Ames W.F., Harrel E.M., Herod J.V. (editors). Differential Equations with Applications to Mathematical Physics
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250
M.
C.
Nucci
1.
automatic packages where you put in your equation and get the
answer without any interaction with the computer. Perhaps
the best example is SPDE by
F.
Schwarz
[25]
which runs with
REDUCE version
3.3
and higher
[la];
2.
interactive programs which require the user
to
make specific
choices at different stages of the computation. An example is
[13]
which also runs with REDUCE.
In the following we emphasize the differences between these two
methods.
For
this purpose we make
a
comparison
of
SPDE and
[13].
The former is able to find classical symmetries of many type
of
differential equations with the following exceptions:
0
equations with arbitrary functions
of
the unknown and its
derivatives,
as
utt
=
[f(u)uZ], [2];
0
overdetermined system
of
equations.
Unfortunately, because SPDE is not reliable all the time, interactive
programs must be used. In
[13]
such programs were developed. They
should be able to handle the above equations and also find their Lie-
Backlund symmetries
[4], [22],
[6],
but:
0
they require
a
good knowledge of LISP
[ll];
0
the non-expert user cannot modify them for his own needs (e.g.
finding the non-classical symmetries
151,
[IS]).
In an effort
to
overcome these problems, we have developed
[17]
easy
to
use interactive programs which do not require an in-depth knowl-
edge of LISP
or
REDUCE. In fact, to use them one only needs
to know
a
single LISP command and have
a
very basic familiar-
ity with REDUCE (version
3.3
or
higher).
The programs labelled
GA automatically construct the determining equations for the clas-
sical symmetries; with only minor modifications,
all
the other
pro-
grams are derived. They calculate the non-classical (SGA), and the
Lie-Backlund
(GS)
symmetries of any differential equation.
When
any of these programs is loaded, REDUCE will automatically run
it and construct the determining equations. At this point the user
Symmetries and Symbolic Computation
251
will begin
to
interact with the computer. In
[17]
we presented sev-
eral computer sessions for calculating classical, non-classical, and
Lie-Backlund symmetries of known equations, including the classical
symmetries of
utt
=
[f(u)uxIx.
Further applications can be found in
various articles
[3],
[18],
[19],
[20],
[21].
Here we show some other
results obtained by using
our
interactive REDUCE programs: the
classical symmetries
of
an overdetermined system in fluid mechanics,
the nonclassical symmetries
of
Burgers' equation, and the third
or-
der Lie-Backlund symmetries of sine-Gordon equation and nonlinear
reaction-diffusion equation
[7].
Although our programs were origi-
nally developed for
a
mainframe IBM
4381,
the following outputs
were obtained by running our programs on
a
SUN
SPARCstation I
of the School of Mathematics
at
Georgia Institute of Technology, At-
lanta (U.S.A). We underline the importance of
a
REDUCE-LaTeX
translator
[lo]
in reporting the results.
2
Classical Symmetries
of
an
Overdetermined System
The following system of
7
first order differential equations with
3
in-
dependent
(t,
5,
y)
and 6 dependent variables
(u,
v,
h, u',
v',
h')
mod-
els the motion of
two
shallow immiscible inviscid incompressible flu-
ids subject
to
the force of gravity and contained in
a
rigid basin
ro-
tating with the Earth. More details can be found in
U.
Ramgulam's
Ph.D. thesis supervised by
C.
Rogers at Loughborough University of
Technology, Loughborough (U.K.).
ut+uux+vuy-
fv+f(Z+h')
P
X
+(I-L)(Z+h)
P
X
=O
(1)
ht
+
(uh)
2
+
(Vh)Y
=
0
U:
+
U'U~
+
V'U&
-
fv'+
(2
+
h')x
=
0
(3)
(4)
252
M.
C.
Nucci
vi
+
u'v:
+
v'vb
+
f
u'
i-
(Z
+
h')
Y
=
0
hi
+
(u'h')
X
t
(v'h')y
=
0
(h'
-
h)
t
+
(u'(h'
-
h))
X
+
(v'(h'
-
h))
Y
=
0
(5)
(6)
(7)
Note that
z
=
Z(x,
y)
is the equation of the basin surface,
f
is the
constant Coriolis parameter,
(u,v)
and
(u',v')
the
x
and
y
compo-
nents of the velocity of the first and second fluid, respectively,
p
and
p'
the density of the first and second fluid, respectively,
h
and
h'
the vertical distance from the free surface to the basin of the first
and second fluid, respectively.
The classical symmetry analysis of
the partial differential equations which model the motion of
a
rotat-
ing shallow liquid in
a
rigid basin was performed in
[15]
by using
a
MACSYMA program
[8].
