One-dimensional Heat Equation Research Articles

Approximate Controllability for Degenerate Heat Equation with Bilinear Control

This paper investigates the nonnegative approximate controllability for the one-dimensional degenerate heat equation governed by bilinear control. Both non-controllability and approximate controllability are studied for the system. If the control is restricted to act on a fixed domain, it is not controllable. If the control is allowed to mobile, it is approximately controllable.

Determination of Compactly Supported Sources for the One-Dimensional Heat Equation

The inverse problem of determining a source in the one-dimensional heat equation in the case of a Dirichlet boundary value problem is investigated. The trace of the solution of the direct problem on straight-line segments inside the domain at the final time is specified as overdetermination (i.e., additional information on the solution of the direct problem). A Fredholm alternative theorem for this problem is proved, and sufficient conditions for its unique solvability are obtained. The inverse problem is considered in classes of smooth functions with derivatives satisfying the Holder condition.

SPDEs with linear multiplicative fractional noise: Continuity in law with respect to the Hurst index

In this article, we consider the one-dimensional stochastic wave and heat equations driven by a linear multiplicative Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index H∈(14,1). We prove that the solution of each of the above equations is continuous in terms of the index H, with respect to the convergence in law in the space of continuous functions. The proof is based on a tightness criterion on the plane and Malliavin calculus techniques in order to identify the limit law.

A central limit theorem for the stochastic heat equation

We consider the one-dimensional stochastic heat equation driven by a multiplicative space–time white noise. We show that the spatial integral of the solution from −R to R converges in total variance distance to a standard normal distribution as R tends to infinity, after renormalization. We also show a functional version of this central limit theorem.

On a new weight tri-diagonal iterative method and its applications

In this article, a new weight tri-diagonal iterative method in solving system of linear equation is proposed and its convergence is discussed. The set of methods in solving linear system generated by high-order scheme for solving one-dimensional Poisson equation and one-dimensional heat equation with periodic boundary are presented. The numerical experiments shows that the proposed method demonstrates a better performance compared with weight Jacobi, successive-over relaxation and alternating group explicit methods.

Nonnegative control of finite-dimensional linear systems

We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations.

Segregation effects and gap formation in cross-diffusion models

In this paper, we extend the results of [8] by proving exponential asymptotic H^1 -convergence of solutions to a one-dimensional singular heat equation with L^2 -source term that describe evolution of viscous thin liquid sheets while considered in the Lagrange coordinates. Furthermore, we extend this asymptotic convergence result to the case of a time inhomogeneous source. This study has also independent interest for the porous medium equation theory.

Long-time behaviour of solutions to a singular heat equation with an application to hydrodynamics

In this paper, we extend the results of [8] by proving exponential asymptotic H^1 -convergence of solutions to a one-dimensional singular heat equation with L^2 -source term that describe evolution of viscous thin liquid sheets while considered in the Lagrange coordinates. Furthermore, we extend this asymptotic convergence result to the case of a time inhomogeneous source. This study has also independent interest for the porous medium equation theory.

Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form sum_{t=1}^{p}A_{t}XB_{t}=C which includes a class of linear matrix equations. The objective of the algorithm is to minimize an error at each iteration by the idea of gradient-descent. We show that the proposed algorithm is widely applied to any problems with any initial matrices as long as such problem has a unique solution. The convergence rate and error estimates are given in terms of the condition number of the associated iteration matrix. Furthermore, we apply the proposed algorithm to sparse systems arising from discretizations of the one-dimensional heat equation and the two-dimensional Poisson’s equation. Numerical simulations illustrate the capability and effectiveness of the proposed algorithm comparing to well-known methods and recent methods.

Predicting the interfacial heat transfer coefficient of cast Mg-Al alloys using Beck’s inverse analysis

Apart from many governing parameters, the interfacial heat transfer coefficient (IHTC) has prime importance for the numerical simulation of casting as it quantifies the heat flux between casting and mold (or chill). Most Mg alloys are based on the Mg-Al system and casting is the commonly used production process. The experimental configuration makes it challenging to measure flux and surface temperatures required to evaluate the IHTC. In this study, the IHTC was predicted for a variety of Mg-Al compositions which were cast using a permanent cylindrical mold. Unidirectional heat flow was ensured in order to replicate the experimental conditions for solving the one-dimensional transient heat equation. The numerically determined mold and surface temperatures, using Beck’s inverse methodology, were in good agreement with the experiments and analytical solution, respectively. Moreover, the heat transfer behavior across the interface depicted in the form of IHTC was analyzed, also various empirical and numerical aspects of the method are discussed.

