Tue Apr 15 00:33:29 2008 run on win32
%*******************************************************************%
% %
% C O N L A W . T S T %
% ------------------- %
% conlaw.tst contains test examples for the programs conlaw0.red %
% conlaw1.red, conlaw2.red, conlaw3.red, conlaw4.red. To run %
% this test read in the files crack.red, conlaw0.red, conlaw1.red, %
% conlaw2.red, conlaw3.red, conlaw4.red or load their compiled %
% version before. %
% %
% Author: Thomas Wolf %
% Date: 15. June 1999, 6. May 2003 %
% %
% Details about the syntax of conlaw1-4 are given in conlaw.tex. %
% To run this file read in or load crack, conlaw0 before. %
% %
% The statement lisp(print_:=nil); suppresses output of the %
% computation. To see details of it do lisp(print_:=50). %
% %
%*******************************************************************%
load crack ;
% ,conlaw0,conlaw1,conlaw2,conlaw3,conlaw4$
lisp(depl!*:=nil)$
% clearing of all dependencies
setcrackflags()$
% standart flags
lisp(print_:=nil)$
% no output of the calculation
%% off batch_mode$
comment -------------------------------------------------------------
The following example calculates all conservation laws of the KdV-
equation with a characteristic function of order not higher than two;
nodepnd {u}$
% deletes all dependencies of u
depend u,x,t$
% declares u to be a function of x,t
conlaw4({{df(u,t) = u*df(u,x)+df(u,x,3)}, {u}, {t,x}},
{0, 2, t, {}, {}} )$
--------------------------------------------------------------------------
This is CONLAW4 - a program for calculating conservation laws of DEs
The DE under investigation is :
u =u + u *u
t 3x x
for the function(s): u(t,x)
======================================================
Currently conservation laws with characteristic
function(s) of order 0 are determined
======================================================
Conservation law:
( u ) * ( u - u - u *u )
t 3x x
=
1 2
df( ---*u , t )
2
+
1 2 1 3
df( - u *u + ---*u - ---*u , x )
2x 2 x 3
======================================================
Conservation law:
( t*u + x ) * ( u - u - u *u )
t 3x x
=
1 2
df( ---*t*u + u*x, t )
2
+
1 2 1 3 1 2
df( - u *t*u - u *x + ---*u *t + u - ---*t*u - ---*u *x, x )
2x 2x 2 x x 3 2
======================================================
Conservation law:
( 1 ) * ( u - u - u *u )
t 3x x
=
df( u, t )
+
1 2
df( - u - ---*u , x )
2x 2
======================================================
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*u - u ) * ( u - u - u *u )
2x t 3x x
=
2 1 3
df( u - ---*u , t )
x 3
+
2 2 1 4
df( - 2*u *u + u + u *u + ---*u , x )
t x 2x 2x 4
======================================================
2
{{{ - 2*df(u,x,2) - u },
2 3
3*df(u,x) - u
{-----------------,
3
2 2 4
- 8*df(u,t)*df(u,x) + 4*df(u,x,2) + 4*df(u,x,2)*u + u
-----------------------------------------------------------}},
4
2
- 2*df(u,x,2) - u
{{1},{u,---------------------}},
2
{{t*u + x},
u*(t*u + 2*x)
{---------------,
2
2 3
( - 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}},
{{u},
2 2 3
u - 6*df(u,x,2)*u + 3*df(u,x) - 2*u
{----,--------------------------------------}}}
2 6
comment -------------------------------------------------------------
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.)
In the following input the equations are solved for the t-derivatives,
so the t-derivatives will be substituted in the conservation-law-
conditions, so the ansatz for q_1 should have no t-derivatives of u
included. Therefore the function r in q_1 below is specified as
depending on t,x,u,v,df(u,x),df(v,x).
In the call of conlaw2 the list of variables is {t,x} and x is
the second of the variables (could equally well be in reverse order).
Therefore df(u,x) takes the form u!`2 when the dependencies of r
are specified (see conlaw.tex);
nodepnd {u,v,r}$
depend u,x,t$
depend v,x,t$
depend r,t,x,u,v,u!`2,v!`2$
q_1:=r*df(u,x,2)$
conlaw2({{df(u,t)=df(v,x),
df(v,t)=df(u,x) }, {u,v}, {t,x}},
{2, 2, t, {r}, {}})$
--------------------------------------------------------------------------
This is CONLAW2 - a program for calculating conservation laws of DEs
The DEs under investigation are :
u =v
t x
v =u
t x
for the function(s): u(x,t), v(x,t)
======================================================
A special ansatz of order 2 for the characteristic
function(s) is investigated.
Conservation law:
( u ) * ( u - v )
2x t x
+
( v ) * ( - u + v )
2x x t
=
1 2 1 2
df( - ---*u - ---*v , t )
2 x 2 x
+
df( u *u - u *v + v *v , x )
t x x x t x
======================================================
{{{df(u,x,2),df(v,x,2)},
2 2
- (df(u,x) + df(v,x) )
{--------------------------,
2
df(u,t)*df(u,x) - df(u,x)*df(v,x) + df(v,t)*df(v,x)}}}
clear q_1$
nodepnd {q_1}$
comment -------------------------------------------------------------
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;
nodepnd {u}$
depend u,x,t$
conlaw1({{df(u,t)=df(u,x,2)+df(u,x)**2/2}, {u}, {t,x}},
{0, 0, t, {}, {}} )$
--------------------------------------------------------------------------
This is CONLAW1 - a program for calculating conservation laws of DEs
The DE under investigation is :
1 2
u =u + ---*u
t 2x 2 x
for the function(s): u(x,t)
======================================================
Currently conservation laws with a conserved density
of order 0 are determined
======================================================
The function c_66(x,t) is not constant!
There are remaining conditions: {c_66 + c_66 }
t 2x
for the functions: c_66(x,t)
Corresponding CLs might not be shown below as they
could be of too low order.
