Problem l1o24
Unknowns |
Inequalities |
Equations |
Solution 1 |
Solution 2 |
Solution 3 |
Solution 4 |
Relevance |
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Unknowns
All solutions for the following 49 unknowns have to be determined:
a1, a2, ..., a10, b1, b2, ..., b38, b39
Inequalities
Each of the following lists represents one inequality which states
that not all unknowns in this list may vanish. These inequalities
filter out solutions which are trivial for the application.
{a1,a6},
{a4,a5},
{a7,a8, a9},
{b1,b19},
{b1,b2,b3,b4,b5,b6,b7,b8, b9,b10,b11,b12,b13,b14,b15,b16,b17,b18},
{b19,b20,b21,b22,b23,b24,b25,b26,b27,b28,
b29,b30,b31,b32,b33,b34,b35,b36,b37,b38,b39}
Equations
All comma separated 123 expressions involving 770 terms have to vanish.
All terms are products of one a- and one b-unknown.
a10*b39,
a3*b23,
a3*b27,
a3*b2,
a3*b4,
a3*b7,
a5*b23,
a10*b31,
2*a10*b18 - a5*b39,
a6*b23 - 2*a8*b19,
3*a3*b29 - a8*b7,
a1*b2 - 2*a2*b1,
a1*b3 - 5*a2*b1,
a2*b7 - 3*a3*b6,
a9*b7 - 2*a3*b30,
a9*b31 + a5*b27,
a4*b20 - a7* b9,
4*a1*b4 - 3*a2*b2 - 24*a3*b1,
2*a1*b5 - 3*a2*b2 - 18*a3*b1,
10*a1*b7 - 3*a3*b4 - 2*a3*b5,
8*a10*b19 + a4*b23 - 4*a6*b31,
12*a10*b19 + a4*b22 - 2*a6*b32,
a9*b16 - a5*b30 + 2*a5*b7,
2*a10*b36 + a5*b36 - a8*b18,
a5*b35 - a7*b18 + 4*a7*b39,
2*a10*b34 + a4*b35 - a7*b17,
a9*b18 - 2* a9*b39 - a5*b38,
2*a10*b10 - a4*b35 + a7*b17,
6*a10*b17 + a4*b18 - 6*a4*b39 - a5*b17,
8* a9*b19 + a2*b23 - 4*a6*b27 + 2*a8*b23,
6* a9*b23 + 2*a2*b27 + 3*a3*b22 - 6*a6*b29,
2*a1*b4 + 4*a1*b5 - 3*a2*b3 - 36*a3*b1,
6*a1*b6 - 2*a2*b4 - 18*a3*b2 - 3*a3*b3,
6*a1*b6 - a2*b5 - 6*a3*b2 - 6*a3*b3,
a1*b20 - a6*b20 - a7*b1 + a7*b19,
a5*b22 - 2*a6*b36 + 2*a7*b31 + 2*a8*b31,
12*a10*b19 + a4*b22 - 2*a6*b31 - 2*a6*b32,
3*a10*b23 + a4*b27 + a5*b22 - 2*a6*b36,
8*a10*b19 + a4*b21 - 2*a6*b34 - a8* b9,
2* a9*b36 + a3*b36 + 3*a5*b29 - a8*b16,
6*a10*b19 + 3*a4*b20 - 2*a6*b33 - a7*b8,
2* a9* b9 - a4*b24 - 2*a5*b20 + a7*b13,
