Sun Jan 3 23:54:59 MET 1999
REDUCE 3.7, 15-Jan-99 ...
1: 1:
2: 2: 2: 2: 2: 2: 2: 2: 2:
3: 3: off echo, dfprint$
If you want to see more details of the following calculation then
change in the file `conlaw.tst' the line
lisp(print_:=nil)$
into
lisp(print_:=10)$
**************************************************************************
The following example calculates all conservation laws of the KdV-
equation with a characteristic function of order not higher than two
--------------------------------------------------------------------------
This is CONLAW3 - a program for calculating conservation laws of DEs
The DE under investigation is :
df(u,t)=df(u,x,3) + df(u,x)*u
for the function(s): {u}
======================================================
Currently conservation laws with characteristic
function(s) of order 0 are determined
======================================================
Conservation law:
( 1 ) * ( df(u,t) - df(u,x,3) - df(u,x)*u )
=
df( u, t )
+
2
- 2*df(u,x,2) - u
df( ---------------------, x )
2
======================================================
Conservation law:
( t*u + x ) * ( df(u,t) - df(u,x,3) - df(u,x)*u )
=
2
t*u + 2*u*x
df( --------------, t )
2
+
2 3
df( ( - 6*df(u,x,2)*t*u - 6*df(u,x,2)*x + 3*df(u,x) *t + 6*df(u,x) - 2*t*u
2
- 3*u *x)/6, x )
======================================================
Conservation law:
( u ) * ( df(u,t) - df(u,x,3) - df(u,x)*u )
=
2
u
df( ----, t )
2
+
2 3
- 6*df(u,x,2)*u + 3*df(u,x) - 2*u
df( --------------------------------------, x )
6
======================================================
Currently conservation laws with characteristic
function(s) of order 1 are determined
======================================================
There is no conservation law of this order.
======================================================
Currently conservation laws with characteristic
function(s) of order 2 are determined
======================================================
Conservation law:
2
( - 2*df(u,x,2) - u ) * ( df(u,t) - df(u,x,3) - df(u,x)*u )
=
2 3
3*df(u,x) - u
df( -----------------, t )
3
+
2 2 4
- 8*df(u,t)*df(u,x) + 4*df(u,x,2) + 4*df(u,x,2)*u + u
df( -----------------------------------------------------------, x )
4
======================================================
**************************************************************************
The next example demonstrates that one can specify an ansatz
for the characteristic function of one or more equations of the
PDE-system. In this example all conservation laws of the wave
equation which is written as a first order system are calculated
such that the characteristic functions of the first of both
equations is proportional to df(u,x,2). (This will include zero
as it is a multiple of df(u,x,2) too.)
--------------------------------------------------------------------------
This is CONLAW2 - a program for calculating conservation laws of DEs
The DEs under investigation are :
df(u,t)=df(v,x)
df(v,t)=df(u,x)
for the function(s): {u,v}
======================================================
A special ansatz of order 2 for the characteristic
function(s) is investigated.
Conservation law:
(df(u,x,2)) * (df(u,t) - df(v,x))
+
(df(v,x,2)) * ( - df(u,x) + df(v,t))
=
2 2
- df(u,x) - df(v,x)
df( ------------------------, t )
2
+
df( df(u,t)*df(u,x) - df(u,x)*df(v,x) + df(v,t)*df(v,x), x )
======================================================
**************************************************************************
For the Burgers equation the following example finds all conservation
laws of zero'th order in the characteristic function up to the solution
of the linear heat equation. This is an example for what happens when not
all conditions could be solved, but it is also an example which shows
that not only characteristic functions of polynomial or rational form
can be found.
--------------------------------------------------------------------------
This is CONLAW1 - a program for calculating conservation laws of DEs
The DE under investigation is :
2
2*df(u,x,2) + df(u,x)
df(u,t)=------------------------
2
for the function(s): {u}
======================================================
Currently conservation laws with a conserved density
of order 0 are determined
======================================================
The function c_132(x,t) is not constant!