Analytical Solution of Homogeneous One-Dimensional Heat Equation with Neumann Boundary Conditions

A partial differential equation is an equation which includes derivatives of an unknown function with respect to two or more independent variables. The analytical solution is needed to obtain the exact solution of partial differential equation. To solve these partial differential equations, the appropriate boundary and initial conditions are needed. The general solution is dependent not only on the equation, but also on the boundary conditions. In other words, these partial differential equations will have different general solution when paired with different sets of boundary conditions. In the present study, the homogeneous one-dimensional heat equation will be solved analytically by using separation of variables method. Our main objective is to determine the general and specific solution of heat equation based on analytical solution. To verify our objective, the heat equation will be solved based on the different functions of initial conditions on Neumann boundary conditions. The results have been compared with different values of initial conditions but the boundary condition remain the same. Based on the results obtained, it can be concluded that increase the number of n will reduce the heat temperature and the time taken. For short length of the rod, the heat temperature quickly converges to zero and take less time to release or reduced the heat temperature when compared to the long length of the rod.

Weak approximation of the complex Brownian sheet from a Lévy sheet and applications to SPDEs

We consider a Lévy process in the plane and we use it to construct a family of complex-valued random fields that we show to converge in law, in the space of continuous functions, to a complex Brownian sheet. We apply this result to obtain weak approximations of the random field solution to a semilinear one-dimensional stochastic heat equation driven by the space–time white noise.

Lunar Regolith Temperature Variation in the Rümker Region Based on the Real-Time Illumination

Chang’E-5 will be China’s first sample−return mission. The proposed landing site is at the late-Eratosthenian-aged Rümker region of the lunar nearside. During this mission, a driller will be sunk into the lunar regolith to collect samples from depths up to two meters. This mission provides an ideal opportunity to investigate the lunar regolith temperature variation, which is important to the drilling program. This study focuses on the temperature variation of lunar regolith, especially the subsurface temperature. Such temperature information is crucial to both the engineering needs of the drilling program and interpretation of future heat-flow measurements at the lunar landing site. Based on the real-time illumination, and particularly the terrain obscuration, a one-dimensional heat equation was applied to estimate the temperature variation over the whole landing region. Our results confirm that while solar illumination strongly affects the surface temperature, such effect becomes weak at increasing depths. The skin depth of diurnal temperature variations is restricted to the uppermost ~5 cm, and the temperature of regolith deeper than ~0.6 m is controlled by the interior heat flow. At such a depth, China’s future lunar exploration is adequate to measure the inner heat flow, considering the drilling depth will be close to 2 m.

Pointwise Controllability for Degenerate Parabolic Equations by the Moment Method

In this paper, we study the pointwise controllability of the one-dimensional degenerate heat equation. Necessary and sufficient conditions for approximate and null controllability are proved. Our approach is mainly based on the moment method developed by Fattorini and Russell.

Modeling of heat transport and exact analytical solutions in thin films with account for constant non-relativistic motion

One-dimensional heat equations beyond Fourier law with account for non-relativistic motion of an observer with respect to the media are considered. Some analytical solutions to these heat transport equations are obtained. The effect of the motion of the observer in the media is studied. Heat transport in thin films is modelled by the system of inhomogeneous differential equations (DE), which involve Guyer–Krumhansl-type and Telegrapher's type equations. Analytical solution to this system is found for the initial Gaussian distribution in the case of Cauchy conditions and for zero initial heat flux. The comparative analysis of the obtained solutions is performed in wide range of Knudsen numbers, Kn⊂[0.1–1]. The effect of the relative media–observer speed on the perceived heat transport is explored.

