Solution 24 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
8
b72=------------------
4
a13
b71=0
3
- 5*a20*a4 *b36
b70=------------------
4
a13
25 3
----*a20*a4 *b36
2
b69=------------------
4
a13
3
5*a20*a4 *b36
b68=---------------
4
a13
b67=0
b66=0
b65=0
b64=0
2
5*a20*a4 *b36
b63=---------------
3
a13
25 2
- ----*a20*a4 *b36
2
b62=---------------------
3
a13
5 2
- ---*a20*a4 *b36
2
b61=--------------------
3
a13
5
---*a20*a4*b36
2
b60=----------------
2
a13
5
---*a20*a4*b36
2
b59=----------------
2
a13
5
- ---*a20*a4*b36
2
b58=-------------------
2
a13
b57=0
b56=0
5
- ---*a20*a4*b36
2
b55=-------------------
2
a13
b54=0
5 4
---*a4 *b36
2
b53=-------------
4
a13
4
10*a4 *b36
b52=------------
4
a13
b51=0
3
- 5*a4 *b36
b50=--------------
3
a13
3
- 10*a4 *b36
b49=---------------
3
a13
3
- 5*a4 *b36
b48=--------------
3
a13
2
- 5*a4 *b36
b47=--------------
2
a13
2
- 20*a4 *b36
b46=---------------
2
a13
25 2
- ----*a4 *b36
2
b45=-----------------
2
a13
2
- 15*a4 *b36
b44=---------------
2
a13
2
- 10*a4 *b36
b43=---------------
2
a13
2
- 5*a4 *b36
b42=--------------
2
a13
b41=0
5*a4*b36
b40=----------
a13
10*a4*b36
b39=-----------
a13
15
----*a4*b36
2
b38=-------------
a13
5
---*a4*b36
2
b37=------------
a13
b35=0
b34=0
5 2 2
---*a20 *a4 *b36
2
b33=------------------
4
a13
5 2 2
---*a20 *a4 *b36
8
b32=------------------
4
a13
5 2
---*a20 *a4*b36
2
b31=-----------------
3
a13
5 2
---*a20 *a4*b36
4
b30=-----------------
3
a13
5 2
---*a20 *a4*b36
4
b29=-----------------
3
a13
b28=0
3
5*a20*a4 *b36
b27=---------------
4
a13
15 3
----*a20*a4 *b36
2
b26=------------------
4
a13
5 2
---*a20*a4 *b36
2
b25=-----------------
3
a13
5 2
---*a20*a4 *b36
2
b24=-----------------
3
a13
2
5*a20*a4 *b36
b23=---------------
3
a13
b22=0
b21=0
5
- ---*a20*a4*b36
2
b20=-------------------
2
a13
5
- ---*a20*a4*b36
2
b19=-------------------
2
a13
- 5*a20*a4*b36
b18=-----------------
2
a13
5
- ---*a20*a4*b36
2
b17=-------------------
2
a13
- 5*a20*a4*b36
b16=-----------------
2
a13
5
- ---*a20*a4*b36
4
b15=-------------------
2
a13
5
- ---*a20*b36
4
b14=----------------
a13
5
- ---*a20*b36
2
b13=----------------
a13
5
- ---*a20*b36
4
b12=----------------
a13
b11=0
15 4
- ----*a4 *b36
2
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
1
b1= - ---*b36
4
a21=0
1
a19= - ---*a20
2
a18=0
a17= - a4
a16= - 2*a4
a15=0
a14=a13
1 2
---*a13
3
a12=----------
a4
a11=0
a10=0
a9= - a20
1
a8= - ---*a20
2
1
- ---*a13*a20
2
a7=----------------
a4
1