a1* b9 - a4*b1 + a4*b19 - a6* b9,
a1*b23 - 2*a6*b22 - a6*b23 + 4*a7*b19 + 6*a8*b19,
a1*b22 - 2*a6*b21 - a6*b22 + 6*a7*b19 + 4*a8*b19,
12* a9*b19 + a2*b22 - 2*a6*b26 - 2*a6*b27 + 2*a8*b22,
2* a9*b22 + 2*a2*b25 + 6*a3*b20 - 2*a6*b28 - a7*b5,
6*a10*b23 + a4*b26 + 2*a5*b21 - 2*a6*b37 - a8*b13,
8*a10*b19 + 4*a4*b20 - 2*a6*b34 - a7* b9 - a7*b8,
2* a9*b13 - 2* a9*b34 - a4*b28 - 2*a5*b24 + a7*b15,
a9*b16 - a9*b38 + a10*b30 - a3*b38 - 2*a5*b30,
2* a9* b9 - a4*b24 - 2*a5*b20 + a5*b2 + a7*b13,
a1* b9 + 3*a1*b8 - 10*a4*b1 - a6* b9 - a6*b8,
2*a1* b9 + a1*b8 - 5*a4*b1 + a4*b19 - a6*b8,
4*a10*b31 + 4*a10*b32 + a4*b36 - 8*a6*b39 - a8*b17,
2*a10*b31 + 2*a10*b34 + a4*b35 - 4*a6*b39 - a7*b17,
4*a10*b33 + 4*a10*b34 + a4*b37 - 8*a6*b39 - 2*a7*b17,
2* a9*b18 + a10*b16 - a3*b18 + a5*b16 - a5*b38,
a1*b21 - 2*a6*b20 - a6*b21 + 4*a7*b19 - a8*b1 + a8*b19,
4* a9*b27 + 3*a2*b29 + 4*a3*b26 - 8*a6*b30 - 2*a8*b29 - a8*b6,
2* a9*b29 - 2* a9*b6 + 4*a2*b30 + 5*a3*b28 - 5*a7*b7 - 4*a8*b30,
24*a10*b19 + 3*a4*b21 - 2*a6*b32 - 4*a6*b33 - 2*a6*b34 - a8*b8,
6*a10*b23 + a4*b26 + 2*a5*b21 - 2*a6*b36 - 2*a6*b37 - a8*b12,
2* a9*b32 + 4*a10*b27 + 3*a4*b29 + 2*a5*b26 - 4*a6*b38 - a8*b15,
2* a9*b15 - 2* a9*b37 - a3*b37 - 4*a4*b30 - 2*a5*b28 + 2*a7*b16,
2* a9*b10 + 2* a9*b13 - a4*b28 - 2*a5*b24 + 2*a5*b4 + a7*b15,
2* a9*b17 - 2*a10*b37 - 2*a4*b38 - 2*a5*b35 - a5*b37 + 4*a7*b18,
4* a9*b22 + 6* a9*b23 + 2*a2*b26 + 6*a3*b21 - 4*a6*b28 - 6*a6*b29 - a8*b5,
6* a9*b19 + a1*b24 + 3*a2*b20 - a6*b24 - 2*a6*b25 + a7*b21 - a7*b3,
4* a9*b31 + 2* a9*b32 + a2*b36 + 2*a5*b26 - 4*a6*b38 + 2*a7*b36 - a8*b14,
2*a10*b20 - a1*b35 - a5*b20 + a6*b35 + a7*b10 - a7*b31 - a7*b34,
2*a4*b25 + 6*a5*b20 - 2*a6*b37 - a7*b11 - a7*b13 + 2*a7*b34 + 2*a8*b34,
2* a9*b8 - 6*a10*b23 - 3*a4*b24 - 6*a5*b20 + 2*a6*b37 + 2*a7*b12 + a7*b13,
2* a9*b12 - 2* a9*b33 - 2*a10*b27 - a4*b28 - 2*a5*b24 + 2*a6*b38 + a7*b15,
2*a1*b10 + 2*a1*b11 + a4*b21 - a4*b3 - 12*a5*b1 - 2*a7*b8 - a8*b8,
