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Regularization of electromagnetic scattering problems via the Abel integral transform.

Elena Vinogradova1, Paul Smith1

  • 1School of Mathematical and Physical Sciences, Macquarie University, Sydney, New South Wales, Australia.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|August 14, 2025
PubMed
Summary
This summary is machine-generated.

The Abel integral transform regularizes ill-posed equations in electromagnetic wave scattering problems. This method ensures stable, accurate numerical solutions for complex scattering scenarios involving apertures.

Keywords:
Abel integral transformdual series equationselectromagnetic wave scatteringintegral equationsmethod of analytical regularizationopen shells with slots and apertures

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Area of Science:

  • Computational Electromagnetics
  • Mathematical Physics

Background:

  • Mixed boundary value problems for Helmholtz and Maxwell equations are challenging.
  • Electromagnetic wave scattering from apertures leads to ill-posed integral equations.

Purpose of the Study:

  • To present a regularization method for ill-posed scattering problems.
  • To demonstrate the effectiveness of the Abel integral transform for electromagnetic wave scattering.

Main Methods:

  • Representing basis functions using Jacobi polynomials.
  • Applying a sequence of Abel integral transforms to series equations.
  • Converting the system to a well-conditioned Fredholm matrix equation.

Main Results:

  • Achieved stable and convergent numerical solutions with guaranteed accuracy.
  • Successfully treated dual and triple series equations in scattering problems.
  • Demonstrated applicability to slotted cylinders and thin-walled shells.

Conclusions:

  • The Abel integral transform provides a robust method for solving complex electromagnetic scattering problems.
  • This approach offers a significant improvement over traditional numerical methods for ill-posed systems.