Abstract
This paper investigates the recovery for time-dependent coefficient and free boundary for heat equation. They are considered under mass/energy specification and Stefan conditions. The main issue with this problem is that the solution is unstable and sensitive to small contamination of noise in the input data. The Crank-Nicolson finite difference method (FDM) is utilized to solve the direct problem, whilst the inverse problem is viewed as a nonlinear optimization problem. The latter problem is solved numerically using the routine optimization toolbox lsqnonlin from MATLAB. Consequently, the Tikhonov regularization method is used in order to gain stable solutions. The results were compared with their exact solution and tested via root mean squares error (RMSE). We found that the numerical results are accurate and stable.
Highlights
The field of inverse problems for heat conduction has very wide applications and physicalIraqi Journal of Science, 2021, Vol 62, No 3, pp: 950-960 background
The mathematical modeling used in many current applied problems in science and technology revealed the need for numerical solutions of inverse problems in mathematical physics, such as the numerical solution of the two-sided
We consider a one-sided free domain problem in one-dimensional space for the parabolic heat equation, with non-homogenous Dirichlet boundary condition when the thermal conductivity is equal to unity
Summary
Iraqi Journal of Science, 2021, Vol 62, No 3, pp: 950-960 background It has been applied in almost all fields of scientific engineering computations, such as power engineering, aerospace engineering, biomedical engineering, etc. We consider a one-sided free domain problem in one-dimensional space for the parabolic heat equation, with non-homogenous Dirichlet boundary condition when the thermal conductivity is equal to unity. This problem contains free boundary depending on time only [2, 3]. The shape of the problem varies with time step marching In this problem, some unknown terms or coefficients are determined by using some additional specified information about their solution, like Stefan condition and zero and first order heat moment conditions. The authors in [6 - 9] investigated the theoretical and numerical aspects of several types of parabolic heat equations, in fixed and free domains of oneand two-dimensions for various types of additional information
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