Solution 8 to problem l1o35
Expressions |
Parameters |
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Back to problem l1o35
Expressions
The solution is given through the following expressions:
3 2
---*a21 *b1
2
b74=-------------
a13*a1
2
3*a21 *b1
b73=-----------
2
a13
b72=0
2
- 6*a13 *a21*b1
b71=------------------
3
a1
- 6*a13*a21*b1
b70=-----------------
2
a1
b69=0
b68=0
2*a21*b1
b67=----------
a1
2*a21*b1
b66=----------
a1
b65=0
b64=0
6*a21*b1
b63=----------
a1
b62=0
3*a21*b1
b61=----------
a1
a21*b1
b60=--------
a13
3*a21*b1
b59=----------
a13
b58=0
b57=0
b56=0
b55=0
5
6*a13 *b1
b54=-----------
5
a1
b53=0
b52=0
b51=0
b50=0
3
- 10*a13 *b1
b49=---------------
3
a1
3
- 10*a13 *b1
b48=---------------
3
a1
b47=0
b46=0
b45=0
b44=0
b43=0
b42=0
b41=0
b40=0
b39=0
b38=0
a13*b1
b37=--------
a1
b36=0
3 3
---*a1*a21 *b1
4
b35=----------------
4
a13
2
- 3*a21 *b1
b34=--------------
a13*a1
2
6*a21 *b1
b33=-----------
2
a13
9 2
---*a21 *b1
2
b32=-------------
2
a13
2
6*a1*a21 *b1
b31=--------------
3
a13
2
4*a1*a21 *b1
b30=--------------
3
a13
2
4*a1*a21 *b1
b29=--------------
3
a13
2
3*a13 *a21*b1
b28=---------------
3
a1
- 12*a13*a21*b1
b27=------------------
2
a1
- 24*a13*a21*b1
b26=------------------
2
a1
- 6*a21*b1
b25=-------------
a1
- 6*a21*b1
b24=-------------
a1
- 12*a21*b1
b23=--------------
a1
a21*b1
b22=--------
a1
- 2*a21*b1
b21=-------------
a1
2*a21*b1
b20=----------
a13
6*a21*b1
b19=----------
a13
8*a21*b1
b18=----------
a13
8*a21*b1
b17=----------
a13
12*a21*b1
b16=-----------
a13
4*a21*b1
b15=----------
a13
a1*a21*b1
b14=-----------
2
a13
4*a1*a21*b1
b13=-------------
2
a13
3*a1*a21*b1
b12=-------------
2
a13
b11=0
4
30*a13 *b1
b10=------------
4
a1
b9=0
b8=0
2
- 10*a13 *b1
b7=---------------
2
a1
2
- 40*a13 *b1
b6=---------------
2
a1
2
- 10*a13 *b1
b5=---------------
2
a1
b4=0
b3=0
b2=0
a1*a21
a20=--------
a13
a19=0
3
- 2*a13
a18=-----------
2
a1
a17=0
a16=0
a15=0
a14=0
a12=0
1 2 2
---*a1 *a21
2
a11=--------------
3
a13
a10= - a21
2*a1*a21
a9=----------
a13
2*a1*a21
a8=----------
a13
2
a1 *a21
a7=---------
2
a13
2
a1 *a21
a6=---------
2
a13
a5=0
2
- 6*a13
a4=-----------
a1
a3=0
a2=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:
