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Homework Help: Linearity for a PDE

  1. Feb 11, 2012 #1
    1. The problem statement, all variables and given/known data
    Is the operator

    Lu = du/dx + u * du/dy

    linear?


    2. Relevant equations

    Linearity occurs for L[u+cv] = L + cL[v]

    3. The attempt at a solution

    I know this isn't linear because of the second term, but I don't understand why I can't write the operator as

    L = (d/dx + u * d/dy)

    which then seems to almost work out, except that I don't know what to make 'u' in the operator when applying the linearity condition since the linearity condition uses two different functions instead of only 'u'.
     
  2. jcsd
  3. Feb 11, 2012 #2
    a function or operator f is linear if f(x+y)=f(x)+f(y) and f(cx)=cf(x), for all x+y and cx in the vector space. in this case it is a vector space of functions such as u(x,y) or u(x,y) + c*v(x,y) and the operator is defined as L(u(x,y))=d/dx(u(x,y) + u(x,y)*d/dy(u(x,y), for arbitrary u(x,y) in the vectorspace.
     
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