Computers & mathematics with applications

Vol. 77, num. 2, p. 407-426

DOI: 10.1016/j.camwa.2018.09.044

Date of publication: 2018-10-11

Abstract:

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 solutions]]>

Computers & mathematics with applications

Vol. 71, num. 4, p. 991-1009

DOI: 10.1016/j.camwa.2016.01.010

Date of publication: 2016-02-06

Abstract:

We study a system modeling thermomechanical deformations for mixtures of thermoelastic solids with two different temperatures, that is, when each component of the mixture has its own temperature. In particular, we investigate the asymptotic behavior of the related solutions. We prove the exponential stability of solutions for a generic class of materials. In case of the coupling matrix View the MathML source being singular, we find that in general the corresponding semigroup is not exponentially stable. In this case we obtain that the corresponding solution decays polynomially as t-1/2 in case of Neumann boundary condition. Additionally, we show that the rate of decay is optimal. For Dirichlet boundary condition, we prove that the rate of decay is t-1/6. Finally, we demonstrate the impossibility of time-localization of solutions in case that two coefficients (related with the thermal conductivity constants) agree.]]>

Computers & mathematics with applications

Vol. 66, num. 1, p. 41-55

DOI: 10.1016/j.camwa.2013.03.022

Date of publication: 2013-08-01

Abstract:

We study a PDE system modeling thermomechanical deformations for a mixture of thermoelastic solids. In particular we investigate the asymptotic behavior of the solutions. First, we identify conditions on the constitutive coefficients to guarantee that the imaginary axis is contained in the resolvent. Subsequently, we find the necessary and sufficient conditions to guarantee the exponential decay of solutions. When the decay is not of exponential type, we prove that the solutions decay polynomially and we find the optimal polynomial decay rate.]]>