˆ
Y
D 0; ‰
X
‰
Y Y
‰
3
Y
D 0: (4.138)
90 4. POINT AND CONTACT TRANSFORMATIONS
Differentiating the second in (4.138) gives
.
‰
Y
/
X
D
‰
Y Y
‰
3
Y
Y
: (4.139)
Show that under the Hodograph transformation (4.136) u becomes
u
D
1
‰
Y
; (4.140)
giving (4.139) as
u
X
D
u
Y
u
3
Y
; (4.141)
where u
D u
1
, another nonlinear diffusion equation! Rogers and Shadwick have
shown that different nonlinear diffusion equations can be linked via reciprocal trans-
formations [71].
4.5. e Harry–Dym equation is given by
u
t
D u
3
u
xxx
:
Show that by introducing the change of variable u D v
1=2
, this equation becomes (to
within scaling of t)
v
t
D
v
x
v
3=2
xx
;
which, under the substitution v D w
x
, becomes
w
t
D
w
xx
w
3=2
x
!
x
: (4.142)
Show under a Hodograph transformation, (4.142) remains unchanged.
4.6. e equations for irrotational steady two-dimensional flows in fluid mechanics are given
by
.u/
x
C .v/
y
D 0; (4.143a)
uu
x
C vu
y
D
p
x
; (4.143b)
uv
x
C vv
y
D
p
y
; (4.143c)
r Ev D 0: (4.143d)
4.4. EXERCISES 91
(i) Show that for 2 - D flows, (4.143d) is u
y
v
x
D 0, and is identically satisfied by
introducing a potential u D
x
and v D
y
.
(ii) Assuming that p D c
2
where c is a constant speed of sound, show that the system
reduces to the single equation
2
x
c
2
xx
C 2
x
y
xy
C
2
y
c
2
yy
D 0: (4.144)
(iii) Show that under a Legendre transformation this equation linearizes.
4.7. e governing equations for highly frictional granular materials is given by
@
xx
@x
C
@
xy
@y
D 0; (4.145a)
@
xy
@x
C
@
yy
@y
D g; (4.145b)
xx
yy
2
xy
D 0; (4.145c)
where
xx
,
yy
and
xy
are normal and shear stresses, respectively, the material
density, and g, the gravity. In (4.145), (4.145a) and (4.145b) are the equilibrium
equations, whereas (4.145c) is the constitutive relation for the material [69].
(i) Show that by introducing a potential u, such that
xx
D u
y
;
xy
D u
x
; then
yy
D
u
2
x
u
y
; (4.146)
Eq. (4.145) reduces to
u
2
y
u
xx
2u
x
u
y
u
xy
C u
2
x
u
yy
C gu
2
y
D 0: (4.147)
(ii) Show that under a Hodograph transformation, this linearizes.
4.8. e shallow water wave equations are given by
u
t
C uu
x
C g
x
D 0; (4.148a)
t
C
Œ
u. h.x//
x
D 0; (4.148b)
where u D u.x; t / the velocity, D .x; t/ the free surface, h D h.x/ the bottom
surface, and g is gravity [70].
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