======================================================
Conservation law:
u/2 1 2
( e *c_66 ) * ( u - u - ---*u )
t 2x 2 x
=
u/2
df( 2*e *c_66, t )
+
u/2 u/2
df( 2*e *c_66 - e *u *c_66, x )
x x
An attempt to factor out linear differential operators:
1 2
eq_1:=u - u - ---*u
t 2x 2 x
u/2
l_1:=e
u/2
e *eq_1 = 2*(l_1 - l_1 )
t 2x
======================================================
u/2
{{{e *c_66},
u/2
{2*e *c_66,
u/2
e *(2*df(c_66,x) - df(u,x)*c_66)}}}
comment -------------------------------------------------------------
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;
nodepnd {u,v}$
depend u,x,t$
depend v,x,t$
conlaw1({{df(u,t)=df(u,x,3)+6*u*df(u,x)+2*v*df(v,x),
df(v,t)=2*df(u,x)*v+2*u*df(v,x) }, {u,v}, {t,x}},
{0, 1, t, {}, {}})$
--------------------------------------------------------------------------
This is CONLAW1 - a program for calculating conservation laws of DEs
The DEs under investigation are :
u =u + 6*u *u + 2*v *v
t 3x x x
v =2*u *v + 2*v *u
t x x
for the function(s): u(x,t), v(x,t)
======================================================
Currently conservation laws with a conserved density
of order 0 are determined
======================================================
Conservation law:
( -1 ) * ( - 2*u *v + v - 2*v *u )
x t x
+
( 0 ) * ( u - u - 6*u *u - 2*v *v )
t 3x x x
=
df( - v, t )
+
df( 2*u*v, x )
======================================================
Conservation law:
( 2*v ) * ( - 2*u *v + v - 2*v *u )
x t x
+
( 2*u ) * ( u - u - 6*u *u - 2*v *v )
t 3x x x
=
2 2
df( u + v , t )
+
2 3 2
df( - 2*u *u + u - 4*u - 4*u*v , x )
2x x
======================================================
Conservation law:
( 0 ) * ( - 2*u *v + v - 2*v *u )
x t x
+
( 1 ) * ( u - u - 6*u *u - 2*v *v )
t 3x x x
=
df( u, t )
+
2 2
df( - u - 3*u - v , x )
2x
======================================================
Currently conservation laws with a conserved density
of order 1 are determined
======================================================
Conservation law:
( - 2*v ) * ( - 2*u *v + v - 2*v *u )
x t x
+
( - 2*u ) * ( u - u - 6*u *u - 2*v *v )
t 3x x x
=
2 2
df( - u - v , t )
+
2 3 2
df( 2*u *u - u + 4*u + 4*u*v , x )
2x x
======================================================
Conservation law:
( - 4*u*v ) * ( - 2*u *v + v - 2*v *u )
x t x
+
2 2
( - 2*u - 6*u - 2*v ) * ( u - u - 6*u *u - 2*v *v )
2x t 3x x x
=
2 3 2
df( u - 2*u - 2*u*v , t )
x
+
2 2 2 4 2 2 4
df( - 2*u *u + u + 6*u *u + 2*u *v + 9*u + 10*u *v + v , x )
t x 2x 2x 2x
======================================================
Conservation law:
2 2
2*v *v - 3*v - 4*u*v
2x x
( -------------------------- ) * ( - 2*u *v + v - 2*v *u )
4 x t x
v
+
4
( --- ) * ( u - u - 6*u *u - 2*v *v )
v t 3x x x
=
2 2
- v + 4*u*v
x
df( -----------------, t )
3
v
+
2 2 2 2 4
- 4*u *v - 4*u *v *v + 2*v *v - 2*v *u - 8*u *v - 8*v
2x x x t x x
df( --------------------------------------------------------------, x )
3
v
======================================================
2 2
2*df(v,x,2)*v - 3*df(v,x) - 4*u*v 4
{{{-------------------------------------,---},
4 v
v
2 2
- df(v,x) + 4*u*v
{----------------------,
3
v
2 2
(2*( - 2*df(u,x,2)*v - 2*df(u,x)*df(v,x)*v + df(v,t)*df(v,x) - df(v,x) *u
2 2 4 3
- 4*u *v - 4*v ))/v }},
2 2
{{ - 4*u*v,2*( - df(u,x,2) - 3*u - v )},
2 3 2
{df(u,x) - 2*u - 2*u*v ,
2 2 2 4
- 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 }},
{{ - 2*v, - 2*u},
2 2
{ - (u + v ),
2 3 2
2*df(u,x,2)*u - df(u,x) + 4*u + 4*u*v }},
2 2
{{0,1},{u, - df(u,x,2) - 3*u - v }},
{{2*v,2*u},
2 2
{u + v ,
2 3 2
- 2*df(u,x,2)*u + df(u,x) - 4*u - 4*u*v }},
{{-1,0},{ - v,2*u*v}}}
comment -------------------------------------------------------------
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;
nodepnd {u}$
depend u,x,t$
conlaw1({{df(u,t)=-df(u,x,5)-a*u**2*df(u,x)
-b*df(u,x)*df(u,x,2)-c*u*df(u,x,3)},
{u}, {t,x}},
{0, 1, t, {a,b,c}, {}})$
--------------------------------------------------------------------------
This is CONLAW1 - a program for calculating conservation laws of DEs
The DE under investigation is :
2
u = - u - u *c*u - u *u *b - u *a*u
t 5x 3x 2x x x
for the function(s): u(x,t)
======================================================
Currently conservation laws with a conserved density
of order 0 are determined
======================================================
The function c_232(x,t) is not constant!
The function c_236(t) is not constant!
There are remaining conditions: {c_232 + c_232 - c_236}
t 5x
for the functions: c_236(t), c_232(x,t)
Corresponding CLs might not be shown below as they
could be of too low order.
======================================================
Conservation law:
c=0,
b=0,
a=0,
( - c_232 ) * ( u + u )
x t 5x
=
df( - c_232 *u, t )
x
+
df( c_232 *u + c_232 *u - c_232 *u + c_232 *u - c_232 *u + c_236*u, x
t 4x x 3x 2x 2x 3x x 4x
)
======================================================
Conservation law:
a=0,
b=3*c,
( 1 ) * ( u + u + u *c*u + 3*u *u *c )
t 5x 3x 2x x
=
df( u, t )
+
2
df( u + u *c*u + u *c, x )
4x 2x x
======================================================
Conservation law:
a=0,
b=3*c,
( - x ) * ( u + u + u *c*u + 3*u *u *c )
t 5x 3x 2x x
=
df( - u*x, t )
+
2
df( - u *x + u - u *c*u*x - u *c*x + u *c*u, x )
4x 3x 2x x x
======================================================
Conservation law:
a=0,
b=3*c,
2
x
( ---- ) * ( u + u + u *c*u + 3*u *u *c )
c t 5x 3x 2x x
=
2
u*x
df( ------, t )
c
+
2 2 2 2 2
u *x - 2*u *x + u *c*u*x + 2*u + u *c*x - 2*u *c*u*x + c*u
4x 3x 2x 2x x x
df( ----------------------------------------------------------------------, x )
c
======================================================
Conservation law:
2
( -6 ) * ( u + u + u *c*u + u *u *b + u *a*u )
t 5x 3x 2x x x
=
df( - 6*u, t )
+
2 2 3
df( - 6*u - 6*u *c*u - 3*u *b + 3*u *c - 2*a*u , x )
4x 2x x x
======================================================
Conservation law:
b=2*c,
2
( - 4*u ) * ( u + u + u *c*u + 2*u *u *c + u *a*u )
t 5x 3x 2x x x
=
2
df( - 2*u , t )
+
2 2 4
df( - 4*u *u + 4*u *u - 2*u - 4*u *c*u - a*u , x )
4x 3x x 2x 2x
======================================================
Conservation law:
b=2*c,
2
( -6 ) * ( u + u + u *c*u + 2*u *u *c + u *a*u )
t 5x 3x 2x x x
=
df( - 6*u, t )
+
2 3
df( - 6*u - 6*u *c*u - 3*u *c - 2*a*u , x )
4x 2x x
======================================================
The function c_254(x,t) is not constant!
