Numerical Theory And Method Research On The Inverse Problem Of A Class Of Wave Equations

In the past twenty years,the inverse problem of mathematical physics has become one of the fields in applied mathematics with high development,and its research has covered many fields of modern research,production and life.The inverse problem of mathematical physics corresponds to the direct problem.It is based on the evolution results of things and explores the internal laws or external effects of things from the observable phenomena.The inverse source problem of wave equation is an important research content in the inverse problem of mathematical physics.This paper mainly focuses on the numerical theories and methods of the inverse source problem of wave equation.Firstly,a finite difference scheme for two-dimensional wave equation is constructed.In order to make the wave not reflect at the edge,the positive problem is calculated by combining the absorbing boundary condition(ABC)and the perfectly matched layer method(PML).The effects of PML decay factor and decay layer thickness on the numerical results are further analyzed.The numerical results show that the absorption effect of the absorption boundary conditions and the perfectly matched layer at the boundary.Secondly,using the numerical results of the positive problem as the source term of the two-dimensional wave equation.An integral operator is obtained by using the eigenfunction expansion method,which establishes the corresponding relationship between the measured data and the source term.In theory,the linearity,boundedness and self-adjointness of the integral operators are proved.Considering the ill-posedness of the inverse problem,Tikhonov regularization method is used to solve the inverse problem numerically,and 1% and 5%random noise perturbations are added to the measured data for the inversion.The position of time-dependent source term and the shape of time-independent source term are obtained,respectively,and the numerical results are analyzed.Finally,the inverse source problem of two-dimensional wave equation is studied with incomplete data.In the case of random perturbation of measured data,the source term of wave equation is further inverted by reducing the measured data.