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A number of problems for the interaction of laser radiation with a heat-conducting half-space and a layer are considered. The obtained solutions are compared with each other and with the solutions of the classic heat equation and the wave equation. A laser impulse is modelled by defining the heat flux at the boundary for the opaque medium, or by defining the distribution of heat sources in the volume for the semitransparent medium. The power of the laser pulse depends on time as the Dirac delta function or as the Heaviside function do. It allows for the simulation of instant and continuous laser exposure on the medium. Temperature distributions are obtained by using Green’s functions for a half-space and a layer with the Dirichlet and Neumann boundary conditions.

PurposeIn this study, the purpose was to introduce two‐dimensional hyperbolic heat conduction equations in order to simulate the fast precooling process of a cylindrically shaped food product with internal heat generation. A modified model for internal heat generation due to respiration in the food product was proposed to take the effect of relaxation time into account. The obtained governing equations were solved numerically using an efficient finite difference technique. The influence of Biot number and heat generation parameters on thermal characteristics was examined and discussed. The results based on hyperbolic model were compared with the classical parabolic heat diffusion model. The present numerical code was validated via comparison with analytical solution and a good agreement was found.Design/methodology/approachThe obtained governing equations were solved numerically using an efficient finite difference technique.FindingsThe influence of Biot number and heat generation parameters on thermal characteristics was examined and discussed. The results based on hyperbolic model were compared with the classical parabolic heat diffusion model. The present numerical code was validated via comparison with analytical solution and a good agreement was found.Originality/valueTwo‐dimensional analysis of fast precooling of cylindrical food product based on hyperbolic heat conduction model has not been investigated yet.

The computability of the solution for the Cauchy Problems of the nonlinear Heat Conduction equation is studied in this paper. A nonlinear map is defined from the initial value to the solution . We used the relevant knowledge of type-2 theory of effectivity, functional analysis and Sobolev space to prove that when , the solution operator of the initial problem for the Heat Conduction equation is computable under certain conditions. The conclusion enriches the theories of computability.

In this paper, the transient temperature field around a partially insulated crack in a half-plane is obtained by applying the dual-phase-lag heat conduction model. Fourier and Laplace transforms are applied and the heat conduction problem is reduced to solving a singular integral equation. The asymptotic fields near the crack tip are obtained in an explicit form and the intensity factors of temperature gradients are determined. Numerical results show that temperature overshooting phenomenon may occur due to the thermal disturbance of crack based on the dual-phase-lag model. The dual-phase-lag model parameters, the heat conductivity of the crack surfaces and the geometric size of the half-plane have significant influence on the dynamic temperature fields.

The problem of structure formation in relativistic dissipative fluids was analyzed in a previous work within Eckart's framework, in which the heat flux is coupled to the hydrodynamic acceleration, additional to the usual temperature gradient term. It was shown that in such case, the pathological behavior of fluctuations leads to the disapperance of the gravitational instability responsible for structure formation. In the present work the problem is revisited now using a constitutive equation derived from relativistic kinetic theory. The new relation, in which the heat flux is not coupled to the hydrodynamic acceleration, leads to a consistent first order in the gradients formalism. In this case the gravitational instability remains, and only relativistic corrections to the Jeans wave number are obtained. In the calculation here shown the non-relativistc limit is recovered, opposite to what happens in Eckart's case., Comment: 10 pages, no figures

The time-nonlocal generalizations of Fourier’s law are analyzed and the equations of the generalized thermoelasticity based on the time-fractional heat conduction equation with the Caputo fractional derivative of order 0 < α ≤ 2 are presented. The equations of thermoelasticity of thin shells are obtained under the assumption of linear dependence of temperature on the coordinate normal to the median surface of a shell. The conditions of Newton’s convective heat exchange between a shell and the environment have been assumed. In the particular case of classical heat conduction (α = 1) the obtained equations coincide with those known in the literature.

