| In science and engineering,the mathematical models of many problems can be represented by partial differential equations.But in most cases,these problems can not be solved analytically,which requires us to find some appropriate numerical methods to obtain numerical solutions.Heat conduction problem and elastic scattering problem are two typical examples of these kind of problems,In this paper,we give a method of noise reduction for heat conduction problem with noise in boundary data.Then,a kind of elastic scattering problem model by coupling of Helmholtz equation and time harmonic Cauchy-Navier equation is obtained.where the acoustic scattering field satisfies Helmholtz equation and the displacement field of elastic body satisfies time harmonic Cauchy-Navier equation,This paper is mainly composed of two parts: the first part is about the numerical noise reduction method related to heat conduction problem,and the second part is about the numerical solution of elastic scattering problem.In this paper,we firstly use the projection integral equation method to solve the problem of heat conduction with noise in a class of boundary data which is related to Laplace equation,among this process the Tikhonov regularization method and Morozov deviation principle is used to filter the noise in the boundary data.The effectiveness,stability and numerical convergence of the method are verified by numerical examplesSince the method of fundamental solutions is derived from the discretization of the integral equation,So after solving the heat conduction problem by the projection integral equation method,we try to use the method of fundamental solutions to solve a kind of elastic scattering problem model obtained by the coupling of Helmholtz equation and Cauchy-Navier equation.In this process,We still use the Tikhonov regularization method and Morozov deviation principle to deal with highly ill-posed matrice.Finally,the effectiveness of our method is verified by numerical examples. |
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