There are remaining conditions: {df(c_132,t) + df(c_132,x,2)}
for the functions: c_132(x,t)
Corresponding CLs might not be shown below as they
could be of too low order.
======================================================
Conservation law:
2
u/2 2*df(u,t) - 2*df(u,x,2) - df(u,x)
( e *c_132 ) * ( ------------------------------------ )
2
=
u/2
df( 2*e *c_132, t )
+
u/2 u/2
df( 2*e *df(c_132,x) - e *df(u,x)*c_132, x )
======================================================
**************************************************************************
In this example all conservation laws of the Ito system are calculated
that have a conserved density of order not higher than one.
This is a further example of non-polynomial conservation laws.
--------------------------------------------------------------------------
This is CONLAW1 - a program for calculating conservation laws of DEs
The DEs under investigation are :
df(u,t)=df(u,x,3) + 6*df(u,x)*u + 2*df(v,x)*v
df(v,t)=2*df(u,x)*v + 2*df(v,x)*u
for the function(s): {u,v}
======================================================
Currently conservation laws with a conserved density
of order 0 are determined
======================================================
Conservation law:
( 0 ) * ( - 2*df(u,x)*v + df(v,t) - 2*df(v,x)*u )
+
( 1 ) * ( df(u,t) - df(u,x,3) - 6*df(u,x)*u - 2*df(v,x)*v )
=
df( u, t )
+
2 2
df( - df(u,x,2) - 3*u - v , x )
======================================================
Conservation law:
( 2*v ) * ( - 2*df(u,x)*v + df(v,t) - 2*df(v,x)*u )
+
( 2*u ) * ( df(u,t) - df(u,x,3) - 6*df(u,x)*u - 2*df(v,x)*v )
=
2 2
df( u + v , t )
+
2 3 2
df( - 2*df(u,x,2)*u + df(u,x) - 4*u - 4*u*v , x )
======================================================
Conservation law:
( -1 ) * ( - 2*df(u,x)*v + df(v,t) - 2*df(v,x)*u )
+
( 0 ) * ( df(u,t) - df(u,x,3) - 6*df(u,x)*u - 2*df(v,x)*v )
=
df( - v, t )
+
df( 2*u*v, x )
======================================================
Currently conservation laws with a conserved density
of order 1 are determined
======================================================
Conservation law:
2 2
- 2*df(v,x,2)*v + 3*df(v,x) + 4*u*v
( ---------------------------------------- ) * ( - 2*df(u,x)*v + df(v,t)
4
v
- 2*df(v,x)*u )
+
- 4
( ------ ) * ( df(u,t) - df(u,x,3) - 6*df(u,x)*u - 2*df(v,x)*v )
v
=
2 2
df(v,x) - 4*u*v
df( -------------------, t )
3
v
+
2 2
df( (4*df(u,x,2)*v + 4*df(u,x)*df(v,x)*v - 2*df(v,t)*df(v,x) + 2*df(v,x) *u
2 2 4 3
+ 8*u *v + 8*v )/v , x )
======================================================
Conservation law:
( - 4*u*v ) * ( - 2*df(u,x)*v + df(v,t) - 2*df(v,x)*u )
+
2 2
( - 2*df(u,x,2) - 6*u - 2*v ) * ( df(u,t) - df(u,x,3) - 6*df(u,x)*u
- 2*df(v,x)*v )
=
2 3 2
df( df(u,x) - 2*u - 2*u*v , t )
+
2 2 2 4
df( - 2*df(u,t)*df(u,x) + df(u,x,2) + 6*df(u,x,2)*u + 2*df(u,x,2)*v + 9*u
2 2 4
+ 10*u *v + v , x )
======================================================
**************************************************************************
In the next example the 5th order Korteweg - de Vries equation is
investigated concerning conservation laws of order 0 and 1 in the
conserved density P_t. Parameters a,b,c in the PDE are determined
such that conservation laws exist. This complicates the problem by
making it non-linear with a number of cases to be considered.
