Solution 16 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
---*a19 *b36
6
b72=--------------
2
a12
b71=0
10
----*a16*a19*b36
9
b70=------------------
2
a12
25
----*a16*a19*b36
18
b69=------------------
2
a12
5
---*a16*a19*b36
3
b68=-----------------
2
a12
b67=0
b66=0
b65=0
b64=0
b63=0
b62=0
b61=0
b60=0
b59=0
10
----*a19*b36
3
b58=--------------
a12
5
---*a19*b36
3
b57=-------------
a12
10
----*a19*b36
3
b56=--------------
a12
5
---*a19*b36
3
b55=-------------
a12
b54=0
5 2
----*a16 *b36
24
b53=---------------
2
a12
5 2
---*a16 *b36
6
b52=--------------
2
a12
b51=0
b50=0
b49=0
b48=0
5
---*a16*b36
6
b47=-------------
a12
10
----*a16*b36
3
b46=--------------
a12
5
---*a16*b36
2
b45=-------------
a12
10
----*a16*b36
3
b44=--------------
a12
10
----*a16*b36
3
b43=--------------
a12
5
---*a16*b36
3
b42=-------------
a12
b41=0
b40=0
b39=0
b38=0
b37=0
b35=0
b34=0
10 2
----*a19 *b36
3
b33=---------------
2
a12
5 2
---*a19 *b36
6
b32=--------------
2
a12
b31=0
b30=0
b29=0
b28=0
5
---*a16*a19*b36
3
b27=-----------------
2
a12
5
---*a16*a19*b36
2
b26=-----------------
2
a12
b25=0
b24=0
b23=0
b22=0
b21=0
10
----*a19*b36
3
b20=--------------
a12
20
----*a19*b36
3
b19=--------------
a12
20
----*a19*b36
3
b18=--------------
a12
5*a19*b36
b17=-----------
a12
20
----*a19*b36
3
b16=--------------
a12
5
---*a19*b36
3
b15=-------------
a12
b14=0
b13=0
b12=0
b11=0
5 2
----*a16 *b36
24
b10=---------------
2
a12
b9=0
b8=0
5
---*a16*b36
6
b7=-------------
a12
10
----*a16*b36
3
b6=--------------
a12
5
---*a16*b36
6
b5=-------------
a12
b4=0
b3=0
b2=0
b1=b36
a21=0
a20=0
a18=0
1
a17=---*a16
2
a15=0
a14=0
a13=0
a11=0
a10=0
a9=2*a19
a8=a19
a7=0
a6=0
a5=0
1
a4=---*a16
2
a3=0
a2=0
a1=a12
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,a16,a19,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
u =u *a12 + u *f(v,v)*a19 + ---*u *a16*u + 2*f(v,v )*a19*u
t 3x x 2 x x
1 2
v =u *a16*u*v + v *a12 + v *f(v,v)*a19 + ---*v *a16*u
t x 3x x 2 x
2 5 5 2
u =(u *a12 *b36 + ---*u *f(v,v)*a12*a19*b36 + ---*u *a12*a16*b36*u
s 5x 3 3x 6 3x
10 20
+ ----*u *u *a12*a16*b36*u + ----*u *f(v,v )*a12*a19*b36
3 2x x 3 2x x
5 3
+ ---*u *a12*a16*b36 + 5*u *f(v ,v )*a12*a19*b36
6 x x x x
20 5 2 2
+ ----*u *f(v,v )*a12*a19*b36 + ---*u *f(v,v) *a19 *b36
3 x 2x 6 x
5 2 5 2 4
+ ---*u *f(v,v)*a16*a19*b36*u + ----*u *a16 *b36*u
2 x 24 x
20 10
+ ----*f(v ,v )*a12*a19*b36*u + ----*f(v,v )*a12*a19*b36*u
3 x 2x 3 3x
10 2 5 3 2
+ ----*f(v,v )*f(v,v)*a19 *b36*u + ---*f(v,v )*a16*a19*b36*u )/a12
3 x 3 x
5 10
v =(---*u *a12*a16*b36*u*v + ----*u *u *a12*a16*b36*v
s 3 3x 3 2x x
10 5 2
+ ----*u *v *a12*a16*b36*u + ---*u *v *a12*a16*b36
3 2x x 2 x x
10 5
+ ----*u *v *a12*a16*b36*u + ---*u *f(v,v)*a16*a19*b36*u*v
3 x 2x 3 x
5 2 3 2 5
+ ---*u *a16 *b36*u *v + v *a12 *b36 + ---*v *f(v,v)*a12*a19*b36
6 x 5x 3 3x
5 2 10
+ ---*v *a12*a16*b36*u + ----*v *f(v,v )*a12*a19*b36
6 3x 3 2x x
5 10
+ ---*v *f(v ,v )*a12*a19*b36 + ----*v *f(v,v )*a12*a19*b36
3 x x x 3 x 2x
5 2 2 25 2
+ ---*v *f(v,v) *a19 *b36 + ----*v *f(v,v)*a16*a19*b36*u
6 x 18 x
5 2 4 10 2 2
+ ----*v *a16 *b36*u + ----*f(v,v )*a16*a19*b36*u *v)/a12
24 x 9 x