Solution 6 to problem l1o35
Expressions |
Parameters |
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Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
2
4*a6 *b53
b73=-----------
2
a13
2
a6 *b53
b72=---------
2
a13
b71=0
- 4*a6*b53
b70=-------------
a13
- 2*a6*b53
b69=-------------
a13
- 4*a6*b53
b68=-------------
a13
b67=0
b66=0
b65=0
b64=0
- 12*a6*b53
b63=--------------
a4
- 6*a6*b53
b62=-------------
a4
- 6*a6*b53
b61=-------------
a4
6*a13*a6*b53
b60=--------------
2
a4
18*a13*a6*b53
b59=---------------
2
a4
6*a13*a6*b53
b58=--------------
2
a4
6*a13*a6*b53
b57=--------------
2
a4
b56=0
b55=0
b54=0
b52=4*b53
b51=0
6*a13*b53
b50=-----------
a4
12*a13*b53
b49=------------
a4
6*a13*b53
b48=-----------
a4
b47=0
b46=0
2
3*a13 *b53
b45=------------
2
a4
2
- 6*a13 *b53
b44=---------------
2
a4
b43=0
2
- 6*a13 *b53
b42=---------------
2
a4
b41=0
b40=0
b39=0
3
- 9*a13 *b53
b38=---------------
3
a4
3
- 9*a13 *b53
b37=---------------
3
a4
b36=0
b35=0
b34=0
2
4*a6 *b53
b33=-----------
2
a13
2
a6 *b53
b32=---------
2
a13
2
- 12*a6 *b53
b31=---------------
a13*a4
2
- 6*a6 *b53
b30=--------------
a13*a4
2
- 6*a6 *b53
b29=--------------
a13*a4
b28=0
- 4*a6*b53
b27=-------------
a13
- 6*a6*b53
b26=-------------
a13
6*a6*b53
b25=----------
a4
6*a6*b53
b24=----------
a4
12*a6*b53
b23=-----------
a4
b22=0
b21=0
6*a13*a6*b53
b20=--------------
2
a4
18*a13*a6*b53
b19=---------------
2
a4
24*a13*a6*b53
b18=---------------
2
a4
24*a13*a6*b53
b17=---------------
2
a4
36*a13*a6*b53
b16=---------------
2
a4
9*a13*a6*b53
b15=--------------
2
a4
2
- 9*a13 *a6*b53
b14=------------------
3
a4
2
- 36*a13 *a6*b53
b13=-------------------
3
a4
2
- 27*a13 *a6*b53
b12=-------------------
3
a4
b11=0
b10=5*b53
b9=0
b8=0
2
- 15*a13 *b53
b7=----------------
2
a4
2
- 60*a13 *b53
b6=----------------
2
a4
2
- 15*a13 *b53
b5=----------------
2
a4
b4=0
b3=0
b2=0
27 4
----*a13 *b53
2
b1=---------------
4
a4
a21=0
2
- ---*a4*a6
3
a20=--------------
a13
1
- ---*a4*a6
3
a19=--------------
a13
a18=0
1
a17=---*a4
3
2
a16=---*a4
3
a15=0
a14=a13
a12=0
a11=0
a10=0
2
- ---*a4*a6
3
a9=--------------
a13
1
- ---*a4*a6
3
a8=--------------
a13
a7=a6
a5=0
a3=0
a2=0
3 2
- ---*a13
2
