| The inverse problem of heat conduction equation is widely used in national defense,military,chemistry,nondestructive testing,biomedicine and other fields,and has become a crucial research interest in the inverse problem field.However,the ill-posed nature of such problems itself makes it difficult to solve them.This article focuses on one-dimensional and two-dimensional heat conduction equations and studies the methods for solving inverse problems of heat conduction equations,provide two types of effective and feasible numerical algorithms.The main research contents are as follows:(1)Aiming at the serious ill posed problem in solving the inverse problem of heat conductioin equation,this paper presents the Conjugate Gradient Least Squares(CGLS)method and the Range Restricted Generalized Minimal Residual(RRGMRES)algorithm based on Krylov subspace principle.CGLS algorithm expands the application range of the traditional conjugate gradient method.RRGMRES algorithm avoids the disadvantage of the Generalized Minimal Residual algorithm that the basis of the search space gradually loses orthogonality due to the existence of errors.(2)Based on the principle of Two-Grid,the Jacobi Preoptimization method and Symmtric Gauss-Seidel Preoptimization method are studied.The Schur Complement Conjugate Gradient is given by combining the principle of Two-Grid with Conjugate Gradient.On the basis of Schur Complement Congugate Gradient,combined with the basic theory of wavelet,its initial approximation is improved and the Fast Wavelet Transform method is obtained.Finally,provided the specific algorithm steps for solving the inverse problem of the heat conduction equation using the above four methods.(3)Based on the diffusion of river pollutants in the field of water environment research,the problem of river pollution transmission is reduced to a mathematical model of onedimensional convective diffusion equation.Combined with the inherent function expansion method for solving the forward problem of heat conduction equation,the stability of the inverse problem of convection diffusion equation is analyzed,and the H?lder type stability estimation is obtained.(4)For the two kinds of algorithms proposed in this paper,corresponding numerical calculation programs have been compiled,which mainly aimed to the calculation of three types of inveres problems of heat conductions equations:initial conditions,boundary conditions,and source terms.The discrete methods include variable separation method,the numerical integration method,the finite difference method,the finite element method and Galerkin method.The validity and feasibility of the algorithms have been verified by the numerical examples of one-dimensional and two-dimensional heat conduction equations and convectiondiffusion equations respectively. |