Controllability of the one-dimensional fractional heat equation under positivity constraints

In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian $ (-d_x^{\,2})^{s}{} $ ($ 0<s<1 $) on the interval $ (-1,1) $. We prove the existence of a minimal (strictly positive) time $ T_{\rm min} $ such that the fractional heat dynamics can be controlled from any initial datum in $ L^2(-1,1) $ to a positive trajectory through the action of a positive control, when $ s>1/2 $. Moreover, we show that in this minimal time constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. We also give some numerical simulations that confirm our theoretical results.

Boundary switch on/off control approach to simultaneous identification of diffusion coefficient and initial state for one-dimensional heat equation

In this paper, we consider simultaneous reconstruction of diffusion coefficient and initial state for a one-dimensional heat equation through boundary control and measurement. The boundary measurement is proposed to make the system approximately observable, and both the coefficient and initial state are shown to be identifiable by this measurement under a boundary switch on/off control. By a Dirichlet series representation for the observation, we can transform the problem into an inverse process of reconstruction of the spectrum and coefficients for the Dirichlet series in terms of observation. This happens to be the reconstruction of spectral data for an exponential sequence with measurement error, and it enables us to develop an algorithm based on the matrix pencil method in signal analysis. A theoretical error analysis for the algorithm concerning the coefficient reconstruction is carried out for the proposed method. The numerical simulations are presented to verify the proposed algorithm.

Output tracking for a 1-D heat equation with non-collocated configurations

In this paper, we are concerned with output tracking for a one-dimensional heat equation where both disturbance and performance output are non-collocated to the controller. The method of trajectory planning is used to overcome the difficulties caused by these non-collocated configurations. An observer based output feedback law is designed by using only the tracking error between performance output and reference signal. Exponential convergence of tracking error and uniform boundedness of state of the whole closed-loop system are obtained. These theoretical results are validated by numerical simulations.

On space-time periodic solutions of the one-dimensional heat equation

We look for solutions \begin{document}$ u\left( x,t\right) $\end{document} of the one-dimensional heat equation \begin{document}$ u_{t} = u_{xx} $\end{document} which are space-time periodic, i.e. they satisfy the property \begin{document}$ u\left( x+a,t+b\right) = u\left( x,t\right) $\end{document} for all \begin{document}$ \left( x,t\right) \in\left( -\infty,\infty\right) \times\left( -\infty,\infty\right), $\end{document} and derive their Fourier series expansions. Here \begin{document}$ a\geq0, b\geq 0 $\end{document} are two constants with \begin{document}$ a^{2}+b^{2}>0. $\end{document} For general equation of the form \begin{document}$ u_{t} = u_{xx}+Au_{x}+Bu, $\end{document} where \begin{document}$ A, B $\end{document} are two constants, we also have similar results. Moreover, we show that non-constant bounded periodic solution can occur only when \begin{document}$ B>0 $\end{document} and is given by a linear combination of \begin{document}$ \cos\left( \sqrt{B}\left( x+At\right) \right) $\end{document} and \begin{document}$ \sin\left( \sqrt{B}\left( x+At\right) \right). $\end{document}

Flutter analysis of sandwich plates with functionally graded face sheets in thermal environment

This paper presents an analytical model for flutter analysis of sandwich plates with functionally graded face sheets in the thermal environment. The material properties of the face sheets are supposed to be temperature-dependent by a nonlinear distribution satisfying one-dimensional heat equation, and vary according to power law distribution in terms of the volume fractions along the thickness of the plate. The vibration of the sandwich plate is modeled on the basis of different plate theories including Mindlin theory, classical theory, exponential theory, third-order theory, sinusoidal theory, hyperbolic theory and fifth-order theory. Modified shear deformation theories applied herein are capable of considering inertia effects and transverse shear stresses. First order piston theory is utilized to model the aerodynamic load due to supersonic flow. The coupled governing equations are derived using Hamilton's principle and, Galerkin approach is applied to obtain vibrational characteristics and critical dynamic pressure of the system. Some comparisons with available results in the literature are performed to validate the present modeling, and excellent agreement is observed. Our attention is focused on analyzing the effects of different parameters such as thickness ratio, aspect ratio, thermal load, the thickness of face sheet and power law index on dynamic stability of the sandwich plate.