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:
b53,a13,a6,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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
3 3 1 2 2 2
u =( - ---*u *a13 - ---*u *f(v,v)*a4 *a6 + u *a13*a4 *u + f(v ,v )*a13*a4*a6
t 2 3x 3 x x x x
2 2
+ f(v,v )*a13*a4*a6 - ---*f(v,v )*a4 *a6*u)/(a13*a4)
2x 3 x
2 2 2 1
v =(u *a13 *v + u *v *a13 + ---*u *a13*a4*u*v - ---*v *f(v,v)*a4*a6
t 2x x x 3 x 3 x
1 2 2
+ ---*v *a13*a4*u - ---*f(v,v )*a4*a6*v)/a13
3 x 3 x
27 6 3 2 4 2 2
u =(----*u *a13 *b53 + 9*u *f(v,v)*a13 *a4 *a6*b53 - 15*u *a13 *a4 *b53*u
s 2 5x 3x 3x
4 2 3 2
- 60*u *u *a13 *a4 *b53*u + 36*u *f(v,v )*a13 *a4 *a6*b53
2x x 2x x
3 4 2 3 2
- 15*u *a13 *a4 *b53 + 24*u *f(v ,v )*a13 *a4 *a6*b53
x x x x
3 2 2 3
+ 24*u *f(v,v )*a13 *a4 *a6*b53 + 12*u *f(v,v )*a13 *a4 *a6*b53*u
x 2x x x
2 4 2 4 2
+ u *f(v,v) *a4 *a6 *b53 - 6*u *f(v,v)*a13*a4 *a6*b53*u
x x
2 4 4 4
+ 5*u *a13 *a4 *b53*u - 27*f(v ,v )*a13 *a4*a6*b53
x 2x 2x
4 3 2
- 36*f(v ,v )*a13 *a4*a6*b53 + 18*f(v ,v )*a13 *a4 *a6*b53*u
x 3x x 2x
3 2 2 3 2
- 6*f(v ,v )*f(v,v)*a13*a4 *a6 *b53 + 6*f(v ,v )*a13 *a4 *a6*b53*u
x x x x
4 3 2
- 9*f(v,v )*a13 *a4*a6*b53 + 6*f(v,v )*a13 *a4 *a6*b53*u
4x 3x
3 2 2 3 2
- 6*f(v,v )*f(v,v)*a13*a4 *a6 *b53 + 6*f(v,v )*a13 *a4 *a6*b53*u
2x 2x
2 3 2 4 2
- 12*f(v,v ) *a13*a4 *a6 *b53 + 4*f(v,v )*f(v,v)*a4 *a6 *b53*u
x x
4 3 2 4
- 4*f(v,v )*a13*a4 *a6*b53*u )/(a13 *a4 )
x
5 5 4
v =( - 9*u *a13 *b53*v - 9*u *v *a13 *b53 - 6*u *a13 *a4*b53*u*v
s 4x 3x x 3x
4 2 2
- 6*u *v *a13 *a4*b53*u - 6*u *f(v,v)*a13 *a4 *a6*b53*v
2x x 2x
3 2 2 2 4 2 3 2
+ 6*u *a13 *a4 *b53*u *v + 3*u *v *a13 *a4*b53 + 12*u *a13 *a4 *b53*u*v
2x x x x
2 2 3 2 2
- 6*u *v *f(v,v)*a13 *a4 *a6*b53 + 6*u *v *a13 *a4 *b53*u
x x x x
2 2 3
- 12*u *f(v,v )*a13 *a4 *a6*b53*v - 4*u *f(v,v)*a13*a4 *a6*b53*u*v
x x x
2 3 3 3
+ 4*u *a13 *a4 *b53*u *v + 6*v *f(v ,v )*a13 *a4*a6*b53
x x x x
3 2 3 2
+ 6*v *f(v,v )*a13 *a4*a6*b53 + v *f(v,v) *a4 *a6 *b53
x 2x x
3 2 2 3 4
- 2*v *f(v,v)*a13*a4 *a6*b53*u + v *a13 *a4 *b53*u
x x
3 3
+ 18*f(v ,v )*a13 *a4*a6*b53*v + 6*f(v,v )*a13 *a4*a6*b53*v
x 2x 3x
3 2 3 2 2 3
+ 4*f(v,v )*f(v,v)*a4 *a6 *b53*v - 4*f(v,v )*a13*a4 *a6*b53*u *v)/(a13 *a4
x x
)