Some of the subcases below can be combined to reduce their number
which currently is not done automatically.
--------------------------------------------------------------------------
This is CONLAW1 - a program for calculating conservation laws of DEs
The DE under investigation is :
2
df(u,t)= - df(u,x,5) - df(u,x,3)*c*u - df(u,x,2)*df(u,x)*b - df(u,x)*a*u
for the function(s): {u}
======================================================
Currently conservation laws with a conserved density
of order 0 are determined
======================================================
Conservation law:
b
c=---,
2
( - 4*u ) * ( (2*df(u,t) + 2*df(u,x,5) + df(u,x,3)*b*u + 2*df(u,x,2)*df(u,x)*b
2
+ 2*df(u,x)*a*u )/2 )
=
2
df( - 2*u , t )
+
2 2
df( - 4*df(u,x,4)*u + 4*df(u,x,3)*df(u,x) - 2*df(u,x,2) - 2*df(u,x,2)*b*u
4
- a*u , x )
======================================================
Conservation law:
b
c=---,
2
( -12 ) * ( (2*df(u,t) + 2*df(u,x,5) + df(u,x,3)*b*u + 2*df(u,x,2)*df(u,x)*b
2
+ 2*df(u,x)*a*u )/2 )
=
df( - 12*u, t )
+
2 3
df( - 12*df(u,x,4) - 6*df(u,x,2)*b*u - 3*df(u,x) *b - 4*a*u , x )
======================================================
Conservation law:
( -6 ) * ( df(u,t) + df(u,x,5) + df(u,x,3)*c*u + df(u,x,2)*df(u,x)*b
2
+ df(u,x)*a*u )
=
df( - 6*u, t )
+
2 2 3
df( - 6*df(u,x,4) - 6*df(u,x,2)*c*u - 3*df(u,x) *b + 3*df(u,x) *c - 2*a*u , x )
======================================================
The function c_320(x,t) is not constant!
The function c_328(t) is not constant!
There are remaining conditions: {df(c_320,t) + df(c_320,x,5) - c_328}
for the functions: c_328(t), c_320(x,t)
Corresponding CLs might not be shown below as they
could be of too low order.
======================================================
Conservation law:
a=0,
b=0,
c=0,
( - 2*u ) * ( df(u,t) + df(u,x,5) )
=
2
df( - u , t )
+
2
df( - 2*df(u,x,4)*u + 2*df(u,x,3)*df(u,x) - df(u,x,2) , x )
======================================================
Conservation law:
a=0,
b=0,
c=0,
( - df(c_320,x) ) * ( df(u,t) + df(u,x,5) )
=
df( - df(c_320,x)*u, t )
+
df( df(c_320,t)*u + df(c_320,x,4)*df(u,x) - df(c_320,x,3)*df(u,x,2)
+ df(c_320,x,2)*df(u,x,3) - df(c_320,x)*df(u,x,4) + c_328*u, x )
======================================================
The function c_303(x,t) is not constant!
The function c_313(t) is not constant!
There are remaining conditions: {df(c_303,t) + df(c_303,x,5) - c_313}
for the functions: c_313(t), c_303(x,t)
Corresponding CLs might not be shown below as they
could be of too low order.
