Solution 19 to problem l1o35
Expressions |
Parameters |
Relevance |
Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
5 2
---*a20 *b36
9
b73=--------------
2
a12
b72=0
b71=0
b70=0
b69=0
b68=0
b67=0
b66=0
b65=0
b64=0
25 2
----*a20 *b36
3
b63=---------------
a12*a7
5 2
---*a20 *b36
3
b62=--------------
a12*a7
10 2
----*a20 *b36
3
b61=---------------
a12*a7
10
----*a20*b36
3
b60=--------------
a12
5*a20*b36
b59=-----------
a12
10
----*a20*b36
3
b58=--------------
a12
5
---*a20*b36
3
b57=-------------
a12
5
---*a20*b36
3
b56=-------------
a12
b55=0
b54=0
b53=0
b52=0
b51=0
b50=0
b49=0
b48=0
b47=0
b46=0
2
10*a20 *b36
b45=-------------
2
a7
b44=0
2
20*a20 *b36
b43=-------------
2
a7
b42=0
b41=0
5*a20*b36
b40=-----------
a7
10*a20*b36
b39=------------
a7
10*a20*b36
b38=------------
a7
5*a20*b36
b37=-----------
a7
b35=0
b34=0
b33=0
5 2
---*a20 *b36
9
b32=--------------
2
a12
5
---*a20*a7*b36
9
b31=----------------
2
a12
10
----*a20*a7*b36
9
b30=-----------------
2
a12
5
---*a20*a7*b36
9
b29=----------------
2
a12
b28=0
b27=0
b26=0
b25=0
b24=0
b23=0
20 2
----*a20 *b36
3
b22=---------------
a12*a7
b21=0
b20=0
b19=0
10*a20*b36
b18=------------
a12
5
---*a20*b36
3
b17=-------------
a12
20
----*a20*b36
3
b16=--------------
a12
10
----*a20*b36
3
b15=--------------
a12
5
---*a7*b36
3
b14=------------
a12
5
---*a7*b36
3
b13=------------
a12
5
---*a7*b36
3
b12=------------
a12
b11=0
b10=0
b9=0
b8=0
2
10*a20 *b36
b7=-------------
2
a7
b6=0
b5=0
5*a20*b36
b4=-----------
a7
10*a20*b36
b3=------------
a7
b2=0
b1=b36
a21=0
a19=0
a18=0
a17=0
a16=0
a15=0
3*a12*a20
a14=-----------
a7
3*a12*a20
a13=-----------
a7
a11=0
a10=0
a9=0
a8=a20
a6=0
a5=0
a4=0
3*a12*a20
a3=-----------
a7
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,a20,a7,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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
2 2
u *a12*a7 + 3*u *a12*a20 + u *f(v,v)*a20*a7 + f(v,v )*a7
3x x x 2x
u =--------------------------------------------------------------
t a7
3*u *a12*a20*v + 3*u *v *a12*a20 + v *a12*a7 + f(v,v )*a20*a7*v
2x x x 3x x
v =-------------------------------------------------------------------
t a7
2 2 2
u =(u *a12 *a7 *b36 + 10*u *u *a12 *a20*a7*b36
s 5x 3x x
10 2 2 2
+ ----*u *f(v,v)*a12*a20*a7 *b36 + 5*u *a12 *a20*a7*b36
3 3x 2x
20 2 3 2 2
+ ----*u *f(v,v )*a12*a20*a7 *b36 + 10*u *a12 *a20 *b36
3 2x x x
20 2 2 5 2
+ ----*u *f(v,v)*a12*a20 *a7*b36 + ---*u *f(v ,v )*a12*a20*a7 *b36
3 x 3 x x x
2 5 2 2 2
+ 10*u *f(v,v )*a12*a20*a7 *b36 + ---*u *f(v,v) *a20 *a7 *b36
x 2x 9 x
5 3 5 3
+ ---*f(v ,v )*a12*a7 *b36 + ---*f(v ,v )*a12*a7 *b36
3 2x 2x 3 x 3x
5 3 5 3
+ ---*f(v ,v )*f(v,v)*a20*a7 *b36 + ---*f(v,v )*a12*a7 *b36
9 x x 3 4x
10 3 5 2 3 2 2
+ ----*f(v,v )*f(v,v)*a20*a7 *b36 + ---*f(v,v ) *a20*a7 *b36)/(a12 *a7 )
9 2x 9 x
2 2
v =(5*u *a12 *a20*a7*b36*v + 10*u *v *a12 *a20*a7*b36
s 4x 3x x
2 2 2
+ 20*u *u *a12 *a20 *b36*v + 10*u *v *a12 *a20*a7*b36
2x x 2x 2x
10 2 2 2 2
+ ----*u *f(v,v)*a12*a20 *a7*b36*v + 10*u *v *a12 *a20 *b36
3 2x x x
2 5 2
+ 5*u *v *a12 *a20*a7*b36 + ---*u *v *f(v,v)*a12*a20 *a7*b36
x 3x 3 x x
25 2 2 2
+ ----*u *f(v,v )*a12*a20 *a7*b36*v + v *a12 *a7 *b36
3 x x 5x
5 2 5 2
+ ---*v *f(v,v )*a12*a20*a7 *b36 + ---*v *f(v ,v )*a12*a20*a7 *b36
3 2x x 3 x x x
10 2 2
+ ----*v *f(v,v )*a12*a20*a7 *b36 + 5*f(v ,v )*a12*a20*a7 *b36*v
3 x 2x x 2x
10 2 5 2 2
+ ----*f(v,v )*a12*a20*a7 *b36*v + ---*f(v,v )*f(v,v)*a20 *a7 *b36*v)/(
3 3x 9 x
2 2
a12 *a7 )