An Inverse Source Problem For One Dimensional Heat Equation

In many natural science and engineering fields, one often encounters the problems of recovering the unknown heat sources in the heat conduction process, which are classical inverse problems for partial differential equations. From physical point of view, if we know the heat source, the boundary status and the initial temperature distribution, we can determine the temperature distribution at a later time, which constitutes the direct problem.However, in several engineering configurations, both the boundary status such as heat flux on the boundary and the heat sources are very hard to measure directly through experiments. In these cases, we have to determine the temperature field for the heat conduction process indirectly by some other extra measurements which can be measured directly. These problems constitute the inverse heat conduction problems. In this paper, we consider the following heat conduction model for temperature field u(x, t) where DT={(x,t):0<x<1,0<t<T}, both f(x,t) and φ(x) are given functions, η>0 is a known constant. Our aim is to identify the heat source strength γ(t) in the heat equation from the given data, x0∈(0,1), or its noisy data hδ(t).We firstly give an overview on the engineering background of the heat conduction problem with unknown parameters to be detected in chapter 1, and the mathematical research status of such kinds of inverse problems are stated. Based on these motivations, we summarize the main contents of this paper.In the second chapter, we introduce the fundamental theory of Tikhonov regulariza-tion together with the choice strategy for the regularizing parameters.In chapter 3, we prove the uniqueness and the conditional stability for the inverse problem concerned. We also propose the regularization scheme solving the inverse prob-lem, with an explicit selection strategy for the regularization parameter.In chapter 4, the numerical algorithms for the inverse problem are proposed, which are tested by several numerical examples.Finally, we make a brief summary of the whole paper, and prospect the possible future work.