In this article, the heat conduction problem of a sector of a finite hollow cylinder is studied as an exact solution approach. The governing equations are in the form of non-homogeneous partial differential equation (PDE) with non-homogeneous boundary conditions. In order to solve the PDE equation, generalized finite Hankel, periodic Fourier, Fourier and Laplace transforms are applied. Three different boundary conditions as case studies for simulations are presented and verified with the result which extracted from finite element method. The results are shown that this approach is suitable and systematic for solving heat conduction problems in cylindrical coordinate.

The problem of fractional heat conduction in a composite medium consisting of two semi-infinite regions being in perfect thermal contact is considered. The heat conduction in each region is described by the time-fractional heat conduction equations with the Caputo derivative of fractional order α and β, respectively. The solution is obtained using the Laplace transform with respect to time and is expressed in terms of the Mittag–Leffler function and Mainardi function. Numerical results are illustrated graphically.

This paper studies the fracture behavior of a thermoelastic cylinder subjected to a sudden temperature change on its outer surface within the framework of non-classical heat conduction. The heat conduction equation is solved by separation of variable technique. Closed form solution for the temperature field and the associated thermal stress are established. The critical parameter governing the level of the transient thermal stress is identified. Exact expression for the transient stress intensity factor is obtained for a crack in the cylinder. The difference between the non-classical solutions and the classical solution are discussed. It is found that the traditional classical heat conduction considerably underestimates the transient thermal stress and thermal stress intensity factor.

Non-Fourier models of heat conduction are increasingly being considered in the modeling of microscale heat transfer in engineering and biomedical heat transfer problems. The dual-phase-lagging model, incorporating time lags in the heat flux and the temperature gradient, and some of its particular cases and approximations, result in heat conduction modeling equations in the form of delayed or hyperbolic partial differential equations. In this work, the application of difference schemes for the numerical solution of lagging models of heat conduction is considered. Numerical schemes for some DPL approximations are developed, characterizing their properties of convergence and stability. Examples of numerical computations are included.

The classical heat equation is characterized both by the instantaneous propagation of a physical interaction and by the lack of a particle that carries the heat. Metals have a particular feature: their atoms form a lattice with electrons traveling through the solid. This paper provides a conduction heat model for metals. The model replaces atoms by potential barrier and assigns the heat transport to electrons and, in some cases, photons. To emulate the transitory solution of Schrodinger's equation, the potential barrier is fragmented in others so that, the electron is losing energy packets while interacting with the sub-barriers. With this approach, the energy distribution is obtained for electrons and, in addition, it is derived a velocity expression for the heat spread through a metal. Last but not least, the classical solution of the heat equation is obtained but now the electron energy limits the heat diffusion avoiding the infinite range of the classical solution.

Rapid processes of heat transfer are not described by the standard heat conduction equation. To take into account a finite velocity of heat transfer, we use the hyperbolic model of heat conduction, which is connected with the relaxation of heat fluxes. In this case, the mathematical model is based on a hyperbolic equation of second order or a system of equations for the temperature and heat fluxes. In this paper we construct for the hyperbolic heat conduction equation the additive schemes of splitting with respect to directions. Unconditional stability of locally one-dimensional splitting schemes is established. New splitting schemes are proposed and studied for a system of equations written in terms of the temperature and heat fluxes.

This contribution investigates the numerical solution of the steady-state heat conduction equation. The finite difference method is applied to simple formulations of heat sources where still analytical solutions can be derived. Thus, the results of the numerical approach can be related to the exact solutions and conclusions on the accuracy obtained. In addition, the numerical implementation of different forms of boundary conditions, i.e. temperature and flux condition, is compared to the exact solution. It is found that the numerical implementation of coordinate dependent sources gives the exact result while temperature dependent sources are only approximately represented. Furthermore, the implementation of the mentioned boundary conditions gives the same results as the analytical reference solution.

