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

Purpose: 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. 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 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. Findings: 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. 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

Simultaneous identification of the right-hand side and time-dependent coefficients in a two-dimensional parabolic equation

This paper investigates the simultaneous identification of time-dependent lowest and source terms in a two-dimensional (2D) parabolic equation from the additional measurements. To investigate the solvability of the inverse problem, we first examine an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. Then, applying the contraction mappings principle existence and uniqueness of the solution of an equivalent problem is proved. Furthermore, using the equivalency, the existence and uniqueness theorem for the classical solution of the original problem is obtained and some discussions on the numerical solutions for this inverse problem are presented including numerical examples

Determination of Unknown Coefficients in the Heat Equation

The purpose of this thesis is to find the numerical solutions of one or multiple unknown coefficient identification problems in the governing heat transfer parabolic equations. These inverse problems are numerically solved subject to various types of overdetermination conditions such as the heat flux, nonlocal observation, mass/energy specification, additional temperature measurement, Cauchy data, general integral type over-determination, Stefan condition and heat momentum of the first, second and third order. The main difficulty associated with solving these inverse problems is that they are ill-posed since small changes in the input data can result in enormous changes in the output solution, therefore traditional techniques fail to provide accurate and stable solutions. Throughout this thesis, the finite-difference method (FDM) with the Crank-Nicolson (C-N) scheme is mainly used as a direct solver except in Chapters 8 and 9 where an alternating direction explicit (ADE) method is employed in order to deal with the two-dimensional heat equation. An explicit forward time central space (FTCS) method is also employed in Chapter 2 for the extension to higher dimensions. The treatment for solving a degenerate parabolic equation, which vanishes at the initial moment of time is discussed in Chapter 6. The inverse problems investigated are discretised using FDM or ADE and recast as nonlinear least-squares minimization problems with lower and upper simple bounds on the unknown coefficients. The resulting optimization problems are numerically solved using the \emph{lsqnonlin} routine from MATLAB optimization toolbox. The stability of the numerical solutions is investigated by introducing random noise into the input data which yields unstable results if no regularization is employed. The regularization method is included (where necessary) in order to reduce the influence of measurement errors on the numerical results. The choice of the regularization parameter(s) is based on the L-curve method, on the discrepancy principle criterion or on trial and error