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Determination of an additive time- and space-dependent coefficient in the heat equation

Mousa Huntul (Department of Applied Mathematics, University of Leeds, Leeds, UK)
Daniel Lesnic (Department of Applied Mathematics, University of Leeds, Leeds, UK)
Tomas Johansson (School of Mathematics, Aston University, Birmingham, UK)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Article publication date: 20 June 2018

Issue publication date: 8 August 2018




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.

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

Practical implications

As 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.


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.



Huntul, M., Lesnic, D. and Johansson, T. (2018), "Determination of an additive time- and space-dependent coefficient in the heat equation", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 28 No. 6, pp. 1352-1373.



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