Spatial behaviour of solutions of the three-phase-lag heat equation

In this paper we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. The theory of three-phase-lag heat conduction leads to a hyperbolic partial differential equation with a fourth-order derivative with respect to time. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under a boundedness restriction on the initial data, an energy estimate is obtained. An upper bound for the amplitude term in this estimate in terms of the initial and boundary data is also established. For the case of zero initial conditions, a more explicit estimate is obtained which shows that solutions decay exponentially along certain spatial-time lines. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T₀ are assumed proportional to their initial values. These results are relevant in the context of the Saint-Venant Principle for heat conduction problems.     

Spatial behaviour of solutions of the dual‐phase‐lag heat equation

In this paper we study the spatial behaviour of solutions of some problems for the dual‐phase‐lag heat equation on a semi‐infinite cylinder. The theory of dual‐phase‐lag heat conduction leads to a hyperbolic partial differential equation with a third derivative with respect to time. First, we investigate the spatial evolution of solutions of an initial boundary‐value problem with zero boundary conditions on the lateral surface of the cylinder. Under a boundedness restriction on the initial data, an energy estimate is obtained. An upper bound for the amplitude term in this estimate in terms of the initial and boundary data is also established. For the case of zero initial conditions, a more explicit estimate is obtained which shows that solutions decay exponentially along certain spatial‐time lines. A class of non‐standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T are assumed proportional to their initial values. Copyright © 2004 John Wiley & Sons, Ltd.     

Spatial behavior for solutions in heat conduction with two delays

In this note, we investigate the spatial behavior of the solutions of the equation proposed to describe a theory for the heat conduction with two delay terms. We obtain an alternative of the Phragmén-Lindelöf type, which means that the solutions either decay or blow-up at infinity, both options in an exponential way. We also describe how to obtain an upper bound for the amplitude term. This is the first contribution on spatial behavior for partial differential equations involving two delay terms. We use energy arguments. The main point of the contribution is the use of an exponentially weighted energy function.     

Analytical solutions to the heat conduction problems for a cylinder and a ball based on determining the temperature perturbation front

Analytical solutions to the heat conduction problems for a cylinder and a ball are obtained by the integral method of heat balance. To improve the accuracy of the solutions, the temperature function is approximated by polynomials of high degrees. Their coefficients are determined via introducing additional boundary conditions, which are found from the governing differential equation and the basic boundary conditions, including those specified at the temperature perturbation front. It is shown that the additional boundary conditions, even in the second approximation, lead to a considerable improvement in the solution accuracy.     

Hyperbolic heat conduction in two semi-infinite bodies in contact

We find, under the viewpoint of the hyperbolic model of heat conduction, the exact analytical solution for the temperature distribution in all points of two semi-infinite homogeneous isotropic bodies that initially are at uniform temperatures T ₀ ¹ and T ₀ ² , respectively, suddenly placed together at time t = 0 and assuming that the contact between the bodies is perfect. We make graphics of the obtained temperature profiles of two bodies at different times and points. And finally, we compare the temperature solution obtained from hyperbolic model to the parabolic or classical solution, for the same problem of heat conduction.This work was partially supported by MEC and FEDER, project MTM-2004-02262 and AVCIT group 03/050.This revised version was published online in April 2005 with a corrected issue number.     

Analysis of dual-phase-lag heat conduction in cylindrical system with a hybrid method

The dual-phase-lag heat transfer model is applied to investigate the transient heat conduction in an infinitely long solid cylinder for an exponentially decaying pulse boundary heat flux and for a short-pulse boundary heat flux. A hybrid application of the Laplace transform method and the control volume scheme is used to obtain the numerical solutions. Comparison between the numerical results and the analytic solution for an exponentially decaying heat flux pulse evidences the accuracy of the present numerical results. Results further show that the present numerical scheme can overcome the mathematical difficulties to analyze such problems. Effects of the thermal lag ratio τ_q/τ_T, the shift time τ_q–τ_T, the function form of heating pulse, and geometry of medium on the behavior of heat transfer are investigated.     

The space localization of unbounded boundary perturbations in nonlinear heat conduction with transfer

We consider the nonlinear parabolic equation u_t = (k(u)u_x)_x + b(u)_x, where u = u(x, t, x ε R¹, t > 0; k(u) ≥ 0, b(u) ≥ 0 are continuous functions as u ≥ 0, b (0) = 0; k, b > 0 as u > 0. At t = 0 nonnegative, continuous and bounded initial value is prescribed. The boundary condition u(0, t) = Ψ(t) is supposed to be unbounded as t → +∞. In this paper, sufficient conditions for space localization of unbounded boundary perturbations are found. For instance, we show that nonlinear equation u_t = (uⁿu_x)_x + (u^β)_x, n ≥ 0, β >; n + 1, exhibits the phenomenon of “inner boundedness,” for arbitrary unbounded boundary perturbations.     

On optimal solutions of general continuous-singular stochastic control problem of McKean-Vlasov type

In this paper, we establish general necessary optimality conditions for stochastic continuous-singular control of McKean-Vlasov type equations. The coefficients of the state equation depend on the state of the solution process as well as of its probability law and the control variable. The coefficients of the system are nonlinear and depend explicitly on the absolutely continuous component of the control. The control domain under consideration is not assumed to be convex. The proof of our main result is based on the first- and second-order derivatives, with respect to measure in Wasserstein space of probability measures, and by using variational method.     

