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Lately, I've been trying to solve differential equations of the form $$f''+k^2 f =g~~,$$ and $g$ is a continuous function on $[0,2\pi]$. A friend mentioned that I check out Green's function. Unfortunately, I found it hard understanding most the materials I found online. I'm hoping someone here will find time to make me understand better.

Let $$D=\frac{\mathrm{d}^2}{\mathrm{d}x^2}+k^2~~~~k\in \mathbf R,$$ how can I find the Green's function for $D$?

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Thanks for clarification. The purpose of Green's function is to give solutions for a boundary value problem. You should specify boundary conditions on $f$ before Green's function can be found. Also, a similar example with Dirichlet boundary conditions is worked out in Wikipedia –  user53153 Jan 12 '13 at 23:37
    
@PavelM: thanks. I'm not really experienced with these things yet. Could suggest appropriate boundary conditions? –  Jonjo Jan 12 '13 at 23:50
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I can't tell what is appropriate in the context of your problem (which is unknown to me). The easiest condition to deal with is the Dirichlet boundary condition: $f(0)=f(2\pi)=0$. It is considered in the Wikipedia example to which I linked. –  user53153 Jan 13 '13 at 0:25
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3 Answers 3

You may want to look here. Your equation is Helmholtz equation in $1$D for which the Green's function is $$\dfrac{i e^{ik\vert x \vert}}{2k}$$

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Thanks for your response. Please can you take me through the derivation? –  Jonjo Jan 12 '13 at 23:20
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$$f''+k^2 f =g$$ Say we call the Green's function $G$ (not to be confused with $g$). Then the solution $f$ of the differential equation should be the convolution $G*g$ of $G$ and $g$, defined as $$ f(x) = (G*g)(x) = \int_{-\infty}^\infty G(x-y)g(y)\,dy = \int_{-\infty}^\infty G(y)g(x-y)\,dy. $$

The Green's function is sometimes defined as the solution of $f''+k f =\delta$, where $\delta$ is Dirac's "delta function". The delta function is the identity element for convolution.

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Thanks.how do I find the Green's function? –  Jonjo Jan 13 '13 at 4:28
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Adding to Michael Hardy, and other above.

For example to get to the solution $ \dfrac{i e^{ik\vert x \vert}}{2k} $ presented above . From the equation $f''+k f =\delta$ you can use the Laplace Transform to get to :

$$ s^2 F(s) - s f(0) - f'(0) + k^2 F(s) = 1 $$ $$ s^2 F(s) + k^2 F(s) = 1 + s f(0) + f'(0) $$ $$ F(s) \left( s^2 + k^2 \right) = 1 + s f(0) + f'(0) $$ $$ F(s) = \frac{1 + s f(0) + f'(0)}{\left( s^2 + k^2 \right)} $$

Then use the initial conditions (green function must also satisfy). The solution above presented uses the Sommerfeld radiation condition (wikipedia also states that). Finally get the inverse to get to the solution of the Green Function.

Also you can find detailed explanation at Butkov Mathematical Physics chapter 12 Green Functions. Or Mathematical Methods Physicists George B. Arfken chapter 10.

Sorry for not getting deep in the derivations but there are multiple ways of finding the green's function as above stated. I really recommend the references.

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