Sun Aug 18 16:53:04 2002 run on Windows
off echo, dfprint$
-------------------------------------------------------
The following runs demonstrate the program LIEPDE for the
computation of infinitesimal symmetries. Times given
below refer to a 8 MB session under LINUX on a 133 MHz
Pentium PC with the CRACK version of April 1998.
-------------------------------------------------------
The first example is a single ODE with a parametric
function f=f(x) for which point symmetries are to be
determined.
(Time ~ 6 sec.)
-------------------------------------------------------
The ODE under investigation is :
2 2 3
df(y,x,2)= - df(f,x)*y - 3*df(y,x)*f - df(y,x)*y - 2*f *y - f*y + y
for the function(s) :
y(x)
The symmetries are:
-------- 1. Symmetry:
int(f,x) 1
xi_x=e *int(-----------,x)
int(f,x)
e
int(f,x) 1
eta_y= - e *int(-----------,x)*f*y - y
int(f,x)
e
-------- 2. Symmetry:
int(f,x)
xi_x= - e
int(f,x)
eta_y=e *f*y
--------
-------------------------------------------------------
The following example demonstrates a number of things.
The Burgers equation is investigated concerning third
order symmetries. The equation is used to substitute
df(u,t) and all derivatives of df(u,t). This computation
also shows that any equations that remain unsolved are
returned, like in this case the heat quation.
(Time ~ 15 sec.)
-------------------------------------------------------
The PDE under investigation is :
2
df(u,t)=df(u,x,2) + df(u,x)
for the function(s) :
u(x,t)
The symmetries are:
-------- 1. Symmetry:
xi_t=0
xi_x=0
2
eta_u=df(u,x,2) + df(u,x)
-------- 2. Symmetry:
xi_t=0
xi_x=0
2 2 2 2
eta_u=4*df(u,x,2)*t + 4*df(u,x) *t + 4*df(u,x)*t*x + 2*t + x
-------- 3. Symmetry:
xi_t=0
xi_x=0
2
eta_u=4*df(u,x,2)*t + 4*df(u,x) *t + 2*df(u,x)*x - 1
-------- 4. Symmetry:
xi_t=0
xi_x=0
3
eta_u=df(u,x,3) + 3*df(u,x,2)*df(u,x) + df(u,x)
-------- 5. Symmetry:
xi_t=0
xi_x=0
2 2 3 2
eta_u=4*df(u,x,3)*t + 12*df(u,x,2)*df(u,x)*t + 4*df(u,x,2)*t*x + 4*df(u,x) *t
2 2
+ 4*df(u,x) *t*x + df(u,x)*x - x
-------- 6. Symmetry:
xi_t=0
xi_x=0
3 3 2
eta_u=8*df(u,x,3)*t + 24*df(u,x,2)*df(u,x)*t + 12*df(u,x,2)*t *x
3 3 2 2 2 2
+ 8*df(u,x) *t + 12*df(u,x) *t *x + 12*df(u,x)*t + 6*df(u,x)*t*x + 6*t*x
3
+ x
-------- 7. Symmetry:
xi_t=0
xi_x=0
eta_u
3 2
=2*df(u,x,3)*t + 6*df(u,x,2)*df(u,x)*t + df(u,x,2)*x + 2*df(u,x) *t + df(u,x) *x
-------- 8. Symmetry:
xi_t=0
xi_x=0
eta_u=df(u,x)
-------- 9. Symmetry:
xi_t=0
xi_x=0
eta_u=2*df(u,x)*t + x
-------- 10. Symmetry:
xi_t=0
xi_x=0
eta_u=1
--------
Further symmetries:
xi_t=0
xi_x=0
c_27 + c_32
eta_u=-------------
u
e
with c_27(x,t), c_32(t)
which still have to satisfy:
0=2*df( - c_27,t) - 2*df( - c_27,x,2) + df( - 2*c_32,t)
-------------------------------------------------------
Now the same equation is investigated, this time only
df(u,x,2) and its derivatives are substituted. As a
consequence less jet-variables (u-derivatives of lower
order) are generated in the process of formulating the
symmetry conditions. Less jet-variables in which the
conditions have to be fulfilled identically means less
overdetermined conditions and more solutions which to
compute takes longer than before.
(Time ~ 85 sec.)
