Problem l1o23
Unknowns | Inequalities |
Equations |
Solution 1 |
Solution 2 |
Solution 3 |
Relevance |
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Unknowns
All solutions for the following 31 unknowns have to be determined:
a1, ..., a10, b1, ..., b21
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,b12},
{b1,b2,b3,b8,b4,b6,b7, b9,b11,b10,b5},
{b13,b14,b16,b12,b15,b19,b17,b20,b21,b18}
Equations
All comma separated 66 expressions involving 341 terms have to vanish.
All terms are products of one a- and one b-unknown.
a3*b15,
a3*b2,
a3*b5,
a5*b15,
4*a10*b11 - a5*b11,
2*a6*b15 - 3*a8*b12,
2*a3*b17 - a8*b5,
2*a1*b2 - 3*a2*b1,
2*a1*b3 - 3*a2*b1,
a2*b5 - 2*a3*b4,
2* a9*b5 - 3*a3*b18,
a4*b13 - a7*b7,
2*a10*b19 - a8*b11,
2* a9*b11 - a5*b21,
2*a1*b4 - a2*b2 - 9*a3*b1,
2*a1*b4 - a2*b3 - 6*a3*b1,
4*a1*b5 - 2*a3*b2 - a3*b3,
6*a10*b12 + a4*b15 - 4*a6*b19,
2* a9*b19 + 2*a5*b17 - a8*b10,
2*a10*b20 + a4*b21 - 4*a7*b11,
6* a9*b12 + a2*b15 - 4*a6*b17 + a8*b15,
a1*b13 - a6*b13 + a7*b12 - a7*b1,
6*a10*b12 + a4*b14 - 2*a6*b20 - a8*b7,
2* a9*b7 - a4*b16 - 2*a5*b13 + a7* b9,
2*a1*b6 + a1*b7 - 3*a4*b1 - a6*b7,
a1*b7 + a4*b12 - a4*b1 - a6*b7,
2* a9*b10 - a3*b10 - 2*a5*b18 + 3*a5*b5,
2*a10*b8 - a2*b11 - a4*b21 + 4*a7*b11,
a1*b15 - 2*a6*b14 - a6*b15 + 3*a7*b12 + 3*a8*b12,
a5*b14 - 2*a6*b21 + 2*a7*b19 + a8*b19 - a8*b8,
6*a10*b12 + a4*b14 - 2*a6*b19 - 2*a6*b20 - a8*b6,
4*a10*b15 + 2*a4*b17 + 2*a5*b14 - 4*a6*b21 - a8* b9,
6*a10*b12 + 3*a4*b13 - 2*a6*b20 - 2*a7*b6 - a7*b7,
2* a9* b9 - 2* a9*b20 - 3*a4*b18 - 2*a5*b16 + 2*a7*b10,
2* a9*b10 - 2* a9*b21 + 2*a10*b18 - a3*b21 - 3*a5*b18,
2* a9*b7 - a4*b16 - 2*a5*b13 + a5*b2 + a7* b9,
2* a9*b7 - 2*a1*b10 - a4*b16 + a5*b3 + 2*a7*b8,
2*a1*b6 + 2*a1*b7 + a4*b12 - 4*a4*b1 - 2*a6*b6,
4* a9*b11 + 2*a10*b10 - 3*a3*b11 + a5*b10 - 2*a5*b21,
a1*b14 - 2*a6*b13 - a6*b14 + 3*a7*b12 + a8*b12 - a8*b1,
6* a9*b12 + a2*b14 - 2*a6*b16 - 2*a6*b17 + a8*b14 - a8*b3,
4* a9*b15 + 2*a2*b17 + 3*a3*b14 - 6*a6*b18 - a8*b17 - a8*b4,
2* a9*b17 - 2* a9*b4 + 3*a2*b18 + 4*a3*b16 - 4*a7*b5 - 3*a8*b18,
2* a9*b6 - 2*a10*b15 - a4*b16 - 2*a5*b13 + 2*a6*b21 + a7* b9,
