Analytical and numerical analysis of time fractional dual-phase-lag heat conduction during short-pulse laser heating

Numerical Algorithms - In this study, we analytically and numerically investigate the non-Fourier heat conduction behavior within a finite medium based on the time fractional dual-phase-lag model....     

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.     

Qualitative properties for solutions of semilinear heat equations with strong absorption

We are concerned with nonnegative solutions to the Cauchy problem

     

Weighted iterative solutions of linear differential equations and heat conduction of ideal gases in moment theory

An iterative method is proposed to find a particular solution of a system of linear differential equations, in the form of a fixed-point problem, with no boundary conditions. To circumvent the unboundedness of differential operators, iterative approximation with gradually decreasing weight is used. Conditions for convergence that can easily be checked in numerical iterations are established. Furthermore, for the numerical iterative scheme, uniqueness and stability theorems are proved. These results are applied to heat conduction of ideal gases in moment theory.     

Weighted iterative solutions of linear differential equations and heat conduction of ideal gases in moment theory

An iterative method is proposed to find a particular solution of a system of linear differential equations, in the form of a fixed-point problem, with no boundary conditions. To circumvent the unboundedness of differential operators, iterative approximation with gradually decreasing weight is used. Conditions for convergence that can easily be checked in numerical iterations are established. Furthermore, for the numerical iterative scheme, uniqueness and stability theorems are proved. These results are applied to heat conduction of ideal gases in moment theory.     

Half-line solutions of a nonlinear heat conduction problem

We solve a half-line problem for a nonlinear diffusion equation with a given time-dependent thermal conductivity at the origin. The problem reduces to a linear Volterra integral equation, which is solvable by Picard’s process of successive approximations. We analyze some explicit examples numerically. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 58–65, July, 2007.     

Qualitative properties of solutions of integral equations

In this paper we study a linear integral equation in which the kernel fails to satisfy standard conditions yielding qualitative properties of solutions. Thus, we begin by following the standard idea of differentiation to obtain . The investigation frequently depends on x^′(t)+C(t,t)x(t)=0 being uniformly asymptotically stable. When that property fails to hold, the investigator must turn to ad hoc methods. We show that there is a way out of this dilemma. We note that if C(t,t) is bounded, then for k>0 the equation resulting from x^′+kx will have a uniformly asymptotically stable ODE part and the remainder can often be shown to be a harmless perturbation. The study is also continued to the pair x^″+kx^′.     

A diffusive-hyperbolic model for heat conduction

The present paper faces the problem of heat conduction within the framework of thermodynamics with internal state variables. A model, in which the heat flux vector depends both on the gradient of the absolute temperature and the gradient of a scalar internal variable, is proposed. Such a model leads to a diffusive-hyperbolic system which in general is parabolic, but also allows to shift to the hyperbolic regime. In the hyperbolic case the propagation of weak discontinuity waves is investigated. The Rankine-Hugoniot and Lax conditions for the propagation of strong shock waves are analyzed as well.     

Uncertain partial differential equation with application to heat conduction

This paper first presents a tool of uncertain partial differential equation, which is a type of partial differential equations driven by Liu processes. As an application of uncertain partial differential equation, uncertain heat equation whose noise of heat source is described by Liu process is investigated. Moreover, the analytic solution of uncertain heat equation is derived and the inverse uncertainty distribution of solution is explored. This paper also presents a paradox of stochastic heat equation.     

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.     

Direct and inverse solutions of the two-dimensional hyperbolic heat conduction problems

A sequential method is proposed to estimate boundary condition of the two-dimensional hyperbolic heat conduction problems. An inverse solution is deduced from a finite difference method, the concept of the future time and a modified Newton–Raphson method. The undetermined boundary condition at each time step is denoted as an unknown variable in a set of non-linear equations, which are formulated from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. The example problem is used to demonstrate the characteristics of the proposed method. In the example, a well-known problem is used to demonstrate the validity of the proposed direct method and then the inverse solutions are evaluated. In the second example, the larger value of the relaxation time is implemented in the direct solutions and the inverse solutions. The close agreement between the exact values and the estimated results is made to confirm the validity and accuracy of the proposed method. The results show that the proposed method is an accurate and stable method to determine the boundary conditions in the two-dimensional inverse hyperbolic heat conduction problems.     

Efficient and accurate spectral method for the time-fractional dual-phase-lag heat transfer model and its parameter estimation

In the current paper, a heat transfer model is suggested based on a time-fractional dual-phase-lag (DPL) model. We discuss the model in two parts, the direct problem and the inverse problem. Firstly, for solving it, the finite difference/Legendre spectral method is constructed. In the temporal direction, we employ the weighted and shifted Grünwald approximation, which can achieve second order convergence. In the spatial direction, the Legendre spectral method is used, it can obtain spectral accuracy. The stability and convergence are theoretically analyzed. For the inverse problem, the Bayesian method is used to construct an algorithm to estimate the four parameters for the model, namely, the time-fractional order α, the time-fractional order β, the delay time τ_T, and the relaxation time τ_q. Next, numerical experiments are provided to demonstrate the effectiveness of our scheme, with the values of τ_q and τ_T for processed meat employed. We also make a comparison with another method. The data obtained for the direct problem are used in the parameter estimation. The paper provides an accurate and efficient numerical method for the time-fractional DPL model.     

On a proposed model for heat conduction

^** Corresponding author. Email: jcsong{at}hanyang.ac.kr A system of partial differential equation for modelling theconduction of heat was proposed by Ghaleb & El-Deen Mohamedein(1989). According to their theory, the initial-value problemfor the temperature is ill-posed. In this paper, two well-posedproblems for the temperature are introduced and investigated.