By introducing relaxation corrections to the heat flux and to the temperature gradient, hyperbolic equations of heat conduction are obtained involving the third and fourth derivatives over the spatial coordinate and time (mixed derivatives). The obtained formula for the heat flux that is used in obtaining the aforementioned hyperbolic equations coincides with the formula by Lykov, which is derived from the Onsager generalized system equations using the hypothesis about the finite diffusion rate of heat and mass. Investigations of the obtained analytic solutions to the hyperbolic equations allow us to conclude about the temperature discontinuities in the vicinity of the boundary of the space coordinate where the boundary condition of the first kind is given. This testifies that instantaneous heating (cooling) of the body up to the temperature of the environment is impossible in no circumstances of external heat exchange.

Three models leading to higher order heat conduction and diffusion equations are considered. A thermodynamic substantiation is suggested for constitutive equations in the framework of extended irreversible thermodynamics. Expressions for generalized temperature and generalized chemical potentials are presented, which are determined by the form of the constitutive laws describing the heat conduction and diffusion processes.

A method for numerical solution of time-domain boundary integral formulations of transient problems governed by the heat equation is presented. The heat conduction problem is analyzed considering homogeneous and non-homogeneous media. In the case of the non-homogeneous media, the conductor material is assumed to be a functionally graded material, i.e., the material properties vary spatially according to known smooth functions. For some specific spatial variations of the material properties, the fundamental solution and the boundary integral equation of the problem are obtained thanks to a change of variables that transforms the original problem to the standard heat conduction problem for homogeneous materials. For the treatment of time-dependent terms, the convolution quadrature method is adopted to approximate numerically the integral equation of the time-domain boundary element method. In the case that the responses are required at a large number of interior points, the convolution performed to calculate them is very time consuming. It is shown that the discrete convolution of the proposed formulation can be computed by means of the fast Fourier transform technique, which considerably reduces the computational complexity. Results for some transient heat conduction examples are presented to validate the numerical techniques studied.

This paper consist inverse thermoelastic problem of heat conduction with the application of partially distributed heat supply and internal heat generation. In this paper unknown temperature, displacement stress functions are determined by using finite Marchi-Zgrablich transform and Fourier cosine transform. The results are obtained in the form of Bessel's function and infinite series.

In this communication 1-D heat conduction in a fractal medium is solved by the local fractional Fourier series method. The solution developed allows relating the basic properties of the fractal medium to the local heat transfer mechanism.

Energy transport within a finite thin film subjected to either symmetric or asymmetric heating at the boundaries is investigated. Non-Gaussian heat sources are modeled as time-varying and spatially-decaying laser incidences. Comparison of heat transfer mechanisms based on the classical diffusion, Cattaneo–Vernotte, simplified thermomass, and dual-phase-lag models is undertaken. This study presents a single generalized analytical temperature profile in a finite thin film in terms of an infinite series, utilizing the superposition technique in conjunction with the solution structure theorems, which is found to be applicable to solutions of all the aforementioned models. The temperature solution for a particular model can be obtained directly by applying appropriate coefficients as they appear in the proposed generalized governing heat conduction equation. Through analyses and numerical examples, the time history of heat transfer behaviors due to the collision of energy from both sides of the thin film is compared among these models. It reveals that the method provides a convenient and efficient solution to the classical heat diffusion equation as well as other forms of hyperbolic heat conduction equations.

Taylor series numerical method (TSNM) is extended to the field of transient heat conduction. Theoretical description of TSNM for transient heat conduction problems is presented. Furthermore, the algorithm is realized and embedded in commercial software ANSYS®. If a lumped mass heat capacity matrix provided, the governing equation of transient heat conduction problems, which is a differential equation, will be solved by a series of recursion calculation of Taylor expanding coefficients. A typical transient heat conduction problem with analytical solution was discussed to verify the TSNM. At last, the TSNM is applied in the transient heat analysis of an all-solid fiber optic gyro (FOG).

