reduce-historical/r38/packages/crack/conlaw.rlg

Tue Feb 10 12:27:18 2004 run on Linux
%*******************************************************************%
%                                                                   %
%                      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: 38149 ms, plus GC time: 610 ms