Hyperbolic Heat Equation Research Articles

Additional Conditions in Boundary Value Problems of Heat Conduction (Review)

A review of studies related to the use of additional boundary conditions (ADBs) and additional sought functions (ADFs) in obtaining analytical solutions to heat conduction problems is presented. ADВs allow the equation to be executed at the boundaries, which leads to its execution inside the domain, excluding direct integration over the spatial coordinate. ADF allows one to reduce a partial differential equation to an ordinary differential equation, from the solution of which the eigenvalues of the boundary value problem are found. Eigenvalues in classical methods are found from the solution of the Sturm–Liouville boundary value problem formulated in the domain of a spatial variable. Consequently, the method used in this work leads to another algorithm for their determination, based on the solution of a temporary differential equation, the order of which is determined by the number of approximations of the resulting solution. In a problem based on determining the front of a temperature disturbance, the equivalence of solutions to the parabolic and hyperbolic heat equations was found. And, in particular, a number of approximations have been found that limit the speed of propagation of a thermal wave in the solution of a parabolic equation to a value equal to its real value for a specific material, at which it coincides with the solution of the hyperbolic equation.

Coupled Axisymmetric Thermoelectroelasticity Problem for a Round Rigidly Fixed Plate

Introduction. To describe the operation of temperature piezoceramic structures, the theory of thermoelectroelasticity is used, in which the mathematical model is formulated as a system of nonself-adjoint differential equations. The complexity of its integration in general leads to the study of problems in an unrelated formulation. This does not allow us to evaluate the effect of electroelastic fields on temperature. The literature does not present studies on these problems in a three-dimensional coupled formulation in which closed solutions would be constructed. At the same time, conducting such studies allows us to understand the interaction picture of mechanical, thermal and electric fields in a structure. To solve this problem, a new closed solution of a coupled problem for a piezoceramic round rigidly fixed plate has been constructed in this research. It provides for qualitative assessment of the cross impact of thermoelectroelastic fields in this electroelastic system.Materials and Methods. The object of the study is a piezoceramic plate. The case of unsteady temperature change on its upper front surface is considered, taking into account the convection heat exchange of the lower plane with the environment (boundary conditions of the 1st and 3rd kind). The electric field induced as a result of the thermal strain generation is fixed by connecting the electrodated surfaces to the measuring device. The thermoelectroelasticity problem includes the equations of equilibrium, electrostatics, and the unsteady hyperbolic heat equation. It is solved by the generalized method of finite biorthogonal transformation, which makes it possible to construct a closed solution of a nonself-adjoint system of equations.Results. A new closed solution of the coupled axisymmetric thermoelectroelasticity problem for a round plate made of piezoceramic material was constructed.Discussion and Conclusion. The obtained solution to the initial boundary value problem made it possible to determine the temperature, electric and elastic fields induced in a piezoceramic element under arbitrary temperature axisymmetric external action. The calculations performed provided determining the dimensions of solid electrodes, which made it possible to increase the functionality of piezoceramic transducers. Numerical analysis of the results enabled us to identify new connections between the nature of external temperature action, the deformation process, and the value of the electric field in a piezoceramic structure. This can validate a proper program of experiments under their designing and significantly reduce the volume of field studies.

Discrete heat conduction equation: Dispersion analysis and continuous limits

The dispersion relation in ω-k space has been derived and analyzed for discrete heat conduction equation (DE). The DE is inherently nonlocal both in time and space and can be used to describe non-Fourier heat transport occurring on ultrashort time and length scales. An important distinction between the DE and the continuum description is that the frequencies and wave numbers available to the discrete system are limited, whereas in the continuum description there are no such limitations. Analytical expressions for the attenuation distance, group and phase velocities have been obtained and analyzed as functions of frequency. In the continuum limit, depending on the basic invariant of the continualization procedure, the dispersion relation for the DE reduces to either dispersion relation for the classical Fourier parabolic equation (PE) or to dispersion relation for the hyperbolic heat equation (HE). However, at the moderate and high frequencies the spectrum of the DE differs significantly from the spectra of the PE and HE. This implies that the continuum-based approaches, including both the PE and HE, unable to describe high-frequency and short-wavelength processes and the DE is preferable. This work provides a relatively simple analytical tool to interpret the transient wave-like behavior of heat transport on ultrashort space and time scales.

