The Robin Coefficient Inversion For Heat Equation

There are many applications for the heat conduction process in industry areas,such as the heat conduction in a homogeneous slab conductor.If the physical parameters of the slab and the heat transfer property via the boundary are known,the conduction process is completely determined.However,either the physical parameters of the slab media or the transfer process with environmental surrounding the strip may be unknown,leading to the determinations of these physical parameters by other measurable data.In view of this background,we consider the inverse problem for heat conduction equation with impedance boundary condition,using the internal measurement data of the temperature to identify the Robin coefficient defined on the boundary of the slab.The inverse problem can be described by the following heat conduction equation system with additional measurement data u(x0,t)for some interior point x0 e(0,π).Here f(x)is the initial temperature,u(x,t)represents the temperature field on the homogeneous strip.We aim to identify the Robin coefficient σ(t)specified at one end x = π from the measurable data of u(x0,t)at interior point x0.The contents of this paper are summarized as follows.Chapter 1 introduces the existing work and the background of our problem.Then in Chapter 2 we give some necessary knowledge in solving the ill-posed problems and some theoretical analysis on our inverse problem,including uniqueness of the problem in L2 space.In chapter 3,the cost functional dealing with this ill-posed problem is established,for which the existence of the minimal element and the convergence of the corresponding functional are proved.Then a regularization scheme is proposed.In the fourth chapter,the numerical solutions for both the direct problem and the inverse problem are given,where the measurement data for inverse problem is simulated for direct problem.The validity of the proposed regularization scheme is illustrated by several numerical experiments.