One-dimensional Heat Equation Research Articles

A two-parameter Tikhonov regularization for a fractional sideways problem with two interior temperature measurements

This paper deals with a fractional sideways problem of determining the surface temperature of a heat body from two interior temperature measurements. Mathematically, it is formulated as a problem for the one-dimensional heat equation with Caputo fractional time derivative of order α∈(0,1], where the data are given at two interior points, namely x=x1 and x=x2, and the solution is determined for x∈(0,L),0<x1<x2≤L. The problem is challenging since it is severely ill-posed for x∉[x1,x2]. For the ill-posed problem, we apply the Tikhonov regularization method in Hilbert scales to construct stable approximation problems. Using the two-parameter Tikhonov regularization, we obtain the order optimal convergence estimates in Hilbert scales by using both a priori and a posteriori parameter choice strategies. Numerical experiments are presented to show the validity of the proposed method.

Application of the heat equation to the study of underground temperature

Modeling underground temperatures provides a practical application of the one-dimensional heat equation. In this work, the one-dimensional heat equation in surface soil is extended to include heat carried by the vertical flow of rainwater through the soil. Analytical solutions, with and without water flow, illustrate the influence of rainwater circulation on the sub-surface propagation of seasonal temperature variations, an important effect that is generally neglected in textbooks. The surface temperature variations are damped by the soil, and this effect was used by the Troglodytae in Egypt or the Petra in South Jordan to insulate against extreme temperatures. For a realistic case of horizontally layered geology, a finite volume Python code was developed for the same purpose. Subsurface temperatures were also measured over a full year at depths up to 1.8 m and used to estimate the thermal skin depth and thermal wavelength. This study provides students with a practical example of how a textbook physics problem can be modified to extract information of contemporary importance in geophysics and global warming.

Identification of Time-Wise Thermal Diffusivity, Advection Velocity on the Free-Boundary Inverse Coefficient Problem

This paper is concerned with finding solutions to free-boundary inverse coefficient problems. Mathematically, we handle a one-dimensional non-homogeneous heat equation subject to initial and boundary conditions as well as non-localized integral observations of zeroth and first-order heat momentum. The direct problem is solved for the temperature distribution and the non-localized integral measurements using the Crank–Nicolson finite difference method. The inverse problem is solved by simultaneously finding the temperature distribution, the time-dependent free-boundary function indicating the location of the moving interface, and the time-wise thermal diffusivity or advection velocities. We reformulate the inverse problem as a non-linear optimization problem and use the lsqnonlin non-linear least-square solver from the MATLAB optimization toolbox. Through examples and discussions, we determine the optimal values of the regulation parameters to ensure accurate, convergent, and stable reconstructions. The direct problem is well-posed, and the Crank–Nicolson method provides accurate solutions with relative errors below 0.006% when the discretization elements are M=N=80. The accuracy of the forward solutions helps to obtain sensible solutions for the inverse problem. Although the inverse problem is ill-posed, we determine the optimal regularization parameter values to obtain satisfactory solutions. We also investigate the existence of inverse solutions to the considered problems and verify their uniqueness based on established definitions and theorems.

A finite volume formulation for one-dimensional Stefan problems

This paper presents a numerical formulation that combines the finite-volume method with a time-dependent boundary-fitted coordinate system to tackle the moving boundary problems for the one-dimensional (1-D) heat equation. Following the formulation process, an algorithm was created with the help of the linear algebra package Eigen to generate the data for the numerical solution. Four benchmark cases of heat conduction were investigated by this approach and compared to the analytical solution: a fixed boundary case, the liquid phase of a melting process, the solid phase of a melting process, and an ablating process. The results demonstrate the capability of the presented numerical formulation in efficiently solving the general 1-D heat problems, especially the one-phase Stefan problems, as well as the consistency of the numerical and analytical solutions. Future works can strive to extend the formulations to Stefan problems of two phases or higher dimensions. The work can then be extended to real-life applications, such as the spacecraft re-entry ablation process, and the process of iceberg melting, demonstrating its critical value in solving practical problems.

Explicit and implicit numerical investigations of one-dimensional heat equation based on spline collocation and Thomas algorithm

Explicit and implicit numerical investigations of one-dimensional heat equation based on spline collocation and Thomas algorithm

On the differential equations of frozen Calogero-Moser-Sutherland particle models

Multivariate Bessel and Jacobi processes describe Calogero-Moser-Sutherland particle models. They depend on a parameter k and are related to time-dependent classical random matrix models like Dysom Brownian motions, where k has the interpretation of an inverse temperature. There are several stochastic limit theorems for k→∞ were the limits depend on the solutions of associated ordinary differential equations (ODEs) where these ODEs admit particular simple solutions which are connected with the zeros of the classical orthogonal polynomials. In this paper we show that these solutions attract all solutions. Moreover we present a connection between the solutions of these multivariate ODEs with associated one-dimensional inverse heat equations. These inverse heat equations are used to compute the expectations of some determinantal formulas for the multivariate Bessel and Jacobi processes.

