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

Abstract

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.

Reconstruction of the thermal properties in a wave-type model of bio-heat transfer

“…The Crank-Nicolson method, which is unconditionally stable and second-order accurate, discretizes (20), (21), (18), (22) and (19) as:…”

“…Concerning inverse problems for bio-heat transfer, a great deal of numerical techniques has been proposed to solve inverse problems for the Pennes’ bio-heat model (Bazán et al , 2017; Cao and Lesnic, 2018; Huntul et al , 2018). However, limited attention has been given to inverse problems for the thermal-wave model (Hsu, 2006; Lee et al , 2013; Yang, 2014).…”

“…The analytical(34) and numerical FDM solutions for the temperature u(x, t) of the direct problem (17)-(19) obtained with various mesh sizes: (a) M = N = 10, (b) M = N = 20, and (c) M = N = 40. The absolute errors between them are also included.…”

Simultaneous Identification of Thermal Conductivity and Heat Source in the Heat Equation

“…Other authors [10] investigated the reconstruction of these coefficients, as well as of the absorption coefficient, using the measurement of the heat moments. The time and space-dependent unknown coefficients from data measurements in the one-dimensional parabolic heat equation were determined elsewhere [11].…”

Recovery of timewise‐dependent heat source for hyperbolic PDE from an integral condition

“…Yang and Fu 5 reconstructed a timewise‐dependent heat source using the temperature measurement, while Yang et al 6 determined a space‐dependent heat source in the inverse problem and obtained the regularization solution using a simplified Tikhonov regularization. Huntul et al 7 determined an additive space and timewise coefficients in the heat equation. Kian and Yamamoto 8 investigated the inverse problem for recovering the time‐ and space‐dependent sources for classical diffusion equation.…”