This paper investigates the inverse problems of simultaneous reconstruction of timedependent thermal conductivity, convection or absorption coefficients in the parabolic heat equation governing transient heat and bio-heat thermal processes. Using initial and boundary conditions, as well as heat moments as over-determination conditions ensure that these inverse problems have a unique solution. However, the problems are still illposed since small errors in the input data cause large errors in the output solution. To overcome this instability we employ the Tikhonov regularization. A discussion of the choice of multiple regularization parameters is provided. The finite-difference method with the Crank-Nicolson scheme is employed as a direct solver. The resulting inverse problems are recast as nonlinear minimization problems and are solved using the lsqnonlin routine from the MATLAB toolbox. Numerical results are presented and discussed.

Purpose -The purpose of this paper is to provide an insight and to solve numerically the identification of an unknown coefficient of radiation/absorption/perfusion appearing in the heat equation from additional temperature measurements. Design/methodology/approach -First, the uniqueness of solution of the inverse coefficient problem is briefly discussed in a particular case. However, the problem is still ill-posed since small errors in the input data cause large errors in the output solution. For the numerical discretisation, the finite difference method combined with a regularized nonlinear minimization is performed using the MATLAB toolbox routine lsqnonlin. Finding -Numerical results presented for three examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.Research limitations/implications -The mathematical formulation is restricted to identify coefficients which separate additively in unknown components dependent individually on time and space, and this may be considered as a research limitation. However, there is no research implication to overcome this since the known input data is also limited to single measurements of temperature at a particular time and space location. Practical implications -Since noisy data are inverted, the study models real situations in which practical measurements are inherently contaminated with noise. Social implications -The identification of the additive time-and space-dependent perfusion coefficient will be of great interest to the bio-heat transfer community and applications. Originality/value -The current investigation advances previous studies which assumed that the coefficient multiplying the lower order temperature term depends on time or space separately. The knowledge of this physical property coefficient is very important in biomedical engineering for understanding the heat transfer in biological tissues. The originality lies in the insight gained by performing for the first time numerical simulations of inversion to find the coefficient additively dependent on time and space in the heat equation from noisy measurements.

The retrieval of the timewise-dependent intensity of a free boundary and the temperature in a two-dimensional parabolic problem is, for the first time, numerically solved. The measurement, which is sufficient to provide a unique solution, consists of the mass/energy of the thermal system. A stability theorem is proved based on the Green function theory and Volterra's integral equations of the second kind. The resulting nonlinear minimization is numerically solved using the lsqnonlin MATLAB optimization routine. The results illustrate the reliability, in terms of accuracy and stability, of the time-dependent free surface reconstruction.

We consider the inverse problem of determining the time-dependent thermal conductivity and the transient temperature satisfying the heat equation with initial data, Dirichlet boundary conditions, and the heat flux as overdetermination condition. This formulation ensures that the inverse problem has a unique solution. However, the problem is still ill-posed since small errors in the input data cause large errors in the output solution. The finite difference method is employed as a direct solver for the inverse problem. The inverse problem is recast as a nonlinear least-squares minimization subject to physical positivity bound on the unknown thermal conductivity. Numerically, this is effectivey solved using the lsqnonlin routine from the MATLAB toolbox. We investigate the accuracy and stablity of results on a few test numerical examples.