Some Results of Stochastic Differential Equations

In this paper, there are two aims: one is Schauder and Sobolev estimates for the one-dimensional heat equation; the other is the stabilization of differential equations by stochastic feedback control based on discrete-time state observations. The nonhomogeneous Poisson stochastic process is used to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The properties of a jump process is used. The stabilization of differential equations by stochastic feedback control is based on discrete-time state observations. Firstly, the stability results of the auxiliary system is established. Secondly, by comparing it with the auxiliary system and using the continuity method, the stabilization of the original system is obtained. Both parts focus on the impact of probability theory.

Internal null controllability for the one-dimensional heat equation with dynamic boundary conditions

Abstract The primary focus of this paper is to establish the internal null controllability for the one-dimensional heat equation featuring dynamic boundary conditions. This achievement is realized by introducing a new Carleman estimate and an observability inequality for the corresponding backward system. In conclusion, the paper includes a set of numerical experiments that serve to confirm the validity of the theoretical findings and underscore the effectiveness of the designed control with a minimal $L^{2}$-norm.

Thermal insulation of flexible pavements utilizing foam glass aggregates to mitigate frost action in cold regions — Development of design tools

This manuscript focuses on the use of foam glass aggregates (FGAs) as insulator to protect pavements against frost action. The ultimate objective of the paper was to develop design tools for the thermal design of pavements insulated with FGAs. A mathematical model was outlined to describe the complex thermal behavior of FGAs and calculate temperature levels within a domain representing a pavement insulated with FGAs. The formulation is based on the one-dimensional heat equation which was discretized with a forward finite-difference technique and subsequently coded in Fortran90. This code was executed to successfully replicate temperature measurements collected within an experimental pavement insulated with FGAs, enabling the validation of the model. The model was then re-executed to perform three different sensitivity analyses and investigate the effects of the pavement geometry, layers humidity, and FGAs’ effective particle size on the maximal frost front penetration. Compared to a pavement without insulation, utilizing FGAs reduced the maximal frost front depth. The sensitivity analyses were summarized in design charts to guide the utilization of FGAs for the thermal design of pavements. The code utilized to generate these charts is freely available on GitHub.

Global boundary null-controllability of one-dimensional semilinear heat equations

Global boundary null-controllability of one-dimensional semilinear heat equations

On Various Space-time Properties of Solutions to the One-dimensional Heat Equation on Semi-infinite Rod

On Various Space-time Properties of Solutions to the One-dimensional Heat Equation on Semi-infinite Rod

Temperature Distribution in a Non-Prismatic Thermoelectric Leg and the Energy Conversion Efficiency

Abstract It has been proposed that asymmetric thermoelectric (TE) legs may enhance the TE energy conversion efficiency. This work presents an analytical model for the temperature field in non-prismatic TE legs and the energy conversion efficiency. Different from the models available in the literature, the present one-dimensional (1D) heat equation for thermal conduction in non-prismatic legs is derived from the general three-dimensional energy and charge balance laws and the thermoelectric constitutive relations considering Joule heating and the Seebeck effect on the heat flow. The temperature field in a TE leg with an arbitrary gradient of the cross sectional area is obtained using the 1D heat equation. The temperature field and energy conversion efficiency are calculated for exponential and quadratic Bi2Te3 legs as well as an exponential PbTe leg. The numerical results indicate that leg tapering has minimal effects on the temperature distribution in and energy efficiency of the non-prismatic leg under the prescribed cold and hot side temperatures boundary conditions provided that the volume of the leg remains the same. The energy efficiency of the tapered leg, however, can be significantly increased under the prescribed hot side heat flux condition. Finally, a simple estimate on the limitation of the 1D models for non-prismatic legs is discussed.

Solutions of fuzzy advection-diffusion and heat equations by natural adomian decomposition method

In this article, we present an algorithm for computing analytical solutions of linear fuzzy advection-diffusion equations and one-dimensional fuzzy heat equations involving an external source. The fuzzy problems can be solved by using the natural transform and Adomian decomposition method. The results obtained through the natural Adomian decomposition method are calculated in a series form that converges rapidly to the exact solution. To enhance the practicality of our work, we provide examples to illustrate our findings.

