Identification of the time-dependent conductivity of an inhomogeneous diffusive material

In this paper, we consider a couple of inverse problems of determining the time-dependent thermal/hydraulic conductivity from Cauchy data in the one-dimensional heat/diffusion equation with space-dependent heat capacity/specific storage. The well-posedness of these inverse problems in suitable spaces of continuously differentiable functions are studied. For the numerical realisation, the problems are discretised using the finite-difference method and recast as nonlinear least-squares minimization problems with a simple positivity lower bound on the unknown thermal/hydraulic conductivity. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. Regularization is included wherever necessary. Numerical results are presented and discussed for several benchmark test examples showing that accurate and stable numerical solutions are achieved. The outcomes of this study will be relevant and of significant importance to the applied mathematics inverse problems community working on thermal/hydraulic property determination in heat transfer and porous media.