边界微分方程及其应用

本文讨论二维Helmholtz方程外边值问题的求解，以较为严格的方式建立了更精准的新的边界微分方程.在贴体坐标系下，Helmholtz方程可转化为非齐次Bessel方程.将Bessel方程的一般解代入Sommerfeld辐射条件可以得到等价于原Helmholtz方程的积分-微分方程，再利用分部积分消去其中积分，即可建立高频问题的边界微分方程.文中通过若干算例对新得到的边界微分方程进行了数值验证.

Boundary differential equations and their applications

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.