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Show that reaction-diffusion PDE is linear

  1. Apr 20, 2004 #1
    du/dt = f(u)+D(laplacian*u)

    where u=u(x,y,t), D>0, f(u)=a*u

    I'm not sure where to start other than substitute f(u) with a*u. Suggestions?
     
  2. jcsd
  3. Apr 21, 2004 #2

    arildno

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    Rewrite your equation in the form:

    L(u)=0, L=d/dt-a-D*Laplacian.
    Here, L is called a differential operator that works on the function u.

    To show that L is linear, you must show that for any functions U,u , and constants c, b that:
    L(c*U+b*u)=c*L(U)+b*L(u)

    We have:
    L(c*U+b*u)=d/dt(c*U+b*u)-a*(c*U+b*u)+D*Laplacian(c*U+b*u)=
    =c*(dU/dt-a*U+D*laplacian(U))+b*(dU/dt-a*U+D*laplacian(U))=
    c*L(U)+b*L(u),
    as required.
    Note that if U and u are both solutions to the original differential equation;
    i.e. L(U)=L(u)=0,
    the linearity of L implies that any sum of U and u also is a solution.
     
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