======================================================
Conservation law:
a=0,
b=0,
c=0,
( - df(c_303,x) ) * ( df(u,t) + df(u,x,5) )
=
df( - df(c_303,x)*u, t )
+
df( df(c_303,t)*u + df(c_303,x,4)*df(u,x) - df(c_303,x,3)*df(u,x,2)
+ df(c_303,x,2)*df(u,x,3) - df(c_303,x)*df(u,x,4) + c_313*u, x )
======================================================
Conservation law:
a=0,
b
c=---,
3
2
- 3*x
( --------- ) * (
b
3*df(u,t) + 3*df(u,x,5) + df(u,x,3)*b*u + 3*df(u,x,2)*df(u,x)*b
----------------------------------------------------------------- )
3
=
2
- 3*u*x
df( -----------, t )
b
+
2 2
df( ( - 3*df(u,x,4)*x + 6*df(u,x,3)*x - df(u,x,2)*b*u*x - 6*df(u,x,2)
2 2 2
- df(u,x) *b*x + 2*df(u,x)*b*u*x - b*u )/b, x )
======================================================
Conservation law:
a=0,
b
c=---,
3
3*df(u,t) + 3*df(u,x,5) + df(u,x,3)*b*u + 3*df(u,x,2)*df(u,x)*b
( - 3*x ) * ( -----------------------------------------------------------------
3
)
=
df( - 3*u*x, t )
+
2
df( - 3*df(u,x,4)*x + 3*df(u,x,3) - df(u,x,2)*b*u*x - df(u,x) *b*x
+ df(u,x)*b*u, x )
======================================================
Conservation law:
a=0,
b
c=---,
3
3*df(u,t) + 3*df(u,x,5) + df(u,x,3)*b*u + 3*df(u,x,2)*df(u,x)*b
( -3 ) * ( ----------------------------------------------------------------- )
3
=
df( - 3*u, t )
+
2
df( - 3*df(u,x,4) - df(u,x,2)*b*u - df(u,x) *b, x )
======================================================
Currently conservation laws with a conserved density
of order 1 are determined
======================================================
The function c_408(t) is not constant!
The function c_372(t,x) is not constant!
There are remaining conditions: {30*df(c_372,t) + 30*df(c_372,x,5) - c_408}
for the functions: c_372(t,x), c_408(t)
Corresponding CLs might not be shown below as they
could be of too low order.
======================================================
Conservation law:
a=0,
b=0,
c=0,
( - 2*df(u,x,2) ) * ( df(u,t) + df(u,x,5) )
=
2
df( df(u,x) , t )
+
2
df( - 2*df(u,t)*df(u,x) - 2*df(u,x,4)*df(u,x,2) + df(u,x,3)
2 2 2
- 2*df(u,x,3)*df(u,x)*c*u - 2*df(u,x,2)*df(u,x) *b - 2*df(u,x) *a*u , x )
======================================================
Conservation law:
a=0,
b
c=---,
3
2
( - 18*df(u,x,2) - 3*b*u ) * (
3*df(u,t) + 3*df(u,x,5) + df(u,x,3)*b*u + 3*df(u,x,2)*df(u,x)*b
----------------------------------------------------------------- )
3
=
2 3
df( 9*df(u,x) - b*u , t )
+
2
df( - 18*df(u,t)*df(u,x) - 18*df(u,x,4)*df(u,x,2) - 3*df(u,x,4)*b*u
2
+ 9*df(u,x,3) + 12*df(u,x,3)*df(u,x)*b*u - 18*df(u,x,3)*df(u,x)*c*u
2 2 2 3
