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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.

The purpose of this paper is to provide an insight and to solve numerically the identification of timewise terms and free boundaries coefficient appearing in the heat equation from over-determination conditions.

The formulated coefficient identification problem is inverse and ill-posed, and therefore, to obtain a stable solution, a nonlinear Tikhonov regularization least-squares approach is used. For the numerical discretization, the finite difference method combined with a regularized nonlinear minimization is performed using the MATLAB subroutine lsqnonlin.

The numerical results presented for two 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 in unknown components dependent on time, and this may be considered as a research limitation. However, there is no research implication to overcome this, as the known input data is also limited to single temperature in heat equation with Stefan conditions, and the first- and second-order heat moments measurements at a particular time location.

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

The identification of the timewise terms and free boundaries will be of great interest in the heat transfer community and related fluid flow applications.

The current investigation advances previous studies, which assumed that the coefficient multiplying the lower order temperature term depends on time. The knowledge of this physical property coefficient is very important in heat transfer and fluid flow. The originality lies in the insight gained by performing for the numerical simulations of inversion to find the timewise terms and free boundaries coefficient dependent on time in the heat equation from noisy measurements.

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.