-------------------------------------------------------
The PDE under investigation is :
2
df(u,x,2)=df(u,t) - df(u,x)
for the function(s) :
u(x,t)
The symmetries are:
-------- 1. Symmetry:
xi_t=0
xi_x=0
eta_u
2
= - 2*df(u,t,x)*df(u,t) - df(u,t,2,x) - df(u,t,2)*df(u,x) - df(u,t) *df(u,x)
-------- 2. Symmetry:
xi_t=0
xi_x=0
2 2 2
eta_u= - 16*df(u,t,x)*df(u,t)*t - 2*df(u,t,x)*x - 8*df(u,t,2,x)*t
2 2 2
- 8*df(u,t,2)*df(u,x)*t - 8*df(u,t,2)*t*x - 8*df(u,t) *df(u,x)*t
2 2
- 8*df(u,t) *t*x - 2*df(u,t)*df(u,x)*x + 2*df(u,t)*x - df(u,x)
-------- 3. Symmetry:
xi_t=0
xi_x=0
4 2 2 4
eta_u= - 32*df(u,t,x)*df(u,t)*t - 24*df(u,t,x)*t *x - 16*df(u,t,2,x)*t
4 3 2 4
- 16*df(u,t,2)*df(u,x)*t - 32*df(u,t,2)*t *x - 16*df(u,t) *df(u,x)*t
2 3 2 2 2
- 32*df(u,t) *t *x - 24*df(u,t)*df(u,x)*t *x + 24*df(u,t)*t *x
3 2 2 4 3
- 8*df(u,t)*t*x + 60*df(u,x)*t + 24*df(u,x)*t*x - df(u,x)*x + 36*t*x + 6*x
-------- 4. Symmetry:
xi_t=0
xi_x=0
5 4 3 2
eta_u= - 64*df(u,t,x)*df(u,t)*t - 160*df(u,t,x)*t - 80*df(u,t,x)*t *x
5 5 4
- 32*df(u,t,2,x)*t - 32*df(u,t,2)*df(u,x)*t - 80*df(u,t,2)*t *x
2 5 2 4 4
- 32*df(u,t) *df(u,x)*t - 80*df(u,t) *t *x - 160*df(u,t)*df(u,x)*t
3 2 3 2 3
- 80*df(u,t)*df(u,x)*t *x - 240*df(u,t)*t *x - 40*df(u,t)*t *x
3 2 2 4 2 3 5
- 120*df(u,x)*t - 120*df(u,x)*t *x - 10*df(u,x)*t*x - 60*t *x - 20*t*x - x
-------- 5. Symmetry:
xi_t=0
xi_x=0
3 2 3
eta_u= - 32*df(u,t,x)*df(u,t)*t - 12*df(u,t,x)*t*x - 16*df(u,t,2,x)*t
3 2 2 3
- 16*df(u,t,2)*df(u,x)*t - 24*df(u,t,2)*t *x - 16*df(u,t) *df(u,x)*t
2 2 2 3
- 24*df(u,t) *t *x - 12*df(u,t)*df(u,x)*t*x + 12*df(u,t)*t*x - 2*df(u,t)*x
2
- 6*df(u,x)*t + 6*df(u,x)*x - 9*x
-------- 6. Symmetry:
xi_t=0
xi_x=0
eta_u= - 4*df(u,t,x)*df(u,t)*t - 2*df(u,t,2,x)*t - 2*df(u,t,2)*df(u,x)*t
2 2
- df(u,t,2)*x - 2*df(u,t) *df(u,x)*t - df(u,t) *x
-------- 7. Symmetry:
xi_t=0
xi_x=0
3
eta_u= - df(u,t,3) - 3*df(u,t,2)*df(u,t) - df(u,t)
-------- 8. Symmetry:
xi_t=0
xi_x=0
2
eta_u= - 8*df(u,t,x)*df(u,t)*t*x + df(u,t,x)*x - 4*df(u,t,3)*t
2
- 4*df(u,t,2,x)*t*x - 12*df(u,t,2)*df(u,t)*t - 4*df(u,t,2)*df(u,x)*t*x
2 3 2 2 2 2
- df(u,t,2)*x - 4*df(u,t) *t - 4*df(u,t) *df(u,x)*t*x - df(u,t) *x
+ df(u,t)*df(u,x)*x
-------- 9. Symmetry:
xi_t=0
xi_x=0
3 2 3
eta_u= - 64*df(u,t,x)*df(u,t)*t *x + 24*df(u,t,x)*t *x - 8*df(u,t,x)*t*x
4 3 4
- 16*df(u,t,3)*t - 32*df(u,t,2,x)*t *x - 48*df(u,t,2)*df(u,t)*t
3 2 2 3 4
- 32*df(u,t,2)*df(u,x)*t *x - 24*df(u,t,2)*t *x - 16*df(u,t) *t