The function c_259(t) is not constant!
There are remaining conditions: {c_254 + c_254 + c_259}
t 5x
for the functions: c_259(t), c_254(x,t)
Corresponding CLs might not be shown below as they
could be of too low order.
======================================================
Conservation law:
c=0,
b=0,
a=0,
( - 2*u ) * ( u + u )
t 5x
=
2
df( - u , t )
+
2
df( - 2*u *u + 2*u *u - u , x )
4x 3x x 2x
======================================================
Conservation law:
c=0,
b=0,
a=0,
( - c_254 ) * ( u + u )
x t 5x
=
df( - c_254 *u, t )
x
+
df( c_254 *u + c_254 *u - c_254 *u + c_254 *u - c_254 *u + c_259*u, x
t 4x x 3x 2x 2x 3x x 4x
)
======================================================
Conservation law:
b=0,
c=0,
- 3 2
( ------ ) * ( u + u + u *a*u )
a t 5x x
=
- 3*u
df( --------, t )
a
+
3
- 3*u - a*u
4x
df( -----------------, x )
a
======================================================
Conservation law:
b=0,
c=0,
2
( - 4*u ) * ( u + u + u *a*u )
t 5x x
=
2
df( - 2*u , t )
+
2 4
df( - 4*u *u + 4*u *u - 2*u - a*u , x )
4x 3x x 2x
======================================================
Currently conservation laws with a conserved density
of order 1 are determined
======================================================
Conservation law:
b=0,
c=0,
- 4*u 2
( -------- ) * ( u + u + u *a*u )
a t 5x x
=
2
- 2*u
df( ---------, t )
a
+
2 4
- 4*u *u + 4*u *u - 2*u - a*u
4x 3x x 2x
df( ---------------------------------------, x )
a
======================================================
Conservation law:
b=3*c,
3 2
( ---- ) * ( u + u + u *c*u + 3*u *u *c + u *a*u )
2 t 5x 3x 2x x x
a
=
3*u
df( -----, t )
2
a
+
2 3
3*u + 3*u *c*u + 3*u *c + a*u
4x 2x x
df( ------------------------------------, x )
2
a
======================================================
The function c_294(x,t) is not constant!
The function c_303(x) is not constant!
There are remaining conditions: {c_294 + c_294 + c_303 }
t 5x 4x
for the functions: c_303(x), c_294(x,t)
Corresponding CLs might not be shown below as they
could be of too low order.
======================================================
Conservation law:
c=0,
b=0,
a=0,
( 2*u ) * ( u + u )
t 5x
=
2
df( u , t )
+
2
df( 2*u *u - 2*u *u + u , x )
4x 3x x 2x
======================================================
Conservation law:
c=0,
b=0,
a=0,
( - 2*u ) * ( u + u )
2x t 5x
=
2
df( u , t )
x
+
2 2 2 2
df( - 2*u *u - 2*u *u + u - 2*u *u *c*u - 2*u *u *b - 2*u *a*u , x )
t x 4x 2x 3x 3x x 2x x x
======================================================
Conservation law:
c=0,
b=0,
a=0,
( - c_294 + c_303 ) * ( u + u )
x t 5x
=
df( - c_294 *u + c_303*u, t )
x
+
df( c_294 *u + c_294 *u - c_294 *u + c_294 *u - c_294 *u - c_303 *u
t 4x x 3x 2x 2x 3x x 4x 3x x
+ c_303 *u - c_303 *u + u *c_303 - u *c*c_294*u - u *u *b*c_294
2x 2x x 3x 4x 3x 2x x
2
- u *a*c_294*u , x )
x
An attempt to factor out linear differential operators:
eq_1:=u + u
t 5x
l_1:=u
x
l_2:= - u
l_3:=u + u
t,x 6x
eq_1 = l_3
x
eq_1 = l_1 - l_2
4x t
======================================================
Conservation law:
a=0,
b=3*c,
( 1 ) * ( u + u + u *c*u + 3*u *u *c )
t 5x 3x 2x x
=
df( u, t )
+
2
df( u + u *c*u + u *c, x )
4x 2x x
======================================================
Conservation law:
a=0,
b=3*c,
( x ) * ( u + u + u *c*u + 3*u *u *c )
t 5x 3x 2x x
=
df( u*x, t )
+
2
df( u *x - u + u *c*u*x + u *c*x - u *c*u, x )
4x 3x 2x x x
======================================================
Conservation law:
a=0,
b=3*c,
2
- x
( ------- ) * ( u + u + u *c*u + 3*u *u *c )
c t 5x 3x 2x x
=
2
- u*x
df( ---------, t )
c
+
2 2 2 2 2
- u *x + 2*u *x - u *c*u*x - 2*u - u *c*x + 2*u *c*u*x - c*u
4x 3x 2x 2x x x
df( -------------------------------------------------------------------------, x
c
)
======================================================
Conservation law:
a=0,
b=3*c,
2
( - 6*u - 3*c*u ) * ( u + u + u *c*u + 3*u *u *c )
2x t 5x 3x 2x x
=
2 3
df( 3*u - c*u , t )
x
+
2 2 2
df( - 6*u *u - 6*u *u - 3*u *c*u + 3*u + 6*u *u *c*u - 6*u *c*u
t x 4x 2x 4x 3x 3x x 2x
2 2 2 3 2 2
- 6*u *u *b + 12*u *u *c - 3*u *c *u - 6*u *a*u , x )
