Solution 22 to problem l1o35
Expressions |
Parameters |
Relevance |
Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
5 2 2
---*a20 *a4 *b36
2
b73=------------------
4
a13
5 2 2
----*a20 *a4 *b36
12
b72=-------------------
4
a13
b71=0
65 3
----*a20*a4 *b36
24
b70=------------------
4
a13
25 3
----*a20*a4 *b36
24
b69=------------------
4
a13
5 3
---*a20*a4 *b36
2
b68=-----------------
4
a13
b67=0
b66=0
b65=0
b64=0
65 2
----*a20*a4 *b36
12
b63=------------------
3
a13
25 2
----*a20*a4 *b36
12
b62=------------------
3
a13
5 2
---*a20*a4 *b36
2
b61=-----------------
3
a13
10
- ----*a20*a4*b36
3
b60=--------------------
2
a13
15
- ----*a20*a4*b36
2
b59=--------------------
2
a13
- 5*a20*a4*b36
b58=-----------------
2
a13
25
- ----*a20*a4*b36
6
b57=--------------------
2
a13
25
- ----*a20*a4*b36
6
b56=--------------------
2
a13
5
- ---*a20*a4*b36
6
b55=-------------------
2
a13
b54=0
5 4
---*a4 *b36
8
b53=-------------
4
a13
5 4
---*a4 *b36
2
b52=-------------
4
a13
b51=0
5 3
---*a4 *b36
2
b50=-------------
3
a13
3
5*a4 *b36
b49=-----------
3
a13
5 3
---*a4 *b36
2
b48=-------------
3
a13
5 2
- ---*a4 *b36
4
b47=----------------
2
a13
2
- 5*a4 *b36
b46=--------------
2
a13
5 2
- ---*a4 *b36
2
b45=----------------
2
a13
2
- 5*a4 *b36
b44=--------------
2
a13
5 2
- ---*a4 *b36
2
b43=----------------
2
a13
5 2
- ---*a4 *b36
2
b42=----------------
2
a13
b41=0
5
- ---*a4*b36
2
b40=---------------
a13
- 5*a4*b36
b39=-------------
a13
- 5*a4*b36
b38=-------------
a13
5
- ---*a4*b36
2
b37=---------------
a13
b35=0
b34=0
5 2 2
---*a20 *a4 *b36
3
b33=------------------
4
a13
5 2 2
----*a20 *a4 *b36
12
b32=-------------------
4
a13
10 2
- ----*a20 *a4*b36
3
b31=---------------------
3
a13
5 2
- ---*a20 *a4*b36
3
b30=--------------------
3
a13
5 2
- ---*a20 *a4*b36
3
b29=--------------------
3
a13
b28=0
5 3
---*a20*a4 *b36
3
b27=-----------------
4
a13
5 3
---*a20*a4 *b36
2
b26=-----------------
4
a13
5 2
- ---*a20*a4 *b36
3
b25=--------------------
3
a13
5 2
- ---*a20*a4 *b36
3
b24=--------------------
3
a13
10 2
- ----*a20*a4 *b36
3
b23=---------------------
3
a13
b22=0
b21=0
5
- ---*a20*a4*b36
3
b20=-------------------
2
a13
10
- ----*a20*a4*b36
3
b19=--------------------
2
a13
- 5*a20*a4*b36
b18=-----------------
2
a13
25
- ----*a20*a4*b36
6
b17=--------------------
2
a13
20
- ----*a20*a4*b36
3
b16=--------------------
2
a13
5
- ---*a20*a4*b36
3
b15=-------------------
2
a13
5
---*a20*b36
3
b14=-------------
a13
5*a20*b36
b13=-----------
a13
10
----*a20*b36
3
b12=--------------
a13
b11=0
15 4
----*a4 *b36
8
b10=--------------
4
a13
b9=0
b8=0
5 2
- ---*a4 *b36
2
b7=----------------
2
a13
2
- 10*a4 *b36
b6=---------------
2
a13
5 2
- ---*a4 *b36
2
b5=----------------
2
a13
b4=0
b3=0
b2=0
b1=b36
a21=0
1
a19=---*a20
3
a18=0
1
a17=---*a4
2
a16=a4
a15=0
a14=a13
2 2
- ---*a13
3
a12=-------------