2 3 2 2 2 2
- 32*df(u,t) *df(u,x)*t *x - 24*df(u,t) *t *x + 24*df(u,t)*df(u,x)*t *x
3 2 4
- 8*df(u,t)*df(u,x)*t*x + 24*df(u,t)*t*x - df(u,t)*x - 24*df(u,x)*t*x
3 2
+ 6*df(u,x)*x - 30*t - 15*x
-------- 10. Symmetry:
xi_t=0
xi_x=0
5 4 3 3
eta_u= - 384*df(u,t,x)*df(u,t)*t *x - 960*df(u,t,x)*t *x - 160*df(u,t,x)*t *x
6 5 6
- 64*df(u,t,3)*t - 192*df(u,t,2,x)*t *x - 192*df(u,t,2)*df(u,t)*t
5 5 4 2
- 192*df(u,t,2)*df(u,x)*t *x - 480*df(u,t,2)*t - 240*df(u,t,2)*t *x
3 6 2 5 2 5
- 64*df(u,t) *t - 192*df(u,t) *df(u,x)*t *x - 480*df(u,t) *t
2 4 2 4 3 3
- 240*df(u,t) *t *x - 960*df(u,t)*df(u,x)*t *x - 160*df(u,t)*df(u,x)*t *x
4 3 2 2 4 3
- 720*df(u,t)*t - 720*df(u,t)*t *x - 60*df(u,t)*t *x - 720*df(u,x)*t *x
2 3 5 3 2 2 4 6
- 240*df(u,x)*t *x - 12*df(u,x)*t*x - 120*t - 180*t *x - 30*t*x - x
-------- 11. Symmetry:
xi_t=0
xi_x=0
4 3 2 3
eta_u= - 160*df(u,t,x)*df(u,t)*t *x + 80*df(u,t,x)*t *x - 40*df(u,t,x)*t *x
5 4 5
- 32*df(u,t,3)*t - 80*df(u,t,2,x)*t *x - 96*df(u,t,2)*df(u,t)*t
4 3 2 3 5
- 80*df(u,t,2)*df(u,x)*t *x - 80*df(u,t,2)*t *x - 32*df(u,t) *t
2 4 2 3 2 3
- 80*df(u,t) *df(u,x)*t *x - 80*df(u,t) *t *x + 80*df(u,t)*df(u,x)*t *x
2 3 3 2 2
- 40*df(u,t)*df(u,x)*t *x + 360*df(u,t)*t + 120*df(u,t)*t *x
4 2 3 5 2
- 10*df(u,t)*t*x + 420*df(u,x)*t *x + 60*df(u,x)*t*x - df(u,x)*x + 120*t
2 4
+ 120*t*x + 10*x
-------- 12. Symmetry:
xi_t=0
xi_x=0
2 3
eta_u= - 48*df(u,t,x)*df(u,t)*t *x + 12*df(u,t,x)*t*x - 2*df(u,t,x)*x
3 2 3
- 16*df(u,t,3)*t - 24*df(u,t,2,x)*t *x - 48*df(u,t,2)*df(u,t)*t
2 2 3 3
- 24*df(u,t,2)*df(u,x)*t *x - 12*df(u,t,2)*t*x - 16*df(u,t) *t
2 2 2 2
- 24*df(u,t) *df(u,x)*t *x - 12*df(u,t) *t*x + 12*df(u,t)*df(u,x)*t*x
3 2
- 2*df(u,t)*df(u,x)*x + 6*df(u,t)*x - 6*df(u,x)*x + 3
-------- 13. Symmetry:
xi_t=0
xi_x=0
eta_u= - 2*df(u,t,x)*df(u,t)*x - 2*df(u,t,3)*t - df(u,t,2,x)*x
3
- 6*df(u,t,2)*df(u,t)*t - df(u,t,2)*df(u,x)*x - 2*df(u,t) *t
2
- df(u,t) *df(u,x)*x
-------- 14. Symmetry:
xi_t=0
xi_x=0
2
eta_u=df(u,t,2) + df(u,t)
-------- 15. Symmetry:
xi_t=0
xi_x=0
2 2 2
eta_u= - 8*df(u,t,x)*t*x - 8*df(u,t,2)*t - 8*df(u,t) *t
2
- 8*df(u,t)*df(u,x)*t*x - 2*df(u,t)*x + 2*df(u,x)*x - 1
-------- 16. Symmetry:
xi_t=0
xi_x=0
3 4 2 4
eta_u= - 32*df(u,t,x)*t *x - 16*df(u,t,2)*t - 16*df(u,t) *t
3 3 2 2 2
- 32*df(u,t)*df(u,x)*t *x - 48*df(u,t)*t - 24*df(u,t)*t *x - 48*df(u,x)*t *x
3 2 2 4
- 8*df(u,x)*t*x - 12*t - 12*t*x - x
-------- 17. Symmetry:
xi_t=0