2x x 2x x 2x x
======================================================
Conservation law:
3 2
a=----*c ,
10
b=2*c,
3 2 2
( -100 ) * ( u + u + u *c*u + 2*u *u *c + ----*u *c *u )
t 5x 3x 2x x 10 x
=
df( - 100*u, t )
+
2 2 3 2
df( - 100*u + 30*u *u *b*c *t*u - 100*u *u *b*x - 60*u *u *c *t*u
4x 2x x 2x x 2x x
2 2 4 2
+ 200*u *u *c*x - 100*u *c*u - 50*u *c + 30*u *a*c *t*u - 100*u *a*u *x
2x x 2x x x x
4 4 2 2 2 3
- 9*u *c *t*u + 30*u *c *u *x - 10*c *u , x )
x x
======================================================
Conservation law:
3 2
a=----*c ,
10
b=2*c,
3 2 2
( - 100*u ) * ( u + u + u *c*u + 2*u *u *c + ----*u *c *u )
t 5x 3x 2x x 10 x
=
2
df( - 50*u , t )
+
2 2 3
df( - 100*u *u + 100*u *u - 50*u + 30*u *u *b*c *t*u - 100*u *u *b*u*x
4x 3x x 2x 2x x 2x x
3 3 2 2 5
- 60*u *u *c *t*u + 200*u *u *c*u*x - 100*u *c*u + 30*u *a*c *t*u
2x x 2x x 2x x
3 4 5 2 3 15 2 4
- 100*u *a*u *x - 9*u *c *t*u + 30*u *c *u *x - ----*c *u , x )
x x x 2
======================================================
Conservation law:
3 2
a=----*c ,
10
b=2*c,
2 3 2 2
( - 1000*u - 300*c*u ) * ( u + u + u *c*u + 2*u *u *c + ----*u *c *u )
2x t 5x 3x 2x x 10 x
=
2 3
df( 500*u - 100*c*u , t )
x
+
2 2
df( - 1000*u *u - 1000*u *u - 300*u *c*u + 500*u + 600*u *u *c*u
t x 4x 2x 4x 3x 3x x
2 2 2 3 4
- 800*u *c*u - 1000*u *u *b + 1400*u *u *c + 90*u *u *b*c *t*u
2x 2x x 2x x 2x x
2 4 4 2 2 2 3
- 300*u *u *b*c*u *x - 180*u *u *c *t*u + 600*u *u *c *u *x - 300*u *c *u
2x x 2x x 2x x 2x
2 2 2 2 2 3 6 4
- 1000*u *a*u + 300*u *c *u + 90*u *a*c *t*u - 300*u *a*c*u *x
x x x x
5 6 3 4 3 5
- 27*u *c *t*u + 90*u *c *u *x - 18*c *u , x )
x x
======================================================
Conservation law:
3 2
a=----*c ,
10
b=2*c,
2 2
- 2000*u *c*t - 600*c *t*u + 2000*x
2x
( ---------------------------------------- ) * ( u + u + u *c*u + 2*u *u *c
c t 5x 3x 2x x
3 2 2
+ ----*u *c *u )
10 x
=
2 2 3
1000*u *c*t - 200*c *t*u + 2000*u*x
x
df( ---------------------------------------, t )
c
+
2 2
df( ( - 2000*u *u *c*t - 2000*u *u *c*t - 600*u *c *t*u + 2000*u *x
t x 4x 2x 4x 4x
2 2 2 2
+ 1000*u *c*t + 1200*u *u *c *t*u - 2000*u - 1600*u *c *t*u
3x 3x x 3x 2x
2 2 2 4 2 4
- 2000*u *u *b*c*t + 2800*u *u *c *t + 90*u *u *b*c *t *u
2x x 2x x 2x x
2 2 2 5 2 4
- 600*u *u *b*c *t*u *x + 1000*u *u *b*x - 180*u *u *c *t *u
2x x 2x x 2x x
3 2 2 3 3
+ 1200*u *u *c *t*u *x - 2000*u *u *c*x - 600*u *c *t*u
2x x 2x x 2x
2 2 2 3 2 2
+ 2000*u *c*u*x - 2000*u *a*c*t*u + 600*u *c *t*u + 1000*u *c*x
2x x x x
4 2 6 2 4 2 2 6 2 6
+ 90*u *a*c *t *u - 600*u *a*c *t*u *x + 1000*u *a*u *x - 27*u *c *t *u
x x x x
4 4 2 2 2 4 5
+ 180*u *c *t*u *x - 300*u *c *u *x - 2000*u *c*u - 36*c *t*u
x x x
2 3
+ 200*c *u *x)/c, x )
======================================================
Conservation law:
1 2 7 3 2
a= - ---*b + ----*b*c - ----*c ,
5 10 10
1 2 2 7 2
( -100 ) * ( u + u + u *c*u + u *u *b - ---*u *b *u + ----*u *b*c*u
t 5x 3x 2x x 5 x 10 x
3 2 2
- ----*u *c *u )
10 x
=
df( - 100*u, t )
+
2 2 2 4
df( - 100*u - 100*u *c*u - 50*u *b + 50*u *c - 20*u *a*b *t*u
4x 2x x x x
4 2 4 2 4 4
+ 70*u *a*b*c*t*u - 30*u *a*c *t*u - 100*u *a*u *x - 4*u *b *t*u
x x x x
3 4 2 2 4 2 2 3 4
+ 28*u *b *c*t*u - 61*u *b *c *t*u - 20*u *b *u *x + 42*u *b*c *t*u
x x x x
2 4 4 2 2 20 2 3 70 3
+ 70*u *b*c*u *x - 9*u *c *t*u - 30*u *c *u *x + ----*b *u - ----*b*c*u
x x x 3 3
2 3
+ 10*c *u , x )
======================================================
Conservation law:
1 2 7 3 2
a= - ---*b + ----*b*c - ----*c ,
5 10 10
2 2
( - 1000*u - 200*b*u + 100*c*u ) * ( u + u + u *c*u + u *u *b
2x t 5x 3x 2x x
1 2 2 7 2 3 2 2
- ---*u *b *u + ----*u *b*c*u - ----*u *c *u )
5 x 10 x 10 x
=
2 200 3 100 3
df( 500*u - -----*b*u + -----*c*u , t )
x 3 3
+
2 2 2
df( - 1000*u *u - 1000*u *u - 200*u *b*u + 100*u *c*u + 500*u
t x 4x 2x 4x 4x 3x
2 2 2
+ 400*u *u *b*u - 200*u *u *c*u - 200*u *b*u - 400*u *c*u - 400*u *u *b
3x x 3x x 2x 2x 2x x
2 3 2 3 2 2
+ 200*u *u *c - 200*u *b*c*u + 100*u *c *u - 1000*u *a*u
2x x 2x 2x x
2 2 2 2 2 2 2 2 3 6
- 200*u *b *u + 700*u *b*c*u - 300*u *c *u - 40*u *a*b *t*u
