Output regulation for a 1-D heat equation with infinite-Dimensional exosystem

Temperature variations may cause solid–liquid phase transition and damage evolution, which requires more refined modelling. In this study, a solid–liquid thermo-mechanical phase field model based on thermodynamically approach is established. A novel predictive equation of temperature-dependent critical strain energy density is firstly derived by combining force-heat equivalent principle and effective heat capacity method. The critical strain energy density of typical brittle materials with narrow phase transition interval (e.g., ice and Al 2 O 3 ) can be successfully captured by the proposed formulation. A simple but effective degradation function associated with phase transition variable is embedded in the phase field model to describe the mechanical degradation within phase transition and avoids the undesirable damage that occurs in liquid-state domain. The established multi-physical framework is implemented through finite element method. In numerical simulations, the phase transition part is verified through the two-phase Stefan’s melting issue preliminarily. Then, the thermo-mechanical module is studied through the shrinkage cracking of a 1D bar. The insensitivity of length scale and fracture toughness degradation under the assumption of small transition interval is proved. The proposed model is subsequently applied to thermal cracking in additive manufacturing and electrothermal de-icing, with its effectiveness and accuracy demonstrated by comparing with experimental results and empirical criterion

Thermal management of electronic chips using microencapsulated phase change material slurry in a taenidia-inspired spiral channel heat exchanger

Quartic variation of the solution to the semilinear stochastic heat equation: Limit behavior and asymptotic independence with respect to the data

Null controllability of the 1D heat equation with interior inverse square potential

Repetitive learning output tracking control for a heat equation with multi-channel periodic disturbances

Abstract In this article, we study the null-controllability and observability properties of a fluid–structure interaction system, governed by the heat equation coupled with the damped beam equation. To do so, we prove a global Carleman estimate for the coupled system, making explicit the dependence on the damping parameter. In the course of the study, we demonstrate several inequalities for the beam equation alone, valid for different damping parameter regimes.

ABSTRACT We investigate the asymptotic behavior of solutions to the defocusing energy‐critical complex Ginzburg‐Landau equation on exterior domains and hyperbolic spaces. Employing the energy method, we establish a rigorous convergence theory for the zero‐dispersion limit from the energy‐critical complex Ginzburg‐Landau equation to the energy‐critical nonlinear heat equation. Moreover, we derive the inviscid limit connecting the energy‐critical complex Ginzburg‐Landau equation to the energy‐critical nonlinear Schrödinger equation.

We study the spectral inequalities of Schrödinger operators in the whole space for different potentials, which can be polynomial type growth or bounded potentials. The spectral inequalities quantitatively depend on the density of the sensor sets with positive measure, growth rate of the potentials and spectrum (or eigenvalues). One important component in the proof is the adaptation of propagation of smallness argument for gradients by Logunov and Malinnikova [ Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited Lectures , World Sci. Publ., Hackensack, NJ, pp. 2391–2411]. As an application, we apply the spectral inequalities to obtain quantitative observability inequalities for heat equations.

The existence and uniqueness for a heat equation with fractional Brownian moving boundaries and Volterra-type Dirichlet boundary term

Wire Arc Additive Manufacturing (WAAM) is a promising technology for producing large, high-performance metallic components, though process stability and geometric control remain critical challenges. The present work investigates the influence of deposition trajectory on the geometry, surface quality, and microhardness of wire arc additive manufactured Inconel 625 walls, produced by a Cold Metal Transfer (CMT) process. A conventional linear double-pass strategy is directly compared with a single-pass triangular weave trajectory under equivalent heat input per unit length, in order to isolate the effect of the torch path from other process variables. Single-layer and multi-layer walls were fabricated and characterized in terms of geometry, dimensional stability, surface waviness, and Vickers microhardness. The results show that the weave trajectory leads to improved geometric consistency, reduced variability, and significantly lower surface waviness compared to the linear strategy, while maintaining comparable mean wall width and microhardness. These findings demonstrate that appropriate trajectory design can enhance geometric stability and near-net-shape capability in WAAM-CMT without altering thermal input or material properties.

In this paper, an initial-boundary value problem for an inhomogeneous heat equation with a piecewise constant argument and Dirichlet boundary conditions is considered. The Fourier method is used to investigate the problem. By expanding the solution in terms of eigenfunctions, the initial-boundary value problem is reduced to the Cauchy problem for an ordinary differential equation with respect to the expansion coefficients with a piecewise continuous argument. The existence and uniqueness of the solution to this problem are proved. As a result, it is shown that the original problem has a unique solution, which is constructed in explicit form.

Global well-posedness for semilinear heat equations with linear or nonlinear boundary dissipation

Convergence analysis of a PINNs-based approach to the inverse source problem of the heat equation with local measurements

Existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term

The aim of this work is to show how we can transform a couple of problems involving equations that describe static processes into dynamic problems with second derivatives with respect to the evolution variable. To demonstrate this effect, we will need to work with the double Laplacian operator. Specifically, we will apply these arguments to the study of a static heat equation of the second gradient type and to the study of thin thermoelastic plates. In both cases, we will demonstrate that we obtain a dissipative analytic semigroup and, consequently, the exponential decay of the solutions. It is worth noting that, through this approximation, we can work with solutions having weaker regularity than the classical solutions, which are the usual type of solutions considered when studying the spatial decay of solutions. Therefore, our approximation also proposes an innovation from this point of view.

Semilinear stochastic heat equation with piecewise constant coefficients: Power variations and parameter estimation

Abstract We consider the quasi-linear stochastic wave and heat equations in $${\mathbb {R}}^d$$ R d with $$d\in \{1,2,3\}$$ d ∈ { 1 , 2 , 3 } and $$d\ge 1$$ d ≥ 1 , respectively, and perturbed by an additive Gaussian noise which is white in time and has a homogeneous spatial correlation with spectral measure $$\mu _n$$ μ n . We allow the Fourier transform of $$\mu _n$$ μ n to be a genuine distribution. Let $$u^n$$ u n be the mild solution to these equations. We provide sufficient conditions on the measures $$\mu _n$$ μ n and the initial data to ensure that $$u^n$$ u n converges in law, in the space of continuous functions, to the solution of our equations driven by a noise with spectral measure $$\mu $$ μ , where $$\mu _n\rightarrow \mu $$ μ n → μ in some sense. We apply our main result to various types of noises, such as the anisotropic fractional noise. We also show that we cover existing results in the literature, such as the case of Riesz kernels and the fractional noise with $$d=1$$ d = 1 .

In this paper, we introduce a novel methodology designed to tackle both domain and boundary disturbances in an unstable stochastic nonlinear heat equation. Active disturbance rejection control (ADRC) has been employed to achieve stabilisation and disturbance rejection in stochastic nonlinear heat equations that are subjected to unknown boundary stochastic disturbances. Firstly, the backstepping transformation is employed to suppress or compensate for the deterministic unstable term present within the domain. Subsequently, by leveraging an averaging signal, the stochastic partial differential equation is converted into a stochastic ordinary differential equation. Following this, the Lyapunov method is applied to elucidate the relationship between the multiplicative noise intensity, induced by white noise within the domain, and the constant incorporated in the backstepping transformation. Moreover, the ADRC method is harnessed to estimate and eliminate unknown boundary disturbances in real-time. Finally, numerical simulations are conducted to validate the effectiveness of the proposed control strategy.

Li-Yau inequality and Liouville property to a semilinear heat equation on Riemannian manifolds