xi_x=0
2 3 2 3
eta_u= - 12*df(u,t,x)*t *x - 8*df(u,t,2)*t - 8*df(u,t) *t
2 2 3
- 12*df(u,t)*df(u,x)*t *x - 6*df(u,t)*t*x + 6*df(u,x)*t*x - df(u,x)*x + 6*t
2
+ 3*x
-------- 18. Symmetry:
xi_t=0
xi_x=0
2
eta_u= - df(u,t,x)*x - 2*df(u,t,2)*t - 2*df(u,t) *t - df(u,t)*df(u,x)*x
-------- 19. Symmetry:
xi_t=0
xi_x=0
eta_u=df(u,t,x) + df(u,t)*df(u,x)
-------- 20. Symmetry:
xi_t=0
xi_x=0
2 2
eta_u= - 4*df(u,t,x)*t - 4*df(u,t)*df(u,x)*t - 4*df(u,t)*t*x + 2*df(u,x)*t
2
- df(u,x)*x + 2*x
-------- 21. Symmetry:
xi_t=0
xi_x=0
3 3 2 2
eta_u= - 8*df(u,t,x)*t - 8*df(u,t)*df(u,x)*t - 12*df(u,t)*t *x - 12*df(u,x)*t
2 3
- 6*df(u,x)*t*x - 6*t*x - x
-------- 22. Symmetry:
xi_t=0
xi_x=0
eta_u= - 4*df(u,t,x)*t - 4*df(u,t)*df(u,x)*t - 2*df(u,t)*x + df(u,x)
-------- 23. Symmetry:
xi_t=0
xi_x=0
eta_u=df(u,t)
-------- 24. Symmetry:
xi_t=0
xi_x=0
2 2
eta_u= - 4*df(u,t)*t - 4*df(u,x)*t*x - 2*t - x
-------- 25. Symmetry:
xi_t=0
xi_x=0
eta_u= - 2*df(u,t)*t - df(u,x)*x + 1
-------- 26. Symmetry:
xi_t=0
xi_x=0
eta_u= - 2*df(u,x)*t - x
-------- 27. Symmetry:
xi_t=0
xi_x=0
eta_u=df(u,x)
-------- 28. Symmetry:
xi_t=0
xi_x=0
eta_u=1
--------
Further symmetries:
xi_t=0
xi_x=0
c_92
eta_u=------
u
e
with c_92(x,t)
which still have to satisfy:
0=df(c_92,t) - df(c_92,x,2)
-------------------------------------------------------
The following example includes the Karpman equations
for three unknown functions in 4 variables.
If point symmetries are to be computed for a single
equation or a system of equations of higher than first
order then there is the option to formulate at first
preliminary conditions for each equation, have CRACK
solving these conditions before the full set of conditions
is formulated and solved. This strategy is adopted if a
lisp flag prelim_ has the value t. The default value
is nil.
Similarly, if a system of equations is to be investigated
and a flag individual_ has the value t then symmetry
conditions are formulated and investigated for each
individual equation successively. The default value is nil.
It is advantageous to split a large set of conditions
into smaller sets to be investigated successively if
each set is sufficiently overdetermined to be solvable
quickly. Then any substitutions are done in the smaller
set and the next set of conditions is shorter. For
example, for the Karpman equations below the speedup for
prelim_:=t; individual_:=t; is a factor of 10.