2*a1*b12 + a1*b13 + a2*b8 - 3*a4*b2 - 6*a5*b1 - a6*b13 - a8*b8,
2* a9*b17 + 2*a10*b14 - a2*b18 - 2*a4*b38 + a5*b14 - 2*a5*b35 + 4*a7*b18,
6*a10* b9 + 6*a10*b8 - 6*a1*b17 + a4*b11 + a4*b12 - 2*a4*b32 - 2*a4*b33,
4*a10* b9 - a1*b17 + a4*b10 - a4*b31 - a4*b34 - a5* b9 + a6*b17,
12* a9*b19 + 2*a1*b27 + a2*b22 - 2*a6*b26 - 2*a6*b27 + 3*a7*b23 - a8*b22 + 3*a8
*b23,
2* a9*b1 - 2* a9*b19 - a1*b24 - a2*b20 + a6*b24 - a7*b23 + a7*b2 + a8*b20,
8* a9*b19 + 2*a1*b25 + 4*a2*b20 - 2*a6*b24 - 2*a6*b25 + a7*b22 - a7*b2 - a7*b3,
a1*b36 + a5*b21 - 2*a6*b35 - a6*b36 + 2*a7*b31 + a7*b32 - a8*b10 + a8*b31,
2* a9*b31 + 2* a9*b34 - 2*a10*b25 + a2*b35 + 2*a5*b25 - 2*a6*b38 - a7*b14 + 2*
a7*b35,
4*a1*b10 + 2*a4*b20 + a4*b21 - a4*b2 - 8*a5*b1 - 3*a7* b9 - a7*b8 - a8* b9,
6* a9* b9 - 2*a1*b14 + 2*a2*b10 - a4*b24 - 2*a4*b25 + a5*b3 + 2*a7*b10 + a7*b11
,
a9*b11 + a9*b13 - 3*a1*b16 + 3*a3*b10 - a4*b28 - a5*b25 + a5*b5 + a7*b14,
2* a9*b14 + 2* a9*b15 - a2*b16 + a3*b14 - 4*a4*b30 - 2*a5*b28 + 3*a5*b6 + 2*a7*
b16,
2* a9*b12 - 2*a1*b16 - a2*b15 - 2*a3*b12 - a4*b29 + a4*b6 + 2*a5*b4 + a8*b15,
a1*b13 + a2* b9 + a4*b23 - a4*b2 - 2*a5*b1 + 2*a5*b19 - a6*b13 - a8* b9,
2* a9*b15 - 2*a2*b16 - a3*b15 - 2*a4*b30 + 5*a4*b7 - 2*a5*b29 + 2*a5*b6 + 2*a8*
b16,
8* a9*b19 + a1*b26 + a2*b21 - 2*a6*b24 - a6*b26 + 3*a7*b23 - a8*b21 + a8*b23 -
a8*b2,
2* a9*b23 - 2* a9*b2 + a1*b28 + a2*b24 + 3*a3*b20 - a6*b28 + a7*b27 - a7*b4 -
a8*b24,
2* a9*b27 - 2* a9*b4 + 4*a1*b30 + a2*b28 + 4*a3*b24 - 4*a6*b30 + a7*b29 - a7*b6
- a8*b28,
2* a9*b26 + 2* a9*b27 - 2* a9*b5 + 3*a2*b28 + 6*a3*b24 + 6*a3*b25 - 12*a6*b30 -
3*a7*b6 - 2*a8*b28,
4*a10*b22 + a4*b26 + 4*a5*b21 - 4*a6*b35 - 4*a6*b36 - 2*a6*b37 + 2*a7*b32 - a8*
b11 + 2*a8*b32,
2*a10*b22 + 2*a4*b25 + 6*a5*b20 - 2*a6*b35 - 2*a6*b37 - a7*b11 - a7*b12 + 2*a7*
b33 + 2*a8*b33,
6*a10* b9 + 2*a10*b8 - 3*a1*b17 + 3*a4*b10 - a4*b31 - 2*a4*b33 - a4*b34 - a5*b8
+ a6*b17,
12*a10* b9 + 4*a10*b8 - 6*a1*b17 + a4*b11 + a4*b13 - 2*a4*b31 - 2*a4*b32 - 2*a4
