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The purpose of this study 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.

First, the uniqueness of solution of the inverse coefficient problem is briefly discussed in a particular case. However, the problem is still ill-posed as small errors in the input data cause large errors in the output solution. For numerical discretization, the finite difference method combined with a regularized nonlinear minimization is performed using the MATLAB toolbox routine lsqnonlin.

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.

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, as the known input data are also limited to single measurements of temperature at a particular time and space location.

As noisy data are inverted, the study models real situations in which practical measurements are inherently contaminated with noise.

The identification of the additive time- and space-dependent perfusion coefficient will be of great interest to the bio-heat transfer community and applications.

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 purpose of the study is to solve numerically the inverse problem of determining the time-dependent convection coefficient and the free boundary, along with the temperature in the two-dimensional convection-diffusion equation with initial and boundary conditions supplemented by non-local integral observations. From the literature, there is already known that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data.

For the numerical discretization, this paper applies the alternating direction explicit finite-difference method along with the Tikhonov regularization to find a stable and accurate numerical solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted.

The numerical results demonstrate that accurate and stable solutions are obtained.

The inverse problem presented in this paper was already showed to be locally uniquely solvable, but no numerical solution has been realized so far; hence, the main originality of this work is to attempt this task.

This study aims to at numerically retrieve five constant dimensional thermo-physical properties of a biological tissue from dimensionless boundary temperature measurements.

The thermal-wave model of bio-heat transfer is used as an appropriate model because of its realism in situations in which the heat flux is extremely high or low and imposed over a short duration of time. For the numerical discretization, an unconditionally stable finite difference scheme used as a direct solver is developed. The sensitivity coefficients of the dimensionless boundary temperature measurements with respect to five constant dimensionless parameters appearing in a non-dimensionalised version of the governing hyperbolic model are computed. The retrieval of those dimensionless parameters, from both exact and noisy measurements, is successfully achieved by using a minimization procedure based on the MATLAB optimization toolbox routine lsqnonlin. The values of the five-dimensional parameters are recovered by inverting a nonlinear system of algebraic equations connecting those parameters to the dimensionless parameters whose values have already been recovered.

Accurate and stable numerical solutions for the unknown thermo-physical properties of a biological tissue from dimensionless boundary temperature measurements are obtained using the proposed numerical procedure.

The current investigation is limited to the retrieval of constant physical properties, but future work will investigate the reconstruction of the space-dependent blood perfusion coefficient.

As noise inherently present in practical measurements is inverted, the paper is of practical significance and models a real-world situation.

The findings of the present paper are of considerable significance and interest to practitioners in the biomedical engineering and medical physics sectors.

In comparison to Alkhwaji et al. (2012), the novelty and contribution of this work are as follows: considering the more general and realistic thermal-wave model of bio-heat transfer, accounting for a relaxation time; allowing for the tissue to have a finite size; and reconstructing five thermally significant dimensional parameters.

The purpose of this paper is to apply the meshless method of fundamental solutions (MFS) to the two‐dimensional time‐dependent heat equation in order to locate an unknown internal inclusion.

The problem is formulated as an inverse geometric problem, using non‐invasive Dirichlet and Neumann exterior boundary data to find the internal boundary using a non‐linear least‐squares minimisation approach. The solver will be tested when locating a variety of internal formations.

The method implemented was proven to be both stable and reasonably accurate when data were contaminated with random noise.

Owing to limited computational time, spatial resolution of internal boundaries may be lower than some similar case investigations.

This research will have practical implications to the modelling and monitoring of crystalline deposit formations within the nuclear industry, allowing development of future designs.

Similar work has been completed in regards to the steady state heat equation, however to the best of the authors' knowledge no previous work has been completed on a time‐dependent inverse inclusion problem relating to the heat equation, using the MFS. Preliminary results presented here will have value for possible future design and monitoring within the nuclear industry