| In this paper, we consider a typical ill-posed problem: inverse heat conduction problem (homogeneous and inhomogeneous initial data). That is to compute the temperature distribution on one side through the temperature distribution of the other side and a temperature measurement on an arbitrary space location. We use the boundary integral equation method and the Tikhonov regularization to solve this problem. We use the given initial boundary data and temperature measurement on a space location, together with the fundamental solution of a given differential equation, to construct the boundary integral equations, and then discretize them. For the particularity of the problem, the coefficient matrix of the discrete equations is often severely ill-conditioned, i.e. the condition number of the coefficient matrix is very large. Therefore, we utilize Tikhonov regularization technique to get the stable numerical results.The paper is organized as follows. In Chapter 1, we introduce the development of the inverse problems of mathematical physics and the regularization theory for solving the inverse problem. In Chapter 2, we give the inverse heat conduction problem for solving the boundary temperature distribution and analyze the ill-posedness of the problem; also, we give the properties of the solution of the corresponding direct problem. In Chapter 3, we describe the boundary integral equation method, which is used to solve the direct problem. Then, in the computation, we will take the numerical results as the measurement data in the inverse problem. In Chapter 4, we solve the inverse problem using the method of boundary integral equation together with the Tikhonov regularization. And we choose the regularization parameter through the GCV rule. In Chapter 5, we respectively give some numerical examples with homogeneous and inhomogeneous initial data. Through the numerical results we show that our method is effective and stable. Finally, we give a conclusion in Chapter 6.
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