- 6*df(u,x,2) *b*u - 6*df(u,x,2)*df(u,x) *b - df(u,x,2)*b *u
2 2
- 18*df(u,x) *a*u , x )
======================================================
Conservation law:
a=0,
b
c=---,
3
3*df(u,t) + 3*df(u,x,5) + df(u,x,3)*b*u + 3*df(u,x,2)*df(u,x)*b
( 6*x ) * ( ----------------------------------------------------------------- )
3
=
2
df( - 3*df(u,x)*x , t )
+
2 2 2
df( 3*df(u,t)*x + 6*df(u,x,4)*x - df(u,x,3)*b*u*x + 3*df(u,x,3)*c*u*x
2 2 2
- 6*df(u,x,3) + 2*df(u,x,2)*b*u*x + 2*df(u,x) *b*x + 3*df(u,x)*a*u *x
- 2*df(u,x)*b*u, x )
======================================================
Conservation law:
a=0,
b
c=---,
3
2
9*x 3*df(u,t) + 3*df(u,x,5) + df(u,x,3)*b*u + 3*df(u,x,2)*df(u,x)*b
( ------ ) * ( -----------------------------------------------------------------
b 3
)
=
3
- 3*df(u,x)*x
df( -----------------, t )
b
+
3 2 3 3
df( (3*df(u,t)*x + 9*df(u,x,4)*x - df(u,x,3)*b*u*x + 3*df(u,x,3)*c*u*x
2 2 2
- 18*df(u,x,3)*x + 3*df(u,x,2)*b*u*x + 18*df(u,x,2) + 3*df(u,x) *b*x
2 3 2
+ 3*df(u,x)*a*u *x - 6*df(u,x)*b*u*x + 3*b*u )/b, x )
======================================================
Conservation law:
2 2
- 2*b + 7*b*c - 3*c
a=------------------------,
10
2 2
( - 1500*df(u,x,2) - 300*b*u + 150*c*u ) * ( (10*df(u,t) + 10*df(u,x,5)
2 2
+ 10*df(u,x,3)*c*u + 10*df(u,x,2)*df(u,x)*b - 2*df(u,x)*b *u
2 2 2
+ 7*df(u,x)*b*c*u - 3*df(u,x)*c *u )/10 )
=
2 3 3
df( 750*df(u,x) - 100*b*u + 50*c*u , t )
+
2
df( - 1500*df(u,t)*df(u,x) - 1500*df(u,x,4)*df(u,x,2) - 300*df(u,x,4)*b*u
2 2
+ 150*df(u,x,4)*c*u + 750*df(u,x,3) + 600*df(u,x,3)*df(u,x)*b*u
2 2
- 300*df(u,x,3)*df(u,x)*c*u - 300*df(u,x,2) *b*u - 600*df(u,x,2) *c*u
2 2 3
- 600*df(u,x,2)*df(u,x) *b + 300*df(u,x,2)*df(u,x) *c - 300*df(u,x,2)*b*c*u
2 3 2 2 2 2 2
+ 150*df(u,x,2)*c *u - 1500*df(u,x) *a*u - 300*df(u,x) *b *u
2 2 2 2 2 3 5 2 5
+ 1050*df(u,x) *b*c*u - 450*df(u,x) *c *u + 12*b *u - 48*b *c*u
2 5 3 5
+ 39*b*c *u - 9*c *u , x )
======================================================
Conservation law:
2 2
- 2*b + 7*b*c - 3*c
a=------------------------,
10
2 2 2
( ( - 1500*df(u,x,2)*b*t + 4500*df(u,x,2)*c*t - 300*b *t*u + 1050*b*c*t*u
2 2
- 450*c *t*u - 1500*x)/(b - 3*c) ) * ( (10*df(u,t) + 10*df(u,x,5)
2 2
+ 10*df(u,x,3)*c*u + 10*df(u,x,2)*df(u,x)*b - 2*df(u,x)*b *u
2 2 2
+ 7*df(u,x)*b*c*u - 3*df(u,x)*c *u )/10 )
=
2 2 2 3 3
df( (750*df(u,x) *b*t - 2250*df(u,x) *c*t - 100*b *t*u + 350*b*c*t*u
2 3
- 150*c *t*u - 1500*u*x)/(b - 3*c), t )
+
df( ( - 1500*df(u,t)*df(u,x)*b*t + 4500*df(u,t)*df(u,x)*c*t
- 1500*df(u,x,4)*df(u,x,2)*b*t + 4500*df(u,x,4)*df(u,x,2)*c*t
2 2 2 2 2
- 300*df(u,x,4)*b *t*u + 1050*df(u,x,4)*b*c*t*u - 450*df(u,x,4)*c *t*u
