The minimum free energy for an electromagnetic conductor

We consider the linear theory of the electromagnetism characterized by a constitutive equation for the current density with memory effects. We find in the frequency domain a first expression of the minimum free energy, which is the maximum recoverable work we can obtain from a given state of the material. By using another equivalent formulation of the minimum free energy, we give the explicit formulae for the particular case of a discrete spectrum model material response.     

The minimum free energy for incompressible viscoelastic fluids

A general closed expression is given in the frequency domain for the isothermal minimum free energy of an incompressible viscoelastic fluid, whose constitutive equation is expressed by a linear functional of the history of strain. Another equivalent form of the minimum free energy is also derived and used to study the particular case of a discrete model material response. Copyright © 2006 John Wiley & Sons, Ltd.     

On the asymptotic behaviour for an electromagnetic system with a dissipative boundary condition

In this work we study some properties of solutions for the systemdescribing a three-dimensional non-homogeneous non-conductingdielectric with a general boundary condition with memory. Wefirst show the existence of the inverse of this boundary condition,which allows us to introduce a boundary free energy, similarto the one considered by Fabrizio & Morro (1996, Arch. Rat.Mech. Anal., 136, 359–381). Then, we prove existence anduniqueness theorems for weak and strong solutions of the evolutiveproblem in a finite time interval. Moreover, following Rivera& Olivera (1997, Boll. U.M.I., 11-A, 115–127), weexamine some dissipative properties of the boundary conditionand of its inverse and we give a useful energy estimate. Finally,when there is no memory in the boundary condition the exponentialdecay of the solution is proved.     

Free energies and asymptotic behaviour for incompressible viscoelastic fluids

The existence, uniqueness and asymptotic stability for an incompressible, linear viscoelastic fluid is studied using a new free energy, the representation of which is based on the concept of a minimal state. A restriction imposed by thermodynamics is also used. Furthermore, an expression for the minimum free energy in the time domain is derived, which shows explicitly its dependence on the minimal state.     

Derivative free two-point methods with and without memory for solving nonlinear equations

Two families of derivative free two-point iterative methods for solving nonlinear equations are constructed. These methods use a suitable parametric function and an arbitrary real parameter. It is proved that the first family has the convergence order four requiring only three function evaluations per iteration. In this way it is demonstrated that the proposed family without memory supports the Kung-Traub hypothesis (1974) on the upper bound 2ⁿ of the order of multipoint methods based on n + 1 function evaluations. Further acceleration of the convergence rate is attained by varying a free parameter from step to step using information available from the previous step. This approach leads to a family of two-step self-accelerating methods with memory whose order of convergence is at least and even in special cases. The increase of convergence order is attained without any additional calculations so that the family of methods with memory possesses a very high computational efficiency. Numerical examples are included to demonstrate exceptional convergence speed of the proposed methods using only few function evaluations.     

The quasi-static problem for an electromagnetic conductor

Viene studiato il problema quasi-statico per un sistema elettromagnetico con una condizione al contorno dissipativa e con memoria. Si dimostra che se il nucleo di memoria soddisfa un'opportuna restrizione, traduzione in termini analitici della dissipatività locale della frontiera, allora il problema quasi-statico ammette una ed una sola soluzione debole.     

Symmetric minimum-norm updates for use in gibbs free energy calculations

This paper examines a type of symmetric quasi-Newton update for use in nonlinear optimization algorithms. The updates presented here impose additional properties on the Hessian approximations that do not result if the usual quasi-Newton updating schemes are applied to certain Gibbs free energy minimization problems. The updates derived in this paper are symmetric matrices that satisfy a given matrix equation and are least squares solutions to the secant equation. A general representation for this class of updates is given. The update in this class that has the minimum weighted Frobenius norm is also presented. This work was done at Sandia National Laboratories and supported by the US Dept. of Energy under contract no. DE-AC04-76DP00789.     

Complex bodies with memory: Linearized setting

The mechanics of complex bodies with memory effects is discussed in linearized setting. The attention is focused on the characterization of free energies in terms of minimum work and maximum recoverable work in the bulk and along a discontinuity surface endowed with its own surface energy, a surface within the body. To this aim, use is made of techniques proposed by Del Piero. Consequences of the Clausius–Duhem inequality are investigated for complex bodies with instantaneous linear elastic response. Copyright © 2008 John Wiley & Sons, Ltd.     

Asymptotic behaviour of the energy for electromagnetic systems with memory

Some linear evolution problems arising in the theory of hereditary electromagnetism are considered here. Making use of suitable Liapunov functionals, existence of solutions as well as asymptotic behaviour, are determined for rigid conductors with electric memory. In particular, we show the polynomially decay of the solutions, when the memory kernel decays exponentially or polynomially. Copyright © 2004 John Wiley & Sons, Ltd.     

Solutions for a nonlocal conservation law with fading memory

Global entropy solutions in for a scalar nonlocal conservation law with fading memory are constructed as the limits of vanishing viscosity approximate solutions. The uniqueness and stability of entropy solutions in are established, which also yield the existence of entropy solutions in while the initial data is only in . Moreover, if the memory kernel depends on a relaxation parameter and tends to a delta measure weakly as measures when , then the global entropy solution sequence in converges to an admissible solution in for the corresponding local conservation law.