Heat transfer in granular materials: effects of nonlinear heat conduction and viscous dissipation

In this paper, we study the heat transfer in a one‐dimensional fully developed flow of granular materials down a heated inclined plane. For the heat flux vector, we use a recently derived constitutive equation that reflects the dependence of the heat flux vector on the temperature gradient, the density gradient, and the velocity gradient in an appropriate frame invariant formulation. We use two different boundary conditions at the inclined surface: a constant temperature boundary condition and an adiabatic condition. A parametric study is performed to examine the effects of the material dimensionless parameters. The derived governing equations are coupled nonlinear second‐order ordinary differential equations, which are solved numerically, and the results are shown for the temperature, volume fraction, and velocity profiles. Copyright © 2013 John Wiley & Sons, Ltd.     

Solution of fractional order heat equation via triple Laplace transform in 2 dimensions

In this article, we introduce the triple Laplace transform for the solution of a class of fractional order partial differential equations. As a consequence, fractional order homogeneous heat equation in 2 dimensions is investigated in detail. The corresponding solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. Numerical plots to the concerned solutions are provided to demonstrate our results.     

Source-type solutions of heat equations with convection in several variables space

In this paper we study the source-type solution for the heat equation with convection: ut = △u + ■· ▽un for (x,t) ∈ ST→ RN × (0,T] and u(x,0) = δ(x) for x ∈ RN, where δ(x) denotes Dirac measure in = RN,N 2,n 0 and b = (b1,...,bN) ∈ RN is a vector. It is shown that there exists a critical number pc = N+2 such that the source-type solution to the above problem exists and is unique if 0 N n < pc and there exists a unique similarity source-type solution in the case n = N+1 , while such a solution does not exist...     

On the existence and uniqueness of time periodic solutions for a semilinear heat equation in the whole space

This paper is concerned with the existence and uniqueness of time periodic solutions in the whole‐space for a heat equation with nonlinear term. The nonlinear term we considered is of this type, |u |^{q ? 1}u + f (x ,t ), with , N > 2. We show that there exists a unique time periodic solution when the source term f is small. In fact, is a critical exponent; when , there is no time periodic solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of travelling wave solutions for the heat equation in infinite cylinders with a nonlinear boundary condition

The existence of travelling wave solutions for the heat equation ∂_t u –Δu = 0 in an infinite cylinder subject to the nonlinear Neumann boundary condition (∂u /∂n) = f (u) is investigated. We show existence of nontrivial solutions for a large class of nonlinearities f. Additionally, the asymptotic behavior at ∞ is studied and regularity properties are established. We use a variational approach in exponentially weighted Sobolev spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration

Nonlinear evolution equations of the fourth order and its partial cases are derived for describing nonlinear pressure waves in a mixture liquid and gas bubbles. Influence of viscosity and heat transfer is taken into account. Exact solutions of nonlinear evolution equation of the fourth order are found by means of the simplest equation method. Properties of nonlinear waves in a liquid with gas bubbles are discussed.     

Determination of material functions through second sound measurements in a hyperbolic heat conduction theory

In a previous paper [1], numerical solutions to initial-boundary value problems for a semi-empirical model of heat conduction were compared with available experimental results.

In the present paper, we modify the model by introducing more realistic approximations of constitutive functions, based on measured heat conductivities and second sound speeds for NaF at low temperatures (10…20° K). We achieve good accordance between measured second sound pulses and numerical solutions in the temperature range covered by experiments, and reasonable behaviour even beyond this interval. Especially, a passage to the diffusive regime of the classical Fourier law is possible.     

Hybrid pseudospectral–finite difference method for solving a 3D heat conduction equation in a submicroscale thin film

This research aims to develop a time‐dependent pseudospectral‐finite difference scheme for solving a 3D dual‐phase‐lagging heat transport equation in a submicroscale thin film. The scheme uses periodic pseudospectral discretization in space and a fully second‐order finite difference discretization in time. The three consecutive time steps model is then solved explicitly, by using a preconditioned conjugate gradient method. The scheme is illustrated by an example which is used to investigate the heat transfer in a gold submicroscale thin film. Comparisons are made with available literature. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Controllability of a class of heat equations with memory in one dimension

This paper addresses a study of the controllability for a class of heat equations with memory in one spacial dimension. Unlike the classical heat equation, a heat equation with memory in general is not null controllable. There always exists a set of initial values such that the property of the null controllability fails. Also, one does not know whether there are nontrivial initial values, which can be driven to zero with a boundary control. In this paper, we give a characterization of the set of such nontrivial initial values. On the other hand, if a moving control is imposed on this system with memory, we prove the null controllability of it in a suitable state space for any initial value. Copyright © 2016 John Wiley & Sons, Ltd.     

Propagation of nonlinear thermoelastic waves in porous media within the theory of heat conduction with memory: physical derivation and exact solutions

In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid‐saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdélyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time‐fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time‐fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.