arXiv subjects

Guillermo Federico Umbricht

Publications and source records attributed to Guillermo Federico Umbricht.

11 recordsLinked to original sources

arXiv2026

Regularization operators for identifying the unknown source in the time-fractional convection-diffusion-reaction equation

This article presents a mathematical study of the problem of identifying a time-dependent source term in transport processes described by a timefractional parabolic equation, based on noisy time-dependent measurements taken at an arbitrary position. The problem is analytically solved using Fourier techniques, and it is shown that the solution is unstable. To address this instability, three one-parameter families of regularization operators are proposed, each designed to counteract the factors responsible for the instability of the inverse operator. Additionally, a new rule for selecting the regularization parameter is introduced, and an error bound is derived for each estimate. Numerical examples with varying characteristics are provided to illustrate the advantages of the proposed strategies.

math.AP↗

arXiv2025

Theoretical analysis of a two-dimensional bilayer convection-diffusion-reaction-source problem

This work investigates the two-dimensional thermal behavior of a bilayer medium subject to both internal and external heat sources. The model incorporates diffusion, advection, and temperature-dependent volumetric heat generation or absorption in each layer, as well as general convective conditions on the external boundaries. The influence of interfacial thermal resistance between the two materials is also considered. An analytical solution is developed using Fourier-based techniques, and a stable and convergent finite difference method is proposed to analyze particular scenarios. The theoretical results are validated against known solutions and numerical simulations, demonstrating consistency with the expected physical behavior. The findings contribute to a deeper understanding of heat transfer phenomena in layered systems and offer potential insights for optimizing thermal performance in engineering applications involving composite materials.

physics.flu-dyn↗

arXiv2025

Analytical and Numerical Study of a Convection-Diffusion-Reaction-Source Problem in Multilayered Materials

In this work, a thermal energy transfer problem in a one-dimensional multilayer body is theoretically analyzed, considering diffusion, advection, internal heat generation or loss linearly dependent on temperature in each layer, as well as heat generation due to external sources. Additionally, the thermal contact resistance at the interfaces between each pair of materials is taken into account. The problem is mathematically modeled, and explicit analytical solutions are derived using Fourier techniques. A convergent finite difference scheme is also formulated to simulate specific cases. The solution is consistent with previous results. A numerical example is provided, demonstrating the coherence between the obtained results and the physical behavior of the problem. This work was recently published for a two-layer body; the generalization to m-layer bodies allows for conclusions that enhance the theoretical understanding of heat transfer in multilayer materials and may contribute to improving the thermal design of multilayer engineering systems.

physics.flu-dyn↗

arXiv2024

Bilayer one-dimensional Convection-Diffusion-Reaction-Source problem. Analytical and numerical solution

This article presents a theoretical analysis of a one-dimensional heat transfer problem in two layers involving diffusion, advection, internal heat generation or loss linearly dependent on temperature in each layer, and heat generation due to external sources. Additionally, the thermal resistance at the interface between the materials is considered. The situation of interest is modeled mathematically, explicit analytical solutions are found using Fourier techniques, and a convergent finite difference scheme is formulated to simulate specific cases. The solution is consistent with previous results. A numerical example is included that shows coherence between the obtained results and the physics of the problem. The conclusions drawn in this work expand the theoretical understanding of two-layer heat transfer and may also contribute to improving the thermal design of multilayer engineering systems.

physics.flu-dyn↗

arXiv2024

Regularization Techniques for Estimating the Source in a Complete Parabolic Equation in $\mathbb{R}^n$

In this article, the problem of identifying the source term in transport processes given by a complete parabolic equation is studied mathematically from noisy measurements taken at an arbitrary fixed time. The problem is solved analytically with Fourier techniques and it is shown that this solution is not stable. Three single parameter families of regularization operators are proposed to dealt with the instability of the solution. Each of them is designed to compensate the factor that causes the instability of the inverse operator. Moreover, a rule of choice for the regularization parameter is included and a Hölder error bound type is obtained for each estimation. Numerical examples of different characteristics are presented to demonstrate the benefits of the proposed strategies.