     

Improved bi-accelerator derivative free with memory family for solving nonlinear equations

The object of the present paper is to accelerate the $R$-order convergence of with memory derivative free family given by Lotfi et al. (2014) without adding any new evaluations. To achieve this goal one more iterative parameter is introduced, which is calculated with the help of Newton''s interpolatory polynomial. It is shown that the $R$-order convergence of the proposed scheme is increased from 12 to 14 without any extra evaluation. Smooth as well as non-smooth examples are presented to confirm theoretical result and significance of the new scheme.     

Variational principles for electromagnetic systems with memory

?Saddle point? or ?mini-max? principles are set up for first order integrodifferential systems arising in the hereditary electromagnetic theory. Variational formulations are obtained directly in terms of electric and magnetic field vectors. In particular, these principles are shown to hold as a consequence of thermodynamic restrictions, when the quasi-static case and the linear evolution problem are considered. Entrata in Redazione il 19 febbraio 1999.     

Non-global existence for an Allen-Cahn-Gurtin equation with logarithmic free energy

We consider a simple Allen-Cahn-Gurtin model with a logarithmic potential. We show that strong solutions exist locally in time, but not globally in general. This is due to a lack of maximum principle.      

A well‐posed problem for electromagnetic media with quadrupoles

In this paper, we consider a non‐local electromagnetic medium for which the quadrupole term in the electric induction is not negligible. After giving an outline of the physical model, the problem of Maxwell equations for such a medium is addressed by proving existence and uniqueness of solutions and a principle of constrained minimum is shown to hold as a consequence of some thermodynamical restrictions. Copyright © 2003 John Wiley & Sons, Ltd.     

Random Attractor for the Nonclassical Diffusion Equation with Fading Memory

In this paper,we consider the stochastic nonclassical diffusion equationwith fading memory on a bounded domain. By decomposition of the solution operator, we give the necessary condition of asymptotic smoothness of the solution to the initial boundary value problem, and then we prove the existence of a random attractor in the space $M_1=D(A^{\frac{1}{2}}) × L^2_μ(R^+, D(A^{\frac{1}{2}}))$, where A=-Δ with Dirichlet boundary condition.     

Practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system with a logarithmic free energy

We propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system with a logarithmic free energy. We use Eyre's convex splitting scheme for the time discretization and a Fourier spectral method for the space variables. Given an absolute temperature, we find composition values that make the total free energy be minimum. Then, we find the splitting parameter value that makes the two split homogeneous free energies be convex on the neighborhood of the local minimum concentrations. For general use, we also propose a sixth‐order polynomial approximation of the minimum concentration and derive a useful formula for the practical estimation of the splitting parameter in terms of the absolute temperature. The numerical tests are phase separation and total energy decrease with different temperature values. The linear stability analysis shows a good agreement between the exact and numerical solutions with an optimal value s. Various computational experiments confirm that the proposed splitting parameter estimation gives stable numerical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Weak solutions for the Falk model system of shape memory alloys in energy class

We show the unique global existence of energy class solutions for the Falk model system of shape memory alloys under the general non‐linearity as well as considered in Aiki (Math. Meth. Appl. Sci. 2000; 23 : 299). Our main tools of the proofs are the Strichartz type estimate for the Boussinesq type equation and the maximal regularity estimate for the heat equation. Copyright © 2005 John Wiley & Sons, Ltd.     

Asymptotic stability in thermoelectromagnetism with memory

A model describing a linear homogeneous dielectric with memory which obeys the Cattaneo–Maxwell law for the heat conduction is presented. The restrictions on the constitutive functionals are found as a direct consequence of the Second Law of Thermodynamics and some free energy potentials exhibited. Such potentials allow to determine a domain of dependence theorem for the first‐order integro‐differential system of equations governing the evolution of the thermoelectromagnetic radiation. The dissipativity due to the memory and to the heat conduction allows to establish some estimates on the asymptotic behaviour and prove the exponential decay of the solution of the system in absence of external sources. Copyright © 1999 John Wiley & Sons, Ltd.     

Asymptotic behavior for coupled abstract evolution equations with one infinite memory

In this paper, we consider two coupled abstract linear evolution equations with one infinite memory acting on the first equation. Our work is motivated by the recent results of [42], where the authors considered the case of two wave equations with one convolution kernel converging exponentially to zero at infinity, and proved the lack of exponential decay. On the other hand, the authors of [42] proved that the solutions decay polynomially at infinity with a decay rate depending on the regularity of the initial data. Under a boundedness condition on the past history data, we prove that the stability of our abstract system holds for convolution kernels having much weaker decay rates than the exponential one. The general and precise decay estimate of solution we obtain depends on the growth of the convolution kernel at infinity, the regularity of the initial data, and the connection between the operators describing the considered equations. We also present various applications to some distributed coupled systems such as wave-wave, Petrovsky-Petrovsky, wave-Petrovsky, and elasticity-elasticity.