- #1

Semo727

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I need to find Green's function for differential operator

[tex]L=a\frac{d^2}{dx^2}+b\frac{d}{dx}+c[/tex]

i.e. find solution for differential equation equation

[tex]LG(x,y)=\delta(x-y)[/tex]

I'm looking for solution in interval [tex][0,1][/tex] and boundary conditions: [tex]G(0,y)=0, \frac{dG(1,y)}{dx}=0[/tex]. Well, my problem is, that I don't know what conditions the derivative [tex]\frac{dG(x,y)}{dx}[/tex] must satisfy for [tex]x\rightarrow y[/tex] from the left and from the right. I just know, that [tex]G(x,y)[/tex] must be continuous function (when we look at it as function in x argument) also in the point y.

On wikipedia there is said, that there should be derivative "jump". i.e.

[tex]\frac{dG(x,y)}{dx}\vert_{y^+}-\frac{dG(x,y)}{dx}\vert_{y^-}=1/p(y)[/tex], where [tex]p(y)[/tex] comes from the Sturm-Liouville differential operator which they consider, i.e. they just examine the case

[tex]L=\frac{d}{dx}(p(x)\frac{d}{dx})+q(x)[/tex]

So my question is: what are the conditions for the derivative jump in case of linear operator above? Or if there are none, how can I find Green function satisfying [tex]LG(x,y)=\delta(x-y)[/tex] as I have just three equations (comming from boundary problems and continuity) for four constants and these equations don't consider the delta function on the right side of the equation?

Thanks for your soon reply and I apologize for my very non-mathematical approach to that problem while formulating my questions, but this is homework problem from my physics classes and they didn't formulated theory of Green's functions very mathematicaly correctly so my all understanding of this problem is only intuitive.