2 2
- 1500*df(u,x,4)*x + 750*df(u,x,3) *b*t - 2250*df(u,x,3) *c*t
2
+ 600*df(u,x,3)*df(u,x)*b *t*u - 2100*df(u,x,3)*df(u,x)*b*c*t*u
2 2 2
+ 900*df(u,x,3)*df(u,x)*c *t*u + 1500*df(u,x,3) - 300*df(u,x,2) *b *t*u
2 2 2
+ 300*df(u,x,2) *b*c*t*u + 1800*df(u,x,2) *c *t*u
2 2 2
- 600*df(u,x,2)*df(u,x) *b *t + 2100*df(u,x,2)*df(u,x) *b*c*t
2 2 2 3
- 900*df(u,x,2)*df(u,x) *c *t - 300*df(u,x,2)*b *c*t*u
2 3 3 3
+ 1050*df(u,x,2)*b*c *t*u - 450*df(u,x,2)*c *t*u - 1500*df(u,x,2)*c*u*x
2 2 2 2 2 3 2
- 1500*df(u,x) *a*b*t*u + 4500*df(u,x) *a*c*t*u - 300*df(u,x) *b *t*u
2 2 2 2 2 2 2
+ 1950*df(u,x) *b *c*t*u - 3600*df(u,x) *b*c *t*u - 750*df(u,x) *b*x
2 3 2 2 2 2
+ 1350*df(u,x) *c *t*u + 750*df(u,x) *c*x - 750*df(u,x)*a*u *x
2 2 2 2 2 2 2 2
- 150*df(u,x)*b *u *x + 525*df(u,x)*b*c*u *x - 225*df(u,x)*c *u *x
4 5 3 5 2 2 5
+ 1500*df(u,x)*c*u + 12*b *t*u - 84*b *c*t*u + 183*b *c *t*u
2 3 3 5 3 4 5 2 3
+ 100*b *u *x - 126*b*c *t*u - 350*b*c*u *x + 27*c *t*u + 150*c *u *x)/(
b - 3*c), x )
======================================================
Conservation law:
a=0,
b
c=---,
3
3*df(u,t) + 3*df(u,x,5) + df(u,x,3)*b*u + 3*df(u,x,2)*df(u,x)*b
( 6*x ) * ( ----------------------------------------------------------------- )
3
=
2
df( - 3*df(u,x)*x , t )
+
2 2 2
df( 3*df(u,t)*x + 6*df(u,x,4)*x - df(u,x,3)*b*u*x + 3*df(u,x,3)*c*u*x
2 2 2
- 6*df(u,x,3) + 2*df(u,x,2)*b*u*x + 2*df(u,x) *b*x + 3*df(u,x)*a*u *x
- 2*df(u,x)*b*u, x )
======================================================
Conservation law:
a=0,
b
c=---,
3
2
9*x 3*df(u,t) + 3*df(u,x,5) + df(u,x,3)*b*u + 3*df(u,x,2)*df(u,x)*b
( ------ ) * ( -----------------------------------------------------------------
b 3
)
=
3
- 3*df(u,x)*x
df( -----------------, t )
b
+
3 2 3 3
df( (3*df(u,t)*x + 9*df(u,x,4)*x - df(u,x,3)*b*u*x + 3*df(u,x,3)*c*u*x
2 2 2
- 18*df(u,x,3)*x + 3*df(u,x,2)*b*u*x + 18*df(u,x,2) + 3*df(u,x) *b*x
2 3 2
+ 3*df(u,x)*a*u *x - 6*df(u,x)*b*u*x + 3*b*u )/b, x )
======================================================
The function c_392(t) is not constant!
The function c_372(t,x) is not constant!
There are remaining conditions: {df(c_372,t) + df(c_372,x,5) - c_392}
for the functions: c_372(t,x), c_392(t)
Corresponding CLs might not be shown below as they
could be of too low order.
======================================================
The function c_443(t) is not constant!
The function c_372(t,x) is not constant!
There are remaining conditions: {20*df(c_372,t) + 20*df(c_372,x,5) - c_443}
for the functions: c_372(t,x), c_443(t)
Corresponding CLs might not be shown below as they
could be of too low order.
