# PDE Linear Operators

• December 29th 2010, 02:55 PM
dwsmith
PDE Linear Operators
$uu_{xy}-u_xu_y=0$

$L(u)+L(v)=uu_{xy}-u_xu_y+vv_{xy}-v_xv_y$

$=uu_{xy}+vv_{xy}-(u_xu_y+v_xv_y)$

$L(u+v)=u(u+v)_{xy}-(u+v)_x(u+v)_y$

$=uu_{xy}+uv_{xy}-[(u_x+v_x)(u_y+v_y)]$

$=uu_{xy}+uv_{xy}-[u_xu_y+u_xv_y+v_xu_y+v_xv_y]$

$=uu_{xy}-u_xu_y+uv_{xy}-u_xv_y-v_xu_y-v_xv_y$

$=L(u)+uv_{xy}-u_xv_y-v_xu_y-v_xv_y$

$L(u)+L(v)\neq L(u+v)$

Not Linear.

Is the progression of $L(u+v)$ done correctly?

Thanks.
• December 29th 2010, 03:27 PM
Ackbeet
Quote:

$L(u+v)=u(u+v)_{xy}-(u+v)_x(u+v)_y$
Should be

$L(u+v)=(u+v)(u+v)_{xy}-(u+v)_x(u+v)_y,$

I think. If you propagate that through, what do you get?