arXiv subjects

Targyn A. Nauryz

Publications and source records attributed to Targyn A. Nauryz.

8 recordsLinked to original sources

arXiv2026

Two-phase source and reaction coefficient Stefan type problems

This paper investigates inverse two-phase Stefan-type problems for parabolic heat equations with unknown time-dependent source and reaction coefficients. Suitable transformations reduce the original free-boundary problems to equivalent equations on fixed spatial domains, and Fourier spectral expansions are used to derive reconstruction formulas for the unknown coefficients. In the first formulation, the source coefficients are identified from integral, pointwise, and nonlocal additional data, leading to Volterra integral equations. In the second formulation, an exponential transformation is applied to recover the reaction coefficients. A principal advantage of the model with two moving boundaries is that each boundary provides its own Stefan condition which sufficient to determine the two unknown time-dependent coefficients without additional overdetermination conditions. Under appropriate assumptions the existence, uniqueness, boundedness and regularity of weak and strong solutions are established. Illustrative examples confirm the applicability of the reconstruction procedure and show that the coefficients remain stable under the noisy data. The proposed approach provides a rigorous base for identifying time-dependent thermal parameters in two-phase phase-change processes.

math-ph↗

arXiv2026

Lie symmetry analysis of the nonlinear generalized heat equation for varying cross-section geometry

We study the nonlinear generalized heat equation $C(u)u_t=\frac{1}{z^ν}\left(K(u)z^νu_z\right)_z$, where $C(u)$ and $K(u)$ are temperature-dependent thermal coefficients and $ν>0$ is a geometric parameter describing the varying cross-section geometry. By applying the classical Lie symmetry method, we derive the determining equations and perform a complete classification of the admitted Lie point symmetries according to the functional dependence between $C(u)$ and $K(u)$. The analysis shows that the symmetry structure splits naturally into two principal cases: $C(u)/K(u)$ non-constant and $C(u)/K(u)=β$ constant. In the first case, only the basic symmetries are admitted for arbitrary coefficients, whereas additional generators appear under special compatibility relations. In the second case, the equation can be transformed to a linear radial heat equation by the substitution $v=\int K(u)du$, yielding an extended symmetry algebra. For each case, we construct the infinitesimal generators, commutator tables, one-parameter transformation groups, and corresponding invariant reductions. Invariant and similarity solutions are obtained and then specialized to several physically relevant subclasses, including power-law, exponential-type, and linear constitutive coefficients. The results provide a unified symmetry-based model for the analysis of generalized nonlinear heat equations in non-Cartesian geometries.

math.AP↗

arXiv2025

Inverse Stefan problems of determining the time-dependent source coefficient and heat flux function

This paper delves into the Inverse Stefan problem, specifically focusing on determining the time-dependent source coefficient in the parabolic heat equation governing heat transfer in a semi-infinite rod. The problem entails the intricate task of uncovering both temperature- and time-dependent coefficients of the source while accommodating Dirichlet and Neumann boundary conditions. Through a comprehensive mathematical model and rigorous theoretical analysis, our study aims to provide a robust methodology for accurately determining the source coefficient from observed temperature and heat flux data in problems with different cases of the source functions. Importantly, we establish the existence and uniqueness, and estimate the continuous dependence of a weak solution upon the given data for some inverse problems, offering a foundational understanding of its solvability.

math.AP↗

arXiv2024

Mathematical modelling of heat transfer in closed electrical contacts and electrical potential field dynamics with Thomson effect

In this study we develop a mathematical model that describe the behavior of electromagnetic fields and heat transfer in closed electrical contacts that arises when instantaneous explosion of the micro-asperity which involves vaporization zone and liquid, solid zones where temperature is defined by a generalized heat equation with Thomson effect. This model account for the nonlinear nature of the thermal coefficients and electrical conductivity depended on temperature. Our proposed solutions are based on similarity transformation which allows us to reduce a Stefan-type problem to a system of nonlinear integral equations whose existence of solution is proved by the fixed point theory in Banach spaces.

math.AP↗

arXiv2023

Mathematical modeling of heat process in a cylindrical domain with nonlinear thermal coefficients and a heat source on the axis

A mathematical model of the heat process in one-dimensional domain governed by a cylindrical heat equation with a heat source on the axis $z=0$ and nonlinear thermal coefficients is considered. The developed model is particularly applicable for analyzing temperature variations on electrical contact surfaces, where precise thermal management is crucial for ensuring optimal performance and preventing overheating. To solve the mathematical model, we employ a solution method based on similarity transformations. This technique allows us to reduce the problem to an ordinary differential problem, which is equivalent to a nonlinear integral equations system. To ensure the existence and uniqueness of the solution, we employ the fixed point theory in a Banach space, providing a rigorous mathematical foundation for our analysis.

math.AP↗

arXiv2023

Mathematical model of thermal phenomena of closure electrical contact with Joule heat source and nonlinear thermal coefficients

The mathematical model describing the dynamics of closed contact heating which involves vaporization of the metal when instantaneous explosion of micro-asperity occurs is presented through a Stefan type problem. The temperature field for metallic vaporization zone is introduced as heat resistance that decreases linearity. Temperature fields for liquid and solid phases of the metal described by spherical heat equations with nonlinear thermal coefficients and Joule heat source have to be determined as well as the free boundaries. Joule heating component depends on space and time variable when alternating current is considered. Solution method of the problem based on similarity variable transformation is applied which enables us to reduce the problem to an ordinary differential equations and nonlinear integral equations. Existence and uniqueness of the integral equations are proved by using fixed point theorem in Banach space.

math.AP↗

arXiv2022

Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients

In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component is considered. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modelled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the solution is provided by using the fixed point Banach theorem.

math.AP↗

arXiv2017

Exact solution of an inverse spherical two phase Stefan problem

In this paper, we represent the exact solution of a two phase inverse spherical Stefan problem, where along with unknown temperature functions heat flux function has to be determined. Suggested solution is obtained from new form of integral error function and its properties which are represented in the form of series whose coefficients have to be determined. The model problem can be used for mathematical modeling and investigation of arc phenomena in electrical contacts.

math-ph↗