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ppauper
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Heat conduction Green's function and delta function?

October 25th, 2017, 6:57 am

Black-Scholes can be transformed to the heat equation (and vice versa).
I'm trying to do something with Green's functions on a half-space but I'm missing something which is probably obvious.

[$]u_{t}=ku_{xx}[$]
If we consider a problem on the half-space [$]x>0[$] and [$]t>0[$] with [$]u(x,0)=0[$] for [$]x>0[$] and [$]u(0,t)=f(t)[$] for [$]t>0[$], there's a Green's function solution
[$]u(x,t)=k\int_{0}^{t}\frac{x}{2\sqrt{\pi}}\frac{1}{(k(t-s))^{3/2}}\exp\left[-\frac{x^{2}}{4k(t-s)}\right]f(s)ds[$]

my question: when [$]x=0[$], pretty obviously we need this formula to reduce to [$]u(0,t)=f(t)[$] so somehow the Green's function reduces to a delta function  in that limit. Anyone able to fill in the gaps in that statement?

I'm also going to need [$]u_{x}(0,t)[$],  which I presume you can get by differentiating the formula for [$]u[$] above and then setting [$]x=0[$], but again there wlll be delta functions, so can someone point the way?
 
frolloos
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Re: Heat conduction Green's function and delta function?

October 25th, 2017, 9:52 am

See the second example (on page 4) in this document. It looks similar to what you're trying to solve. Maybe it helps?

I think the method used (for solving on the half line) is the reflection method.
 
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ppauper
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Re: Heat conduction Green's function and delta function?

October 25th, 2017, 10:27 am

See the second example (on page 4) in this document. It looks similar to what you're trying to solve. Maybe it helps?

I think the method used (for solving on the half line) is the reflection method.
thanks, but it looks like he's left unsaid the part I need.
Reflection method is used, you're correct, and that's where the green's function comes from. The formula I've used comes from polynanin's book on linear PDEs (handbook of linear partial differential equations, there's 2 or  russian guys, polynanin is one, have written a series of texts with a whole bunch of formulae in, including tables of solvable ODEs (sort of an updated Kamke) and tables of solvable integral  equations).
In that example, all he's done is shift the boundary condition to a forcing term in the PDE which doesn't really help me
 
frolloos
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Re: Heat conduction Green's function and delta function?

October 25th, 2017, 10:47 am

I re-read your question, and maybe I mis-read it first time: so it's not that you're really concerned about the solution, or how to obtain that solution, but more with whether and how the Green's function reduces to the Dirac delta function? Is that right?
 
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Paul
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Re: Heat conduction Green's function and delta function?

October 25th, 2017, 10:57 am

Just a suitable change of variables and a bit of asymptotics, no? Maybe integrate by parts first.
 
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ppauper
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Re: Heat conduction Green's function and delta function?

October 25th, 2017, 11:14 am

I re-read your question, and maybe I mis-read it first time: so it's not that you're really concerned about the solution, or how to obtain that solution, but more with whether and how the Green's function reduces to the Dirac delta function? Is that right?
yes
 
frolloos
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Re: Heat conduction Green's function and delta function?

October 25th, 2017, 11:43 am

There are several limit representations of the delta function, Gaussian is one of them.

Will try to find a good link/paper for you on this later today.
 
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Paul
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Re: Heat conduction Green's function and delta function?

October 25th, 2017, 11:52 am

Oh, come on! Any 11-year old could do this without having to google anything!
 
frolloos
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Re: Heat conduction Green's function and delta function?

October 25th, 2017, 12:00 pm

Oh, come on! Any 11-year old could do this without having to google anything!
Maybe, but not while travelling / on the road. LaTex is not journey-friendly!

But the general idea is integrate, take the limit and something else.
 
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Paul
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Re: Heat conduction Green's function and delta function?

October 25th, 2017, 12:36 pm

I'm travelling too. Steering the wheel with my teeth as I type.

Forget int by parts. Just change variables to get exp to look normal, and without the x.
 
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Alan
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Re: Heat conduction Green's function and delta function?

October 25th, 2017, 3:08 pm

It follows immediately if you solve the original problem by a Laplace transform in t. If [$]\alpha[$] is the transform variable, so as not to confuse with [$]s[$] already used, and hats denote the transform, the delta function property amounts to showing that 

[$]\lim_{x \rightarrow 0^+} \hat{G} (x,\alpha) =\lim_{x \rightarrow 0^+}  \int_0^{\infty} e^{-\alpha t} G(x,t) \,  dt = 1[$], independent of [$]\alpha[$].

But since, for x>0,  [$]\hat{G}(x,\alpha) = e^{-\sqrt{\alpha} \,  x}[$], the result is immediate.

(I take [$]k=1[$] for simplicity).   

p.s. Personally, I would say [$]G(0,t) = \delta^+(t)[$] to emphasize the "function" is the limit of a class of true functions with support on [$]t > 0[$].
Last edited by Alan on October 25th, 2017, 4:04 pm, edited 15 times in total.
 
frolloos
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Re: Heat conduction Green's function and delta function?

October 25th, 2017, 3:11 pm

I think Alan just wrote the solution down. But, going back to Paul's comment, I don't think the variable transform

[$] y = \frac{x}{2 \sqrt{k(t-s)}} [$]

actually proves that the Dirac delta is a limit of Gaussian, does it? But maybe this is not the transform Paul meant?
 
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Paul
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Re: Heat conduction Green's function and delta function?

October 25th, 2017, 3:48 pm

You Taylor Series the f(.) after the change of variables.