5.3. THE MARTIN EQUATION 107
5.3 THE MARTIN EQUATION
e governing equations of an inviscid, one-dimensional nonsteady gas, neglecting heat condu-
tion and heat radiation are
t
C
.
u
/
x
D 0; (5.125a)
u
t
C uu
x
D
P
x
; (5.125b)
or, in conservative form
t
C
.
u
/
x
D 0; (5.126a)
.
u
/
C
u
2
C P
x
D 0: (5.126b)
Normally, a third equation is given, conservation of energy, or an equation of state, but here we
will leave the system underdetermined. If we introduce stream functions and
N
such that
D
x
; u D
t
; (5.127a)
u D
N
x
; u
2
C P D
N
t
; (5.127b)
then (5.126a) and (5.126b) are automatically satisfied. If we introduce a new variable such that
N
D tP; (5.128)
then (5.127b) becomes
u D
x
tP
x
; u
2
D
t
C tP
t
: (5.129)
Following Martin [74], we choose new independent variables . ; P /, so that (5.127a)
and (5.129) become
D
t
P
J
; (5.130a)
u D
x
P
J
; (5.130b)
u D
P
t
t
P
t t
J
; (5.130c)
u
2
D
P
x
x
P
tx
J
; (5.130d)
where J D t
x
P
t
P
x
. From which (5.130a) and (5.130b) we deduce
u D
x
P
t
P
: (5.131)
Eliminating and u from (5.130c) and (5.130d) using (5.130a) and (5.131) gives
x
P
D t t
C t
P
t
P
; (5.132a)
x
2
P
t
P
D tx
C x
P
x
P
; (5.132b)
108 5. FIRST INTEGRALS
from which we deduce
.
t
P
/
t
x
P
t
p
x
D 0: (5.133)
us,
t D
P
; (5.134)
as we are assuming that t
x
P
t
p
x
¤ 0. From (5.132) we obtain
x
P
D
PP
; (5.135)
and from (5.134) and (5.135) gives (5.131) as
u D
: (5.136)
We return to (5.130a) and with (5.134) this becomes
x
D
P
C ; (5.137)
where D
1
. Eliminating x from (5.135) and (5.137) gives the Monge–Ampere equation
PP
2
P
D
P
: (5.138)
With a renaming of variables, Martin [75] asked which forms of the Monge–Ampere equation
u
xx
u
yy
u
2
xy
C
2
D 0; D X.x/Y.y/; (5.139)
admit first integrals. He was able to deduce the following forms:
D Y .y/; D
x
m1
y
mC1
; (5.140)
where m is an arbitrary constant and Y .y/ an arbitrary function of its argument and identified
the special cases
D y
2
; D x
1
y
1
; D e
x
e
y
: (5.141)
We now consider this same problem, however, we will remove the assumption that is
separable. From the section on first integrals, if a first integral F.x; y; u; p; q/ exists, then F
satisfies
F
x
C pF
u
.x; y/F
q
D 0; (5.142a)
F
y
C qF
u
C .x; y/F
p
D 0:
(5.142b)
Requiring that these two be compatible gives rise to the additional equation
F
u
1
2
x
F
p
1
2
y
F
q
D 0: (5.143)
5.3. THE MARTIN EQUATION 109
If we further require that (5.142) and (5.143) be compatible, we obtain
xx
x
3
2
x
F
p
C
xy
x
3
2
y
F
q
D 0; (5.144a)
xy
y
3
2
x
F
p
C
yy
y
3
2
y
F
q
D 0: (5.144b)
Since F
p
F
q
¤ 0, from (5.144) we obtain
xx
x
3
2
x
yy
y
3
2
y
xy
x
3
2
y
xy
y
3
2
x
D 0; (5.145)
or
2
xx
3
2
x
2
yy
3
2
y
2
xy
3
x
y
2
D 0: (5.146)
Melshenko [76] also obtains (5.146) and states that this equation admits solutions of the form
D H
.
c
1
x C c
2
y
/
and D .y Cc
2
/
2
H
x C c
1
y C c
2
; (5.147)
for arbitrary constants c
1
; c
2
and arbitrary function H . Remarkably, under the substitution
D
1
!
2
; (5.148)
Eq. (5.146) becomes the homogeneous Monge–Ampere equation
!
xx
!
yy
!
2
xy
D 0; (5.149)
which is know to admit the solutions ([77])
!
D
f
.
c
1
x C c
2
y
/
C c
3
x C c
4
y C c
5
; (5.150)
for arbitrary constants c
1
c
5
and arbitrary function f ,
! D
.
c
1
x C c
2
y C c
3
/
f
c
4
x C c
5
y C c
6
c
1
x C c
2
y C c
3
C c
7
x C c
8
y C c
9
; (5.151)
with arbitrary constants c
1
c
9
and arbitrary function f , and the parametric solutions
! D tx C f .t/y Cg.t/; x C f
0
.t/y C g
0
.t/ D 0; (5.152)
with arbitrary functions f and g noting that (5.147) are special cases of (5.150) and (5.151).
One can also show that (5.149) also admits solutions of the form
! D
.
c
1
x C c
2
y C c
3
/
f
c
4
x C c
5
y C c
6
c
7
x C c
8
y C c
9
C c
10
x C c
11
y C c
12
; (5.153)
with arbitrary constants c
1
c
12
and arbitrary function f , but with the constraint that c
1
c
5
c
9
C
c
2
c
6
c
7
C c
3
c
4
c
8
c
1
c
6
c
8
c
2
c
4
c
9
c
3
c
5
c
7
D 0.
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