One-dimensional Heat Equation Research Articles

On the asymptotic spatial behaviour in the theory of mixtures of thermoelastic solids

This paper is concerned with the study of asymptotic spatial behaviour of solutions in a mixture consisting of two thermoelastic solids. A second-order differential inequality for an adequate volumetric measure and the maximum principle for solutions of the one-dimensional heat equation are used to establish a spatial decay estimate of solutions in an unbounded body occupied by the mixture. For a fixed time, the result in question proves that the mechanical and thermal effects are controlled by an exponential decay estimate in terms of the square of the distance from the support of the external given data. The decay constant depends only on the thermal constitutive coefficients of the mixture.

Adjoint-based optimization of PDEs in moving domains

In this investigation we address the problem of adjoint-based optimization of PDE systems in moving domains. As an example we consider the one-dimensional heat equation with prescribed boundary temperatures and heat fluxes. We discuss two methods of deriving an adjoint system necessary to obtain a gradient of a cost functional. In the first approach we derive the adjoint system after mapping the problem to a fixed domain, whereas in the second approach we derive the adjoint directly in the moving domain by employing methods of the noncylindrical calculus. We show that the operations of transforming the system from a variable to a fixed domain and deriving the adjoint do not commute and that, while the gradient information contained in both systems is the same, the second approach results in an adjoint problem with a simpler structure which is therefore easier to implement numerically. This approach is then used to solve a moving boundary optimization problem for our model system.

Pulsed-laser-induced dewetting in nanoscopic metal films: Theory and experiments

Hydrodynamic pattern formation (PF) and dewetting resulting from pulsed laser induced melting of nanoscopic metal films have been used to create spatially ordered metal nanoparticle arrays with monomodal size distribution on SiO_{\text{2}}/Si substrates. PF was investigated for film thickness h\leq7 nm < laser absorption depth \sim11 nm and different sets of laser parameters, including energy density E and the irradiation time, as measured by the number of pulses n. PF was only observed to occur for E\geq E_{m}, where E_{m} denotes the h-dependent threshold energy required to melt the film. Even at such small length scales, theoretical predictions for E_{m} obtained from a continuum-level lumped parameter heat transfer model for the film temperature, coupled with the 1-D transient heat equation for the substrate phase, were consistent with experimental observations provided that the thickness dependence of the reflectivity of the metal-substrate bilayer was incorporated into the analysis. The spacing between the nanoparticles and the particle diameter were found to increase as h^{2} and h^{5/3} respectively, which is consistent with the predictions of the thin film hydrodynamic (TFH) dewetting theory. These results suggest that fast thermal processing can lead to novel pattern formation, including quenching of a wide range of length scales and morphologies.

Carleman estimates for the one-dimensional heat equation with non-smooth coefficients and applications

We study the observability and some of its consequences for one-dimensional heat equations ∂ ± ∂x(c∂x with non-smooth coefficients c. The observability, for a linear equation, is obtained by a Carleman-type estimate. In a first step, we derive global Carleman estimates in the case of a piecewise ℒ1-coefficient. In a second step, we treat the case of a coefficient with bounded variations by approximating c by piecewise regular coefficients, cϵ, and passing to the limit in the Carleman estimates associated to the operators defined with cϵ. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient.

Equivalence of Conservation Laws and Equivalence of Potential Systems

We study conservation laws and potential symmetries of (systems of) differential equations applying equivalence relations generated by point transformations between the equations. A Fokker-Planck equation and the Burgers equation are considered as examples. Using reducibility of them to the one-dimensional linear heat equation, we construct complete hierarchies of local and potential conservation laws for them and describe, in some sense, all their potential symmetries. Known results on the subject are interpreted in the proposed framework. This paper is an extended comment on the paper of J.-q. Mei and H.-q. Zhang [Internat. J. Theoret. Phys., 2006, in press].

The numerical solution of the one-dimensional heat equation by using third degree B-spline functions

In this paper, the boundary value problem for the one-dimensional heat equation with a nonlocal initial condition is examined by using the third degree B-splines functions. The numerical solution of the equations are discussed and illustrated with an example. The results of numerical testing show that the numerical method based on the given techniques produce good results.

A modified method for determining the surface heat flux of IHCP

We consider the determination of the surface heat flux of a body from a measured temperature history at a fixed location inside the body. Mathematically, it is formulated as a problem for the one-dimensional heat equation in a quarter plane with data given along the line x = 1, and the gradient of the solution is determined for 0≤x<1. This problem is of practical interest in some engineering contexts and it is severely ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. In this article, we employ a fourth-order modified method to solve the problem. Some stability estimates are given. Numerical examples show that the modified method works very well.

Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem

We study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise C 1 ). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient.

Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients

We derive global Carleman estimates for one-dimensional linear parabolic operators ∂ t ± ∂ x ( c ∂ x ) with a coefficient c with bounded variations. These estimates are obtained by approximating c by piecewise regular coefficients, c ε , and passing to the limit in the Carleman estimates associated to the operators defined with c ε . Such estimates yield results of controllability to the trajectories for a class of semilinear parabolic equations. To cite this article: J. Le Rousseau, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

The Heat Equation with Initial Data Corrupted by Measurement Error and Missing Data

In this work the Cauchy problem for the one-dimensional heat equation is considered. In contrast to existing literature it is assumed that the initial state f is unknown and that information regarding f is obtained by some process of measurement. To enhance realism, both measurement errors and missing data are allowed for. Under assumptions on f in the Fourier-domain first an approximation to f is derived from the data by means of a novel uncertainty principle. Then, it is studied how this perturbation in the initial state propagates with time.

Direct simulation monte carlo of pulsed laser ablation of metals with clusterization processes in vapor

A mathematical model is proposed to describe laser ablation of metals in vacuum under the action of nanosecond laser pulses of moderate intensity taking into account the processes of cluster formation and decay in the vapor cloud. To describe the laser radiation absorption and metal heating, the thermal model based on the unsteady one-dimensional heat equation with a volume heat source is used, and the method of statistical modelling is employed for modelling vapor expansion and the processes of cluster formation. The efficiency of the proposed complex model is considered by the example of pulsed laser ablation of a niobium target.

Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients

We derive global Carleman estimates for one-dimensional linear parabolic equations ∂ t ± ∂ x ( c ∂ x ) with a coefficient of bounded variations. These estimates are obtained by approximating c by piecewise constant coefficients, c ε , and passing to the limit in the Carleman estimates associated to the operators defined with c ε . Such estimates yields observability inequalities for the considered linear parabolic equation, which, in turn, yield controllability results for classes of semilinear equations.

Rothe time-discretization method for the semilinear heat equation subject to a nonlocal boundary condition

This paper is devoted to prove, in a nonclassical function space, the weak solvability of a mixed problem which combines a Neumann condition and an integral boundary condition for the semilinear one-dimensional heat equation. The investigation is made by means of approximation by the Rothe method which is based on a semidiscretization of the given problem with respect to the time variable.

Heat distribution in an infinite rod

We construct an asymptotic expansion of the solution of the Cauchy problem for the one-dimensional heat equation for the case in which the initial function at infinity has power asymptotics.

Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient, and applications

We study the observability and some of its consequences for the one-dimensional heat equation with a discontinuous coefficient (piecewise C 1 ). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields results of controllability to the trajectories for semilinear equations. It also yields a stability result for the inverse problem of the identification of the diffusion coefficient. To cite this article: A. Benabdallah et al., C. R. Mecanique 334 (2006).

Saddle points for maximin investment problems with observable but non-predictable parameters: solution via heat equation†

We study optimal investment problem for a market model where the evolution of risky assets prices is described by Ito's equations. The risk-free rate, the appreciation rates, and the volatility of the stocks are all random; they depend on a random parameter that is not adapted to the driving Brownian motion. The distribution of this parameter is unknown. The optimal investment problem is stated in a maximin setting to ensure that a strategy is found such that the minimum of expected utility over all possible distributions of parameters is maximal. We show that a saddle point exists and can be found via solution of the standard one-dimensional heat equation with a Cauchy condition defined via one-dimensional minimization. This solution even covers models with unknown solution for a given distribution of the market parameters.

Transient temperature and elastic response of a space-based mirror in the radiation-conduction environment

The problem of the one-dimensional heat equation with radiation boundary conditions is studied. Steady state and transient solutions for a space-based mirror are obtained under a broad class of boundary specification. Associated elastic stress and displacement are investigated. Time-varying heat flux at the boundary is also considered. The method requires solving equations of the typetan⁡λ=(Bi1+Bi2)λ/(λ2−Bi1Bi2)\tan \lambda =\left (Bi_1+Bi_2\right )\lambda /\!\left (\lambda ^2-Bi_1Bi_2\right )for the eigenvaluesλ\lambda, for which complete asymptotic solutions are given. Selected transient response results for temperature and stresses are presented.

Carleman estimates for one-dimensional degenerate heat equations

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain.

Surface-Absorption Assumption for Radiant Heating and Ignition of Energetic Solids

In this note a simple analytical model is described for investigating the surface absorption assumption. The analysis is based on the one-dimensional heat equation with in-depth, volumetric absorption. The analysis also sets forth a new nondimensional conduction/radiation parameter for radiant ignition of solids that can be used to evaluate the validity of the surface-absorption assumption

On a time-discretization method for a semilinear heat equation with purely integral conditions in a nonclassical function space

In this paper, we construct a semidiscrete approximate solution to a semilinear one-dimensional heat equation subject to integral boundary conditions by means of the Rothe discretization in time method. The convergence of the approximation scheme obtained is proved, yielding the well-posedness of the problem considered.