This thesis aims to determine the numerical solutions to various types of inverse problems associated with hyperbolic thermal-wave models of bio-heat transfer. Knowledge of the thermo-physical properties of biological tissues has become vital in biomedical applications, and therefore, it is the focus of this thesis to identify such essential properties simultaneously. Such inverse problems are, in general, ill-posed because their solutions might not be unique or, even if they are unique they could be unstable under small perturbations in the input measured data. Hence, traditional numerical methods cannot produce stable and accurate solutions.
The contributions of this thesis lie in the mathematical formulations and numerical solutions of a Cauchy problem, inverse source problems, and inverse coefficient identification problems associated with thermal-wave models of bio-heat transfer. These problems are solved subject to various types of additional information in the form of boundary, integral type, time-average, upper-base or internal temperature measurements.
First, these inverse problems are reduced to nonlinear least-squares minimization problems. For the numerical discretization, the Crank-Nicolson finite-difference method (FDM) and the alternating direction implicit (ADI) method are used as direct solvers in one-dimension and two-dimensions, respectively. The minimization problems are solved using the conjugate gradient method (CGM) or the {\it lsqnonlin} routine from the MATLAB optimization toolbox. The stability of the numerical solutions is examined by including Gaussian random noise in the input data, and to ensure stability the discrepancy principle or the Tikhonov regularization method are employed. The regularization parameter associated with the Tikhonov regularization method is chosen according to the discrepancy principle or by simple trial and error. The numerical results associated with the numerical examples considered verify that the proposed methods can produce stable and accurate solutions.
