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Math Help - PDE Linear Operators

  1. #1
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    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.
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  2. #2
    A Plied Mathematician
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    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?
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