Inverse Problem Of A Class Of Parabolic Equation With Robin Boundary

We consider the reconstruction of the boundary Robin coefficient for the heat conduction system in 2-dimensional spatial domain from the nonlocal boundary mea-surement data. By applying the potential theory to the heat conduction system, we propose to find the boundary Robin coefficient numerically from a nonlinear op-timization problem with respect to the density function and the Robin coefficient as two arguments with regularizing term. The well-posedness of this optimization problem is established for the existence and convergence of the minimizer. Then the alternative iteration scheme is proposed to solve this nonlinear optimization problem. The numerical implementations are carried out to support our theoretical analysis.Under some assumptions, the heat conduction problem can be mathematically described by a boundary value problem for the two-dimensional heat conduction equation with Robin boundary condition, namely,where D is a bounded domain in R2 with piecewise smooth boundary 3D. Through-out the paper, u represents the temperature, φ(x,t) is the additional heat flux, and σ(x)> 0 is the heat transfer coefficient on the boundary (?)D. The inverse problem considered in this thesis is to reconstruct σ(x) in the above system from the following nonlocal boundary measurement dataOur thesis consists of four parts.In Chapter 1, we give a brief introduction to the mathematical model for heat conduction problem and then describe the physical background of the inverse heat conduction problem. The main results of our thesis are also presented.In Chapter 2, we give some preliminaries, containing potential theory of the heat conduction system and the related Nystrom scheme and collocation method for computing the integrals.In Chapter 3, we study the single-layer potential scheme for solving the forward heat conduction problem with Robin boundary condition. By applying the potential expression, the solution to the forward heat conduction problem is transformed into a boundary integral equation with respect to the density function. It is a Fredholm integral equation of the second kind. Noticing the weak singularity in the potential expression, we propose to attain the numerical solution by combining the Nystrom method and collocation method together. Such a scheme is also required in our iterative solution for the inverse problem by finding the numerical solution from a nonlinear cost functional.In Chapter 4, we consider the reconstruction of boundary Robin coefficient from the average temperature measurement with respect to time t. We solve this problem numerically by considering a nonlinear optimization problem. We prove the exis-tence and convergence of the minimizer theoretically. Then an alternative iteration algorithm is proposed to get its efficient numerical realization. We decompose the nonlinear optimization functional with two arguments into two easy-to-solved quadric optimization functionals with one argument in the iteration. For given regulariza-tion parameter a and the m-th iteration values σm(x), we solve the corresponding Tikhononv regularization functional with respect to one argument q(x, t) to obtain the minimizer qm(x, t). Then for fixed qm(x, t), we get the approximation of σm+1(x) by least squares method. Numerical results are presented to illustrate the efficiency and robustness of the proposed algorithms.