Utilization of Cattaneo–Christov theory to study heat transfer in Powell–Eyring fluid of hyperbolic heat equation

In this article, the two-dimensional, steady flow, and incompressible flow of Powell–Eyring fluid (non-Newtonian fluid) is considered toward a stretching surface. Applications of this emerging challenge include heating meals, grilling, baking, roasting, treating wounds, inflammation, and injuries with thermotherapy and cryotherapy, as well as steam turbines, industrial operations, casting, smelting, cooling aviation engines, and other things. Motion of particles is induced using variable wall velocity involving constant magnetic field. The characteristics of Lorentz force (magnetic field) and stagnation point flow are analyzed while direction of Lorentz is implemented toward y-direction of a bidirectional stretching surface under consideration of low magnetic Reynolds parameter. The phenomena regarding transfer of thermal energy have been addressed subject to thermal radiation, non-Fourier’s law and Joule heating. The PD-equations (partial differential equations) associated with heat transfer and flow are re-framed in forms of OD-equations (ordinary differential equations) using similarity variables. The influences based on various parameters against variation of motion and temperature field are analyzed in tables and graphical forms. The features of divergent velocity and temperature gradient are tabulated in Table 1. It is noted that temperature field is decreased against argument values of the Prandtl number whereas the same impact is noticed in thickness regarding thermal layers. The fluid parameter brings maximum acceleration into motion of fluid particles but the opposite trend is noticed in motion of fluid particles versus enhancement of Ratio of expansion rates number and Powell–Eyring fluid parameter. Table 1. Wall stresses and temperature gradient for different values of and when and HAM [37] Numerical [37] Present HAM [37] Numerical [37] Present

Wave propagation in thermo-poroelasticity: A finite-element approach

We have developed continuous and discrete-time finite-element (FE) methods to solve an initial boundary-value problem for the thermo-poroelasticity wave equation based on the combined Biot/Lord-Shulman (LS) theories to describe the porous and thermal effects, respectively. In particular, the LS model, which includes a Maxwell-Vernotte-Cattaneo relaxation term, leads to a hyperbolic heat equation, thus avoiding infinite signal velocities. The FE methods are formulated on a bounded domain with absorbing boundary conditions at the artificial boundaries. The dynamical equations predict four propagation modes, a fast P (P1) wave, a Biot slow (P2) wave, a thermal (T) wave, and a shear (S) wave. The spatial discretization uses globally continuous bilinear polynomials to represent solid displacements and temperature, whereas the vector part of the Raviart-Thomas-Nedelec of zero order is used to represent fluid displacements. First, a priori optimal error estimates are derived for the continuous-time FE method, and then an explicit conditionally stable discrete-time FE method is defined and analyzed. The explicit FE algorithm is implemented in one dimension to analyze the behavior of the P1, P2, and T waves. The algorithms can be useful for a better understanding of seismic waves in hydrocarbon reservoirs and crustal rocks, whose description is mainly based on the assumption of isothermal wave propagation.

Wave propagation in double-porosity thermoelastic media

The Lord-Shulman thermoporoelasticity theory couples the Biot and hyperbolic heat equations to describe wave propagation, modeling explicitly the effects of heat and fluid flows. We have extended the theory to the case of double porosity by taking into account the local heat flow (LHF) and local fluid flow (LFF) due to wave propagation. The plane-wave analysis finds the presence of the classical P and S waves and three slow P waves, namely, the slow (Biot) P1, the slow (Biot) P2, and a thermal slow P wave (or T wave). The frequency-dependent attenuation curves find that these slow waves manifest as Zener-like relaxation peaks, which are loosely related to the LFF, Biot, and LHF loss mechanisms. The viscosity and thermoelasticity properties can lead to the diffusive behavior of the three slow P modes. However, the S wave is considered to be independent of fluid and temperature influence. We examine the effect of thermophysical properties (e.g., thermal conductivity and relaxation time) on the wave velocity and attenuation of different modes. It is confirmed that the T wave is prone to be observed in media with high thermal conductivities and high homogeneity at high frequencies. Our double-porosity thermoelastic model reasonably explains laboratory measurements and well-log data from ultradeep fractured carbonates at high temperatures.

Anomalous thermal diffusion in two-layer system: The temperature profile and photoacoustic signal for rear light incidence

We analyze the influence of the generalizations of hyperbolic (Cattaneo) heat equations (GCE) on temperature profile and photoacoustic signal for the two-layer samples. The GCE extensions employ fractional-order derivatives related to the anomalous thermal diffusion in solid. We compare the superdiffusive (GCEII) and subdiffusive (GCEIII) generalizations with the classical and hyperbolic results. To show the fractional-order influence on results, we analyze opaque layers of aluminum with dye by considering only the last one with the anomalous characteristic. The promising results show that minor variations in fractional order and relaxation time can influence the photoacoustic signal.