x x x x
2 6 2 6 4 3 6
+ 160*u *a*b *c*t*u - 130*u *a*b*c *t*u - 200*u *a*b*u *x + 30*u *a*c *t*u
x x x x
4 5 6 4 6 3 2 6
+ 100*u *a*c*u *x - 8*u *b *t*u + 60*u *b *c*t*u - 150*u *b *c *t*u
x x x x
3 4 2 3 6 2 4 4 6
- 40*u *b *u *x + 145*u *b *c *t*u + 160*u *b *c*u *x - 60*u *b*c *t*u
x x x x
2 4 5 6 3 4 3 5 2 5
- 130*u *b*c *u *x + 9*u *c *t*u + 30*u *c *u *x + 8*b *u - 32*b *c*u
x x x
2 5 3 5
+ 26*b*c *u - 6*c *u , x )
======================================================
Conservation law:
1 2 7 3 2
a= - ---*b + ----*b*c - ----*c ,
5 10 10
2 2 2 2 2
( ( - 2000*u *b*t + 6000*u *c*t - 400*b *t*u + 1400*b*c*t*u - 600*c *t*u
2x 2x
1 2 2
- 2000*x)/(b - 3*c) ) * ( u + u + u *c*u + u *u *b - ---*u *b *u
t 5x 3x 2x x 5 x
7 2 3 2 2
+ ----*u *b*c*u - ----*u *c *u )
10 x 10 x
=
2 2 400 2 3 1400 3 2 3
df( (1000*u *b*t - 3000*u *c*t - -----*b *t*u + ------*b*c*t*u - 200*c *t*u
x x 3 3
- 2000*u*x)/(b - 3*c), t )
+
df( ( - 2000*u *u *b*t + 6000*u *u *c*t - 2000*u *u *b*t + 6000*u *u *c*t
t x t x 4x 2x 4x 2x
2 2 2 2 2
- 400*u *b *t*u + 1400*u *b*c*t*u - 600*u *c *t*u - 2000*u *x
4x 4x 4x 4x
2 2 2
+ 1000*u *b*t - 3000*u *c*t + 800*u *u *b *t*u - 2800*u *u *b*c*t*u
3x 3x 3x x 3x x
2 2 2 2
+ 1200*u *u *c *t*u + 2000*u - 400*u *b *t*u + 400*u *b*c*t*u
3x x 3x 2x 2x
2 2 2 2 2
+ 2400*u *c *t*u - 800*u *u *b *t + 2800*u *u *b*c*t
2x 2x x 2x x
2 2 2 3 2 3
- 1200*u *u *c *t - 400*u *b *c*t*u + 1400*u *b*c *t*u
2x x 2x 2x
3 3 2 2 2 2
- 600*u *c *t*u - 2000*u *c*u*x - 2000*u *a*b*t*u + 6000*u *a*c*t*u
2x 2x x x
2 3 2 2 2 2 2 2 2 2
- 400*u *b *t*u + 2600*u *b *c*t*u - 4800*u *b*c *t*u - 1000*u *b*x
x x x x
2 3 2 2 4 2 6 3 2 6
+ 1800*u *c *t*u + 1000*u *c*x - 40*u *a*b *t *u + 280*u *a*b *c*t *u
x x x x
2 2 2 6 2 4 3 2 6
- 610*u *a*b *c *t *u - 400*u *a*b *t*u *x + 420*u *a*b*c *t *u
x x x
4 4 2 6 2 4
+ 1400*u *a*b*c*t*u *x - 90*u *a*c *t *u - 600*u *a*c *t*u *x
x x x
2 2 6 2 6 5 2 6 4 2 2 6
- 1000*u *a*u *x - 8*u *b *t *u + 84*u *b *c*t *u - 330*u *b *c *t *u
x x x x
4 4 3 3 2 6 3 4
- 80*u *b *t*u *x + 595*u *b *c *t *u + 560*u *b *c*t*u *x
x x x
2 4 2 6 2 2 4 2 2 2
- 495*u *b *c *t *u - 1220*u *b *c *t*u *x - 200*u *b *u *x
x x x
5 2 6 3 4 2 2
+ 189*u *b*c *t *u + 840*u *b*c *t*u *x + 700*u *b*c*u *x
x x x
6 2 6 4 4 2 2 2
- 27*u *c *t *u - 180*u *c *t*u *x - 300*u *c *u *x + 2000*u *c*u
x x x x
4 5 3 5 2 2 5 400 2 3
+ 16*b *t*u - 112*b *c*t*u + 244*b *c *t*u + -----*b *u *x
3
3 5 1400 3 4 5 2 3
- 168*b*c *t*u - ------*b*c*u *x + 36*c *t*u + 200*c *u *x)/(b - 3*c), x
3
)
======================================================
Conservation law:
b=2*c,
2
( 4*u ) * ( u + u + u *c*u + 2*u *u *c + u *a*u )
t 5x 3x 2x x x
=
2
df( 2*u , t )
+
2 2 4
df( 4*u *u - 4*u *u + 2*u + 4*u *c*u + a*u , x )
4x 3x x 2x 2x
======================================================
Conservation law:
b=2*c,
2
( 6 ) * ( u + u + u *c*u + 2*u *u *c + u *a*u )
t 5x 3x 2x x x
=
df( 6*u, t )
+
2 3
df( 6*u + 6*u *c*u + 3*u *c + 2*a*u , x )
4x 2x x
======================================================
Conservation law:
a=0,
b=2*c,
( 2 ) * ( u + u + u *c*u + 2*u *u *c )
t 5x 3x 2x x
=
df( 2*u, t )
+
2
df( 2*u + 2*u *c*u + u *c, x )
4x 2x x
======================================================
Conservation law:
a=0,
b=2*c,
( 2*u ) * ( u + u + u *c*u + 2*u *u *c )
t 5x 3x 2x x
=
2
df( u , t )
+
2 2
df( 2*u *u - 2*u *u + u + 2*u *c*u , x )
4x 3x x 2x 2x
======================================================
Conservation law:
- 6 2
( ------ ) * ( u + u + u *c*u + u *u *b + u *a*u )
a t 5x 3x 2x x x
=
- 6*u
df( --------, t )
a
+
2 2 3
- 6*u - 6*u *c*u - 3*u *b + 3*u *c - 2*a*u
4x 2x x x
df( ---------------------------------------------------, x )
a
======================================================
Conservation law:
a=0,
( 2 ) * ( u + u + u *c*u + u *u *b )
t 5x 3x 2x x
=
df( 2*u, t )
+
2 2
df( 2*u + 2*u *c*u + u *b - u *c, x )
4x 2x x x
======================================================
{{{2},
{2*u,
2 2
2*df(u,x,4) + 2*df(u,x,2)*c*u + df(u,x) *b - df(u,x) *c}},
- 6
{{------},
a
- 6*u
{--------,
a
2 2 3
- 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
