**1. Introduction**

In recent years, some studies have focused on the deviation from the classical Fourier heat transfer equation. In the classical theory of conduction heat transfer based on Fourier's law, the thermal heat flux is a linear relationship with the gradient of temperature and the heat wave propagation speed is assumed to be unlimited (Equation (1)).

$$q^\* = -k\nabla T^\* \tag{1}$$

where *k* is the coe fficient of thermal conductivity and *T*∗ is temperature. When this equation is combined into the balance equation of energy,

$$\nabla \cdot q^\* = -\rho c\_p \frac{\partial T^\*}{\partial t^\*} \tag{2}$$

The classical parabolic heat conduction equation is derived,

$$\frac{\partial T^\*}{\partial t^\*} = \alpha \nabla^2 T^\* \tag{3}$$

where α = *k* ρ*cP* , ρ, and *cP* are thermal di ffusivity, density, and specific heat, respectively. Also, the e ffect of any thermal disturbance on the medium is instantaneously sensed through the entire molecular

network. In the majority of engineering applications such as material processing (welding, cutting, forming, etc.), applying high power laser radiation, cryogenic applications, and materials which experience the high heat transfer rates [1–4], this equation is very useful. However, at very low times and very high thermal fluxes and very low temperatures close to absolute zero, the Fourier law has poor accuracy, and considering the e ffects of non-Fourier in describing the heat dissipation process and prediction of temperature distribution, non-Fourier are more reliable in these situations. Fourier's failure to exactly predict the temperature field in su fficiently high heat flux and low temperature engineering usages is because it assumes that thermal energy transport is occurring at an infinite propagation speed [2,5]. As a result, a more advanced method, in the situation of the thermal waves finite propagation speed, is required to analyze the high temperature gradients. Usually, when the infinite propagation speed was assumed, the temperature was calculated more than its actual values and it causes some errors in temperature prediction.

A modified equation of non-Fourier heat flux has been developed by Cattaneo [6] and Vernotte [7] in the present form,

$$q^\* + \tau\_0 \frac{\partial q^\*}{\partial t^\*} = -k \nabla T^\* \tag{4}$$

where τ0 is known thermal relaxation time. If the relaxation time ignores, τ0 = 0, the law of the non-Fourier model is converted to the Fourier law. The energy equation is derived as follow,

$$-\frac{\partial \overrightarrow{q}^\*}{\partial r^\*} + \dot{\mathbf{g}}^{\prime\prime} = \rho \mathbf{c}\_p \frac{\partial T^\*}{\partial t^\*} \tag{5}$$

In the Equation (5), . *g* expresses the rate of internal generation of energy. Inserting Equation (4) into Equation (5), the equation of hyperbolic heat transfer, containing source the term, derived,

$$
\Delta a \nabla^2 T^\* = \frac{\partial T^\*}{\partial t^\*} + \tau\_0 \frac{\partial^2 T^\*}{\partial t^{\*2}} + Q(r, t) \tag{6}
$$

where the source term is *Q*(*<sup>r</sup>*, *t*). Di fferent solution procedures for Equation (6) with di fferent boundary and initial conditions for finite media can be found in literature.

The analytical, numerical, and experimental methods were used in many researches for analysis and calculating the rate of heat transfer in applied physics problems [8–10]. Using the heat sources such as lasers and microwaves at very small times of applying the heat or high frequencies has a grea<sup>t</sup> deal of application in analytical physics, applied sciences, and engineering. In fast and short processes and rapid and concentrated conduction heat transfer, the order of space and time is very small. So, the law of Fourier equation which assumed that heat propagates at an infinite velocity, cannot be used [11].

Ozisik and Vick [12] studied the heat propagation in a semi-infinite body with a volumetric source of energy by solving the thermal wave equation. They found that the classical Fourier's law was no longer suitable in obtaining the temperature field at short times. Jiang [13] applied the method of Laplace transform to study the hyperbolic conduction heat transfer in a hollow sphere whose boundaries are a ffected by a sudden change in temperature. Moosaie [14] solved, analytically, the equation of non-Fourier heat conduction for a finite body with an arbitrary initial condition and insulated boundaries. His results showed that the time needed for reaching steady state situation is enhanced with rising the relaxation time τ0. Moosaie [15] investigated a finite body subjected to an arbitrary non-periodic surface disturbance. Their obtained solution is such that for a given non-periodic disturbance, analytically if possible, but in general numerically is a straight forward computational task. Ahmadikia and Riesmanian [16] presented an analytical method for solving the hyperbolic heat transfer in a blade under periodic boundary conditions applying the Laplace transform approach. The findings showed that in small blades under rapid phenomena, temperature behavior is the non-Fourier wave form. Bamdad et al. [17] studied the non-fluoride e ffects at extended surfaces. Their results showed that for all fins at the start times, the point of discontinuity is time, relaxation time, and cross-section of the dependent fin. In addition, the cross-sectional e ffects on the amplitude of the heat wave reflecting from the tip of the fin are such that there is no reflected heat wave in the fins with concave shape. Lam and Fang [18] presented an analytical method to investigate the heat conduction in a slab applied by various boundary conditions. They indicated that the solution accuracy depends on the terms number which were applied in the Fourier series expansion process.

Liu et al. [19] surveyed the non-Fourier heat conduction characteristics in the oil/water emulsions experimentally. Their results showed that in the ratio of time lag less than one, no thermal waves exist for oil/water emulsions.

The analysis of non-Fourier heat conduction in infinite hollow cylinders subjected to a heat source, which is a function of time, was investigated by Daneshjou et al. [20]. They used the Laplace transform method and demonstrated that their approach is valuable in correctness and exactness. Ma et al. [21] studied the C-V wave model for a plate which is irradiated by a non-Gaussian laser pulse. The method of mode superposition was applied for solving the equation. They discussed the dependence of the wave velocity on relaxation time. The influence of scanning and wave velocity on the temperature field was also presented. Wankhade et al. [22] investigated the heat transfer response of wet fins using the models of Fourier and non-Fourier. The method of separation of variables was applied, and the results showed a considerable deviation in temperature response using the non-Fourier heat conduction compared to the Fourier model. Also, the e ffect of fin surface conditions was studied. Han and Peddieson [23] investigated the Non-Fourier one-dimensional unsteady equation in a body for medium speeds less than (sub-critical), equal to (critical), and greater than (super-critical) the thermal wave speed. Liu et al [24] studied a hyperbolic lattice Boltzmann method (HLBM) compare to the parabolic lattice Boltzmann method (PLBM) to survey the non-Fourier e ffects. The results show that the electron temperatures simulated by the two-step HLBM/PLBM and two-temperature models are not much di fferent from each other and both of them coincide with the experimental data.

Review of this research indicated that various solutions are studied in di fferent works. In each study, di fferent features of non-Fourier heat conduction were investigated and, therefore, various results were established which could be useful in its position. However, the shortage of a general problem with nonzero initial conditions and boundary conditions with internal heat generation in these studies is observed. In this study, an exact solution is presented to non-Fourier heat conduction in a cylinder with nonzero initial and boundary conditions. As it will be mentioned later, analytical solutions of this problem were obtained by applying the theory of solution structure combined with the superposition method. This approach can be used for obtaining the heat transfer in many physical applications which solved by di fferent methods [25–31].
