Finite Difference Method And Convergence Analysis For The Source Term Inversion Of A Class Of Heat Conduction Equation

The source identification problem of heat conduction equations plays a significant role in many engineering applications and scientific research fields.This paper focuses on the inverse problem of a heat conduction equation for a class of reconstructed time-dependent source terms.This inverse problem aims to reconstruct the time-dependent source terms from the non-local measurement data of the temperature field.Based on the finite difference method of numerical solutions of partial differential equations,four different finite-difference schemes for reconstructing source terms are proposed.The convergence and error of the schemes are given,combined with the mollification regularization method of the data.The regularization method gives the method of stabilizing the reconstruction source term.The first chapter briefly introduces the main research content and significance of this article.In the second chapter,the method of smoothing regularization of data is introduced,and the error estimation is given.In the third chapter,the main research results of this paper are given for the heat conduction equation in the one-dimensional space domain.Using the additional non-local observation data,based on the finite difference method of numerical solutions of partial differential equations,four finite difference schemes are proposed to reconstruct the time-dependent unknowns in the heat conduction equation source term.That is,based on the difference schemes of the four inversion source terms of the Crank-Nicolson scheme,the backward Euler scheme,and the forward Euler scheme,the existence,uniqueness and convergence of the solution of the inversion difference scheme are proved,error estimation of the numerical solution of the source term is given for the measurement data with errors.Numerical example verifies the convergence of the difference scheme.Combined with the mollification regularization strategy of the data,it shows that the proposed difference scheme is valid.In the fourth chapter,we extend the research results of the third chapter to the inverse problem of heat source equation in two-dimensional space domain.Numerical examples show that the proposed finite difference scheme is effective for inverse problem of high dimensional heat conduction equation.The fifth chapter makes a summary of the full text and looks forward to the future research.