A general theory of heat conduction with finite wave speeds

Gurtin, Morton E.; Pipkin, A. C.

doi:10.1007/bf00281373

Cited by 1,091 publications

(600 citation statements)

References 16 publications

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“…For k 1 = 0, Gurtin and Pipkin [25] introduced the equation as a model. Integrals of this type for E = 0 are used in the Boltzman theory of linear viscoelasticity to express the present value of stress in terms of past values of strain.…”

Section: Heat Conduction In a Solidmentioning

confidence: 99%

A general class of evolutionary equations

Hale

2007

Ukr Math J

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Using observable quantities and state variable of a dynamical process, a general evolutionary equation is defined which unifies classical ordinary differential equations, partial differential equations, and hereditary systems of retarded and neutral type. Specific illustrations are given using transmission lines nearest neighbor coupled at the boundary and the theory of heat transfer in solids. Some spectral theory for linearization of the equations also is discussed.Za dopomohog spostereΩuvanyx velyçyn ta zminno] stanu dynamiçnoho procesu vyznaçeno zahal\ne evolgcijne rivnqnnq, wo uzahal\ng[ klasyçni zvyçajni dyferencial\ni rivnqnnq, dyferencial\ni rivnqnnq z çastynnymy poxidnymy ta spadkovi systemy iz zapiznennqm i systemy nejtral\noho typu. Navedeno specyfiçni ilgstraci] z vykorystannqm linij transmisi] iz zçeplennqm ,,najblyΩçyx susidiv" na meΩi ta teori] teploperenosu u tverdyx tilax. Rozhlqnuto takoΩ pevnu spektral\nu teorig dlq linearyzaci] rivnqn\. Introduction.Motivated by the fact that a dynamical system may evolve through an observable quantity rather than the state of the system, a general class of evolutionary equations is defined. This class includes standard ordinary and partial differential equations as well as functional differential equations of retarded and neutral type. In this way, the theory serves as a unification of these classical problems.Included in this general formulation is a general theory for the evolution of temperature in a solid material. In the general case, temperature is transmitted as waves with a finite speed of propagation. Special cases include a theory of delayed diffusion.We describe also in some detail a lattice on a circle where each point on the lattice is a transmission line for current and voltage whose dynamics is governed by a linear hyperbolic equation on [0, 1] with dynamic boundary conditons given by the circuitry on the line. The systems are coupled to their nearest neighbor at the end point 1 through resistors. A limiting process letting the distance between the lattice points approach zero leads to an interesting set of partial differential equations on [0, 1] × S 1 with a hyperbolic equation on [0, 1] and a parabolic equation on S 1 . These equations have not been analyzed in detail. However, we can show that the voltage at 1 satisfies a partial neutral functional differential equation. We analyze some properties of these equations including synchronization and the behavior of solutins near periodic orbits.There are other applications which involve partial differential equations on lattices for which the dynamics on each lattice point is governed by a partial differential equation on a bounded domain Ω. These systems could be coupled to neighbors through interaction on some subset of the boundary ∂Ω of Ω. If certain limiting processes are justified, one can obtain a partial differential equation on Ω together with another partial differential equation on another domain Ω 1 (determined by the nature of the lattice). We do not discuss this in the text, but ...

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Section: Heat Conduction In a Solidmentioning

confidence: 99%

A general class of evolutionary equations

Hale

2007

Ukr Math J

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“…Another possibility would be to consider an integral expression for the heat flux in a medium with memory. 38 The heat flux vector, however, is known to vanish with ٌT ͑Fourier's Law͒. Thus expanding the flux J T in ٌT and disregarding terms of the order higher than the first one we obtain the regular expression for the heat flux, J T ϭϪ ٌT, where the thermal conductivity may be a function of T and .…”

Section: ͑30͒mentioning

confidence: 99%

Thermal effects in dynamics of interfaces

Umantsev

2002

The Journal of Chemical Physics

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Dynamical Ginzburg-Landau theory is applied to the study of thermal effects of motion of interfaces that appear after different phase transitions. These effects stem from the existence of the surface thermodynamic properties and temperature gradients in the interfacial transition region. Thermal effects may be explained by the introduction of a new thermodynamic force exerted on the interface, called here Gibbs-Duhem force, and the internal energy density flux through the interface. The evolution equations for the interfacial motion are derived. For the experimental verification of the thermal effects during continuous ordering the expression is derived for the amplitude of temperature waves.

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“…Indeed, let ϑ : Ω× R + → R be the absolute temperature of the conductor, ϑ 0 ∈ R + the uniform equilibrium temperature and u(x, t) = ϑ(x, t) − ϑ 0 ϑ 0 the temperature variation field relative to the equilibrium reference value. According to the well-established theory due to Gurtin and Pipkin [9], we consider only small variations of the absolute temperature and the temperature gradient from equilibrium, namely, |u| 1 and 1 ϑ 0 |∇ϑ| = |∇u| 1 and we suppose that the internal energy e : Ω × R → R and the heat flux vector q : Ω × R → R 3 are described by the following linear constitutive equations:…”

Section: K(σ)∆u(x T − σ) Dσ + G(u(x T)) = F (X T)mentioning

confidence: 99%

Asymptotic behavior of a nonlinear hyperbolic heat equation with memory

Giorgi

Pata

2001

NoDEA, Nonlinear differ. equ. appl.

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Abstract. In this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of an integro-differential equation describing the heat flow in a rigid heat conductor with memory. This model arises matching the energy balance, in presence of a nonlinear time-dependent heat source, with a linearized heat flux law of the Gurtin-Pipkin type. Existence and uniqueness of solutions for the corresponding semilinear system (subject to initial history and Dirichlet boundary conditions) is provided. Moreover, under proper assumptions on the heat flux memory kernel and the magnitude of nonlinearity, the existence of a uniform absorbing set is achieved.

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A general theory of heat conduction with finite wave speeds

Cited by 1,091 publications

References 16 publications

A general class of evolutionary equations

A general class of evolutionary equations

Thermal effects in dynamics of interfaces

Asymptotic behavior of a nonlinear hyperbolic heat equation with memory

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