In order to obtain the numerical solution of the heat conduction equation in one dimensional nanometer materials, we discrete the demand equation by Galerkin-Legendre spectral method and obtain a system of order ordinary differential equations, then we solve this system by using a fifth order five stage A-stable new diagonally implicit Runge-Kutta method. Numerical results are presented to demonstrate the accuracy and efficiency of this method.

a b s t r a c t We investigate the existence of pseudo almost periodic solutions for a class of integrodifferential equation with nonlocal initial conditions arising in the study of heat conduction in materials with memory. We first establish some existence theorems for abstract semilinear integro-differential equations with nonlocal initial conditions, and then, we apply our abstract results to the addressed integro-differential equation arising in the study of heat conduction in materials with memory.

The classical Fourier heat conduction law gives sufficient accuracy for many practical engineering applications. However, it cannot exactly reflect the real physical mechanism of heat conduction by highly-varying thermal load, at very low temperatures, or at nanoscale. The hyperbolic heat conduction equation can better explain heat conduction in solids. However, such equation is very difficult to solve. This paper studies the temperature field for a 1-D plate and a 1-D cylinder. The associated thermal stress for a 1-D plate is also given. Only in these simple situations, closed form solutions are possible and are given. The results can be used in the future analysis of thermal shock cracking and reliability analysis of materials in modern science and technology.

In this paper we propose a numerical algorithm based on the method of fundamental solutions for recovering a space-dependent heat source and the initial data simultaneously in an inverse heat conduction problem. The problem is transformed into a homogeneous backward-type inverse heat conduction problem and a Dirichlet boundary value problem for Poisson's equation. We use an improved method of fundamental solutions to solve the backward-type inverse heat conduction problem and apply the finite element method for solving the well-posed direct problem. The Tikhonov regularization method combined with the generalized cross validation rule for selecting a suitable regularization parameter is applied to obtain a stable regularized solution for the backward-type inverse heat conduction problem. Numerical experiments for four examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed algorithm.

This paper investigates a numerical solution of 2D transient heat conduction in an anisotropic cylinder, subjected to a prescribed temperature over the two end sections and a convective boundary condition over the whole lateral surface. The analysis of this anisotropic heat conduction problem is tedious because the corresponding partial differential equation contains a mixed-derivative. In order to overcome this difficulty, a linear coordinate transformation is used to reduce the anisotropic cylinder heat conduction problem to an equivalent isotropic one, without complicating the boundary conditions but with a more complicated geometry. The alternating-direction implicit finite-difference method (ADI) is used to integrate the isotropic equation combined with boundary conditions. Inverse transformation provides profile temperature in the anisotropic cylinder for full-field configuration. The numerical code is validated by the analytical heat conduction solutions available in the literature such as transient isotropic solution and steady-state orthotropic solution. The aim of this paper is to study the effect of cross-conductivity on the temperature profile inside an axisymmetrical anisotropic cylinder versus time, radial Biot number (Bir), and principal conductivities. The results show that cross-conductivity promotes the effect of Bir according to the principal conductivities. Furthermore, the anisotropy increases the time required to achieve the steady-state heat conduction.

The aim of the work is to present a detailed numerical study of the steady state heat conduction and temperature distribution in a rectangular plate Subjected to different boundary conditions. Steady-state heat conduction equation is very slowly convergent series for temperatures for different boundary conditions. The steady-state heat conduction equation is first converted into an algebraic equation by using the finite difference method (FDM). Finite Difference method used here is second order accurate and solution of algebraic equation is obtained through FOTRAN CODE. Temperature is one of the most important and fundamental parameter to be measured in the field of engineering. This is basically because the properties and behavior of materials often show temperature dependence and therefore, measurement of temperature and its comprehensive understanding are cardinal for development and processing of materials. Since the temperature state of a material which is undergoing heating or cooling transiently, is often required to monitor not only the temperature at a certain point but also its distribution in a certain area of the material. The results of our work can be useful in the field of education and for teaching steady-state heat conduction, Result can also be useful for design and development of fins and Engines. Contour obtained from The temperature contour of FOTRAN CODE is validated through the ANSYS Classics.