Here we perform the classical symmetry
analysis of
(1-7)
by using our interactive REDUCE program,
and
after having solved the system
(1-7)
algebraically for
ut,
vt,
ht,
ui,
vb,
v:,
and
hi,
i.e.:
p'
az
az
P
ax
P
ax
p'
a2
p'
az
P
aY
P
aY
Ut
=
-(u%
+
Vuy
-
fv+
-(-
+
h:)
+-
(1
-
f)
(-
+
h,))
(8)
Vt
=
-
(UVX
t
vvy
+
fu
+
-(-
+hi)
+
(1
-
-) (-
+
hy))
(9)
The classical symmetry analysis consists of looking for the Lie group
of infinitesimal transformation which leaves
(1-7)
invariant. We find
Symmetries and Symbolic Computation
253
that for
a
basin of general form
z
=
Z(Z,Y)
the generator of the Lie
group
is
given by the following operator:
Q
=
fiat
+
V2az
+
Gay
+
GI&
+
G2av
+
G3&
+
G4&
+
G5&
+
GG&’
(15)
with:
v,
=
a
(16)
(17)
(18)
G1
=
(j-px
-
-U
+
-Y
f
+
2-
+
2~~1+
~af
+
2~~2)/2 (19)
da
da
v2
=
((z
+
2Cl)Z
+
(af
+
2C2)Y
+
2P)/2
v3
=
((x
+
2Cl)Y
-
(af
+
2C2)Z
+
27)/2
d2a da da dP
dt dt dt
dt
Ga=
(=g--v-
d2a da
Zxf
da
+2--uaf
d7
-2uc2+2vc1)/2 (20)
G4
=
(wx
d2a
-
zu
da
I
+
-yf
da
+
2-
dP
+
221‘c1+
v‘af
+
2v’cz) /2 (22)
(21)
da
G3
=
-(-
-
2q)h
dt
dt dt
dy
2v‘c1)/2 (23)
G5
=
-((v’+~f)z+af+2~2u’-
da
-y-2--
d2a
dt2 dt
(24)
da
GG
=
-(-
-
2cl)h’
dt
where
c1
and
c2
are constants, whereas
a
=
a(t),
7
=
7(t)
are functions of time, subject to the constraints:
=
p(t), and
?f!)z
=
0
$y
=
0
(25)
da
az
with:
$
=
2[
(af
+
2C2)Z
-
-p
-
2yc1
-
271
Y
254
M.
C.
Nucci
3
Non-Classical Symmetries
of
Burgers’
Equation
The non-classical symmetry analysis
of
the Burgers’ equation
[l]:
Ut
=
Uxz
+
2121,
consists in adding another “restriction” on
u
given by the invariant
surface condition:
(27)
Vi(t,x,U)Ut
+
Vz(t,X,u)uz
-
G(t,x,u)
=
0
Vi
at
t
Vzaz
+
G&
(28)
(29)
which is associated with the symmetry generator, i.e.:
By using
our
interactive
REDUCE
programs, we obtain three cases
in addition to the classical one:
CASE
1
v,
=
1
15
=
-u
G=O (30)
CASE
2
v,
=
1;
v,
=
(.
+
24
12;
G
=
-
(2
+
2u2a
-
4up
-
47)
14
(31)
where
ct,p,y
are functions of
(t,z)
which must satisfy the Burgers-
heat system
[5]:
aa
a2a
aCY
ap
at
ax,
ax
dX
aa
ap
azp
ay
2-0
+
-
-
-
-
-
ax
at
ax2
ax
aa
a7
a27
ax
at
ax2
+
2-CY
+
2-
=
0
=o
2-7
+
-
-
-
=
0
CASE
3
(33)
(34)
v,
=
0
V,
=
1
G
=
G(~,x,u)
(35)
(36)
where
G
must satisfy:
2GG,,
+
G’G,,
t
G,,
+
uG,
-
Gt
+
G2
=
0
Symmetries and Symbolic Computation
255
4
Third Order Lie-Backlund Symmetries
4.1
Sine-Gordon Equation
Here we compute the third order Lie-Backlund symmetries
of
the
sine-Gordon equation:
The corresponding infinitesimal generator is given by:
uzt
=
sin(u)
(37)
(
38)
w,
t,
u,
Ut,
uz,
Utt,
uzz,
wit,
uzzz)&
By using
[17]
we obtain:
=
(2tutC4
-
2XUzC4
+
U:C2
+
2utC5
+
+2uzc3
+
2UtttC2
+
2uzzzc1)
/2
(39)
where
c;
(i
=
1,2,3,4)
are arbitrary constants. This Lie-Backlund
symmetry generator contains as particular cases those found by
Kumei
[14].
In
[22]
(Exercise
5.21,
pag.
372),
Olver noticed that the
explicit dependence on the independent variables is required if one
wants
to
generate
a
non-trivial conservation law
of
the sine-Gordon
equation.
4.2
Nonlinear Reaction-Diffusion Equation
Here we find the conditions under which third order Lie-Backlund
symmetries exist for nonlinear reaction-diffusion equation
of
the type:
ut
=
[H(u)~z]z
+
J(u)uz
+
l((x,t,u)
w,
t,
U,
uz,
Uzz,
uzzz)&
(40)
(41)
The symmetry generator
is
given by:
The third order Lie-Backlund symmetries of
(39)
were found in
[7]
in the case where
J'
=
0.
By using
[17]
we find that third order
Lie-Backlund symmetries
of
(39)
exist if
H,
J,
Ii'
are
of
the following
form:
2
H
=
1/(uA1
+
A2)
(42)
256
M.
C.
Nucci
J
=
((.A1
+
A2)'A4
+
A3)/(UA1+ Az)~
(43)
da da
K
=
-(2a2/3eA3(A4t+x)A3~2
-
-uA1
dt
-
-A2)/(3aA1)
dt
(44)
where
A;
(i
=
1,2,3,4)
and
c2
are constants, and
a
=
a(t)
is
a
function
of
time. The symmetry operator
(40)
is given by:
Symmetries
and
Symbolic
Computation
257
258
M.
C.
Nucci
+~~$~Gu,u,,cY*A:A~
-
12
GuXC~~A~A~ A:
-G$~Gu,cY~A~A~A~
-
~$‘ZU,QA~A:C~
-18fi~,,~~~A3AlAi
-
6*u,,~ypAlAi
5
-
G~~,,,~~A~A~)/(G~(uA~
+
A~)
a~~)
(45)
where
c1
is
a
constant and
/3
=
P(t)
is
a
function of time which must
satisfv:
dcr dP
2-/3
-
3-CY
-
~cYA~c~
=
0
dt
dt
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