math.AP↗

arXiv2021

Optimal Estimation of Thermal Diffusivity in an Energy Transfer Problem

This work focuses on determining the coefficient of thermal diffusivity in a one-dimensional heat transfer process along a homogeneous and isotropic bar, embedded in a moving fluid with heat generation. A first type (Dirichlet) condition is imposed on one boundary and a third type (Robin) condition is considered at the other one. The parameter is estimated by minimizing the squared errors where noisy observations are numerically simulated at different positions and instants. The results are evaluated by means of the relative errors for different levels of noise. In order to enhance the estimation performance, an optimal design technique is chosen to select the most informative data. Finally, the improvement of the estimate is discussed when an optimal design is used.

physics.flu-dyn↗

arXiv2021

Identification of the Source for Full Parabolic Equations

In this work, we consider the problem of identifying the time independent source for full parabolic equations in $\mathbb{R}^n$ from noisy data. This is an ill-posed problem in the sense of Hadamard. To compensate the factor that causes the instability, a family of parametric regularization operators is introduced, where the rule to select the value of the regularization parameter is included. This rule, known as regularization parameter choice rule, depends on the data noise level and the degree of smoothness that it is assumed for the source. The proof for the stability and convergence of the regularization criteria is presented and a Hölder type bound is obtained for the estimation error. Numerical examples are included to illustrate the effectiveness of this regularization approach.

math.NA↗

arXiv2021

Estimación y Análisis de Sensibilidad para el Coeficiente de Difusividad en un Problema de Conducción de Calor

The aim of this article is to discuss the estimation of the diffusivity coefficient of a homogeneous metal rod from temperature values at a fixed point in the bar for different time instants. The time-dependent problem of heat conduction is analyzed in an insulated conductor wire of length l considering constant boundary conditions. The problem is modeled by a parabolic partial differential equation, imposing Dirichlet boundary conditions. We consider simulated temperature values at a point of the bar for different time instants and estimate the coefficient of diffusivity using usual techniques for solving inverse problems. For the discretization of the equation we consider a finite difference centered scheme. We include an analytical and numerical study of the sensitivity of the temperature function with respect to the coefficient of diffusivity. Numerical experiments show very good accuracy in the estimates.

math.NA↗

arXiv2021

Método de Regularización para Identificar una Fuente en una Ecuación Elíptica

The aim of this paper is to numerically study the performance of a method of regularization. This technique was developed to solve the illposed problem of estimating a source-dimensional Poisson equation for two dimensions from measurements taken over a line inside the domain. The proposed method consists in adding a regularization term to the equation that depends on a parameter, which is called regularization parameter. In this paper we show the results for different values of this parameter as well as for different levels of noise in the data used for estimation. After analyzing the results some considerations on its effective implementation are discussed.

math.AP↗

arXiv2021

Estimation of a Thermal Conductivity in a Stationary Heat Transfer Problem with a Solid-Solid Interface

An inverse problem for a stationary heat transfer process is studied for a totally isolated bar on its lateral surface, made up of two consecutive sections of different, isotropic and homogeneous materials, perfectly assembly, where one of the materials, that is unreachable and unknown, has to be identified. The length of the bar is assumed to be much greater that the diameter so that a 1D heat transfer process is considered. A constant temperature is assumed at the end of the unknown part of the rod while the other end is let free for convection. We propose a procedure to identify the unknown material of the bar based on a noisy flow measurement at the opposite end. Necessary and sufficient conditions are derived together with a bound for the estimation error. Moreover, elasticity analysis is performed to study the influence of the data in the conductivity estimation and numerical examples are included to illustrate the proposed ideas and show the estimation performance.

math.AP↗

arXiv2021

Estimation Technique for a Contact Point Between two Materials in a Stationary Heat Transfer Problem

An inverse problem for a stationary heat transfer process is studied for a totally isolated bar on its lateral surface, of negligible diameter, made up of two consecutive sections of different, isotropic and homogeneous materials. At the left boundary, a Dirichlet type condition is imposed that represents a constant temperature source while a Robin type condition that models the heat dissipation by convection is considered at the right one. Many articles in the literature focus on thermal and stress analysis at the interface but no one is dedicated to the estimation of the contact point location between the two materials. In this work, it is assumed that the interface position is unknown. A technique to determine it from a unique noisy flow measurement at the right boundary is introduced. Necessary and sufficient conditions are derived in order to obtain the estimation of the interface point from a heat flux measured at the right boundary. Numerical solutions are obtained together with an expression for the estimation error. Moreover, an elasticity analysis is included to study the influence of data errors. The results show that our approach is useful for determining the location of the materials interface.

math.AP↗