Determination of Unknown Coefficients in the Heat Equation

The purpose of this thesis is to find the numerical solutions of one or multiple unknown coefficient identification problems in the governing heat transfer parabolic equations. These inverse problems are numerically solved subject to various types of overdetermination conditions such as the heat flux, nonlocal observation, mass/energy specification, additional temperature measurement, Cauchy data, general integral type over-determination, Stefan condition and heat momentum of the first, second and third order. The main difficulty associated with solving these inverse problems is that they are ill-posed since small changes in the input data can result in enormous changes in the output solution, therefore traditional techniques fail to provide accurate and stable solutions. Throughout this thesis, the finite-difference method (FDM) with the Crank-Nicolson (C-N) scheme is mainly used as a direct solver except in Chapters 8 and 9 where an alternating direction explicit (ADE) method is employed in order to deal with the two-dimensional heat equation. An explicit forward time central space (FTCS) method is also employed in Chapter 2 for the extension to higher dimensions. The treatment for solving a degenerate parabolic equation, which vanishes at the initial moment of time is discussed in Chapter 6. The inverse problems investigated are discretised using FDM or ADE and recast as nonlinear least-squares minimization problems with lower and upper simple bounds on the unknown coefficients. The resulting optimization problems are numerically solved using the \emph{lsqnonlin} routine from MATLAB optimization toolbox. The stability of the numerical solutions is investigated by introducing random noise into the input data which yields unstable results if no regularization is employed. The regularization method is included (where necessary) in order to reduce the influence of measurement errors on the numerical results. The choice of the regularization parameter(s) is based on the L-curve method, on the discrepancy principle criterion or on trial and error.