(Time ~ 1 min.)
-------------------------------------------------------
Time: 210863 ms plus GC time: 4817 ms
The PDE-system under investigation is :
2 2 2 2 2
df(v,x,2)=( - 4*df(f,t)*a2*r - 2*df(f,x) *a2*r *s1 - 2*df(f,y) *a2*r *s1
2 2 2 2
- 2*df(f,z) *a2*r *s2 - 4*df(f,z)*a2*r *w1 - 2*df(r,x) *a2*s1
2
- 2*df(r,y) *a2*s1 - 2*df(r,z,2)*a2*r*s1 + 2*df(r,z,2)*a2*r*s2
2 2
- 2*df(r,z) *a2*s1 + df(v,t,2)*s1 - df(v,y,2)*s1*w2
2 2 2
- df(v,z,2)*s1*w2 - 4*a1*a2*r *v)/(s1*w2 )
2 2 2
df(r,x,2)=(2*df(f,t)*r + df(f,x) *r*s1 + df(f,y) *r*s1 + df(f,z) *r*s2
+ 2*df(f,z)*r*w1 - df(r,y,2)*s1 - df(r,z,2)*s2 + 2*a1*r*v)/s1
df(f,x,2)=( - 2*df(f,x)*df(r,x)*s1 - df(f,y,2)*r*s1 - 2*df(f,y)*df(r,y)*s1
- df(f,z,2)*r*s2 - 2*df(f,z)*df(r,z)*s2 - 2*df(r,t) - 2*df(r,z)*w1)/
(r*s1)
for the function(s) :
r(t,z,y,x), f(t,z,y,x), v(t,z,y,x)
=============== Initializations
time for initializations: 1312 ms GC time : 0 ms
=============== Preconditions for the 1. equation
time to formulate conditions: 7342 ms GC time : 0 ms
CRACK needed : 28442 ms GC time : 0 ms
=============== Preconditions for the 2. equation
=============== Preconditions for the 3. equation
time to formulate conditions: 2814 ms GC time : 0 ms
CRACK needed : 6637 ms GC time : 1152 ms
=============== Full conditions for the 1. equation
time to formulate conditions: 2193 ms GC time : 0 ms
CRACK needed : 54725 ms GC time : 1312 ms
=============== Full conditions for the 2. equation
time to formulate conditions: 691 ms GC time : 0 ms
CRACK needed : 2504 ms GC time : 0 ms
=============== Full conditions for the 3. equation
time to formulate conditions: 1111 ms GC time : 0 ms
CRACK needed : 3545 ms GC time : 0 ms
The symmetries are:
-------- 1. Symmetry:
xi_x=0
xi_y=0
xi_z=0
xi_t=0
eta_r=0
- t
eta_f=-------
s1*s2
1
eta_v=----------
a1*s1*s2
-------- 2. Symmetry:
xi_x=0
xi_y=0
xi_z=0
xi_t=0
eta_r=0
2
- t
eta_f=-------
s1*s2
2*t
eta_v=----------
a1*s1*s2
-------- 3. Symmetry:
xi_x=0
xi_y=0
xi_z=0
xi_t=0
eta_r=0
1
eta_f=-------
s1*s2
eta_v=0
-------- 4. Symmetry:
xi_x=0
xi_y=0
xi_z=0
xi_t=1
eta_r=0
eta_f=0
eta_v=0
-------- 5. Symmetry:
xi_x=0
xi_y=0
1
xi_z=----
s1
xi_t=0
eta_r=0
- w1
eta_f=-------
s1*s2
eta_v=0
-------- 6. Symmetry:
xi_x=0
1
xi_y=-------
s1*s2
xi_z=0
xi_t=0
eta_r=0
eta_f=0
eta_v=0
-------- 7. Symmetry:
y
xi_x=-------
s1*s2
- x
xi_y=-------
s1*s2
xi_z=0
xi_t=0
eta_r=0
eta_f=0
eta_v=0
-------- 8. Symmetry:
1
xi_x=-------
s1*s2
xi_y=0
xi_z=0
xi_t=0
eta_r=0
eta_f=0
eta_v=0
--------
Time: 118073 ms plus GC time: 2464 ms
-------------------------------------------------------
In the following example a system of two equations (by
V.Sokolov) is investigated concerning a special ansatz for
4th order symmetries. The ansatz for the symmetries includes
two unknown functions f,g. Because x is the second variable
in the list of variables {t,x}, the name u!`2 stands for
df(u,x).
