Solution 21 to problem l1o35
Expressions |
Parameters |
Relevance |
Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
5 2
---*a20 *b36
3
b73=--------------
2
a12
5 2
---*a20 *b36
6
b72=--------------
2
a12
b71=0
b70=0
b69=0
b68=0
b67=0
b66=0
b65=0
b64=0
5 2
- ---*a20 *b36
2
b63=-----------------
a12*a7
5 2
- ---*a20 *b36
2
b62=-----------------
a12*a7
5 2
- ---*a20 *b36
4
b61=-----------------
a12*a7
5
---*a20*b36
4
b60=-------------
a12
15
----*a20*b36
4
b59=--------------
a12
25
----*a20*b36
6
b58=--------------
a12
25
----*a20*b36
6
b57=--------------
a12
5*a20*b36
b56=-----------
a12
5
---*a20*b36
3
b55=-------------
a12
b54=0
b53=0
b52=0
b51=0
b50=0
b49=0
b48=0
b47=0
b46=0
15 2
----*a20 *b36
8
b45=---------------
2
a7
b44=0
15 2
----*a20 *b36
8
b43=---------------
2
a7
b42=0
b41=0
5
- ---*a20*b36
2
b40=----------------
a7
15
- ----*a20*b36
4
b39=-----------------
a7
25
- ----*a20*b36
8
b38=-----------------
a7
15
- ----*a20*b36
16
b37=-----------------
a7
10 2
----*a20 *a7*b36
27
b35=------------------
3
a12
b34=0
b33=0
5 2
- ---*a20 *b36
6
b32=-----------------
2
a12
5
---*a20*a7*b36
3
b31=----------------
2
a12
10
----*a20*a7*b36
3
b30=-----------------
2
a12
b29=0
b28=0
b27=0
b26=0
b25=0
b24=0
b23=0
b22=0
b21=0
b20=0
b19=0
5
- ---*a20*b36
2
b18=----------------
a12
5
---*a20*b36
2
b17=-------------
a12
5
- ---*a20*b36
6
b16=----------------
a12
5
- ---*a20*b36
6
b15=----------------
a12
5
---*a7*b36
4
b14=------------
a12
5
- ---*a7*b36
3
b13=---------------
a12
5
----*a7*b36
12
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
16
b4=-----------------
a7
5
- ---*a20*b36
8
b3=----------------
a7
b2=0
1
b1=----*b36
16
a21=0
a19=a20
a18=0
a17=0
a16=0
a15=0
3
- ---*a12*a20
2
a14=----------------
a7
3
- ---*a12*a20
4
a13=----------------
a7
1
---*a20*a7
3
a11=------------
a12
a10=0
a9=0
a8=0
a6= - a7
a5=0
a4=0
3
- ---*a12*a20
4
a3=----------------
a7
a2=0
1
a1=---*a12
4
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 2 2
u =(---*u *a12 *a7 - ---*u *a12 *a20 - f(v ,v )*a12*a7 + f(v,v )*a12*a7
t 4 3x 4 x x x 2x
1 2 2
+ ---*f(v,v) *a20*a7 )/(a12*a7)
3
3 3
v =( - ---*u *a12*a20*v - ---*u *v *a12*a20 + v *a12*a7 + v *f(v,v)*a20*a7
t 4 2x 2 x x 3x x
+ f(v,v )*a20*a7*v)/a7
x
1 3 2 5 3
u =(----*u *a12 *a7 *b36 - ---*u *u *a12 *a20*a7*b36
s 16 5x 8 3x x
5 2 2 5 2 3
- ---*u *f(v,v)*a12 *a20*a7 *b36 - ----*u *a12 *a20*a7*b36
6 3x 16 2x
5 2 2 5 3 3 2
- ---*u *f(v,v )*a12 *a20*a7 *b36 + ---*u *a12 *a20 *b36
6 2x x 8 x
5 2 2 5 2 2
+ ---*u *f(v ,v )*a12 *a20*a7 *b36 - ---*u *f(v,v )*a12 *a20*a7 *b36
2 x x x 2 x 2x
5 2 2 2 5 2 3
- ---*u *f(v,v) *a12*a20 *a7 *b36 + ----*f(v ,v )*a12 *a7 *b36
6 x 12 2x 2x
5 2 3 5 2 3
- ---*f(v ,v )*a12 *a7 *b36 + ---*f(v,v )*a12 *a7 *b36
3 x 3x 4 4x
10 3 5 2 3
+ ----*f(v,v )*f(v,v)*a12*a20*a7 *b36 + ---*f(v,v ) *a12*a20*a7 *b36
3 2x 3 x
10 3 2 3 3 2
+ ----*f(v,v) *a20 *a7 *b36)/(a12 *a7 )
27
15 2 25 2
v =( - ----*u *a12 *a20*a7*b36*v - ----*u *v *a12 *a20*a7*b36
s 16 4x 8 3x x
15 2 2 15 2
+ ----*u *u *a12 *a20 *b36*v - ----*u *v *a12 *a20*a7*b36
8 2x x 4 2x 2x
5 2 15 2 2 2
- ---*u *f(v,v)*a12*a20 *a7*b36*v + ----*u *v *a12 *a20 *b36
4 2x 8 x x
5 2 5 2
- ---*u *v *a12 *a20*a7*b36 - ---*u *v *f(v,v)*a12*a20 *a7*b36
2 x 3x 2 x x
5 2 2 2
- ---*u *f(v,v )*a12*a20 *a7*b36*v + v *a12 *a7 *b36
2 x x 5x
5 2 2
+ ---*v *f(v,v)*a12*a20*a7 *b36 + 5*v *f(v,v )*a12*a20*a7 *b36
3 3x 2x x
25 2 25 2
+ ----*v *f(v ,v )*a12*a20*a7 *b36 + ----*v *f(v,v )*a12*a20*a7 *b36
6 x x x 6 x 2x
5 2 2 2 15 2
+ ---*v *f(v,v) *a20 *a7 *b36 + ----*f(v ,v )*a12*a20*a7 *b36*v
6 x 4 x 2x
5 2 5 2 2 2
+ ---*f(v,v )*a12*a20*a7 *b36*v + ---*f(v,v )*f(v,v)*a20 *a7 *b36*v)/(a12
4 3x 3 x
2
*a7 )