Boundary Differential Equations and Their Applications to Scattering Problems

TL;DR

This paper introduces new boundary differential equations for 2D exterior scattering problems, simplifying the Helmholtz equation to facilitate high-frequency problem analysis with demonstrated numerical validity.

Contribution

The paper derives novel boundary differential equations from the Helmholtz equation, reducing complexity for scattering problems in a body-fitted coordinate system.

Findings

Derived boundary differential equations for scattering problems

Reduced Helmholtz equation to inhomogeneous Bessel's equation

Validated equations through numerical examples

Abstract

In this paper, new boundary differential equations for the two-dimensional exterior scattering problem have been derived. It has been shown that the Helmholtz equation can be reduced to an inhomogeneous Bessel's equation in a body-fitted coordinate system. By imposing the Sommerfeld radiation condition to the general solution of the Bessel's equation, an integro-differential equation, which is equivalent to the original Helmholtz equation, can be obtained. The boundary differential equation can then be established by use of the integration by parts to get rid of the integral in the integro-differential equation for high frequency problems. Numerical examples have been presented to demonstrate the validity of the new boundary differential equations.