Because higher order symmetries are investigated we have
to set prelim_:=nil. The symmetries to be calculated are
lengthy and therefore conditions are not very overdetermined.
In that case CRACK can take long to solve a single
subset of conditions. The complete set of conditions would
have been more overdetermined and easier to solve. Therefore
the advantage of first formulating all conditions and then
solving them together with one CRACK call is that having
more equations, the chance of finding short integrable
equations among then is higher, i.e. CRACK has more freedom
in optimizing the computation. Therefore individual_:=nil
is more appropriate in this example.
Because 4th order conditions are to be computed the
`binding stack size' is increased.
(Time ~ 5 min.)
-------------------------------------------------------
The PDE-system under investigation is :
df(u,t)=df(u,x,2) + df(u,x)*u + df(u,x)*v + df(v,x)*u
df(v,t)=df(u,x)*v - df(v,x,2) + df(v,x)*u + df(v,x)*v
for the function(s) :
u(t,x), v(t,x)
The symmetries are:
-------- 1. Symmetry:
xi_t=0
xi_x=0
eta_u=(2*df(u,x,4) + 4*df(u,x,3)*u + 4*df(u,x,3)*v + 10*df(u,x,2)*df(u,x)
2
+ 6*df(u,x,2)*df(v,x) + 3*df(u,x,2)*u + 9*df(u,x,2)*u*v
2 2 2
+ 3*df(u,x,2)*v + 2*df(u,x,2) + 6*df(u,x) *u + 9*df(u,x) *v
+ 4*df(u,x)*df(v,x,2) + 9*df(u,x)*df(v,x)*u + 6*df(u,x)*df(v,x)*v
3 2 2
+ df(u,x)*u + 9*df(u,x)*u *v + 9*df(u,x)*u*v + 2*df(u,x)*u
3 3
+ df(u,x)*v + 2*df(u,x)*v + 2*df(v,x,3)*u + 3*df(v,x)*u
2 2
+ 9*df(v,x)*u *v + 3*df(v,x)*u*v + 2*df(v,x)*u)/2
eta_v=(2*df(u,x,3)*v + 4*df(u,x,2)*df(v,x) + 6*df(u,x)*df(v,x,2)
2
- 6*df(u,x)*df(v,x)*u - 9*df(u,x)*df(v,x)*v + 3*df(u,x)*u *v
2 3
+ 9*df(u,x)*u*v + 3*df(u,x)*v + 2*df(u,x)*v - 2*df(v,x,4)
2
+ 4*df(v,x,3)*u + 4*df(v,x,3)*v + 10*df(v,x,2)*df(v,x) - 3*df(v,x,2)*u
2 2
- 9*df(v,x,2)*u*v - 3*df(v,x,2)*v - 2*df(v,x,2) - 9*df(v,x) *u
2 3 2 2
- 6*df(v,x) *v + df(v,x)*u + 9*df(v,x)*u *v + 9*df(v,x)*u*v
3
+ 2*df(v,x)*u + df(v,x)*v + 2*df(v,x)*v)/2
-------- 2. Symmetry:
xi_t=0
xi_x=0
eta_u=(2*df(u,x,4) + 4*df(u,x,3)*u + 4*df(u,x,3)*v + 10*df(u,x,2)*df(u,x)
2
+ 6*df(u,x,2)*df(v,x) + 4*df(u,x,2)*t + 3*df(u,x,2)*u + 9*df(u,x,2)*u*v