-------------------------------------------------------------------------}},
a
{{2*u},
2
{u ,
2 2
2*df(u,x,4)*u - 2*df(u,x,3)*df(u,x) + df(u,x,2) + 2*df(u,x,2)*c*u }},
{{2},
{2*u,
2
2*df(u,x,4) + 2*df(u,x,2)*c*u + df(u,x) *c}},
{{6},
{6*u,
2 3
6*df(u,x,4) + 6*df(u,x,2)*c*u + 3*df(u,x) *c + 2*a*u }},
{{4*u},
2
{2*u ,
2 2 4
4*df(u,x,4)*u - 4*df(u,x,3)*df(u,x) + 2*df(u,x,2) + 4*df(u,x,2)*c*u + a*u }
},
2 2 2
{{(200*( - 10*df(u,x,2)*b*t + 30*df(u,x,2)*c*t - 2*b *t*u + 7*b*c*t*u
2 2
- 3*c *t*u - 10*x))/(b - 3*c)},
2 2 2 3 3 2 3
{(200*(15*df(u,x) *b*t - 45*df(u,x) *c*t - 2*b *t*u + 7*b*c*t*u - 3*c *t*u
- 30*u*x))/(3*(b - 3*c)),
( - 6000*df(u,t)*df(u,x)*b*t + 18000*df(u,t)*df(u,x)*c*t
- 6000*df(u,x,4)*df(u,x,2)*b*t + 18000*df(u,x,4)*df(u,x,2)*c*t
2 2 2 2 2
- 1200*df(u,x,4)*b *t*u + 4200*df(u,x,4)*b*c*t*u - 1800*df(u,x,4)*c *t*u
2 2
- 6000*df(u,x,4)*x + 3000*df(u,x,3) *b*t - 9000*df(u,x,3) *c*t
2
+ 2400*df(u,x,3)*df(u,x)*b *t*u - 8400*df(u,x,3)*df(u,x)*b*c*t*u
2 2 2
+ 3600*df(u,x,3)*df(u,x)*c *t*u + 6000*df(u,x,3) - 1200*df(u,x,2) *b *t*u
2 2 2
+ 1200*df(u,x,2) *b*c*t*u + 7200*df(u,x,2) *c *t*u
2 2 2
- 2400*df(u,x,2)*df(u,x) *b *t + 8400*df(u,x,2)*df(u,x) *b*c*t
2 2 2 3
- 3600*df(u,x,2)*df(u,x) *c *t - 1200*df(u,x,2)*b *c*t*u
2 3 3 3
+ 4200*df(u,x,2)*b*c *t*u - 1800*df(u,x,2)*c *t*u - 6000*df(u,x,2)*c*u*x
2 2 2 2 2 3 2
- 6000*df(u,x) *a*b*t*u + 18000*df(u,x) *a*c*t*u - 1200*df(u,x) *b *t*u
2 2 2 2 2 2 2
+ 7800*df(u,x) *b *c*t*u - 14400*df(u,x) *b*c *t*u - 3000*df(u,x) *b*x
2 3 2 2 4 2 6
+ 5400*df(u,x) *c *t*u + 3000*df(u,x) *c*x - 120*df(u,x)*a*b *t *u
3 2 6 2 2 2 6
+ 840*df(u,x)*a*b *c*t *u - 1830*df(u,x)*a*b *c *t *u
2 4 3 2 6
- 1200*df(u,x)*a*b *t*u *x + 1260*df(u,x)*a*b*c *t *u
4 4 2 6
+ 4200*df(u,x)*a*b*c*t*u *x - 270*df(u,x)*a*c *t *u
2 4 2 2 6 2 6
- 1800*df(u,x)*a*c *t*u *x - 3000*df(u,x)*a*u *x - 24*df(u,x)*b *t *u
5 2 6 4 2 2 6 4 4
+ 252*df(u,x)*b *c*t *u - 990*df(u,x)*b *c *t *u - 240*df(u,x)*b *t*u *x
3 3 2 6 3 4
+ 1785*df(u,x)*b *c *t *u + 1680*df(u,x)*b *c*t*u *x
2 4 2 6 2 2 4
- 1485*df(u,x)*b *c *t *u - 3660*df(u,x)*b *c *t*u *x
2 2 2 5 2 6 3 4
- 600*df(u,x)*b *u *x + 567*df(u,x)*b*c *t *u + 2520*df(u,x)*b*c *t*u *x
2 2 6 2 6 4 4
+ 2100*df(u,x)*b*c*u *x - 81*df(u,x)*c *t *u - 540*df(u,x)*c *t*u *x
2 2 2 4 5 3 5
- 900*df(u,x)*c *u *x + 6000*df(u,x)*c*u + 48*b *t*u - 336*b *c*t*u
2 2 5 2 3 3 5 3
+ 732*b *c *t*u + 400*b *u *x - 504*b*c *t*u - 1400*b*c*u *x
4 5 2 3
+ 108*c *t*u + 600*c *u *x)/(3*(b - 3*c))}},
2 2
{{100*( - 10*df(u,x,2) - 2*b*u + c*u )},
2 3 3
100*(15*df(u,x) - 2*b*u + c*u )
{-----------------------------------,
3
2
- 1000*df(u,t)*df(u,x) - 1000*df(u,x,4)*df(u,x,2) - 200*df(u,x,4)*b*u
2 2
+ 100*df(u,x,4)*c*u + 500*df(u,x,3) + 400*df(u,x,3)*df(u,x)*b*u
2 2
- 200*df(u,x,3)*df(u,x)*c*u - 200*df(u,x,2) *b*u - 400*df(u,x,2) *c*u
2 2 3
- 400*df(u,x,2)*df(u,x) *b + 200*df(u,x,2)*df(u,x) *c - 200*df(u,x,2)*b*c*u
2 3 2 2 2 2 2
+ 100*df(u,x,2)*c *u - 1000*df(u,x) *a*u - 200*df(u,x) *b *u
2 2 2 2 2 3 6
+ 700*df(u,x) *b*c*u - 300*df(u,x) *c *u - 40*df(u,x)*a*b *t*u
2 6 2 6 4
+ 160*df(u,x)*a*b *c*t*u - 130*df(u,x)*a*b*c *t*u - 200*df(u,x)*a*b*u *x
3 6 4 5 6
+ 30*df(u,x)*a*c *t*u + 100*df(u,x)*a*c*u *x - 8*df(u,x)*b *t*u
4 6 3 2 6 3 4
+ 60*df(u,x)*b *c*t*u - 150*df(u,x)*b *c *t*u - 40*df(u,x)*b *u *x
2 3 6 2 4 4 6
+ 145*df(u,x)*b *c *t*u + 160*df(u,x)*b *c*u *x - 60*df(u,x)*b*c *t*u
2 4 5 6 3 4 3 5
- 130*df(u,x)*b*c *u *x + 9*df(u,x)*c *t*u + 30*df(u,x)*c *u *x + 8*b *u
2 5 2 5 3 5
- 32*b *c*u + 26*b*c *u - 6*c *u }},
{{-100},
{ - 100*u,
2 2
( - 300*df(u,x,4) - 300*df(u,x,2)*c*u - 150*df(u,x) *b + 150*df(u,x) *c
2 4 4 2 4
- 60*df(u,x)*a*b *t*u + 210*df(u,x)*a*b*c*t*u - 90*df(u,x)*a*c *t*u
2 4 4 3 4
- 300*df(u,x)*a*u *x - 12*df(u,x)*b *t*u + 84*df(u,x)*b *c*t*u
2 2 4 2 2 3 4
- 183*df(u,x)*b *c *t*u - 60*df(u,x)*b *u *x + 126*df(u,x)*b*c *t*u
2 4 4 2 2 2 3
+ 210*df(u,x)*b*c*u *x - 27*df(u,x)*c *t*u - 90*df(u,x)*c *u *x + 20*b *u
3 2 3
- 70*b*c*u + 30*c *u )/3}},
2 2
200*( - 10*df(u,x,2)*c*t - 3*c *t*u + 10*x)
{{----------------------------------------------},
