Solution 17 to problem l1o35
Expressions |
Parameters |
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Back to problem l1o35
Expressions
The solution is given through the following expressions:
b74=0
20 2
----*a19 *b36
9
b73=---------------
2
a12
10 2
----*a19 *b36
9
b72=---------------
2
a12
b71=0
20
----*a16*a19*b36
9
b70=------------------
2
a12
20
----*a16*a19*b36
9
b69=------------------
2
a12
20
----*a16*a19*b36
9
b68=------------------
2
a12
b67=0
b66=0
b65=0
b64=0
b63=0
b62=0
b61=0
5
---*a19*b36
3
b60=-------------
a12
10
----*a19*b36
3
b59=--------------
a12
5*a19*b36
b58=-----------
a12
10
----*a19*b36
3
b57=--------------
a12
5*a19*b36
b56=-----------
a12
5
---*a19*b36
3
b55=-------------
a12
b54=0
10 2
----*a16 *b36
9
b53=---------------
2
a12
20 2
----*a16 *b36
9
b52=---------------
2
a12
b51=0
b50=0
b49=0
b48=0
5
---*a16*b36
3
b47=-------------
a12
5*a16*b36
b46=-----------
a12
10
----*a16*b36
3
b45=--------------
a12
5*a16*b36
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
20 2
----*a19 *b36
9
b33=---------------
2
a12
10 2
----*a19 *b36
9
b32=---------------
2
a12
b31=0
b30=0
b29=0
b28=0
20
----*a16*a19*b36
9
b27=------------------
2
a12
40
----*a16*a19*b36
9
b26=------------------
2
a12
b25=0
b24=0
b23=0
b22=0
b21=0
5
---*a19*b36
3
b20=-------------
a12
10
----*a19*b36
3
b19=--------------
a12
5*a19*b36
b18=-----------
a12
10
----*a19*b36
3
b17=--------------
a12
5*a19*b36
b16=-----------
a12
5
---*a19*b36
3
b15=-------------
a12
b14=0
b13=0
b12=0
b11=0
10 2
----*a16 *b36
3
b10=---------------
2
a12
b9=0
b8=0
10
----*a16*b36
3
b7=--------------
a12
40
----*a16*b36
3
b6=--------------
a12
10
----*a16*b36
3
b5=--------------
a12
b4=0
b3=0
b2=0
b1=b36
a21=0
a20=a19
a18=0
a17=a16
a15=0
a14=0
a13=0
a11=0
a10=0
a9=a19
a8=a19
a7=0
a6=0
a5=0
a4=2*a16
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:
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
2
u =u *a12 + u *f(v,v)*a19 + 2*u *a16*u + f(v,v )*a19*u
t 3x x x x
2
v =u *a16*u*v + v *a12 + v *f(v,v)*a19 + v *a16*u + f(v,v )*a19*v
t x 3x x x x
2 5 10 2
u =(u *a12 *b36 + ---*u *f(v,v)*a12*a19*b36 + ----*u *a12*a16*b36*u
s 5x 3 3x 3 3x
40
+ ----*u *u *a12*a16*b36*u + 5*u *f(v,v )*a12*a19*b36
3 2x x 2x x
10 3 10
+ ----*u *a12*a16*b36 + ----*u *f(v ,v )*a12*a19*b36
3 x 3 x x x
10 2 2
+ 5*u *f(v,v )*a12*a19*b36 + ----*u *f(v,v) *a19 *b36
x 2x 9 x
40 2 10 2 4
+ ----*u *f(v,v)*a16*a19*b36*u + ----*u *a16 *b36*u
9 x 3 x
10 5
+ ----*f(v ,v )*a12*a19*b36*u + ---*f(v,v )*a12*a19*b36*u
3 x 2x 3 3x
20 2 20 3 2
+ ----*f(v,v )*f(v,v)*a19 *b36*u + ----*f(v,v )*a16*a19*b36*u )/a12
9 x 9 x
5 10
v =(---*u *a12*a16*b36*u*v + ----*u *u *a12*a16*b36*v + 5*u *v *a12*a16*b36*u
s 3 3x 3 2x x 2x x
10 2
+ ----*u *v *a12*a16*b36 + 5*u *v *a12*a16*b36*u
3 x x x 2x
20 20 2 3 2
+ ----*u *f(v,v)*a16*a19*b36*u*v + ----*u *a16 *b36*u *v + v *a12 *b36
9 x 9 x 5x
5 5 2
+ ---*v *f(v,v)*a12*a19*b36 + ---*v *a12*a16*b36*u
3 3x 3 3x
10
+ 5*v *f(v,v )*a12*a19*b36 + ----*v *f(v ,v )*a12*a19*b36
2x x 3 x x x
10 2 2
+ 5*v *f(v,v )*a12*a19*b36 + ----*v *f(v,v) *a19 *b36
x 2x 9 x
20 2 10 2 4
+ ----*v *f(v,v)*a16*a19*b36*u + ----*v *a16 *b36*u
9 x 9 x
10 5
+ ----*f(v ,v )*a12*a19*b36*v + ---*f(v,v )*a12*a19*b36*v
3 x 2x 3 3x
20 2 20 2 2
+ ----*f(v,v )*f(v,v)*a19 *b36*v + ----*f(v,v )*a16*a19*b36*u *v)/a12
9 x 9 x