**2. Formulation**

#### *2.1. Hyperbolic Heat Conduction*

According to the Figure 1, we assume a cylinder composed of a homogenous heat conducting material with di fferent boundary conditions at both sides: A symmetry boundary condition in the cylinder's central line and convection in the cylinder surface (*r* = *R*) with ambient.

The heat is conducted through the cylinder in r-direction, where one dimensional heat conduction dominates. This problem was solved for L/r > 10 and a one-dimensional assumption for this problem is reasonable. For simplifying the solution, by using the following parameters, we can non-dimensionalize the governing equations,

$$r = \frac{cr^\*}{2a'}, \mathbf{t} = \frac{c^2 t^\*}{2a}, \ T = \frac{kcT^\*}{af\_r}, \ T\_{\text{oo}} = \frac{kcT\_{\text{oo}}}{af\_r}, \ q = \frac{q^\*}{f\_r}, \ g = \frac{4ag^{\prime\prime\prime}}{cf\_r} \tag{7}$$

**Figure 1.** A schematic illustration of the problem.

Combining the Equations (4)–(6) with Equation (7), we can derive the non-dimensional form of non-Fourier heat conduction equations as follows,

$$\frac{\partial q}{\partial t} + \frac{\partial T}{\partial r} = -2q\tag{8}$$

$$\frac{\partial T}{\partial t} + \frac{\partial q}{\partial r} = \frac{\mathcal{g}}{2} \tag{9}$$

$$2\frac{\partial^2 T}{\partial t^2} + 2\frac{\partial T}{\partial t} = \frac{\partial^2 T}{\partial r^2} + f(r, t) \tag{10}$$

where *f*(*<sup>r</sup>*, *t*) as a total internal heat generation in system can be determined as follows,

$$f(r,t) = \frac{1}{2}\frac{\partial g}{\partial t} + \text{g} \tag{11}$$

For the problem situation, the boundary conditions are defined as follows,

$$\frac{\partial T(0,t)}{\partial r} = 0\tag{12}$$

$$-\frac{\partial T(1,t)}{\partial r} = m(T - T\_{\infty}),\ m = \frac{-2ah}{kc} \tag{13}$$

*T*∞ is the dimensionless ambient temperature and *m* is the convection heat transfer coefficient. The initial conditions are considered to be,

$$T(r,0) = \varphi(r) \tag{14}$$

$$\frac{\partial T(r,0)}{\partial t} = \psi(r) = \frac{\mathcal{G}\_0}{2} \tag{15}$$

These conditions can be derived from Equation (9). In Equations (14) and (15), ϕ and ψ are dimensionless initial condition function and dimensionless initial rate of temperature change function, respectively.

In this paper, *g* as an internal energy generation will be defined as [11],

$$\lg(r) = \lg\_0 \exp(-\mu r) \tag{16}$$

or

$$\lg(r) = \lg\_0 \exp(-\mu r) \exp(-t) \tag{17}$$

where

$$g\_0 = \frac{2l\_0\mu(1-R)}{f\_r} \tag{18}$$

where μ is the absorption coefficient, *I*0 is the amplitude of laser density, and *R* is the solid surface reflectivity. This model assumes no spatial variations of *g*0 in the plane perpendicular to the laser beam.

#### *2.2. Solution Structure Theorems and Superposition Approach*

One of the famous and most widely used techniques for solving some types of heat conduction equations is the superposition method. This method can be used for solving the linear heat transfer problems with non-homogenous conditions. In this method, an origin problem is split into di fferent easier subproblems which can be integrated to take a solution to the original problem. The method of superposition relies upon the assumption that the original problem (Equation (10)) can be divided into three subproblems by setting the heat generation term (Equation (11)), the initial conditions (Equations (14) and (15)), and the boundary conditions (Equations (12) and (13)) to di fferent values in each subproblem:

$$f(r,t) = q(r) = 0\tag{19}$$

$$f(r,t) = \psi(r) = 0\tag{20}$$

$$
\varphi(r) = \psi(r) = 0 \tag{21}
$$

Solutions to these subproblems are assigned as *T*1, *T*2, and *T*3 sequentially. Therefore, the general solution to the first equation (Equation (10)) is the sum of subproblems one through three, which is

