波动方程反移动源问题  

Inverse moving source problems for the wave equation

作　　者：郭军 于群意 李瑞红 GUO Jun;YU Qunyi;LI Ruihong(College of Mathematics and Statistics,South-Central Minzu University,Wuhan 430074,China)

机构地区：[1]中南民族大学数学与统计学学院,武汉430074

出　　处：《中南民族大学学报（自然科学版）》2024年第3期428-432,共5页Journal of South-Central University for Nationalities：Natural Science Edition

基　　金：中南民族大学大学生创新训练计划资助项目(XCX2253)。

摘　　要：考虑三维波动方程的反移动源问题,其移动源项为F(x,t)=f(x-a(t))g(t).波场在可测量球面上的Dirichlet数据已知,利用Fourier变换将波动方程问题转化为频域的Helmholtz方程,建立了源项与观测数据的积分等式.利用Fourier逆变换和一阶微分方程解的存在唯一性定理,证明了轨迹函数a(t)的存在唯一性.最后,利用积分不等式来分析反演a(t)的稳定性.The inverse moving source problem of three-dimensional wave equation is investigated,and F(x,t)=f(x-a(t))g(t)is the moving source term.The Dirichlet data of the wave field in the measurement sphere is known,then by using the Fourier transform,the problem of wave equation can be transformed into the Helmholtz equation on frequencydomain,which enables us to establish the integral equality of the source term and the observation data.Based on the inverse Fourier transform as well as the existence and uniqueness theorem of first-order ODE,the existence and uniqueness of orbit function a(t)is demonstrated.Finally,by the integral inequality,the stability can be proved.

关 键 词：反移动源问题 FOURIER变换 唯一性 稳定性 

分 类 号：O175.27[理学—数学]