======================================================
Conservation law:
a=0,
b=0,
c=0,
( - 2*df(u,x,2) ) * ( df(u,t) + df(u,x,5) )
=
2
df( df(u,x) , t )
+
2
df( - 2*df(u,t)*df(u,x) - 2*df(u,x,4)*df(u,x,2) + df(u,x,3)
2 2 2
- 2*df(u,x,3)*df(u,x)*c*u - 2*df(u,x,2)*df(u,x) *b - 2*df(u,x) *a*u , x )
======================================================
Conservation law:
2
3*b
a=------,
40
b
c=---,
2
2
( - 4000*df(u,x,2) - 600*b*u ) * ( (40*df(u,t) + 40*df(u,x,5)
2 2
+ 20*df(u,x,3)*b*u + 40*df(u,x,2)*df(u,x)*b + 3*df(u,x)*b *u )/40 )
=
2 3
df( 2000*df(u,x) - 200*b*u , t )
+
2
df( - 4000*df(u,t)*df(u,x) - 4000*df(u,x,4)*df(u,x,2) - 600*df(u,x,4)*b*u
2
+ 2000*df(u,x,3) + 3200*df(u,x,3)*df(u,x)*b*u - 4000*df(u,x,3)*df(u,x)*c*u
2 2 2 3
- 1600*df(u,x,2) *b*u - 1200*df(u,x,2)*df(u,x) *b - 300*df(u,x,2)*b *u
2 2 2 2 2 3 5
- 4000*df(u,x) *a*u + 300*df(u,x) *b *u - 9*b *u , x )
======================================================
Conservation law:
2
3*b
a=------,
40
b
c=---,
2
2 2
- 4000*df(u,x,2)*b*t - 600*b *t*u + 8000*x
( ---------------------------------------------- ) * ( (40*df(u,t)
b
2 2
+ 40*df(u,x,5) + 20*df(u,x,3)*b*u + 40*df(u,x,2)*df(u,x)*b + 3*df(u,x)*b *u
)/40 )
=
2 2 3
2000*df(u,x) *b*t - 200*b *t*u + 8000*u*x
df( --------------------------------------------, t )
b
+
df( ( - 4000*df(u,t)*df(u,x)*b*t - 4000*df(u,x,4)*df(u,x,2)*b*t
2 2 2
- 600*df(u,x,4)*b *t*u + 8000*df(u,x,4)*x + 2000*df(u,x,3) *b*t
2
+ 3200*df(u,x,3)*df(u,x)*b *t*u - 4000*df(u,x,3)*df(u,x)*b*c*t*u
2 2
- 2000*df(u,x,3)*b*u*x + 4000*df(u,x,3)*c*u*x - 8000*df(u,x,3)
2 2 2 2
- 1600*df(u,x,2) *b *t*u - 1200*df(u,x,2)*df(u,x) *b *t
3 3 2 2
- 300*df(u,x,2)*b *t*u + 4000*df(u,x,2)*b*u*x - 4000*df(u,x) *a*b*t*u
2 3 2 2 2 2
+ 300*df(u,x) *b *t*u + 2000*df(u,x) *b*x + 4000*df(u,x)*a*u *x
2 2 2 4 5 2 3
- 300*df(u,x)*b *u *x - 4000*df(u,x)*b*u - 9*b *t*u + 200*b *u *x)/b, x
)
======================================================
Conservation law:
b=0,
c=0,
3 2
( --- ) * ( df(u,t) + df(u,x,5) + df(u,x)*a*u )
a
=
- 3*df(u,x)*x
df( ----------------, t )
a
+
df(
3
3*df(u,t)*x + 3*df(u,x,4) + 3*df(u,x,3)*c*u*x + 3*df(u,x,2)*df(u,x)*b*x + a*u
--------------------------------------------------------------------------------
a
, x )
======================================================
The function c_425(t) is not constant!
The function c_372(t,x) is not constant!
There are remaining conditions: {2*df(c_372,t) + 2*df(c_372,x,5) - c_425}
for the functions: c_372(t,x), c_425(t)
Corresponding CLs might not be shown below as they
could be of too low order.
======================================================
**************************************************************************
CONLAWi can also be used to determine first integrals of ODEs.
The generality of the ansatz is not just specified by the order.
For example, the Lorentz system below is a first order system
therefore any first integrals are zero order expressions.