*b34 + 2*a6*b17,
24* a9*b19 + a1*b26 + 3*a2*b21 - 2*a6*b24 - 4*a6*b25 - 3*a6*b26 + 2*a7*b22 + a8
*b21 + a8*b22 - a8*b3,
6* a9*b23 + 3*a1*b29 + a2*b26 + 3*a3*b21 - 2*a6*b28 - 3*a6*b29 + 2*a7*b27 - a8*
b26 + a8*b27 - a8*b4,
2*a1*b10 + 2*a1*b11 + a4*b21 + a4*b22 - a4*b3 - 12*a5*b1 - 2*a6*b10 - 3*a7* b9
- a7*b8 - 3*a8* b9,
6* a9* b9 + 2* a9*b8 - 4*a1*b14 - 2*a1*b15 - 2*a4*b25 - a4*b26 + a4*b5 + 4*a5*
b3 + 2*a7*b11 + a8*b11,
3*a1*b11 + 4*a1*b12 + 2*a1*b13 + a2*b8 + a4*b22 - 3*a4*b3 - 24*a5*b1 - a6*b11 -
2*a7*b8 - 3*a8*b8,
a1*b11 + 2*a1*b13 + a2* b9 + a4*b22 + a4*b23 - a4*b3 - 8*a5*b1 - a6*b11 - 2*a7*
b9 - 3*a8* b9,
2*a1*b12 + 2*a1*b13 + a2* b9 + a2*b8 + a4*b23 - 4*a4*b2 - 8*a5*b1 - 2*a6*b12 -
a8* b9 - a8*b8,
2* a9* b9 - a1*b15 - a2*b13 - 3*a3* b9 - a4*b27 + a4*b4 - 2*a5*b23 + 2*a5*b2 +
a6*b15 + a8*b13,
2* a9*b13 - 2*a1*b16 - a2*b15 - 2*a3*b13 - a4*b29 + a4*b6 - 2*a5*b27 + 2*a5*b4
+ 2*a6*b16 + a8*b15,
4*a10*b11 - 8*a1*b18 - a2*b17 + a4*b14 - 4*a4*b35 - a4*b36 - a4*b37 + 4*a5*b10
+ 4*a7*b17 + a8*b17,
2*a10*b12 + 2*a10*b13 - 4*a1*b18 - a2*b17 + a4*b14 - a4*b36 - a4*b37 + 2*a5*b10
- 2*a5*b33 + a8*b17,
2*a10*b12 + 2*a10*b13 - 4*a1*b18 - a2*b17 + a4*b15 - a4*b36 - a4*b37 + a5*b11 -
a5*b32 + a8*b17,
a1*b37 + a4*b24 + 6*a5*b20 - 4*a6*b35 - a6*b37 - 2*a7*b10 - a7*b11 + a7*b32 + 2
*a7*b33 + 2*a7*b34 + a8*b34,
2* a9*b14 - 2* a9*b35 - 2* a9*b36 - 2* a9*b37 + 2*a10*b28 - 2*a2*b38 - 3*a3*b35
- 3*a5*b28 + 3*a7*b16 - 2*a7*b38 + 2*a8*b38,
2* a9*b8 - 3*a1*b15 - 2*a2*b12 - a2*b13 - 3*a3*b8 - a4*b27 + 3*a4*b4 + 6*a5*b2
+ a6*b15 + 2*a8*b12 + a8*b13,
4*a10*b13 - 4*a1*b18 - a2*b17 + a4*b14 - a4*b36 - a4*b37 + 2*a5*b10 - 2*a5*b31
- 2*a5*b34 + 4*a6*b18 + a8*b17,
4* a9*b17 + 4*a10*b15 - 4*a2*b18 - 3*a3*b17 + 3*a4*b16 - 4*a4*b38 + 2*a5*b14 +
a5*b15 - 2*a5*b36 - 2*a5*b37 + 4*a8*b18,
2* a9*b10 - 2* a9*b31 - 2* a9*b34 + 2*a10*b24 - 2*a1*b38 - a2*b35 - 2*a5*b24 +
2*a6*b38 + a7*b14 - a7*b36 - a7*b37 + a8*b35,
2* a9*b11 - 2* a9*b32 - 4* a9*b33 - 4* a9*b34 - a2*b37 - 2*a4*b28 - 4*a5*b24 -