The Dirichlet problem for the time-fractional heat conduction equation in a half-line domain is studied with the boundary value of temperature varying harmonically in time. The Caputo fractional derivative is employed. The Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate are used. Different formulations of the considered problem for the classical heat conduction equation and for the wave equation describing ballistic heat conduction are discussed.

In this paper, the change of water temperature in the bathtub is mainly studied. Through the simulation of the water temperature distribution in time and space and the construction of the thermal conduction model, we discuss the water injection scheme. According to the Fourier's law and Newton cooling law, we establish a one-dimensional heat conduction differential model with water temperature, bath depth and time as variables. Then the model is classified into two parts because of the heat source's existence. By the evaporation model we can conclude the relationship between the water temperature and time, then solve the partial differential equation in the usage of implicit format of finite difference method. Equation of the water speed and temperature and the optimal scheme can be determined through the Improved Genetic Algorithm on the base of law of conservation of energy. The judging equation can be obtained after the analysis of the various influence factors of the strategy plan. Finally, we draw a conclusion that: the shape and volume of bathtub infect the strategy most.

Symmetry properties of the causal (hyperbolic) heat equations are studied. Symmetry groups of two variants of hyperbolic equations are calculated. The results obtained are analysed and compared with known properties of the classical heat equation. These findings may be considered as possible arguments in favour of one of the causal heat conduction models.

Heat transfer of the wave character is considered in linear and nonlinear media on the basis of a new law of the heat conduction with allowance not only for the first and second order temporal derivatives of the heat flux, but the higher order derivatives as well. This allowed us to reduce the heat transfer that was studied on the basis of a hyperbolic equation to a task about the heat transfer described by a parabolic equation with a retarded time argument. It is shown that under certain conditions for the heat transfer at free boundaries, the discontinuities of the first kind arising in the temperature distribution in nonlinear media are trans-formed to compression wave heat fluxes. The results obtained here confirm the assumptions made. The results can be used in designing the heat transfer in hypersonic vehicles under conditions of high-intensity aero-gas-dynamic and radiation heating.

Separation-of-variables method is one of the analytical solution methods to solve unsteady state heat conduction problems. But unsteady state conduction with heat source is an inhomogeneous problem. It can not be solved by separation-of-variables method directly. The combination of variables division method and separation-of-variables method was applied in this paper to deal with heat conduction with heat source in an infinite plate wall. The problem was divided to a steady state and inhomogeneous heat conduction problem, and an unsteady state and homogeneous heat conduction problem by variables division method. The steady state and inhomogeneous problem can be integrated directly. The unsteady state and homogeneous problem can be transformed to the problem that can be solved by separation-of-variables method directly through variable substitution. The unsteady state temperature field in the infinite plate wall was then obtained.

The problem of heat slip flow along solid walls is investigated within the framework of modern thermodynamics. The underlying idea is to elevate the heat flux at the boundary to the status of independent variable. General boundary conditions are obtained from the constraint imposed by the second law of thermodynamics expressing that the rate of entropy production is non-negative. In parallel, evolution equations for the heat flux inside the bulk of the system are also formulated.

The effect of periodic Ohm’s heating on hyperbolic heat conduction in porous media is studied analytically with the objective of identifying the thermal resonance conditions. Local thermal equilibrium conditions are assumed to apply. The paper focuses initially on the temperature solution and looks at the conditions required for resonating the temperature signal. The heat flux solution is then evaluated. While a discrete infinite set of modes can be resonated, it is shown that in practice the resonance in the temperature signal is felt starting from moderately small values of Fourier numbers and becomes too small to be noticed if the Fourier number is extremely small. The temperature solution is shown to represent a standing wave the amplitude of which is strongly affected by the Fourier number. While the heat flux solution is shown to differ from the one obtained for the temperature, it also shows similar features such as the standing wave behavior the amplitude of which is strongly affected by the Fourier number.