The ansatz to be investigated below looks for first integrals of
the form a1(x,1)+a2(y,t)+a3(x,t)=const. and determines parameters
s,b,r such that first integrals exist.
--------------------------------------------------------------------------
This is CONLAW1 - a program for calculating conservation laws of DEs
The DEs under investigation are :
df(x,t)= - s*x + s*y
df(y,t)=r*x + x*z - y
df(z,t)= - b*z + x*y
for the function(s): {x,y,z}
======================================================
A special ansatz of order 0 for the conserved current is investigated.
The function c_584(x) is not constant!
======================================================
Conservation law:
s=0,
r=0,
b=1,
2*t
( - 2*e *z ) * ( df(z,t) - x*y + z )
+
2*t
( 2*e *y ) * ( df(y,t) - x*z + y )
+
( 0 ) * ( df(x,t) )
=
2*t 2 2*t 2
df( e *y - e *z , t )
======================================================
Conservation law:
s=0,
r=0,
b=1,
( 0 ) * ( df(z,t) - x*y + z )
+
( 0 ) * ( df(y,t) - x*z + y )
+
( df(c_584,x) ) * ( df(x,t) )
=
df( c_584, t )
======================================================
The function c_584(x) is not constant!
======================================================
Conservation law:
s=0,
r=0,
( 0 ) * ( df(z,t) + b*z - x*y )
+
( 0 ) * ( df(y,t) - x*z + y )
+
( df(c_584,x) ) * ( df(x,t) )
=
df( c_584, t )
======================================================
The function c_586(x) is not constant!
======================================================
Conservation law:
s=0,
( 0 ) * ( df(z,t) + b*z - x*y )
+
( 0 ) * ( df(y,t) - r*x - x*z + y )
+
( df(c_586,x) ) * ( df(x,t) )
=
df( c_586, t )
======================================================
Conservation law:
1
s=---,
2
r=0,
b=1,
t
( - e ) * ( df(z,t) - x*y + z )
+
( 0 ) * ( df(y,t) - x*z + y )
+
t 2*df(x,t) + x - y
( 2*e *x ) * ( ------------------- )
2
=
t 2 t
df( e *x - e *z, t )
======================================================
Conservation law:
1
s=---,
2
r=0,
b=1,
2*t
( - 2*e *z ) * ( df(z,t) - x*y + z )
+
2*t
( 2*e *y ) * ( df(y,t) - x*z + y )
+
2*df(x,t) + x - y
( 0 ) * ( ------------------- )
2
=
2*t 2 2*t 2
df( e *y - e *z , t )
======================================================
Conservation law:
r=0,
b=1,
2*t
( - 2*e *z ) * ( df(z,t) - x*y + z )
+
2*t
( 2*e *y ) * ( df(y,t) - x*z + y )
+
( 0 ) * ( df(x,t) + s*x - s*y )
=
2*t 2 2*t 2
df( e *y - e *z , t )
======================================================
Conservation law:
s=1,
b=1,
2*t
( - 2*e *z ) * ( df(z,t) - x*y + z )
+
2*t
( 2*e *y ) * ( df(y,t) - r*x - x*z + y )
+
2*t
( - 2*e *r*x ) * ( df(x,t) + x - y )
=
2*t 2 2*t 2 2*t 2
df( - e *r*x + e *y - e *z , t )
======================================================
Conservation law:
b=2*s,
2*s*t
( - 2*e ) * ( df(z,t) + 2*s*z - x*y )
+
( 0 ) * ( df(y,t) - r*x - x*z + y )
+
2*s*t
2*e *x
( ------------ ) * ( df(x,t) + s*x - s*y )
s
=
2*s*t 2*s*t 2
- 2*e *s*z + e *x
df( -----------------------------, t )
s
======================================================
4: 4: 4: 4: 4: 4: 4: 4: 4:
Time for test: 216760 ms, plus GC time: 22500 ms
5: 5:
Quitting
Sun Jan 3 23:59:02 MET 1999