2 2 2
+ 3*df(u,x,2)*v + 6*df(u,x) *u + 9*df(u,x) *v + 4*df(u,x)*df(v,x,2)
+ 9*df(u,x)*df(v,x)*u + 6*df(u,x)*df(v,x)*v + 4*df(u,x)*t*u
3 2 2
+ 4*df(u,x)*t*v + df(u,x)*u + 9*df(u,x)*u *v + 9*df(u,x)*u*v
3
+ df(u,x)*v + 2*df(u,x)*x + 2*df(v,x,3)*u + 4*df(v,x)*t*u
3 2 2
+ 3*df(v,x)*u + 9*df(v,x)*u *v + 3*df(v,x)*u*v + 2*u)/2
eta_v=(2*df(u,x,3)*v + 4*df(u,x,2)*df(v,x) + 6*df(u,x)*df(v,x,2)
- 6*df(u,x)*df(v,x)*u - 9*df(u,x)*df(v,x)*v + 4*df(u,x)*t*v
2 2 3
+ 3*df(u,x)*u *v + 9*df(u,x)*u*v + 3*df(u,x)*v - 2*df(v,x,4)
+ 4*df(v,x,3)*u + 4*df(v,x,3)*v + 10*df(v,x,2)*df(v,x) - 4*df(v,x,2)*t
2 2 2
- 3*df(v,x,2)*u - 9*df(v,x,2)*u*v - 3*df(v,x,2)*v - 9*df(v,x) *u
2 3
- 6*df(v,x) *v + 4*df(v,x)*t*u + 4*df(v,x)*t*v + df(v,x)*u
2 2 3
+ 9*df(v,x)*u *v + 9*df(v,x)*u*v + df(v,x)*v + 2*df(v,x)*x + 2*v)/2
-------- 3. Symmetry:
xi_t=0
xi_x=0
eta_u=(2*df(u,x,4) + 4*df(u,x,3)*u + 4*df(u,x,3)*v + 8*df(u,x,3)
2
+ 10*df(u,x,2)*df(u,x) + 6*df(u,x,2)*df(v,x) + 3*df(u,x,2)*u
2
+ 9*df(u,x,2)*u*v + 12*df(u,x,2)*u + 3*df(u,x,2)*v + 12*df(u,x,2)*v
2 2 2
+ 6*df(u,x) *u + 9*df(u,x) *v + 12*df(u,x) + 4*df(u,x)*df(v,x,2)
+ 9*df(u,x)*df(v,x)*u + 6*df(u,x)*df(v,x)*v + 12*df(u,x)*df(v,x)
3 2 2 2
+ df(u,x)*u + 9*df(u,x)*u *v + 6*df(u,x)*u + 9*df(u,x)*u*v
3 2
+ 24*df(u,x)*u*v + df(u,x)*v + 6*df(u,x)*v + 2*df(v,x,3)*u
3 2 2 2
+ 3*df(v,x)*u + 9*df(v,x)*u *v + 12*df(v,x)*u + 3*df(v,x)*u*v
+ 12*df(v,x)*u*v)/2
eta_v=(2*df(u,x,3)*v + 4*df(u,x,2)*df(v,x) + 6*df(u,x)*df(v,x,2)
- 6*df(u,x)*df(v,x)*u - 9*df(u,x)*df(v,x)*v - 12*df(u,x)*df(v,x)
2 2 3
+ 3*df(u,x)*u *v + 9*df(u,x)*u*v + 12*df(u,x)*u*v + 3*df(u,x)*v
2
+ 12*df(u,x)*v - 2*df(v,x,4) + 4*df(v,x,3)*u + 4*df(v,x,3)*v
2
+ 8*df(v,x,3) + 10*df(v,x,2)*df(v,x) - 3*df(v,x,2)*u - 9*df(v,x,2)*u*v
2 2
- 12*df(v,x,2)*u - 3*df(v,x,2)*v - 12*df(v,x,2)*v - 9*df(v,x) *u
2 2 3 2
- 6*df(v,x) *v - 12*df(v,x) + df(v,x)*u + 9*df(v,x)*u *v
2 2 3
+ 6*df(v,x)*u + 9*df(v,x)*u*v + 24*df(v,x)*u*v + df(v,x)*v
2
+ 6*df(v,x)*v )/2
-------- 4. Symmetry:
xi_t=0
xi_x=0
eta_u=(2*df(u,x,4) + 8*df(u,x,3)*t + 4*df(u,x,3)*u + 4*df(u,x,3)*v
+ 10*df(u,x,2)*df(u,x) + 6*df(u,x,2)*df(v,x) + 12*df(u,x,2)*t*u
