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

Abstract

<jats:sec> <jats:title content-type="abstract-subheading">Purpose</jats:title> <jats:p>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.</jats:p> </jats:sec> <jats:sec> <jats:title content-type="abstract-subheading">Design/methodology/approach</jats:title> <jats:p>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.</jats:p> </jats:sec> <jats:sec> <jats:title content-type="abstract-subheading">Findings</jats:title> <jats:p>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.</jats:p> </jats:sec> <jats:sec> <jats:title content-type="abstract-subheading">Research limitations/implications</jats:title> <jats:p>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.</jats:p> </jats:sec> <jats:sec> <jats:title content-type="abstract-subheading">Practical implications</jats:title> <jats:p>As noisy data are inverted, the study models real situations in which practical measurements are inherently contaminated with noise.</jats:p> </jats:sec> <jats:sec> <jats:title content-type="abstract-subheading">Social implications</jats:title> <jats:p>The identification of the timewise terms and free boundaries will be of great interest in the heat transfer community and related fluid flow applications.</jats:p> </jats:sec> <jats:sec> <jats:title content-type="abstract-subheading">Originality/value</jats:title> <jats:p>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.</jats:p> </jats:sec>

An inverse problem of determining the time-dependent potential in a higher-order Boussinesq-Love equation from boundary data

“…Recently, Huntul et al (2021) investigated the time-dependent potential coefficient in a fourth-order pseudo-parabolic equation with non-local initial data, non-local boundary conditions and the temperature measurement as overdetermination condition. Huntul et al (2018) determined an additive time-and space-dependent terms in the parabolic equation, while Huntul and Tamsir (2020) determined time-dependent heat source. Additionally, Alshomrani et al (2017) presented a numerical algorithm based on modified cubic trigonometric B-spline functions to solve the hyperbolic-type wave equations, while Kumar et al (2013) proposed a polynomial differential quadrature method to solve quasilinear hyperbolic equations.…”