In this article, heat conduction in one-dimensional functionally graded media is investigated based on the dual-phase-lag theory to consider the microstructural interactions in the fast transient process of heat conduction. All material properties of the media are assumed to vary continuously according to a power-law formulation with arbitrary non-homogeneity indices except the phase lags which are taken constant for simplicity. The one-dimensional heat conduction equations based on the dual-phase-lag theory are derived in a unified form which can be used for Cartesian, cylindrical, and spherical coordinates. A semi-analytical solution for temperature and heat flux is presented using the Laplace transform to eliminate the time dependency of the problem. The results in the time domain are then given by employing a numerical Laplace inversion technique. The semi-analytical solution procedure leads to exact expressions for the thermal wave speed in one-dimensional functionally graded media with different geometries based on the dual-phase-lag and hyperbolic heat conduction theories. The transient temperature distributions have been found for various types of dynamic thermal loading. The numerical results are shown to reveal the effects of phase lags, non-homogeneity indices, and thermal boundary conditions on the thermal responses for different temporal disturbances. The results are verified with those reported in the literature for hyperbolic heat conduction in cylindrical and spherical coordinates.

The variation law of the average temperature with time in general case is derived by the differential equation of heat conduction which it is the reflection of the conservation of energy principle. The expression of the average temperature under the second boundary condition is given by the integral form of initial and boundary conditions. And what can be also derived are that the average temperature has a linear relationship with time when the boundary heat flux is constant, and it does not change with time under the adiabatic boundary condition.

Diffusion only, two dimensional heat conduction has been described on partial differential equation. Based on Finite Volume Method, Discretized algebraic Equation of partial differential equation have been deduced. different coefficients and source terms have been discussed under different boundary conditions, which include prescribed heat flux, prescribed temperature, convection and insulated. Transient heat conduction analysises of infinite plate with uniform thickness and two dimensional rectangle region have been realized by programming using MATLAB. It is useful to make the heat conduction equation more understandable by its solution with graphical expression, feasibility and stability of numerical method have been demonstrated by running result.

This article deals with the analysis of transient one-dimensional heat conduction in both Cartesian and cylindrical geometry by employing the polynomial approximation method (PAM). Four different models such as specified heat flux for both slab and tube and heat generation in both slab and tube have been analyzed. The transient temperature is found to depend on various model parameters, namely, Biot number, heat source parameter and time. With the use of PAM, it has been possible to derive a unified relation for the transient thermal behavior of solid (slab and tube) with both internal generation and boundary heat flux. Present prediction is found to be in good agreement with other analytical results reported in the literature.

In this paper, the transient temperature field around a thermally insulated crack in a substrate bonded to a coating is obtained using the hyperbolic heat conduction model. Fourier and Laplace transforms are applied, and the thermal conduction problem is reduced to solving a singular integral equation. Numerical results show that the hyperbolic heat conduction parameters, the heat conductivity of the substrate and coating, and the geometric size of the composite have significant influence on the transient temperature field. In the case of very small time scales, the results predicted by the hyperbolic model are more conservative than that by the parabolic model.

