Free boundary problems with nonlinear diffusion

Tamsir

2020

Purpose 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. Design/methodology/approach 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. Findings 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. Research limitations/implications 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. 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 timewise terms and free boundaries will be of great interest in the heat transfer community and related fluid flow applications. Originality/value 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.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Simultaneous identification of timewise terms and free boundaries for the heat equation

Tamsir

2020

“…Free boundary problems for parabolic equations occur naturally in many branches of physics, engineering, chemistry, biology and other areas, see [5,9,12] to mention only a few. For instance, the simultaneous identification of transient coefficients and multiple unknown free boundaries were recently investigated by the authors in [15], whilst in [17], free boundary problems with nonlinear diffusion were considered.…”

Section: Introductionmentioning

confidence: 99%

Determination of a Time-Dependent Free Boundary in a Two-Dimensional Parabolic Problem

Int. J. Appl. Comput. Math

Lesnic

2019

The retrieval of the timewise-dependent intensity of a free boundary and the temperature in a two-dimensional parabolic problem is, for the first time, numerically solved. The measurement, which is sufficient to provide a unique solution, consists of the mass/energy of the thermal system. A stability theorem is proved based on the Green function theory and Volterra's integral equations of the second kind. The resulting nonlinear minimization is numerically solved using the lsqnonlin MATLAB optimization routine. The results illustrate the reliability, in terms of accuracy and stability, of the time-dependent free surface reconstruction.

“…In [2,12], the authors investigated free boundary problems with nonlinear diffusion. The numerical solution of inverse Stefan problems based on the method of fundamental solutions has been investigated in [13,14].…”

Section: Introductionmentioning

confidence: 99%

Time-Dependent Reaction Coefficient Identification Problems with a Free Boundary

International Journal for Computational Methods in Engineering

Lesnic

2019

The determination of time-dependent reaction coefficients in free boundary heat transfer problems is, for the first time, numerically investigated. The additional data which provides a unique solution is given by the Stefan boundary condition and the heat moments. The finite difference method with the Crank-Nicolson scheme combined with a regularized nonlinear optimization are employed. The resulting nonlinear system of equations is solved numerically using the MATLAB toolbox routine lsqnonlin for minimizing the Tikhonov regularization functional. A discussion of the choice of regularization parameters is provided. Numerical results are presented and discussed.