$$T(r,t) = T\_1(r,t) + T\_2(r,t) + T\_3(r,t) \tag{22}$$

It can be seen that *T*1, *T*2, and *T*3 illustrated the independent contributions of the initial rate of temperature variation, initial condition, and internal heat generation to the temperature field, respectively. Subproblems one to three can be simply solved by applying the solution structure theorems [18] once the solution to subproblem one is known. By using the solution structure theorems, the solutions of subproblems one to three can be determined as follows:

$$T\_1(r, t) = F(r, t, \psi(r))\tag{23}$$

$$T\_2(r, t) = (2 + \frac{\partial}{\partial t}) F(r, t, \varphi(r)) \tag{24}$$

$$T\_3(r,t) = \bigcup\_{\xi=0}^t F(r, t - \xi, f(r, \xi))d\xi \tag{25}$$

where *<sup>T</sup>*1(*<sup>r</sup>*, *t*) will be obtained by using the Fourier method. *<sup>T</sup>*2(*<sup>r</sup>*, *t*) and *<sup>T</sup>*3(*<sup>r</sup>*, *t*) can be simply derived by using the solution structure theorem. It means that only the solution of subproblem one is needed to obtain the solutions to subproblems two and three. Finally, the general solution to the first heat conduction equation is the sum of subproblems one to three.

#### *2.3. Formulation of the Problem*

In the literature, there are a few non-Fourier heat conduction problems with different boundary conditions but most of them are limited to boundaries with zero temperature or the insulated boundaries. In this study, we obtain the general analytical solution to hyperbolic heat conduction in a cylinder composed of a homogenous heat conducing material with different boundary conditions at both sides: A symmetry boundary condition in the cylinder's central line and the convection boundary condition in the cylinder surface (*r* = R) with ambient.

Let us first consider the solution to subproblem one. This subproblem is solved with the condition *f*(*<sup>r</sup>*, *t*) = ϕ(*r*) = 0. As a result, the equation of this subproblem and the initial conditions and boundary conditions are as bellow:

$$\frac{\partial^2 T\_1}{\partial t^2} + 2\frac{\partial T\_1}{\partial t} = \frac{\partial^2 T\_1}{\partial r^2} \tag{26}$$

$$\frac{\partial T\_1(0,t)}{\partial r} = 0\tag{27}$$

$$1 - \frac{\partial T(1, t)}{\partial r} = m(T - T\_{\infty}), \quad m = \frac{-2ah}{kc} \tag{28}$$

$$T\_1(r,0) = 0\tag{29}$$

$$\frac{\partial T\_1(r,0)}{\partial t} = \psi(r) \tag{30}$$

Using the Fourier series expansion theory, the general form of solution of equation is,

$$\begin{split} T\_1(r, t) &= T\_{\infty} + \frac{1}{2} \Big[ 1 - e^{-2t} \Big] \Big|\_{0}^{1} (\psi(\zeta) - T(\infty)) d\zeta + \\ &\sum\_{n=0}^{\infty} \frac{e^{-2t} \Big[ \int\_{0}^{1} \zeta(\psi(\zeta) - T(\infty)) l\_0(\lambda\_n \zeta) d\zeta \Big]}{\frac{\lambda\_n^2 + m^2}{2\lambda\_n^2} h\_0^{-2}(\lambda\_n)} \Big] \sin(\chi\_n t) f\_0(\lambda\_n r) \end{split} \tag{31}$$

Now, by using the solution structure theorems, we have

$$\begin{aligned} T\_2(r,t) &= T\_{\infty} + \int\_0^1 \varphi(\zeta) - T(\infty) d\zeta + \sum\_{n=1}^{\infty} \left[ \frac{\int\_0^1 r\varphi(\zeta) - T(\infty) f\_0(\lambda\_n r) d\zeta}{\frac{\lambda\_n^2 - n^2}{2\lambda\_n^2} f\_0^2(\lambda\_n)} \right] \times \\ &\left[ \frac{\zeta^\perp}{\gamma\_n} [\sin(\gamma\_n t) + \gamma\_n \cos(\gamma\_n t)] \right] \mathbf{j}\_0(\lambda\_n r) \end{aligned} \tag{32}$$