Temperature distributions recorded by thermocouples in a solid body (slab) subject to surface heating are used in a mathematical model of two-dimensional heat conduction. The corresponding Dirichlet problem for a holomorphic function (complex potential), involving temperature and a heat stream function, is solved in a strip. The Zhukovskii function is reconstructed through singular integrals, involving an auxiliary complex variable. The complex potential is mapped onto an auxiliary half-plane. The flow net (orthogonal isotherms and heat lines) of heat conduction is compared with the known Carslaw–Jaeger solution and shows a puzzling topology of three regimes of energy fluxes for temperature boundary conditions common in passive thermal insulation. The simplest regime is realized if cooling of a shaded zone is mild and heat flows in a slightly distorted “resistor model” flow tube. The second regime emerges when cooling is stronger and two disconnected separatrices demarcate the back-flow of heat from a relatively hot segment of the slab surface to the atmosphere through relatively cold parts of this surface. The third topological regime is characterized by a single separatrix with a critical point inside the slab, where the thermal gradient is nil. In this regime the back-suction of heat into the atmosphere is most intensive. The closed-form solutions obtained can be used in assessment of efficiency of thermal protection of buildings.

The transient thermal stresses around a crack in a thermo-elastic half-plane are obtained under a thermal shock using the hyperbolic heat conduction theory. Fourier, Laplace transforms and singular integral equations are applied to solve the temperature and thermal stress fields consecutively. The integral equations are solved numerically and the asymptotic fields around the crack tip are obtained. Numerical results show that the hyperbolic heat conduction have significant influence on the dynamic temperature and stress field. It is suggested that to design materials and structures against fracture under thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.

Diffusion only, two dimensional heat conduction has been described on partial differential equation. Based on Finite Volume Method, Discretized algebraic Equation of partial differential equation have been deduced. different coefficients and source terms have been discussed under different boundary conditions, which include prescribed heat flux, prescribed temperature, convection and insulated.. Transient heat conduction analysis of infinite plate with uniform thickness and two dimensional rectangle region are realized by programming using MATLAB. It is useful to make the heat conduction equation more understandable by its solution with graphical expression, feasibility and stability of numerical method have been demonstrated by running result.

This paper develops a finite element code based on the hyperbolic heat conduction equation including the non-Fourier effect in heat conduction. The finite element space discretization is used to obtain a system of differential equations for time. The time-related responses are obtained by solving the system of differential equations via finite difference technique. A relationship for the time step length and the element size was obtained to ensure that numerical oscillation in temperature be suppressed. Temperature-dependent material properties are taken into account in the proposed analysis model. In addition to the temperature field, the thermal stresses are also obtained from the developed method. The thermal stresses associated with the non-classical heat conduction are found to be considerably difference from those associated with classical heat conduction.

The discrete Boltzmann transport equation is firstly formulated by accounting for the impact of particle collisions on the distribution function in the discrete Liouville equation. Based on this equation, we re-derive the improved dual-phase-lagging heat conduction model with lagging effect in time and nonlocal effect in space. By taking into account the contribution of the higher order moments of the distribution function to the heat flux, we show that the discrete Boltzmann transport equation can give rise to the well-known Guyer–Krumhansl heat conduction model.

In this paper, the transient temperature and thermal stresses around a partially insulated crack in a thermoelastic strip under a temperature impact are obtained using the hyperbolic heat conduction theory. Fourier and Laplace transforms are applied and the thermal and mechanical problems are reduced to solving singular integral equations. Numerical results show that the hyperbolic heat conduction parameters, the thermal conductivity of crack faces, and the geometric size of the strip have significant influence on the dynamic temperature and stress field. The results based on hyperbolic heat conduction show much higher temperature and much more dynamic thermal stress concentrations in the very early stage of impact loading comparing to the Fourier heat conduction model. It is suggested that to design materials and structures against fracture under transient thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.

The momentary state of a semiconductor device of heat conduction is described by a system of four nonlinear partial differential equations. One elliptic equation is for the electrostatic, two parabolic equations are for the electron concentration and the hole concentration, and one heat exchange equation is for the temperature. According to the necessary of practical numerical simulations and based on the balance equation, finite difference schemes for two-dimensional transient behavior of a semiconductor device of heat conduction on composite triangular grids are constructed. Studying their stability and convergence properties, the error estimate in the energy norm is obtained. Finally, a numerical example is given.