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Confusion over the definition of a Green's function

  1. May 7, 2014 #1
    This is how I learned about Green's functions:

    For the 1-D problem with the linear operator L and the inner product,
    [itex](\cdot,\cdot)[/itex],
    [itex]Lu(x) = f(x) \rightarrow u=(f(x),G(\xi,x))[/itex]

    if the Green's function G is defined such that

    [itex]L^*G(\xi,x) = \delta(\xi-x)[/itex]

    I understand how to arrive at this algebraically. However, most articles I read define the Green's function backwards (?) like this:

    [itex]LG(x,\xi)=\delta(x-\xi)[/itex]

    How do I arrive at this definition? As in, how do I work through the algebra to show that the green's function can be defined like this? I am assuming that the swapping of the variables indicates an equivalence between the two definitions, but I do not immediately see it, and it has been confusing me for quite a bit. Does it have something to do with whether we're on the interval [a,b] in [itex]\xi[/itex] vs [itex]x[/itex]? Could someone walk me through the steps? No, this is not a homework or coursework question. I'm just confused with the definition.
     
  2. jcsd
  3. May 7, 2014 #2

    strangerep

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    Part of the difficulty might be that you haven't indicated explicitly which variable ##L## acts on in each case.

    So re-write your earlier equation as
    $$u(\xi) ~=~ \Big(f(x),G(\xi,x)\Big)$$
    and then apply your operator ##L## to it. But take care: the "x" is a dummy integration variable in the inner product, so your ##L## would need to act on the ##\xi## variable.

    You'll still have to work through it to get understanding... :biggrin:
     
  4. May 7, 2014 #3
    How do you specify what variable an operator acts on? I thought an operator was implicit, and how it behaves is defined based on the function it's operating on, for example, if

    [itex]L=\frac{\partial}{\partial x}[/itex]

    Then wouldn't the result of that operation be apparent depending on what u is? If u is independent of x, then the Lu would just be 0. That's why I didn't think that I would need to define what variables L is operating on. Also, in the definitions and examples I've worked through, I've never had to be explicit about what variables L was acting on. I had to be explicit about the functions L was acting on, but not the variables, since that depends on a further restriction of L. Wouldn't I lose generality if I explicitly define L like that?

    Also, my first equation should be $$u(x) ~=~ \Big(f(x),G(\xi,x)\Big)$$ with [itex]x[/itex] instead of [itex]\xi[/itex] (Line 3 in my first post). So, it's all in functions of x, with the dummy variable of integration in the inner product being [itex]\xi[/itex]
     
    Last edited: May 8, 2014
  5. May 9, 2014 #4

    strangerep

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    It depends on the details on the operator. See below.

    That depends on the details of the operator. Integral operators are a bit trickier...
    Suppose L is just the operator of differentiation. Then you can of course write ##f' = L f## in abstract notation. You could also use concrete notation, e.g., $$f'(x) ~=~ \frac{d}{dx} \; f(x)$$ which contains essentially the same information as $$f'(z) ~=~ \frac{d}{dz} \; f(z) ~,$$ (functionally speaking).

    That wouldn't make sense, since in that case you could pull ##f(x)## outside the integral, i.e.,
    $$\int d\xi \, f(x) G(\xi,x) ~=~ f(x) \int d\xi\, G(\xi,x) ~.$$But what's actually needed is to "contract" ##G## with ##f##.

    To explain what I mean by "contract", here's another way to think about Green's functions is as a continuous-index generalization of ordinary matrices. Consider a column vector with components ##a_i##, and a 2x2 matrix with components ##M_{ij}##. The action of M on ##a## produces a new vector ##b## as follows:$$b_i~=~ \sum_j M_{ij} a_j ~.$$Now think of a function ##f## as a column vector with a continuous index ##x##. So I'll write ##f_x := f(x)##, etc. The action of the Green's function is $$u_x \equiv u(x) ~=~ \int d\xi \, G_{x\xi} f_\xi ~\equiv~ \int d\xi \, G(x,\xi) \, f(\xi) ~,$$where I hope you can see that the Greens function is analogous to the earlier matrix M, but here we're doing a "contraction" over the "index" ##\xi##, implemented as integration instead of discrete summation. A purist might even think of G as an integral operator (instead of just the kernel function of an integral operator as above), and write ##u = Gf## and then the notation exactly parallels the matrix/vector case -- provided one can keep track of the types of all the symbols.
     
  6. May 9, 2014 #5
    Oh, that's just a brilliant explanation. Yes, I've dealt with tensors in indicial notation a ton when talking about solid mechanics. What you just explained to me thoroughly unified so many concepts in my head. Thanks! I see where I was erring in my thought process now. I need to make sure to define my contracted variable when specifying the inner product. Thanks!
     
  7. May 9, 2014 #6

    strangerep

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    I felt the same way when I first saw this. It means that we can think of
    $$L u = f$$ as being solved by $$u = L^{-1} f ~,$$and ##G## is just a concrete implementation of ##L^{-1}##.

    Happy gardening.
     
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