Stability estimate for an inverse problem of a hyperbolic heat equation from boundary measurement

Stability estimate for an inverse problem of a hyperbolic heat equation from boundary measurement

Numerical modeling of laser photothermal heating in organic and inorganic structures

In this paper, we discuss flaws with static heating models as well as flaws with diffusive transport models such as the Fourier-Biot equation, the Cattaneo-Vernotte thermal wave model, and the hyperbolic heat equation and how the Dual-Phase-Lag (DPL) heat transfer model corrects for these flaws. We present novel numerical solutions and coding methodology in detail for solutions of the DPL heat equations, which take into account nanoscale heating effects such as phonon and electron interactions, for both organic and inorganic media for the laser heating of various structures and boundary conditions. We then use the models presented to validate our results against the experimental and theoretical results of various others for time scales ranging from 100 fs to 100 s and from spatial scales all the way from macroscale down to nanometers in both organic and inorganic media and systems of different kinds to show the validity of not only the DPL equation but also our solution methodology to a wide variety of problems.

Asymptotic preserving schemes on conical unstructured 2D meshes

AbstractIn this article, we consider the hyperbolic heat equation. This system is linear hyperbolic with stiff source terms and satisfies a diffusion limit. Some finite volume numerical schemes have been proposed which reproduce this diffusion limit. Here, we extend such schemes, originally defined on polygonal meshes, to conical meshes (using rational quadratic Bezier curves). We obtain really new schemes that do not reduce to the polygonal version when the conical edges tend to straight lines. Moreover, these schemes can handle curved unstructured meshes so that geometric error on initial data representation is reduced and geometry of the domain is improved. Extra flux coming from conical edge (through his midedge point) has a deep impact on the stabilization when compared with the original polygonal scheme. Cross‐stencil phenomenon of polygonal scheme has disappeared, and issue of positivity for the diffusion problem (although unresolved on distorted mesh and/or with varying cross‐section) has been in some sense improved.

Green’s function for the anisotropic hyperbolic heat equation in bounded spatial and temporal domains

Green’s function for the anisotropic hyperbolic heat equation in bounded spatial and temporal domains

Solution method for the time‐fractional hyperbolic heat equation

In this article, we propose a method to solve the time‐fractional hyperbolic heat equation. We first formulate a boundary value problem for the standard hyperbolic heat equation in a finite domain and provide an analytical solution by means of separation of variables and Fourier series. Then, we consider the same boundary value problem for the fractional hyperbolic heat equation. The fractional problem is solved using three different definitions of the fractional derivative: the Caputo fractional derivative and two recently defined alternative versions of this derivative, the Caputo–Fabrizio and the Atangana–Baleanu. A closed form of the solution is provided for each case. Finally, we compare the solutions of the fractional and the standard problem and show numerically that the solution of the standard hyperbolic heat equation can be retrieved from the solution of the fractional equation in the limit γ→2, where γ represents the exponent of the fractional derivative.

The relativistic heat equation on flat Friedman spacetime

The exact solutions of the relativistic hyperbolic heat equation on a flat Friedman spacetime metric are obtained. These solutions are obtained in three cases of flat Friedman model which correspond to radiation, matter and dark energy-dominated universe models. In particular, in the dark energy-dominated universe case, a closed form solution is obtained for which, using appropriate boundary conditions, an interesting exact solution is derived. This solution suggests that the temperature of the universe drops as it expands by a scale of an exponential function.

Green’s function for the one-dimensional hyperbolic heat equation: remarks for global regularity

We compute the Green’s function for the Dirichlet problem associated with the hyperbolic heat equation in the spatial interval [0, 1]. This result is used as a counterexample about the global regularity of the solutions of the hyperbolic heat equation, showing that its Green’s functions need not be continuous in all points of their domain. We apply this result to solve some mixed boundary value problems in hyperbolic heat theory.

The Specification Property for $$C_0$$-Semigroups

We study one of the strongest versions of chaos for continuous dynamical systems, namely the specification property. We extend the definition of specification property for operators on a Banach space to strongly continuous one-parameter semigroups of operators, that is, $$C_0$$ -semigroups. In addition, we study the relationships of the specification property for $$C_0$$ -semigroups (SgSP) with other dynamical properties: mixing, Devaney’s chaos, distributional chaos, and frequent hypercyclicity. Concerning the applications, we provide several examples of semigroups which exhibit the SgSP with particular interest on solution semigroups to certain linear PDEs, which range from the hyperbolic heat equation to the Black–Scholes equation.

Change of entropy for the one-dimensional ballistic heat equation: Sinusoidal initial perturbation.