Comparative Analysis of a Numerical Method and Machine Learning Methods of Temperature Determination of a Doped Lubricating Layer with Experimental Data

This article compares machine learning methods and a numerical method of determination of the doped lubricating layer with experimental data. Based on the sweep method, the one-dimensional Fourier heat equation with boundary and initial conditions is solved. As a result of comparing numerical and predictive data with experiments, it can be concluded that machine learning models are better at predicting results compared to numerical data

One-dimensional heat and advection-diffusion equation based on improvised cubic B-spline collocation, finite element method and Crank-Nicolson technique

In this study, we numerically investigate the solution of one-dimensional heat and advection-diffusion equation by employing improvised quartic order cubic B-spline using Crank-Nicolson method and finite element method (FEM). The numerical technique discretizes the spatial derivatives variables using Crank-Nicolson method and time derivatives using finite element method. It is demonstrated that the method is unconditionally stable with Von-Neumann process and accurate to convergence order Oh4+∆t2. The numerical results are simulated and error bounds are calculated using finite difference scheme on a piecewise uniform mesh. Finally, to support our claim, five test problems are considered and the experimental results also compared to existing methods using MATLAB as well as MATHEMATICA software tools. The error norms L2,L∞,root mean square (RMS) error with computational time and order of convergence are investigated for each example. The method is computationally efficient with less memory storage. The accuracy and efficiency of present scheme is measured in terms of absolute errors and also compared with analytical and published results in literature.

Nonlinear vibration analysis of graphene platelets reinforced nanocomposite doubly curved concrete panels subjected to thermal shock: Development of an artificial intelligence technique

When the temperature gradient across a component quickly changes, a phenomenon known as thermal shock occurs that exposes the component to fast variable thermal stresses and large-scale strains. One of the goals of this abrupt loading is to increase the capacity of different systems to withstand thermal shock. The nonlinear thermally induced vibration of doubly curved graphene nanoplatelet reinforced nanocomposite panels subjected to abrupt thermal shock is being studied right now. The bottom portion of the curved panel system is held at a reference temperature while the top portion is thermally shocked. The Crank-Nicholson technique and generalized differential quadrature (GDQ) are used to build up and solve the one-dimensional transient heat equation. The double-curved panel’s thermomechanical characteristics are temperature dependent; hence this equation is nonlinear. Based on the law of decoupled thermoelasticity, Kant’s theory, and von Karman’s assumption of some kind of geometric nonlinearity, the overall function of the doubly curved panel is established. After modeling the current system mathematically and comparing the current approach’s findings to those of earlier research, we revalidated the findings in a Python environment by comparing them to those of the Extreme Gradient Boosting (XGBoost) machine learning algorithm. This approach is founded on the “boosting” idea, which combines additive training techniques with the identification of weak learners to produce strong learners. The findings demonstrate that certain structural geometrical and physical factors are significant contributors to the double-curved GPLRC panel’s nonlinear frequency under thermal shock.

On Simultaneous Determination of Thermal Conductivity and Volume Heat Capacity of Substance

The study of nonlinear problems associated with heat transfer in substance is important for practice. Earlier, the authors proposed an efficient algorithm for determining the thermal conductivity from experimental observations of the dynamics of the temperature field in an object. In this work, we explore the possibility of extending the algorithm to the numerical solution of the problem of simultaneous identification of the temperature-dependent volume heat capacity and the thermal conductivity of the substance under study. The consideration is based on the Dirichlet boundary value problem for the one-dimensional nonstationary heat equation. The coefficient inverse problem in question is reduced to a variational problem, which is solved by applying gradient methods based on the fast automatic differentiation technique. The uniqueness of the solution to the inverse problem is analyzed.

Distributed controllability of one-dimensional heat equation in unbounded domains: The Green’s function approach

Distributed controllability of one-dimensional heat equation in unbounded domains: The Green’s function approach

A Feynman–Kac approach for the spatial derivative of the solution to the Wick stochastic heat equation driven by time homogeneous white noise

We consider the (unique) mild solution [Formula: see text] of a one-dimensional stochastic heat equation on [Formula: see text] driven by time-homogeneous white noise in the Wick–Skorokhod sense. The main result of this paper is the computation of the spatial derivative of [Formula: see text], denoted by [Formula: see text], and its representation as a Feynman–Kac type closed form. The chaos expansion of [Formula: see text] makes it possible to find its (optimal) Hölder regularity especially in space.

Speeds of Decay of Temperature Distribution in a Laterally Insulated Copper Bar

Employing one-dimensional heat equation, speeds of decay of the temperature distribution in a laterally insulated copper bar are analyzed. Eigenfunctions and their corresponding Eigenvalues of the model equation are derived and analyzed with different period of time. The temperature distribution with given initial temperatures on copper bar in period of time are also analyzed. One-dimensional heat equation of a laterally insulated copper bar is also solved for examining and analyzing its temperature distribution Eigenfunction and corresponding Eigenvalues. Further, finite difference method is used to solve the temperature distribution numerically, and comparing its results with analytical solution.