Hi!This is a quite sophisticated problem, but it’s interesting and challenging!(adsbygoogle = window.adsbygoogle || []).push({});

Consider the following case: Let’s say we have a 3-dimensional disk with a radius [itex]r_{2}[/itex] and a thickness [itex]d[/itex] (so it actually is a cylinder with a quite short height compared to radius). We’re interested in solving the (complex) vectorfield [itex]E_{z}[/itex] directed in the [itex]\hat{z}[/itex] direction for this disk. The PDE for this field is:

[itex]\nabla^2 E_{z}+k \sigma E_{z}=0[/itex]

where [itex]\sigma\geq0[/itex] and [itex]k[/itex] is a pure imaginary number, with a real part 0 and a negative imaginary part. This disk has cylindrical rotation symmetry so [itex]E_{z}[/itex] does not depend on [itex]\phi[/itex]. If we choose our cylindrical coordinate system so that the z-axis passes through the center and that z=0 at one of the circular planes of the disk; then the boundary values on the disk are the following:

1. [itex]E_{z}(\rho,z=0)=0[/itex] for all [itex]\rho\in[0,r_{2}][/itex].

2. [itex]\int_0^{r_{2}}E_{z}(\rho,z' )\rho\,d\rho=0[/itex] for all [itex]z' \in(0,d)[/itex].

3. [itex]\int_{r_{1}}^{r_{2}}E_{z}(\rho,z=d)\rho\,d\rho=-I/\sigma[/itex], where [itex]0 \leq r_{1} \leq r_{2}[/itex] is a constant and [itex]I[/itex] is a complex constant.

4. [itex]E_{z}(r' ,z=d)=0[/itex] for all [itex]r' \in[r_{0},r_{1}][/itex], where [itex]r_{0}[/itex] is a constant such that [itex]0 \leq r_{0} \leq r_{1} \leq r_{2}[/itex].

5. [itex]\int_0^{r_{0}}E_{z}(\rho,z=d)\rho d \rho=I/\sigma[/itex].

6. Obviously [itex]E_z[/itex] must also be finite for all points in the disk.

I would be very thankful for any insight or idea on how to solve this problem (full solutions not necessary acquired!). So if you can help me in any way I owe you a huge amount of thankfulness and respect!

(I have posted this problem in the classical physics forum aswell because the background to the problem is in electromagnetism, but the problem here is mainly mathematical)

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# PDE Helmholtz eq. in 3D + boundary value)

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