- ---*a13*a20
2
a6=----------------
a4
a5=0
a3=0
a2=0
1 2
- ---*a13
6
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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
1 2 1 2 2 1
u =( - ---*u *a13 - ---*u *f(v,v)*a20*a4 + u *a4 *u - ---*f(v ,v )*a13*a20
t 6 3x 2 x x 2 x x
1
- ---*f(v,v )*a13*a20 - f(v,v )*a20*a4*u)/a4
2 2x x
2 1 2
v =(u *a13*a4*v + u *v *a13*a4 - 2*u *a4 *u*v + ---*v *a13
t 2x x x x 3 3x
1 2 2
- ---*v *f(v,v)*a20*a4 - v *a4 *u + f(v,v )*a20*a4*v)/a4
2 x x x
1 4 5 2
u =( - ---*u *a13 *b36 - ---*u *f(v,v)*a13 *a20*a4*b36
s 4 5x 4 3x
5 2 2 2 2 2
+ ---*u *a13 *a4 *b36*u + 10*u *u *a13 *a4 *b36*u
2 3x 2x x
2 5 3 2 2
- 5*u *f(v,v )*a13 *a20*a4*b36 + ---*u *a13 *a4 *b36
2x x 2 x
5 2 2
- ---*u *f(v ,v )*a13 *a20*a4*b36 - 5*u *f(v,v )*a13 *a20*a4*b36
2 x x x x 2x
2 5 2 2 2
+ 5*u *f(v,v )*a13*a20*a4 *b36*u + ---*u *f(v,v) *a20 *a4 *b36
x x 8 x
15 3 2 15 4 4
+ ----*u *f(v,v)*a20*a4 *b36*u - ----*u *a4 *b36*u
2 x 2 x
5 3 5 3
- ---*f(v ,v )*a13 *a20*b36 - ---*f(v ,v )*a13 *a20*b36
4 2x 2x 2 x 3x
5 2 5 2
- ---*f(v ,v )*a13 *a20*a4*b36*u + ---*f(v ,v )*f(v,v)*a13*a20 *a4*b36
2 x 2x 4 x x
5 2 2 5 3
+ ---*f(v ,v )*a13*a20*a4 *b36*u - ---*f(v,v )*a13 *a20*b36
2 x x 4 4x
5 2 5 2
- ---*f(v,v )*a13 *a20*a4*b36*u + ---*f(v,v )*f(v,v)*a13*a20 *a4*b36
2 3x 4 2x
5 2 2 5 2 2
+ ---*f(v,v )*a13*a20*a4 *b36*u + ---*f(v,v ) *a13*a20 *a4*b36
2 2x 2 x
5 2 2 3 3 4
+ ---*f(v,v )*f(v,v)*a20 *a4 *b36*u + 5*f(v,v )*a20*a4 *b36*u )/a13
2 x x
5 3 15 3 2 2
v =(---*u *a13 *a4*b36*v + ----*u *v *a13 *a4*b36 - 5*u *a13 *a4 *b36*u*v
s 2 4x 2 3x x 3x
2 2 3
- 10*u *u *a13 *a4 *b36*v + 10*u *v *a13 *a4*b36
2x x 2x 2x
2 2 5 2
- 15*u *v *a13 *a4 *b36*u - ---*u *f(v,v)*a13*a20*a4 *b36*v
2x x 2 2x
3 2 25 2 2 2
- 5*u *a13*a4 *b36*u *v - ----*u *v *a13 *a4 *b36
2x 2 x x
2 3 3 2 2
- 10*u *a13*a4 *b36*u*v + 5*u *v *a13 *a4*b36 - 20*u *v *a13 *a4 *b36*u
x x 3x x 2x
25 2 3 2
- ----*u *v *f(v,v)*a13*a20*a4 *b36 - 5*u *v *a13*a4 *b36*u
2 x x x x
2 3
+ 5*u *f(v,v )*a13*a20*a4 *b36*v + 5*u *f(v,v)*a20*a4 *b36*u*v
x x x
4 3 4 5 2
+ 10*u *a4 *b36*u *v + v *a13 *b36 - ---*v *f(v,v)*a13 *a20*a4*b36
x 5x 2 3x
2 2 2 5 2
- 5*v *a13 *a4 *b36*u - ---*v *f(v,v )*a13 *a20*a4*b36
3x 2 x 2x
5 2 2 2 25 3 2
+ ---*v *f(v,v) *a20 *a4 *b36 + ----*v *f(v,v)*a20*a4 *b36*u
8 x 2 x
5 4 4 5 2
+ ---*v *a4 *b36*u + ---*f(v ,v )*a13 *a20*a4*b36*v
2 x 2 x 2x
5 2 5 2 2
+ ---*f(v,v )*a13 *a20*a4*b36*v - ---*f(v,v )*f(v,v)*a20 *a4 *b36*v
2 3x 2 x
3 2 4
- 5*f(v,v )*a20*a4 *b36*u *v)/a13
x