#green&039;s function

[peeterjoot]

Summary of some gradient related Green's functions

[Click here for a PDF version of this post] Here is a summary of Green’s functions for a number of gradient related differential operators (many of which are of interest for electrodynamics, and most of them have been derived recently in blog posts.) These Green’s functions all satisfy \begin{equation}\label{eqn:deltaFunctions:120} \delta(\Bx – \Bx’) = L G(\Bx, \Bx’). \end{equation} Let \( \Br =…

[peeterjoot]

Green's function for the 3D wave equation.

[Click here for a PDF version of this and related posts] We’ve now evaluated the 1D Green’s function and 2D Green’s function for the wave equation. For the sake of completeness, now let’s evaluate the Green’s function for the 3D wave equation operator. Again with \( \Br = \Bx – \Bx’, \tau = t – t’ \) we want the \( \epsilon \rightarrow 0 \) limit…

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[peeterjoot]

Derivation of the 2D Green's function for the wave equation operator.

[Click here for a PDF version of this post] While it was difficult to attempt to verify the 2D Green’s function, it actually turns out to be fairly easy to derive it, provided we pick an alternate pole displacement from the 1D evaluation to make our lives easier. With \( \Br = \Bx – \Bx’ \), and \( \tau = t – t’ \), and \( \epsilon > 0 \), we can…

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[peeterjoot]

Green's function for the wave equation: 1D and 2D cases.

[Click here for a PDF version of this post] The Green’s function(s) \( G(\Br, \tau) \) for the 3D wave equation \begin{equation}\label{eqn:waveEquationGreens:40} \lr{ \spacegrad^2 – \inv{c^2}\frac{\partial^2}{\partial t^2} } G(\Br, \tau) = \delta(\Br) \delta(\tau), \end{equation} where \begin{equation}\label{eqn:waveEquationGreens:20} \begin{aligned} \Br &= \Bx – \Bx’ \\ r &= \Abs{\Br} \\ \tau &=…

[peeterjoot]

A trilogy in 7+ parts: A better check of the 1D Helmholtz Green's function.

[Click here for a PDF version of this post, and the previous posts in this series.] We used a complicated limiting argument to show that the \( \mathrm{sgn}(x – x’) \) factor in the contour integral derivation of the Helmholtz operator Green’s function was wrong. Having discovered, even if slightly by accident, what the correct form of that Green’s function is, we can check it more directly. This…

[peeterjoot]

A trilogy in six+ parts: 1D Laplacian Green's function

[Click here for a PDF version of this post, and the others in this series.] Having blundered our way to what appears to be the correct Green’s function for the 1D Helmholtz operator, let’s further validate that by deriving the Green’s function for the 1D Laplacian. We should also be able to verify that it has the correct delta function semantics. The 1D Laplacian Green’s function. Expanding the…

[peeterjoot]

A trilogy in five+ parts: Confirming an error in the derived 1D Helmholtz Green's function.

[Click here for a PDF version of this, and previous, posts in this series]. The discontinuity in the derived 1D Helmholtz Green’s function is somewhat surprising. Let’s try to verify that this works or find what does. The first think to check is that \begin{equation}\label{eqn:helmholtzGreens:1220} \lr{ \spacegrad^2 + k^2} G(x,x’) = 0, \end{equation} at locations where \( x \ne x’ \). Since we…

[peeterjoot]

A trilogy in four+ parts: The 2D Laplacian Green's function.

[Click here for a PDF version of this post] I was questioning the correctness of the 1D and 2D Helmholtz Green’s functions derived above, since they are both seemingly malformed for \( k \rightarrow 0 \). Let’s try to carefully expand the 2D Green’s function in the neighbourhood of \( k = 0 \) to validate that result, and as a side effect, obtain the Green’s function for the 2D Laplacian. In [1],…

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