a4
a11=0
a10=0
2
a9=---*a20
3
1
a8=---*a20
3
2
- ---*a13*a20
3
a7=----------------
a4
2
- ---*a13*a20
3
a6=----------------
a4
a5=0
a3=0
a2=0
2 2
- ---*a13
3
a1=-------------
a4
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,a13,a4
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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
2 2 1 2 2 2
u =( - ---*u *a13 + ---*u *f(v,v)*a20*a4 + u *a4 *u - ---*f(v ,v )*a13*a20
t 3 3x 3 x x 3 x x
2 2
- ---*f(v,v )*a13*a20 + ---*f(v,v )*a20*a4*u)/a4
3 2x 3 x
2 2 2
v =(u *a13*a4*v + u *v *a13*a4 + u *a4 *u*v - ---*v *a13
t 2x x x x 3 3x
1 1 2 2
+ ---*v *f(v,v)*a20*a4 + ---*v *a4 *u + f(v,v )*a20*a4*v)/a4
3 x 2 x x
4 5 2 5 2 2 2
u =(u *a13 *b36 - ---*u *f(v,v)*a13 *a20*a4*b36 - ---*u *a13 *a4 *b36*u
s 5x 3 3x 2 3x
2 2 20 2
- 10*u *u *a13 *a4 *b36*u - ----*u *f(v,v )*a13 *a20*a4*b36
2x x 3 2x x
5 3 2 2 25 2
- ---*u *a13 *a4 *b36 - ----*u *f(v ,v )*a13 *a20*a4*b36
2 x 6 x x x
2 10 2
- 5*u *f(v,v )*a13 *a20*a4*b36 - ----*u *f(v,v )*a13*a20*a4 *b36*u
x 2x 3 x x
5 2 2 2 5 3 2
+ ----*u *f(v,v) *a20 *a4 *b36 + ---*u *f(v,v)*a20*a4 *b36*u
12 x 2 x
15 4 4 10 3
+ ----*u *a4 *b36*u + ----*f(v ,v )*a13 *a20*b36
8 x 3 2x 2x
3 10 2
+ 5*f(v ,v )*a13 *a20*b36 - ----*f(v ,v )*a13 *a20*a4*b36*u
x 3x 3 x 2x
5 2 5 2 2
- ---*f(v ,v )*f(v,v)*a13*a20 *a4*b36 - ---*f(v ,v )*a13*a20*a4 *b36*u
3 x x 3 x x
5 3 5 2
+ ---*f(v,v )*a13 *a20*b36 - ---*f(v,v )*a13 *a20*a4*b36*u
3 4x 3 3x
5 2 5 2 2
- ---*f(v,v )*f(v,v)*a13*a20 *a4*b36 - ---*f(v,v )*a13*a20*a4 *b36*u
3 2x 3 2x
10 2 2 5 2 2
- ----*f(v,v ) *a13*a20 *a4*b36 + ---*f(v,v )*f(v,v)*a20 *a4 *b36*u
3 x 3 x
5 3 3 4
+ ---*f(v,v )*a20*a4 *b36*u )/a13
3 x
5 3 3 5 2 2
v =( - ---*u *a13 *a4*b36*v - 5*u *v *a13 *a4*b36 - ---*u *a13 *a4 *b36*u*v
s 2 4x 3x x 2 3x
5 2 2 3
- ---*u *u *a13 *a4 *b36*v - 5*u *v *a13 *a4*b36
2 2x x 2x 2x
2 2 5 2
- 5*u *v *a13 *a4 *b36*u + ---*u *f(v,v)*a13*a20*a4 *b36*v
2x x 2 2x
5 3 2 5 2 2 2
+ ---*u *a13*a4 *b36*u *v - ---*u *v *a13 *a4 *b36
2 2x 2 x x
2 3 5 3 2 2
+ 5*u *a13*a4 *b36*u*v - ---*u *v *a13 *a4*b36 - 5*u *v *a13 *a4 *b36*u
x 2 x 3x x 2x
25 2 5 3 2
+ ----*u *v *f(v,v)*a13*a20*a4 *b36 + ---*u *v *a13*a4 *b36*u
12 x x 2 x x
65 2 5 3
+ ----*u *f(v,v )*a13*a20*a4 *b36*v + ---*u *f(v,v)*a20*a4 *b36*u*v
12 x x 2 x
5 4 3 4 5 2
+ ---*u *a4 *b36*u *v + v *a13 *b36 - ---*v *f(v,v)*a13 *a20*a4*b36
2 x 5x 6 3x
5 2 2 2 25 2
- ---*v *a13 *a4 *b36*u - ----*v *f(v,v )*a13 *a20*a4*b36
4 3x 6 2x x
25 2 2
- ----*v *f(v ,v )*a13 *a20*a4*b36 - 5*v *f(v,v )*a13 *a20*a4*b36
6 x x x x 2x
5 2 2 2 25 3 2
+ ----*v *f(v,v) *a20 *a4 *b36 + ----*v *f(v,v)*a20*a4 *b36*u
12 x 24 x
5 4 4 15 2
+ ---*v *a4 *b36*u - ----*f(v ,v )*a13 *a20*a4*b36*v
8 x 2 x 2x
10 2 5 2 2
- ----*f(v,v )*a13 *a20*a4*b36*v + ---*f(v,v )*f(v,v)*a20 *a4 *b36*v
3 3x 2 x
65 3 2 4
+ ----*f(v,v )*a20*a4 *b36*u *v)/a13
24 x