This work presents a thermodynamic analysis of the ballistic heat equation from two viewpoints: classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT). A formula for calculating the entropy within the framework of EIT for the ballistic heat equation is derived. The entropy is calculated for a sinusoidal initial temperature perturbation by using both approaches. The results obtained from CIT show that the entropy is a non-monotonic function and that the entropy production can be negative. The results obtained for EIT show that the entropy is a monotonic function and that the entropy production is nonnegative. A comparison between the entropy behaviors predicted for the ballistic, for the ordinary Fourier-based, and for the hyperbolic heat equation is made. A crucial difference of the asymptotic behavior of the entropy for the ballistic heat equation is shown. It is argued that mathematical time reversibility of the partial differential ballistic heat equation is not consistent with its physical irreversibility. The processes described by the ballistic heat equation are irreversible because of the entropy increase.

Hyperbolic heat/mass transport and stochastic modelling - Three simple problems

Starting from the correspondence between the Cattaneo hyperbolic heat equation and the stochastic formulation based on Poisson-Kac processes, that holds solely for one-dimensional spatial models, this article analyzes three paradigmatic problems in the hyperbolic theory of heat and mass transport. The problems considered involve unbounded, semi-bounded and bounded domains, and are aimed at : (i) highlighting analogies and differences between the two approaches (Cattaneo vs Poisson-Kac), (ii) addressing the role of a bounded propagation velocity in order to regularize the properties of the solutions of heat/mass transport problems. A typical example of the latter phenomenology is expressed by boundary-layer regulatization of interfacial fluxes. The case of transport in bounded domains permits to pinpoint unambiguously the need of a stochastic interpretation of the transport equation in order to unveil the occurrence of physical inconsistencies that may occur in the linear Cattaneo hyperbolic model in some range of parameter values.

Comparison of approximate solutions to the phonon Boltzmann transport equation with the relaxation time approximation: Spherical harmonics expansions and the discrete ordinates method

The phonon Boltzmann transport equation is used to analyze model problems in one and two spatial dimensions, under transient and steady-state conditions. New, explicit solutions are obtained by using the P1 and P3 approximations, based on expansions in spherical harmonics, and are compared with solutions from the discrete ordinates method. For steady-state energy transfer, it is shown that analytic expressions derived using the P1 and P3 approximations agree quantitatively with the discrete ordinates method, in some cases for large Knudsen numbers, and always for Knudsen numbers less than unity. However, for time-dependent energy transfer, the PN solutions differ qualitatively from converged solutions obtained by the discrete ordinates method. Although they correctly capture the wave-like behavior of energy transfer at short times, the P1 and P3 approximations rely on one or two wave velocities, respectively, yielding abrupt, step-changes in temperature profiles that are absent when the angular dependence of the phonon velocities is captured more completely. It is shown that, with the gray approximation, the P1 approximation is formally equivalent to the so-called “hyperbolic heat equation.” Overall, these results support the use of the PN approximation to find solutions to the phonon Boltzmann transport equation for steady-state conditions. Such solutions can be useful in the design and analysis of devices that involve heat transfer at nanometer length scales, where continuum-scale approaches become inaccurate.

Comparative assessment of deterministic approaches to modeling quasi-ballistic phonon heat conduction in multi-dimensional geometry

When the mean free path of the dominant energy carriers in semiconductor materials, namely phonons, is comparable to or larger than the characteristic length scale, scattering of phonons is rare, and the equilibrium Bose-Einstein distribution is not restored locally. Four different models for modeling non-equilibrium (or quasi-ballistic) heat transport in multi-dimensional geometry are critically assessed in this study. These include (1) the hyperbolic heat equation (HHCE), (2) the first-order method of spherical harmonics or P1 approximation (P1), (3) the frequency-dependent phonon Boltzmann Transport Equation (BTE), and (4) a hybrid approach in which the BTE is hybridized with the P1 approximation (Hybrid). In addition, the Fourier law (Fourier) is also considered to assess its regime of validity. A two-dimensional sub-micron heat conduction problem is solved using all of the aforementioned models, and their accuracy and efficiency are assessed. The full BTE solution with high spatial and angular resolution is considered the benchmark solution for comparison. Both steady and unsteady computations are conducted. It is found that at steady state, the accuracy of the models go hand in hand with the degree of diffuse approximation incorporated into the model. At low temperature, where the degree of non-equilibrium (ballistic nature) is dominant, the Fourier model completely fails. For transient computations, at short times, all models other than the BTE exhibit significant error. Also, the HHCE is found to be significantly superior to the Fourier model at short times.

Existence and uniqueness of solutions of Robin's problem for the anisotropic hyperbolic heat equation with non regular data

Existence and uniqueness of solutions of Robin's problem for the anisotropic hyperbolic heat equation with non regular data