2 2
+ 12*df(u,x,2)*t*v + 3*df(u,x,2)*u + 9*df(u,x,2)*u*v + 3*df(u,x,2)*v
2 2 2
+ 4*df(u,x,2)*x + 12*df(u,x) *t + 6*df(u,x) *u + 9*df(u,x) *v
+ 4*df(u,x)*df(v,x,2) + 12*df(u,x)*df(v,x)*t + 9*df(u,x)*df(v,x)*u
2
+ 6*df(u,x)*df(v,x)*v + 6*df(u,x)*t*u + 24*df(u,x)*t*u*v
2 3 2 2
+ 6*df(u,x)*t*v + df(u,x)*u + 9*df(u,x)*u *v + 9*df(u,x)*u*v
3
+ 4*df(u,x)*u*x + df(u,x)*v + 4*df(u,x)*v*x + 16*df(u,x)
2 3
+ 2*df(v,x,3)*u + 12*df(v,x)*t*u + 12*df(v,x)*t*u*v + 3*df(v,x)*u
2 2 2
+ 9*df(v,x)*u *v + 3*df(v,x)*u*v + 4*df(v,x)*u*x + 2*u + 6*u*v)/2
eta_v=(2*df(u,x,3)*v + 4*df(u,x,2)*df(v,x) + 6*df(u,x)*df(v,x,2)
- 12*df(u,x)*df(v,x)*t - 6*df(u,x)*df(v,x)*u - 9*df(u,x)*df(v,x)*v
2 2 2
+ 12*df(u,x)*t*u*v + 12*df(u,x)*t*v + 3*df(u,x)*u *v + 9*df(u,x)*u*v
3
+ 3*df(u,x)*v + 4*df(u,x)*v*x - 2*df(v,x,4) + 8*df(v,x,3)*t
+ 4*df(v,x,3)*u + 4*df(v,x,3)*v + 10*df(v,x,2)*df(v,x)
2
- 12*df(v,x,2)*t*u - 12*df(v,x,2)*t*v - 3*df(v,x,2)*u - 9*df(v,x,2)*u*v
2 2 2
- 3*df(v,x,2)*v - 4*df(v,x,2)*x - 12*df(v,x) *t - 9*df(v,x) *u
2 2 2
- 6*df(v,x) *v + 6*df(v,x)*t*u + 24*df(v,x)*t*u*v + 6*df(v,x)*t*v
3 2 2
+ df(v,x)*u + 9*df(v,x)*u *v + 9*df(v,x)*u*v + 4*df(v,x)*u*x
3 2
+ df(v,x)*v + 4*df(v,x)*v*x + 6*u*v + 2*v )/2
-------- 5. Symmetry:
xi_t=0
xi_x=0
eta_u=(2*df(u,x,4) + 4*df(u,x,3)*u + 4*df(u,x,3)*v + 10*df(u,x,2)*df(u,x)
2
+ 6*df(u,x,2)*df(v,x) + 3*df(u,x,2)*u + 9*df(u,x,2)*u*v
2 2 2
+ 3*df(u,x,2)*v + 6*df(u,x) *u + 9*df(u,x) *v + 4*df(u,x)*df(v,x,2)
3
+ 9*df(u,x)*df(v,x)*u + 6*df(u,x)*df(v,x)*v + df(u,x)*u
2 2 3
+ 9*df(u,x)*u *v + 9*df(u,x)*u*v + df(u,x)*v + 2*df(u,x)
3 2 2
+ 2*df(v,x,3)*u + 3*df(v,x)*u + 9*df(v,x)*u *v + 3*df(v,x)*u*v )/2
eta_v=(2*df(u,x,3)*v + 4*df(u,x,2)*df(v,x) + 6*df(u,x)*df(v,x,2)
2
- 6*df(u,x)*df(v,x)*u - 9*df(u,x)*df(v,x)*v + 3*df(u,x)*u *v
2 3
+ 9*df(u,x)*u*v + 3*df(u,x)*v - 2*df(v,x,4) + 4*df(v,x,3)*u
2
+ 4*df(v,x,3)*v + 10*df(v,x,2)*df(v,x) - 3*df(v,x,2)*u
2 2 2
- 9*df(v,x,2)*u*v - 3*df(v,x,2)*v - 9*df(v,x) *u - 6*df(v,x) *v
3 2 2 3
+ df(v,x)*u + 9*df(v,x)*u *v + 9*df(v,x)*u*v + df(v,x)*v + 2*df(v,x))
/2
--------
Time for test: 1062859 ms, plus GC time: 13951 ms