Solution 3 to problem l1o35
Expressions |
Parameters |
Relevance |
Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
10 6
- ----*a19 *b36
9
b73=------------------
2 4
a17 *a7
5 6
---*a19 *b36
9
b72=--------------
2 4
a17 *a7
b71=0
5 5
---*a19 *b36
9
b70=--------------
4
a17*a7
25 5
----*a19 *b36
9
b69=---------------
4
a17*a7
20 5
----*a19 *b36
9
b68=---------------
4
a17*a7
b67=0
b66=0
b65=0
b64=0
5 4
- ---*a19 *b36
9
b63=-----------------
3
a17*a7
25 4
- ----*a19 *b36
9
b62=------------------
3
a17*a7
10 4
- ----*a19 *b36
9
b61=------------------
3
a17*a7
10 3
----*a19 *b36
9
b60=---------------
2
a17*a7
5 3
---*a19 *b36
3
b59=--------------
2
a17*a7
20 3
- ----*a19 *b36
9
b58=------------------
2
a17*a7
5 3
- ---*a19 *b36
9
b57=-----------------
2
a17*a7
5 3
- ---*a19 *b36
3
b56=-----------------
2
a17*a7
5 3
- ---*a19 *b36
3
b55=-----------------
2
a17*a7
b54=0
5 4
---*a19 *b36
9
b53=--------------
4
a7
20 4
----*a19 *b36
9
b52=---------------
4
a7
b51=0
10 3
- ----*a19 *b36
9
b50=------------------
3
a7
20 3
- ----*a19 *b36
9
b49=------------------
3
a7
10 3
- ----*a19 *b36
9
b48=------------------
3
a7
5 2
- ---*a19 *b36
3
b47=-----------------
2
a7
20 2
- ----*a19 *b36
3
b46=------------------
2
a7
2
- 5*a19 *b36
b45=---------------
2
a7
50 2
- ----*a19 *b36
9
b44=------------------
2
a7
50 2
- ----*a19 *b36
9
b43=------------------
2
a7
20 2
- ----*a19 *b36
9
b42=------------------
2
a7
b41=0
5
---*a19*b36
3
b40=-------------
a7
10
----*a19*b36
3
b39=--------------
a7
25
----*a19*b36
9
b38=--------------
a7
10
----*a19*b36
9
b37=--------------
a7
b35=0
b34=0
20 6
----*a19 *b36
9
b33=---------------
2 4
a17 *a7
5 6
---*a19 *b36
9
b32=--------------
2 4
a17 *a7
20 5
----*a19 *b36
9
b31=---------------
2 3
a17 *a7
10 5
----*a19 *b36
9
b30=---------------
2 3
a17 *a7
10 5
----*a19 *b36
9
b29=---------------
2 3
a17 *a7
b28=0
20 5
----*a19 *b36
9
b27=---------------
4
a17*a7
10 5
----*a19 *b36
3
b26=---------------
4
a17*a7
10 4
----*a19 *b36
9
b25=---------------
3
a17*a7
10 4
----*a19 *b36
9
b24=---------------
3
a17*a7
20 4
----*a19 *b36
9
b23=---------------
3
a17*a7
b22=0
b21=0
20 3
- ----*a19 *b36
9
b20=------------------
2
a17*a7
10 3
- ----*a19 *b36
3
b19=------------------
2
a17*a7
10 3
- ----*a19 *b36
3
b18=------------------
2
a17*a7
5 3
- ---*a19 *b36
3
b17=-----------------
2
a17*a7
20 3
- ----*a19 *b36
9
b16=------------------
2
a17*a7
5 3
- ---*a19 *b36
9
b15=-----------------
2
a17*a7
10 2
- ----*a19 *b36
9
b14=------------------
a17*a7
25 2
- ----*a19 *b36
9
b13=------------------
a17*a7
5 2
- ---*a19 *b36
3
b12=-----------------
a17*a7
b11=0
5 4
- ---*a19 *b36
9
b10=-----------------
4
a7
b9=0
b8=0
5 2
---*a19 *b36
9
b7=--------------
2
a7
20 2
----*a19 *b36
9
b6=---------------
2
a7
5 2
---*a19 *b36
9
b5=--------------
2
a7
5
---*a19*b36
9
b4=-------------
a7
5
---*a19*b36
9
b3=-------------
a7
b2=0
1
b1= - ---*b36
9
a21=0
a20= - a19
a18=0
a16=2*a17
a15=0
- a17*a7
a14=-----------
a19
- a17*a7
a13=-----------
a19
2
