Recently the dual-phase-lag (DPL) heat conduction model has stimulated considerable interest in the heat transfer community, by offering alternative interpretations and new perspectives to a large body of non-Fourier thermal behaviors in energy transportation process under special considerations, such as heat conduction in biological materials, heat transport in amorphous media, layered-film heating in superconductors, fins and reactor walls. and many commonly used devices, such as personal computers or cellular phones. Needless to say, numerous efforts have been invested to the development of an explicit mathematical solution to the heat conduction equation under the DPL model. Most of these analytical solutions to the DPL heat conduction problems in the literature were formulated ad hoc, only applicable to specific formulations of initial-boundary conditions. Other than the notoriously annoying fictitious numerical oscillations frequently encountered in solving hyperbolic partial differential equations (HPDE), the intrinsic complexity of the DPL heat conduction equation alone (highorder mixed derivative with respect to space and time which dramatically alter the fundamental characteristics of the solution) poses a tremendous hindering obstacle against a general solution [5]. In the present work highorder mixed derivative with respect to space and time is treated in special manner in the dual-phase-lag heat equation in order to construct a general solution applicable to wide range of dual-phase-lag heat transfer problems of general initial-boundary conditions using Green’s function solution method.

The definition of Green’s functions for a wave-type conduction equation and a general form of the Green’s function solution method for finite bodies is introduced by Haji-Sheikh and Beck [6]. Loureiro et al. [7] studied the hyperbolic bioheat conduction equation using the explicit Green’s approach method. The dual-phase-lag heat equation was used to generalize macroscopic model in treating the transient heat conduction in finite slabs irradiated by short pulse laser using Green’s function method by [8,9]. For powerful reviewing of construction of several Green’s functions for different boundary and initial condition of various physical equations, the reader is referred to [10].

The present work is devoted to define a generalized Green’s function solution for the dual-phase-lag model in homogeneous materials. Also, the Green’s function for a finite medium subjected to arbitrary heat source and arbitrary initial and boundary conditions is constructed. To examine the applicability of the present method, calculations are performed on four different previously solved researches [1-4]. The obtained results show good agreement with these researches.

Let be an open bounded domain with smooth boundary where d is the number of space dimensions and let be the time domain with the dual-phase lag model (DPL), given by Tzou [11], which allows either the temperature gradient (cause) to precede the heat flux vector (effect) or the heat flux vector (cause) to precede the temperature gradient (effect) in the transient process, can be represented, mathematically, by

where and are the temperature and heat flux distributions at position at time respectively. is the phase lag (relaxation time) of the heat flux vector, is the phase lag (relaxation time) of the temperature gradient, is the thermal conductivity. Combining Equation (1) with the energy conservation law,

leads to the energy transport equation (the dual-phase-lag heat equation) in the form

where is specific heat at constant pressure, is the density, is the heat generation per unit volume and is the thermal diffusivity. The highorder mixed derivative with respect to space and time is dramatically alter the fundamental characteristics of the solution and completely destroys the wave structure resulting from the wavy term, the second-order derivative term with respect to time, and the energy equation is parabolic in nature. It predicts a higher temperature level in the heat-penetration zone than diffusion but does not have a sharp wavefront in heat propagation.

The smooth boundary can be imposed on either prescribed temperature or prescribed heat flux. In addition to the prescribed boundary values, the initial condition on temperature may be also specified as below

while according to the conservation law (2), with the consideration that the initial value of the heat flux the initial value of the time derivative of the temperature distribution may takes the form

The Green’s functions are an important tool in solving partial differential equations since the solution of the problem subjected to any kind of initial conditions, boundary conditions and internal heat generation can be obtained through integral equations once the Green’s function is known. The Green’s function for finite or semi-infinite medium of constant physical properties with arbitrary initial and boundary conditions which correspond to the dual-phase-lag heat conduction Equation (3) is defined as the solution of

For convenience of subsequent analysis, the following dimensionless variables are defined

Using the above dimensionless variables, Equations (3) and (6) are expressed as

where is Dirac delta function. For convenience of algebra, Equation (9) can be reduced to a simpler form. To accomplish this task, one can define a Green’s function so that

Examining the above two equations, one can hypothesize that this acceptable since both and have homogeneous boundary conditions and their initial conditions, including all time derivatives, are zero. To show this relation between and, simply substitute for in Equation (10b) and get

that reduces to the equation,

Notice that any function that satisfies Equation (10a) also satisfy Equation (10b). Therefore, instead of solving for from Equation (9), it is sufficient to solve a simpler Equation (10a), and then utilize the relation

Changing the spatial variables in Equation (10a) to “prime” space and time from to yields

Moreover, the dual-phase-lag heat equation in space is

Multiplying Equation (15) by and Equation (14) by, then subtracting the results to produce equation

Both sides of Equation (16) are integrated, over volume and from 0 to, where is a small positive number. Then, following the application of the Green’s theorem and after letting to go to zero, one gets

where the source contribution to the temperature distribution is given by

while the boundary conditions contribution to temperature distribution is

where is the boundary of the volume and is the unit vector outward normal to the boundary. Notice that due to the causality principle one has and for and consequently, the initial conditions contribution to the temperature distribution is

Thus, the temperature distribution can be expressed as

For the hyperbolic model, i.e. the temperature distribution can be expressed as

which cionside with that given by [6], with the consideration that the present formulae is dimensionless one.

Green’s function for finite medium can be derived by solving Equation (10a) for with homogeneous initial and boundary conditions. Applying a suitable finite transform to Equation (10a) using homogeneous boundary conditions either of Nueman or Dirrichlet kind or even radiation boundary conditions, yields to

where are the eigen values corresponding to the eigen functions and

where is the orthogonality constant.

Now, solving Equation (23) with homogeneous initial conditions and then using the inversion Formula (24b), the first component of the Green’s function can be expressed as

where

Then using equation, the second component of the Green’s function can be expressed as

where

Thus the Green’s function can be written in the form

where

(27b)

Note that

Note that the above postulated Green’s function can be modified to a semi-infinite medium by extending to a semi-open domain and consequently the integral transform (24a) and its inversion (24b) should be modified.

Since the lagging behavior is a special response to time, the consideration of one-dimensional problems in space is sufficient to illustrate its fundamental characteristics. In addition, from a mathematical point of view, the lagging behavior introduces the highest order differentials in the energy equation, reflected by the mixed-derivative and the wave term. These terms characterize the fundamental solutions of the energy equation employing the dual-phase-lag model. Consideration of multidimensional problem will not alter the qualitative behavior depicted by the one-dimensional problem.

With the objective of showing the applicability and generality of the given Green’s function method to deal with any heat generation and any kind of initial and boundary conditions, three one-dimensional examples and one two dimensional example are discussed.

Hence, the purpose of the present paper is to describe the analytical solution of the dual-phase-lag heat equation in a unified manner by Green’s function method. The highorder mixed derivative with respect to space and time which reflect the lagging behavior is treated in special manner in the dual-phase-lag heat equation in order to construct a general solution applicable to wide range of dual-phase-lag heat transfer problems of general initialboundary conditions using Green’s function solution method. Also, the Green’s function for a finite medium subjected to arbitrary heat source and arbitrary initial and boundary conditions is constructed. Since the lagging behavior is a special response to time, the consideration of one-dimensional problems in space is sufficient to illustrate its fundamental characteristics. Therefore, three one dimensional examples and one two dimensional example of different physical situations are analyzed in order to illustrate the accuracy and potentialities of the proposed unified method. The obtained results show good agreement with works of [1-4].

The author would like to thank the scientific deanship of University of Dammam for its generous support of this work through the project No. 2011082.