c
2 2 3
200*(5*df(u,x) *c*t - c *t*u + 10*u*x)
{-----------------------------------------,
c
( - 2000*df(u,t)*df(u,x)*c*t - 2000*df(u,x,4)*df(u,x,2)*c*t
2 2 2
- 600*df(u,x,4)*c *t*u + 2000*df(u,x,4)*x + 1000*df(u,x,3) *c*t
2 2 2
+ 1200*df(u,x,3)*df(u,x)*c *t*u - 2000*df(u,x,3) - 1600*df(u,x,2) *c *t*u
2 2 2
- 2000*df(u,x,2)*df(u,x) *b*c*t + 2800*df(u,x,2)*df(u,x) *c *t
4 2 4 2 2
+ 90*df(u,x,2)*df(u,x)*b*c *t *u - 600*df(u,x,2)*df(u,x)*b*c *t*u *x
2 5 2 4
+ 1000*df(u,x,2)*df(u,x)*b*x - 180*df(u,x,2)*df(u,x)*c *t *u
3 2 2
+ 1200*df(u,x,2)*df(u,x)*c *t*u *x - 2000*df(u,x,2)*df(u,x)*c*x
3 3 2 2
- 600*df(u,x,2)*c *t*u + 2000*df(u,x,2)*c*u*x - 2000*df(u,x) *a*c*t*u
2 3 2 2 4 2 6
+ 600*df(u,x) *c *t*u + 1000*df(u,x) *c*x + 90*df(u,x)*a*c *t *u
2 4 2 2 6 2 6
- 600*df(u,x)*a*c *t*u *x + 1000*df(u,x)*a*u *x - 27*df(u,x)*c *t *u
4 4 2 2 2
+ 180*df(u,x)*c *t*u *x - 300*df(u,x)*c *u *x - 2000*df(u,x)*c*u
4 5 2 3
- 36*c *t*u + 200*c *u *x)/c}},
2
{{100*( - 10*df(u,x,2) - 3*c*u )},
2 3
{100*(5*df(u,x) - c*u ),
2
- 1000*df(u,t)*df(u,x) - 1000*df(u,x,4)*df(u,x,2) - 300*df(u,x,4)*c*u
2 2
+ 500*df(u,x,3) + 600*df(u,x,3)*df(u,x)*c*u - 800*df(u,x,2) *c*u
2 2
- 1000*df(u,x,2)*df(u,x) *b + 1400*df(u,x,2)*df(u,x) *c
3 4 2
+ 90*df(u,x,2)*df(u,x)*b*c *t*u - 300*df(u,x,2)*df(u,x)*b*c*u *x
4 4 2 2
- 180*df(u,x,2)*df(u,x)*c *t*u + 600*df(u,x,2)*df(u,x)*c *u *x
2 3 2 2 2 2 2
- 300*df(u,x,2)*c *u - 1000*df(u,x) *a*u + 300*df(u,x) *c *u
3 6 4 5 6
+ 90*df(u,x)*a*c *t*u - 300*df(u,x)*a*c*u *x - 27*df(u,x)*c *t*u
3 4 3 5
+ 90*df(u,x)*c *u *x - 18*c *u }},
{{ - 100*u},
2
{ - 50*u ,
2
( - 200*df(u,x,4)*u + 200*df(u,x,3)*df(u,x) - 100*df(u,x,2)
2 3
+ 60*df(u,x,2)*df(u,x)*b*c *t*u - 200*df(u,x,2)*df(u,x)*b*u*x
3 3
- 120*df(u,x,2)*df(u,x)*c *t*u + 400*df(u,x,2)*df(u,x)*c*u*x
2 2 5 3
- 200*df(u,x,2)*c*u + 60*df(u,x)*a*c *t*u - 200*df(u,x)*a*u *x
4 5 2 3 2 4
- 18*df(u,x)*c *t*u + 60*df(u,x)*c *u *x - 15*c *u )/2}},
{{-100},
{ - 100*u,
2 2
- 100*df(u,x,4) + 30*df(u,x,2)*df(u,x)*b*c *t*u - 100*df(u,x,2)*df(u,x)*b*x
3 2
- 60*df(u,x,2)*df(u,x)*c *t*u + 200*df(u,x,2)*df(u,x)*c*x
2 2 4
- 100*df(u,x,2)*c*u - 50*df(u,x) *c + 30*df(u,x)*a*c *t*u
2 4 4 2 2 2 3
- 100*df(u,x)*a*u *x - 9*df(u,x)*c *t*u + 30*df(u,x)*c *u *x - 10*c *u }},
2
{{3*( - 2*df(u,x,2) - c*u )},
2 3
{3*df(u,x) - c*u ,
2 2
3*( - 2*df(u,t)*df(u,x) - 2*df(u,x,4)*df(u,x,2) - df(u,x,4)*c*u + df(u,x,3)
2 2
+ 2*df(u,x,3)*df(u,x)*c*u - 2*df(u,x,2) *c*u - 2*df(u,x,2)*df(u,x) *b
2 2 3 2 2
+ 4*df(u,x,2)*df(u,x) *c - df(u,x,2)*c *u - 2*df(u,x) *a*u )}},
2
- x
{{-------},
c
2
- u*x
{---------,
c
2 2
( - df(u,x,4)*x + 2*df(u,x,3)*x - df(u,x,2)*c*u*x - 2*df(u,x,2)
2 2 2
- df(u,x) *c*x + 2*df(u,x)*c*u*x - c*u )/c}},
{{x},
{u*x,
2
df(u,x,4)*x - df(u,x,3) + df(u,x,2)*c*u*x + df(u,x) *c*x - df(u,x)*c*u}},
{{1},
{u,
2
df(u,x,4) + df(u,x,2)*c*u + df(u,x) *c}},
{{ - df(c_294,x) + c_303},
{u*( - df(c_294,x) + c_303),
df(c_294,t)*u + df(c_294,x,4)*df(u,x) - df(c_294,x,3)*df(u,x,2)
+ df(c_294,x,2)*df(u,x,3) - df(c_294,x)*df(u,x,4) - df(c_303,x,3)*df(u,x)
+ df(c_303,x,2)*df(u,x,2) - df(c_303,x)*df(u,x,3) + df(u,x,4)*c_303
2
- df(u,x,3)*c*c_294*u - df(u,x,2)*df(u,x)*b*c_294 - df(u,x)*a*c_294*u }},
{{ - 2*df(u,x,2)},
2
{df(u,x) ,
2
- 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 }},
{{2*u},
2
{u ,
2
2*df(u,x,4)*u - 2*df(u,x,3)*df(u,x) + df(u,x,2) }},
3
{{----},
2
a
3*u
{-----,
2
a
2 3
3*df(u,x,4) + 3*df(u,x,2)*c*u + 3*df(u,x) *c + a*u
-----------------------------------------------------}},
2
a
- 4*u
{{--------},
a
2
- 2*u
{---------,
a
2 4
- 4*df(u,x,4)*u + 4*df(u,x,3)*df(u,x) - 2*df(u,x,2) - a*u
--------------------------------------------------------------}},
a
{{ - 4*u},
2
{ - 2*u ,
2 4
- 4*df(u,x,4)*u + 4*df(u,x,3)*df(u,x) - 2*df(u,x,2) - a*u }},
- 3
{{------},
a
3
- 3*u - 3*df(u,x,4) - a*u
{--------,-----------------------}},
a a
{{ - df(c_254,x)},
{ - df(c_254,x)*u,
df(c_254,t)*u + df(c_254,x,4)*df(u,x) - df(c_254,x,3)*df(u,x,2)
+ df(c_254,x,2)*df(u,x,3) - df(c_254,x)*df(u,x,4) + c_259*u}},
{{ - 2*u},
2
{ - u ,
2
- 2*df(u,x,4)*u + 2*df(u,x,3)*df(u,x) - df(u,x,2) }},