4*a5*b25 + 8*a6*b38 + 2*a7*b14 + 2*a7*b15 - 2*a7*b37,
4* a9*b8 - 2*a1*b14 - 4*a1*b15 - a2*b11 - a2*b12 - a4*b26 + 2*a4*b5 + 6*a5*b2 +
2*a5*b3 + 2*a7*b12 + a8*b11 + 2*a8*b12,
2* a9*b21 + 2* a9*b22 + 6* a9*b23 - 2* a9*b3 + 2*a1*b28 + 4*a2*b24 + 2*a2*b25 +
18*a3*b20 - 6*a6*b28 + a7*b26 - 2*a7*b4 - 2*a7*b5 - a8*b24 - 2*a8*b25,
6* a9* b9 + 2* a9*b8 - 4*a1*b14 - 2*a2*b10 - 2*a4*b24 - a4*b26 + 2*a4*b4 - 2*a5
*b21 + 6*a5*b2 + 2*a5*b3 + 2*a7*b12 + 2*a7*b13 + 2*a8*b10 + a8*b13,
2* a9*b11 + 4* a9*b12 + 4* a9*b13 - 12*a1*b16 - 2*a2*b14 - a2*b15 - 2*a4*b28 -
3*a4*b29 + 3*a4*b6 - 2*a5*b26 + 4*a5*b4 + 4*a5*b5 + 2*a7*b15 + 2*a8*b14 + a8*
b15,
6* a9* b9 + 2* a9*b8 - 2*a1*b14 - 4*a1*b15 - a2*b11 - a2*b13 - a4*b26 - 2*a4*
b27 + 2*a4*b5 - 2*a5*b22 + 6*a5*b2 + 2*a5*b3 + 2*a6*b14 + 2*a7*b13 + a8*b11 + 2
*a8*b13
Relevance
In the following evolutionary system u=u(t,x) is a scalar function,
v=v(t,x) is a vector function and f(..,..) stands for the scalar
product of both vector arguments of f.
3
u = u *a1 + u *a2*u + f(v,v )*a4 + f(v,v)*a5*u + a3*u
t 2x x x
2
v = u *a7*v + v *a6 + v *a8*u + f(v,v)*a10*v + a9*u *v
t x 2x x
For any solution of the above algebraic conditions for the
a's and b's this system has the following symmetry:
2 2
u = u *b1 + u *b2*u + u *u *b3 + u *f(v,v)*b10 + u *b4*u + u *b5*u
t 4x 3x 2x x 2x 2x x
3
+ u *f(v,v )*b11 + u *f(v,v)*b14*u + u *b6*u + f(v ,v )*b8
x x x x x 2x
+ f(v ,v )*b12*u + f(v,v )* b9 + f(v,v )*b13*u + f(v,v )*f(v,v)*b17
x x 3x 2x x
2 2 3 5
+ f(v,v )*b15*u + f(v,v) *b18*u + f(v,v)*b16*u + b7*u
x
2
v = u *b20*v + u *v *b21 + u *b24*u*v + u *b25*v + u *v *b22
t 3x 2x x 2x x x 2x
2
+ u *v *b26*u + u *f(v,v)*b35*v + u *b28*u *v + v *b19 + v *b23*u
x x x x 4x 3x
2
+ v *f(v,v)*b31 + v *b27*u + v *f(v,v )*b32 + v *f(v,v)*b36*u
2x 2x x x x
3
+ v *b29*u + f(v ,v )*b33*v + f(v,v )*b34*v + f(v,v )*b37*u*v
x x x 2x x
2 2 4
+ f(v,v) *b39*v + f(v,v)*b38*u *v + b30*u *v