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