{{-6},
{ - 6*u,
2 3
- 6*df(u,x,4) - 6*df(u,x,2)*c*u - 3*df(u,x) *c - 2*a*u }},
{{ - 4*u},
2
{ - 2*u ,
2 2
- 4*df(u,x,4)*u + 4*df(u,x,3)*df(u,x) - 2*df(u,x,2) - 4*df(u,x,2)*c*u
4
- a*u }},
{{-6},
{ - 6*u,
2 2 3
- 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 }},
2
x
{{----},
c
2
u*x
{------,
c
2 2
(df(u,x,4)*x - 2*df(u,x,3)*x + df(u,x,2)*c*u*x + 2*df(u,x,2)
2 2 2
+ df(u,x) *c*x - 2*df(u,x)*c*u*x + c*u )/c}},
{{ - x},
{ - u*x,
2
- df(u,x,4)*x + df(u,x,3) - df(u,x,2)*c*u*x - df(u,x) *c*x + df(u,x)*c*u}},
{{1},
{u,
2
df(u,x,4) + df(u,x,2)*c*u + df(u,x) *c}},
{{ - df(c_232,x)},
{ - df(c_232,x)*u,
df(c_232,t)*u + df(c_232,x,4)*df(u,x) - df(c_232,x,3)*df(u,x,2)
+ df(c_232,x,2)*df(u,x,3) - df(c_232,x)*df(u,x,4) + c_236*u}}}
comment -------------------------------------------------------------
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;
nodepnd {x,y,z,a1,a2,a3,b,s,r}$
depend x,t$
depend y,t$
depend z,t$
depend a1,x,t$
depend a2,y,t$
depend a3,z,t$
p_t:=a1+a2+a3$
conlaw1({{df(x,t) = - s*x + s*y,
df(y,t) = x*z + r*x - y,
df(z,t) = x*y - b*z},
{x,y,z},{t}
},
{0,0,t,{a1,a2,a3,s,r,b},{}})$
--------------------------------------------------------------------------
This is CONLAW1 - a program for calculating conservation laws of DEs
The DEs under investigation are :
x = - s*x + s*y
t
y =r*x + x*z - y
t
z = - b*z + x*y
t
for the function(s): x(t), y(t), z(t)
======================================================
A special ansatz of order 0 for the conserved current is investigated.
Conservation law:
1
s=---*b,
2
b*t
( - e ) * ( z + b*z - x*y )
t
+
( 0 ) * ( y - r*x - x*z + y )
t
+
b*t
2*e *x 1 1
( ---------- ) * ( x + ---*b*x - ---*b*y )
b t 2 2
=
b*t b*t 2
- e *b*z + e *x
df( -----------------------, t )
b
======================================================
The function c_473(x) is not constant!
======================================================
Conservation law:
s=0,
( 0 ) * ( z + b*z - x*y )
t
+
( 0 ) * ( y - r*x - x*z + y )
t
+
( c_473 ) * ( x )
x t
=
df( c_473, t )
======================================================
Conservation law:
b=1,
s=1,
2*t
( - 2*e *z ) * ( z - x*y + z )
t
+
2*t
( 2*e *y ) * ( y - r*x - x*z + y )
t
+
2*t
( - 2*e *r*x ) * ( x + x - y )
t
=
2*t 2 2*t 2 2*t 2
df( - e *r*x + e *y - e *z , t )
======================================================
Conservation law:
b=1,
r=0,
2*t
( - 2*e *z ) * ( z - x*y + z )
t
+
2*t
( 2*e *y ) * ( y - x*z + y )
t
+
( 0 ) * ( x + s*x - s*y )
t
=
2*t 2 2*t 2
df( e *y - e *z , t )
======================================================
Conservation law:
b=1,
r=0,
1
s=---,
2
t
( - e ) * ( z - x*y + z )
t
+
( 0 ) * ( y - x*z + y )
t
+
t 1 1
( 2*e *x ) * ( x + ---*x - ---*y )
t 2 2
=
t 2 t
df( e *x - e *z, t )
======================================================
Conservation law:
b=1,
r=0,
1
s=---,
2
2*t
( - 2*e *z ) * ( z - x*y + z )
t
+
2*t
( 2*e *y ) * ( y - x*z + y )
t
+
1 1
( 0 ) * ( x + ---*x - ---*y )
t 2 2
=
2*t 2 2*t 2
df( e *y - e *z , t )
======================================================
The function c_489(x) is not constant!
======================================================
Conservation law:
b=1,
r=0,
s=0,
2*t
( - 2*e *z ) * ( z - x*y + z )
t
+
2*t
( 2*e *y ) * ( y - x*z + y )
t
+
( 0 ) * ( x )
t
=
2*t 2 2*t 2
df( e *y - e *z , t )
======================================================
Conservation law:
b=1,
r=0,
s=0,
( 0 ) * ( z - x*y + z )
t
+
( 0 ) * ( y - x*z + y )
t
+
( c_489 ) * ( x )
x t
=
df( c_489, t )
======================================================
{{{0,0,df(c_489,x)},{c_489}},
2*t 2*t
{{ - 2*e *z,2*e *y,0},
2*t 2 2
{e *(y - z )}},
2*t 2*t
{{ - 2*e *z,2*e *y,0},
2*t 2 2
{e *(y - z )}},
t t
{{ - e ,0,2*e *x},
t 2
{e *(x - z)}},
2*t 2*t
{{ - 2*e *z,2*e *y,0},
2*t 2 2
{e *(y - z )}},
2*t
{{ - 2*e *z,
2*t
2*e *y,
2*t
- 2*e *r*x},
2*t 2 2 2
{e *( - r*x + y - z )}},
{{0,0,df(c_473,x)},{c_473}},
b*t
b*t 2*e *x
{{ - e ,0,----------},
b
b*t 2
e *( - b*z + x )
{--------------------}}}
b
clear p_t$
nodepnd {u,v,r,p_t,x,y,z,a1,a2,a3,b,s,r}$
end$
Time for test: 43198 ms, plus GC time: 1337 ms