2born (2born) wrote,
2born
2born

Важное по методу изображений!

Laplace's equation for a point source near a sphere: improved internal solution using spheroidal harmonics: https://arxiv.org/abs/1711.09551

s shown recently [Phys. Rev. E 95, 033307 (2017)], spheroidal harmonics expansions are well suited for the external solution of Laplace's equation for a point source outside a spherical object. Their intrinsic singularity matches the line singularity of the analytic continuation of the solution and the series solution converges much faster than the standard spherical harmonic solution. Here we extend this approach to internal potentials using the Kelvin transformation, ie. radial inversion, of the spheroidal coordinate system. This transform converts the standard series solution involving regular solid spherical harmonics into a series of irregular spherical harmonics. We then substitute the expansion of irregular spherical harmonics in terms of transformed irregular spheroidal harmonics into the potential. The spheroidal harmonic solution fits the image line singularity of the solution exactly and converges much faster. We also discuss why a solution in terms of regular spheroidal harmonics cannot work, even though these functions are finite everywhere in the sphere. We also present the analogous solution for an internal point source, and two new relationships between the spherical and spheroidal harmonics.
Tags: КлЭД, наука, разгребая arXiv'ы
Subscribe

Recent Posts from This Journal

  • Post a new comment

    Error

    default userpic

    Your reply will be screened

    Your IP address will be recorded 

    When you submit the form an invisible reCAPTCHA check will be performed.
    You must follow the Privacy Policy and Google Terms of use.
  • 5 comments