- a17*a7
a12=------------
2
a19
a11=0
a10=0
a9=2*a19
a8=a19
a6=a7
a5=0
a4=0
a3=0
a2=0
a1=0
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,a17,a7,a19
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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
u =u *f(v,v)*a19 + f(v ,v )*a7 + f(v,v )*a7 + 2*f(v,v )*a19*u
t x x x 2x x
2 2
v =( - u *a17*a19*a7*v - u *v *a17*a19*a7 + 2*u *a17*a19 *u*v - v *a17*a7
t 2x x x x 3x
3 2 2 3 2
+ v *f(v,v)*a19 + v *a17*a19 *u - f(v,v )*a19 *v)/a19
x x x
1 2 4 5 2 3
u =( - ---*u *a17 *a7 *b36 + ---*u *u *a17 *a19*a7 *b36
s 9 5x 9 3x x
5 3 2 5 2 2 2 2
- ---*u *f(v,v)*a17*a19 *a7 *b36 + ---*u *a17 *a19 *a7 *b36*u
9 3x 9 3x
5 2 2 3 20 2 2 2
+ ---*u *a17 *a19*a7 *b36 + ----*u *u *a17 *a19 *a7 *b36*u
9 2x 9 2x x
20 3 2 5 3 2 2 2
- ----*u *f(v,v )*a17*a19 *a7 *b36 + ---*u *a17 *a19 *a7 *b36
9 2x x 9 x
5 3 2 10 3 2
- ---*u *f(v ,v )*a17*a19 *a7 *b36 - ----*u *f(v,v )*a17*a19 *a7 *b36
3 x x x 3 x 2x
20 4 5 2 6
+ ----*u *f(v,v )*a17*a19 *a7*b36*u + ---*u *f(v,v) *a19 *b36
9 x x 9 x
10 5 2 5 2 4 4
+ ----*u *f(v,v)*a17*a19 *b36*u - ---*u *a17 *a19 *b36*u
3 x 9 x
5 2 3 25 2 3
- ---*f(v ,v )*a17*a19 *a7 *b36 - ----*f(v ,v )*a17*a19 *a7 *b36
3 2x 2x 9 x 3x
10 3 2 10 5
- ----*f(v ,v )*a17*a19 *a7 *b36*u + ----*f(v ,v )*f(v,v)*a19 *a7*b36
3 x 2x 9 x x
10 4 2 10 2 3
+ ----*f(v ,v )*a17*a19 *a7*b36*u - ----*f(v,v )*a17*a19 *a7 *b36
9 x x 9 4x
20 3 2 10 5
- ----*f(v,v )*a17*a19 *a7 *b36*u + ----*f(v,v )*f(v,v)*a19 *a7*b36
9 3x 9 2x
10 4 2 20 2 5
+ ----*f(v,v )*a17*a19 *a7*b36*u + ----*f(v,v ) *a19 *a7*b36
9 2x 9 x
20 6 20 5 3 2 4
+ ----*f(v,v )*f(v,v)*a19 *b36*u + ----*f(v,v )*a17*a19 *b36*u )/(a17 *a7 )
9 x 9 x
10 2 3 25 2 3
v =(----*u *a17 *a19*a7 *b36*v + ----*u *v *a17 *a19*a7 *b36
s 9 4x 9 3x x
20 2 2 2 50 2 2 2
- ----*u *a17 *a19 *a7 *b36*u*v - ----*u *u *a17 *a19 *a7 *b36*v
9 3x 9 2x x
10 2 3 50 2 2 2
+ ----*u *v *a17 *a19*a7 *b36 - ----*u *v *a17 *a19 *a7 *b36*u
3 2x 2x 9 2x x
10 4 10 2 3 2
- ----*u *f(v,v)*a17*a19 *a7*b36*v - ----*u *a17 *a19 *a7*b36*u *v
9 2x 9 2x
2 2 2 2 20 2 2 3
- 5*u *v *a17 *a19 *a7 *b36 - ----*u *a17 *a19 *a7*b36*u*v
x x 9 x
5 2 3 20 2 2 2
+ ---*u *v *a17 *a19*a7 *b36 - ----*u *v *a17 *a19 *a7 *b36*u
3 x 3x 3 x 2x
25 4 10 2 3 2
- ----*u *v *f(v,v)*a17*a19 *a7*b36 - ----*u *v *a17 *a19 *a7*b36*u
9 x x 9 x x
5 4 20 5
- ---*u *f(v,v )*a17*a19 *a7*b36*v + ----*u *f(v,v)*a17*a19 *b36*u*v
9 x x 9 x
20 2 4 3 2 4
+ ----*u *a17 *a19 *b36*u *v + v *a17 *a7 *b36
9 x 5x
5 3 2 5 2 2 2 2
- ---*v *f(v,v)*a17*a19 *a7 *b36 - ---*v *a17 *a19 *a7 *b36*u
3 3x 3 3x
5 3 2 5 3 2
- ---*v *f(v,v )*a17*a19 *a7 *b36 - ---*v *f(v ,v )*a17*a19 *a7 *b36
3 2x x 9 x x x
20 3 2 5 2 6
- ----*v *f(v,v )*a17*a19 *a7 *b36 + ---*v *f(v,v) *a19 *b36
9 x 2x 9 x
25 5 2 5 2 4 4
+ ----*v *f(v,v)*a17*a19 *b36*u + ---*v *a17 *a19 *b36*u
9 x 9 x
5 3 2 10 3 2
+ ---*f(v ,v )*a17*a19 *a7 *b36*v + ----*f(v,v )*a17*a19 *a7 *b36*v
3 x 2x 9 3x
10 6 5 5 2 2 4
- ----*f(v,v )*f(v,v)*a19 *b36*v + ---*f(v,v )*a17*a19 *b36*u *v)/(a17 *a7
9 x 9 x
)