b1,a13,a21,a1
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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
3 2 2 2 5 2
u =(u *a13 *a1 + 2*u *f(v,v)*a13 *a1 *a21 - 6*u *a13 *u
t 3x x x
3 3 2 2
+ f(v ,v )*a13*a1 *a21 + f(v,v )*a13*a1 *a21 + 2*f(v,v )*a13 *a1 *a21*u
x x 2x x
1 2 3 2 3 2 3
+ ---*f(v,v) *a1 *a21 - f(v,v)*a13 *a1*a21*u )/(a13 *a1)
2
2 2 3 2 4 3
u *a13 *a1 *v + f(v,v )*a1 *a21*v + f(v,v)*a13*a1 *a21*u*v - 2*a13 *u *v
2x x
v =---------------------------------------------------------------------------
t 2
a13*a1
4 4 3 4 6 2 2
u =(u *a13 *a1 *b1 + 4*u *f(v,v)*a13 *a1 *a21*b1 - 10*u *a13 *a1 *b1*u
s 5x 3x 3x
6 2 3 4
- 40*u *u *a13 *a1 *b1*u + 12*u *f(v,v )*a13 *a1 *a21*b1
2x x 2x x
4 3 3 6 2
- 2*u *f(v,v)*a13 *a1 *a21*b1*u - 10*u *a13 *a1 *b1
2x x
2 4 3 3 4
+ u *f(v,v)*a13 *a1 *a21*b1 + 8*u *f(v ,v )*a13 *a1 *a21*b1
x x x x
3 4 4 3
+ 8*u *f(v,v )*a13 *a1 *a21*b1 - 12*u *f(v,v )*a13 *a1 *a21*b1*u
x 2x x x
9 2 2 4 2 5 2 2
+ ---*u *f(v,v) *a13 *a1 *a21 *b1 - 24*u *f(v,v)*a13 *a1 *a21*b1*u
2 x x
8 4 2 5
+ 30*u *a13 *b1*u + 3*f(v ,v )*a13 *a1 *a21*b1
x 2x 2x
2 5 3 4
+ 4*f(v ,v )*a13 *a1 *a21*b1 + 6*f(v ,v )*a13 *a1 *a21*b1*u
x 3x x 2x
5 2 4 3 2
+ 4*f(v ,v )*f(v,v)*a13*a1 *a21 *b1 - 6*f(v ,v )*a13 *a1 *a21*b1*u
x x x x
2 5 3 4
+ f(v,v )*a13 *a1 *a21*b1 + 2*f(v,v )*a13 *a1 *a21*b1*u
4x 3x
5 2 4 3 2
+ 4*f(v,v )*f(v,v)*a13*a1 *a21 *b1 - 6*f(v,v )*a13 *a1 *a21*b1*u
2x 2x
2 5 2 2 4 2
+ 6*f(v,v ) *a13*a1 *a21 *b1 + 6*f(v,v )*f(v,v)*a13 *a1 *a21 *b1*u
x x
5 2 3 3 3 5 3
- 12*f(v,v )*a13 *a1 *a21*b1*u + ---*f(v,v) *a1 *a21 *b1
x 4
2 3 3 2 2 6 4 4 4
- 3*f(v,v) *a13 *a1 *a21 *b1*u + 3*f(v,v)*a13 *a1*a21*b1*u )/(a13 *a1 )
3 4 2 4 5 2 2
v =(u *a13 *a1 *b1*v + 3*u *f(v,v)*a13 *a1 *a21*b1*v - 10*u *a13 *a1 *b1*u *v
s 4x 2x 2x
2 5 2 2 4
- 10*u *a13 *a1 *b1*u*v + 6*u *f(v,v )*a13 *a1 *a21*b1*v
x x x
5 2 4
+ 3*f(v ,v )*a13*a1 *a21*b1*v + 2*f(v ,v )*a13 *a1 *a21*b1*u*v
x 2x x x
5 2 4
+ f(v,v )*a13*a1 *a21*b1*v + 2*f(v,v )*a13 *a1 *a21*b1*u*v
3x 2x
5 2 3 3 2
+ 3*f(v,v )*f(v,v)*a1 *a21 *b1*v - 6*f(v,v )*a13 *a1 *a21*b1*u *v
x x
3 2 4 2 4 2 3
+ ---*f(v,v) *a13*a1 *a21 *b1*u*v - 6*f(v,v)*a13 *a1 *a21*b1*u *v
2
7 5 2 5
+ 6*a13 *b1*u *v)/(a13 *a1 )