Mittag-Leffler函数、Vekua变换与障碍物散射反问题
Mittag-Leffler's function, Vekua transform and an inverse obstacle scattering problem
摘要：
本文研究了一个以二维Helmholtz方程为控制方程的障碍物散射反问题的原型。给出了一种从固定波数的远场算子中提取未知障碍物位置和形状信息的显式方法。该方法基于：一个明确的建设修改的米塔格-莱弗勒的功能通过Vekua变换和研究的渐近行为;一个明确的密度在Herglotz波函数近似修改的米塔格-莱弗勒的功能在有界域周围未知的障碍;一个系统的不等式来自基尔希的因式分解公式的远场运营商。然后引入了一个指示函数，它可以由远场算子作用于密度计算得到。它示出的指示函数的渐近行为产生的信息的外部的障碍物的可见部分。
英文摘要：
This paper studies a prototype of inverse obstacle scattering problems whose governing equation is the Helmholtz equation in two dimensions. An explicit method to extract information about the location and shape of unknown obstacles from the far field operator with a fixed wave number is given. The method is based on: an explicit construction of a modification of Mittag-Leffler's function via the Vekua transform and the study of the asymptotic behaviour; an explicit density in the Herglotz wave function that approximates the modification of Mittag-Leffler's function in the bounded domain surrounding unknown obstacles; a system of inequalities derived from Kirsch's factorization formula of the far field operator. Then an indicator function which can be calculated from the far field operator acting on the density is introduced. It is shown that the asymptotic behaviour of the indicator function yields information about the visible part of the exterior of the obstacles.
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