4*a10*b6 + 4*a10*b7 - 8*a1*b11 + a4* b9 - 2*a4*b19 - 2*a4*b20,
6* a9*b12 + a1*b16 + 3*a2*b13 - 3*a6*b16 + a7*b14 - a7*b2 - 2*a7*b3,
2*a10*b13 - a1*b21 - a5*b13 + a6*b21 - a7*b19 - a7*b20 + a7*b8,
a4*b16 + 4*a5*b13 - 4*a6*b21 - a7* b9 + 2*a7*b20 - 2*a7*b8 + a8*b20,
2* a9* b9 + 2* a9*b8 - a2*b10 - 3*a4*b18 - 2*a5*b16 + 2*a5*b4 + 2*a7*b10,
2*a10*b6 + 2*a10*b7 - 4*a1*b11 - a4*b19 - a4*b20 + a4*b8 - a5*b6,
4*a10*b7 - 4*a1*b11 - a4*b19 - a4*b20 + a4*b8 - a5*b7 + 4*a6*b11,
2* a9*b12 - 2* a9*b1 + a1*b16 + a2*b13 - a6*b16 + a7*b15 - a7*b2 - a8*b13,
2* a9*b14 + 2* a9*b15 - 2* a9*b3 + 2*a2*b16 + 6*a3*b13 - 6*a6*b18 - 2*a7*b4 -
a8*b16,
4*a1*b8 + 2*a4*b13 + a4*b14 - a4*b2 - 6*a5*b1 - 2*a7*b6 - 2*a7*b7 - a8*b7,
2*a1* b9 + 2*a1*b8 + a2*b6 + a4*b14 - 2*a4*b3 - 6*a5*b1 - 2*a7*b6 - 2*a8*b6,
2* a9*b6 - 2*a1*b10 - a2* b9 - 3*a3*b6 - a4*b17 + a4*b4 + 2*a5*b2 + a8* b9,
a1* b9 + a2*b7 + a4*b15 - a4*b2 + 2*a5*b12 - 2*a5*b1 - a6* b9 - a8*b7,
a9* b9 - a2*b10 - a3* b9 - a4*b18 + 2*a4*b5 - a5*b17 + a5*b4 + a8*b10,
2*a10* b9 - 2*a2*b11 + a4*b10 - 2*a4*b21 - a5*b19 - a5*b20 + a5*b8 + 2*a8*b11,
6* a9*b12 + 2*a1*b17 + a2*b14 - 2*a6*b16 - 2*a6*b17 + 2*a7*b15 - a8*b14 + a8*
b15 - a8*b2,
2* a9*b15 - 2* a9*b2 + 3*a1*b18 + a2*b16 + 3*a3*b13 - 3*a6*b18 + a7*b17 - a7*b4
- a8*b16,
2* a9*b19 + 2* a9*b20 - 2* a9*b8 - 2*a10*b16 + a2*b21 + 2*a5*b16 - 2*a7*b10 + 2
*a7*b21 - a8*b21,
3*a1* b9 + 2*a2*b6 + a2*b7 + a4*b15 - 3*a4*b2 - 6*a5*b1 - a6* b9 - 2*a8*b6 - a8
*b7,
2*a1* b9 + 2*a1*b8 + a2*b7 + a4*b14 + a4*b15 - 2*a4*b3 - 6*a5*b1 - 2*a6*b8 - 2*
a7*b7 - 2*a8*b7,
2* a9*b7 - 2*a1*b10 - a2* b9 - 3*a3*b7 - a4*b17 + a4*b4 - 2*a5*b15 + 2*a5*b2 +
2*a6*b10 + a8* b9,
4* a9*b6 + 4* a9*b7 - 8*a1*b10 - a2* b9 - 2*a2*b8 - 2*a4*b16 - 2*a4*b17 + 2*a4*
b4 - 2*a5*b14 + 4*a5*b2 + 4*a5*b3 + 2*a7* b9 + a8* b9 + 2*a8*b8
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 *b3 + u *f(v,v)*b8 + u *b4*u + f(v ,v )*b6
t 3x 2x x x x x x
2 2 4
+ f(v,v )*b7 + f(v,v )* b9*u + f(v,v) *b11 + f(v,v)*b10*u + b5*u
2x x
v = u *b13*v + u *v *b14 + u *b16*u*v + v *b12 + v *b15*u + v *f(v,v)*b19
t 2x x x x 3x 2x x
2 3
+ v *b17*u + f(v,v )*b20*v + f(v,v)*b21*u*v + b18*u *v
x x