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High-order asymptotic methods provide accurate, analytic solutions to intractable potential problems.

Alexander W Wray1, Madeleine R Moore2

  • 1Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow, G1 1XH, UK. alexander.wray@strath.ac.uk.

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Summary

A new asymptotic solution accurately determines the density and capacity of potential source arrays. This method works for non-circular footprints and solves previously inaccessible physical problems rapidly.

Keywords:
Asymptotic methodsElectrostaticsEvaporationPotential problems

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Area of Science:

  • Physics
  • Applied Mathematics
  • Engineering

Background:

  • The classical problem of determining the density and capacity of potential source arrays is fundamental across various scientific disciplines.
  • Existing methods often struggle with complex source geometries and arbitrary array configurations.
  • Applications include electrostatic capacitance, elastostatic stress analysis, and fluid droplet evaporation.

Purpose of the Study:

  • To develop a novel asymptotic solution for calculating the density and capacity of potential source arrays.
  • To provide a method applicable to sources with arbitrary, non-circular footprints, including polygonal shapes.
  • To enable rapid and accurate solutions for a wider range of classical physical problems.

Main Methods:

  • Derivation of a new asymptotic solution.
  • Extensive validation against experimental data.
  • Extensive validation against numerical simulations.

Main Results:

  • The derived asymptotic solution demonstrates excellent accuracy for arrays of sources with non-circular and polygonal footprints.
  • The solution successfully models diverse physical phenomena, including electrostatic capacitance and fluid dynamics.
  • The method allows for the rapid and accurate analysis of previously intractable problems.

Conclusions:

  • The novel asymptotic solution offers a powerful and versatile tool for analyzing potential source arrays.
  • This approach significantly expands the scope of solvable problems in areas like electrostatics and fluid mechanics.
  • The validated accuracy and speed of the solution facilitate new scientific investigations.