Solution 20 to problem l1o35
Expressions |
Parameters |
Relevance |
Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
b73=0
5 2
- ----*a20 *b36
36
b72=------------------
2
a12
b71=0
b70=0
b69=0
b68=0
b67=0
b66=0
b65=0
b64=0
5 2
- ---*a20 *b36
6
b63=-----------------
a12*a7
5 2
---*a20 *b36
6
b62=--------------
a12*a7
b61=0
5
---*a20*b36
6
b60=-------------
a12
5
---*a20*b36
3
b59=-------------
a12
5
---*a20*b36
6
b58=-------------
a12
5
---*a20*b36
4
b57=-------------
a12
5
---*a20*b36
3
b56=-------------
a12
b55=0
b54=0
b53=0
b52=0
b51=0
b50=0
b49=0
b48=0
b47=0
b46=0
5 2
---*a20 *b36
8
b45=--------------
2
a7
b44=0
5 2
---*a20 *b36
4
b43=--------------
2
a7
b42=0
b41=0
5
- ---*a20*b36
2
b40=----------------
a7
- 5*a20*b36
b39=--------------
a7
15
- ----*a20*b36
4
b38=-----------------
a7
5
- ---*a20*b36
4
b37=----------------
a7
b35=0
b34=0
b33=0
5 2
- ----*a20 *b36
36
b32=------------------
2
a12
5
----*a20*a7*b36
18
b31=-----------------
2
a12
5
---*a20*a7*b36
9
b30=----------------
2
a12
5
----*a20*a7*b36
18
b29=-----------------
2
a12
b28=0
b27=0
b26=0
b25=0
b24=0
b23=0
5 2
---*a20 *b36
6
b22=--------------
a12*a7
b21=0
b20=0
b19=0
5
- ---*a20*b36
2
b18=----------------
a12
5
- ----*a20*b36
12
b17=-----------------
a12
5
- ---*a20*b36
3
b16=----------------
a12
5
- ---*a20*b36
6
b15=----------------
a12
5
---*a7*b36
6
b14=------------
a12
5
---*a7*b36
6
b13=------------
a12
5
---*a7*b36
6
b12=------------
a12
b11=0
b10=0
b9=0
b8=0
5 2
- ---*a20 *b36
8
b7=-----------------
2
a7
b6=0
b5=0
5
---*a20*b36
8
b4=-------------
a7
5
---*a20*b36
4
b3=-------------
a7
b2=0
1
b1= - ---*b36
4
a21=0
a19=0
a18=0
a17=0
a16=0
a15=0
3
- ---*a12*a20
2
a14=----------------
a7
3
- ---*a12*a20
2
a13=----------------
a7
1
---*a20*a7
6
a11=------------
a12
a10=0
a9=0
a8= - a20
1
a6=---*a7
2
a5=0
a4=0
3
---*a12*a20
4
a3=-------------
a7
a2=0
1
a1= - ---*a12
2
Parameters
Apart from the condition that they must not vanish to give
a non-trivial solution and a non-singular solution with
non-vanishing denominators, the following parameters are free:
b36,a20,a7,a12
Relevance for the application:
The solution given above tells us that the system {u_s, v_s}
is a higher order symmetry for the lower order system {u_t,v_t}
where u=u(t,x) is a scalar function, v=v(t,x) is a vector
function of arbitrary dimension and f(..,..) is the scalar
product between two such vectors:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
1 2 3 2 2
u =( - ---*u *a12 *a7 + ---*u *a12 *a20 - u *f(v,v)*a12*a20*a7
t 2 3x 4 x x
1 2 2 1 2 2
+ ---*f(v ,v )*a12*a7 + f(v,v )*a12*a7 + ---*f(v,v) *a20*a7 )/(a12*a7)
2 x x 2x 6
3 3
- ---*u *a12*a20*v - ---*u *v *a12*a20 + v *a12*a7 + f(v,v )*a20*a7*v
2 2x 2 x x 3x x
v =--------------------------------------------------------------------------
t a7
1 2 2 5 2
u =( - ---*u *a12 *a7 *b36 + ---*u *u *a12 *a20*a7*b36
s 4 5x 4 3x x
5 2 5 2 2
- ---*u *f(v,v)*a12*a20*a7 *b36 + ---*u *a12 *a20*a7*b36
6 3x 8 2x
5 2 5 3 2 2
- ---*u *f(v,v )*a12*a20*a7 *b36 - ---*u *a12 *a20 *b36
3 2x x 8 x
5 2 2 5 2
+ ---*u *f(v,v)*a12*a20 *a7*b36 - ----*u *f(v ,v )*a12*a20*a7 *b36
6 x 12 x x x
5 2 5 2 2 2
- ---*u *f(v,v )*a12*a20*a7 *b36 - ----*u *f(v,v) *a20 *a7 *b36
2 x 2x 36 x
5 3 5 3
+ ---*f(v ,v )*a12*a7 *b36 + ---*f(v ,v )*a12*a7 *b36
6 2x 2x 6 x 3x
5 3 5 3
+ ----*f(v ,v )*f(v,v)*a20*a7 *b36 + ---*f(v,v )*a12*a7 *b36
18 x x 6 4x
5 3 5 2 3 2 2
+ ---*f(v,v )*f(v,v)*a20*a7 *b36 + ----*f(v,v ) *a20*a7 *b36)/(a12 *a7 )
9 2x 18 x
5 2 15 2
v =( - ---*u *a12 *a20*a7*b36*v - ----*u *v *a12 *a20*a7*b36
s 4 4x 4 3x x
5 2 2 2
+ ---*u *u *a12 *a20 *b36*v - 5*u *v *a12 *a20*a7*b36
4 2x x 2x 2x
5 2 2 2 5 2
+ ---*u *v *a12 *a20 *b36 - ---*u *v *a12 *a20*a7*b36
8 x x 2 x 3x
5 2 5 2
+ ---*u *v *f(v,v)*a12*a20 *a7*b36 - ---*u *f(v,v )*a12*a20 *a7*b36*v
6 x x 6 x x
2 2 5 2
+ v *a12 *a7 *b36 + ---*v *f(v,v )*a12*a20*a7 *b36
5x 3 2x x
5 2 5 2
+ ---*v *f(v ,v )*a12*a20*a7 *b36 + ---*v *f(v,v )*a12*a20*a7 *b36
4 x x x 6 x 2x
5 2 2 2 5 2
- ----*v *f(v,v) *a20 *a7 *b36 + ---*f(v ,v )*a12*a20*a7 *b36*v
36 x 3 x 2x
5 2 2 2
+ ---*f(v,v )*a12*a20*a7 *b36*v)/(a12 *a7 )
6 3x