On the regularity and stability of the dual-phase-lag equation

In this note we consider the following linear partial different equation which is usually seen as an approximation to the dual-phase-lag heat equation proposed by Tzou. \dot {T}+\tau_q \ddot {T}+\frac{\tau_{q}^{2}}{2}\dddot{T}=\kappa \triangle T+\kappa\tau_{T}\triangle \dot {T}+ \kappa \frac{\tau_{T}^{2}}{2}\triangle \ddot {T} on a bounded domain $\Omega$ in $R^n$ with smooth boundary. We obtain analyticity for the associated $C_0-$semigroup. Moreover, we also obtain exponential stability of the solutions by spectrum analysis and Hurwitz criterion under one of the following conditions: (1) $\displaystyle{\frac{\tau_{T}}{\tau_{q}}>2-\sqrt{3}};$ (2) $2-\sqrt{3} \ge \displaystyle{\frac{\tau_{T}}{\tau_{q}} >\frac{\sqrt{(1+\kappa\tau_{T}\lambda_1)^{2}+(\kappa\tau_{T}\lambda_1)^{2}+(\kappa\tau_{T}\lambda_1)^{3}}-(1+\kappa\tau_{T}\lambda_1)}{\kappa\tau_{T}\lambda_{1}(1+\kappa\tau_{T}\lambda_1)}},$ where $\lambda_1$ is the smallest eigenvalue of the negative Laplacian on $\Omega$ with Dirichlet boundary condition.Peer ReviewedPostprint (author's final draft

Numerical resolution of an exact heat conduction model with a delay term

In this paper we analyze, from the numerical point of view, a dynamic thermoelastic problem. Here, the so-called exact heat conduction model with a delay term is used to obtain the heat evolution. Thus, the thermomechanical problem is written as a coupled system of partial differential equations, and its variational formulation leads to a system written in terms of the velocity and the temperature fields. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A priori error estimates are proved, from which the linear convergence of the algorithm could be derived under suitable additional regularity conditions. Finally, a two-dimensional numerical example is solved to show the accuracy of the approximation and the decay of the discrete energy.Peer ReviewedPostprint (published version

Numerical analysis of some dual-phase-lag models

In this paper we analyse, from the numerical point of view, two dual-phase-lag models appearing in the heat conduction theory. Both models are written as linear partial differential equations of third order in time. The variational formulations, written in terms of the thermal acceleration, lead to linear variational equations, for which existence and uniqueness results, and energy decay properties, are recalled. Then, fully discrete approximations are introduced for both models using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. Discrete stability properties are proved, and a priori error estimates are obtained, from which the linear convergence of the approximations is derived. Finally, some numerical simulations are described in one and two dimensions to demonstrate the accuracy of the approximations and the behaviour of the solutionsPeer ReviewedPostprint (author's final draft

Numerical analysis of a thermoelastic problem with dual-phase-lag heat conduction

In this paper we study, from the numerical point of view, a thermoelastic problem with dual-phase-lag heat conduction. The variational formulation is written as a coupled system of hyperbolic linear variational equations. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability result is proved and a priori error estimates are obtained, from which the linear convergence of the algorithm is deduced under suitable additional regularity conditions. Finally, some two-dimensional numerical simulations are presented to demonstrate the accuracy of the approximation and the behaviour of the solution.Peer ReviewedPostprint (author's final draft

Some inverse source problems of determining a space dependent source in fractional-dual-phase-lag type equations

The dual-phase-lag heat transfer models attract a lot of interest of researchers in the last few decades. These are used in problems arising from non-classical thermal models, which are based on a non-Fourier type law. We study uniqueness of solutions to some inverse source problems for fractional partial differential equations of the Dual-Phase-Lag type. The source term is supposed to be of the formh(t)f(x)with a known functionh(t). The unknown space dependent sourcef(x)is determined from the final time observation. New uniqueness results are formulated in Theorem 1 (for a general fractional Jeffrey-type model). Here, the variational approach was used. Theorem 2 derives uniqueness results under weaker assumptions onh(t)(monotonically increasing character ofh(t)was removed) in a case ofdominant parabolicbehavior. The proof technique was based on spectral analysis. Section Modified Model for tau q>tau Tshows that an analogy of Theorem 2 fordominant hyperbolicbehavior (fractional Cattaneo-Vernotte equation) is not possible

Numerical analysis of a dual-phase-lag model involving two temperatures

In this paper, we numerically analyse a phase-lag model with two temperatures which arises in the heat conduction theory. The model is written as a linear partial differential equation of third order in time. The variational formulation, written in terms of the thermal acceleration, leads to a linear variational equation, for which we recall an existence and uniqueness result and an energy decay property. Then, using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives, fully discrete approximations are introduced. A discrete stability property is proved, and a priori error estimates are obtained, from which the linear convergence of the approximation is derived. Finally, some one-dimensional numerical simulations are described to demonstrate the accuracy of the approximation and the behaviour of the solution.Peer ReviewedPostprint (author's final draft

Existence and uniqueness of a weak solution to fractional single-phase-lag heat equation

In this article, we study the existence and uniqueness of a weak solution to the fractional single-phase lag heat equation. This model contains the terms $\cal{D}_t^\alpha(u_t)$ and $\cal{D}_t^\alpha u$ (with $\alpha \in(0,1)$), where $\cal{D}_t^\alpha$ denotes the Caputo fractional derivative in time of constant order $\alpha\in(0,1)$. We consider homogeneous Dirichlet boundary data for the temperature. We rigorously show the existence of a unique weak solution under low regularity assumptions on the data. Our main strategy is to use the variational formulation and a semidiscretisation in time based on Rothe's method. We obtain a priori estimates on the discrete solutions and show convergence of the Rothe functions to a weak solution. The variational approach is employed to show the uniqueness of this weak solution to the problem. We also consider the one-dimensional problem and derive a representation formula for the solution. We establish bounds on this explicit solution and its time derivative by extending properties of the multinomial Mittag-Leffler function.Comment: 27 page

Universal far-from-equilibrium Dynamics of a Holographic Superconductor

Symmetry breaking phase transitions are an example of non-equilibrium processes that require real time treatment, a major challenge in strongly coupled systems without long-lived quasiparticles. Holographic duality provides such an approach by mapping strongly coupled field theories in D dimensions into weakly coupled quantum gravity in D+1 anti-de Sitter spacetime. Here, we use holographic duality to study formation of topological defects -- winding numbers -- in the course of a superconducting transition in a strongly coupled theory in a 1D ring. When the system undergoes the transition on a given quench time, the condensate builds up with a delay that can be deduced using the Kibble-Zurek mechanism from the quench time and the universality class of the theory, as determined from the quasinormal mode spectrum of the dual model. Typical winding numbers deposited in the ring exhibit a universal fractional power law dependence on the quench time, also predicted by the Kibble-Zurek Mechanism.Comment: 33 pages; 8 figure