$$\begin{aligned} T\_3(r, t) &= T\_{\infty} + \int\_0^t \frac{1}{2} (1 - e^{-2(t - \xi)}) \left[ \int\_0^1 f(\zeta) d\zeta \right] d\zeta + \\ \int\_0^t \begin{Bmatrix} \frac{\zeta^{-(t - \xi)}}{\mathcal{I}^n} \left[ \frac{\int\_0^1 r f(\zeta) f\_0(\lambda\_n r) d\zeta}{\frac{\lambda\_n^2 + m^2}{2\lambda\_n^2} f\_0^2(\lambda\_n)} \right] \sin(\gamma\_n (t - \xi)) f\_0(\lambda\_n r) \end{Bmatrix} d\xi \end{aligned} \tag{33}$$

where

$$\frac{J\_0(\lambda\_n)}{J\_1(\lambda\_n)} = \frac{\lambda\_n}{m} \tag{34}$$

*J*0 and *J*1 are the first order Bessel functions and was obtained by solving the Equation (34). Also,

$$
\gamma\_n = \sqrt{\lambda\_n^2 - 1} \tag{35}
$$

Ultimately, the temperature field within the slab can be obtained as,

$$T(\mathbf{x}, t) = T\_1(r, t) + T\_2(r, t) + T\_3(r, t),$$

#### **3. Results and Discussions**

In this paper, the temperature field in a one dimensional cylinder with non-zero initial and boundary condition are examined. The values *g*0 = 100, μ = 5, *T*0 = 2.5, and *T* ∞ = 1.8 were selected. It can be concluded that all the profiles of temperature from *<sup>T</sup>*1(*<sup>r</sup>*, *t*) to *<sup>T</sup>*3(*<sup>r</sup>*, *t*) are in the infinite series form. By using a relative error test, we can write:

$$\frac{\left|T\_{n+1}(r,t) - T\_n(r,t)\right|}{\left|T\_{n+1}(r,t)\right|} < \varepsilon \tag{36}$$

where *Tn*+<sup>1</sup>(*<sup>r</sup>*, *t*) and *Tn*(*<sup>r</sup>*, *t*) are two consecutive partial sums for the temperature, and ε = 10−<sup>6</sup> is the relative error for this study. We showed that the original partial di fferential equation is split into three subproblems, subproblems one through three demonstrating the contributions of initial rate of change in temperature, initial condition, and internal heat generation, which are given by Equations (31) to (35), respectively.

It can be noted from these equations that all the temperature profiles from *<sup>T</sup>*1(*<sup>r</sup>*, *t*) to *<sup>T</sup>*3(*<sup>r</sup>*, *t*) are in the form of an infinite series.

Figure 2 shows the contribution of di fferent components of temperature at various times for a one dimensional cylinder with non-zero initial and boundary condition. According to the Figure 2a, it can be seen that at small times (*t* = 0.1) up to about *r* = 0.4, the contribution of *T*1 and *T*3 dominate compared to *T*2 which contributes little to the overall temperature. But at *r* > 0.4, all three temperature components will have the same role and less impact on the overall temperature.

Figure 2b shows that *T*2 still does not have much e ffect on the overall temperature and acts approximately uniformly with a constant value. *T*1 and *T*3 still have a downward trend, but *T*3, because of is related to the temperature component of internal heat generation, is dominant.

The downward trend of *T*1 and *T*3 will be continued to *r* = 0.5 and *r* = 0.8, respectively. After these points, significant variations were not seen. Also, in the areas near the center of the cylinder, where the source term generates the energy, the overall temperature has larger values compared to its values at *t* = 0.1.

**Figure 2.** *Cont*.

**Figure 2.** Contribution of temperature components at *t* = 0.1 (**a**), *t* = 0.5 (**b**), and *t* = 1 (**c**).

As time increases Figure 2c, the contribution of *T*2 to the overall temperature is less. Also, the effects of *T*1 on the overall temperature near to the center of the cylinder are decreased. This means that the effects of initial rate of temperature change have been remarkable, only, at the small times and with increasing the time the effects were not impressive. Due to the fact that the internal heat generation is time and space dependent, the values of *T*3 in areas close to the center of the cylinder have increased dramatically and will have the greatest impact on the overall temperature (*T*).

Figure 3a illustrates the temperature temporal trend at low times. As time arises, temperatures of cylinder's center (*r* = 0) enhance because of absorption of more energy compared to the other places. However, the boundary of cylinder surface remains relatively unchanged and keeps its initial temperature. According to the Figure 3b, with enhancing the time, the surface boundary temperature became affected by the entering energy, hence increasing the temperatures at a slower rate occurred. The temperature throughout the cylinder will continue to increase toward equilibrium between the center and the surface of the cylinder. This trend was shown in Figure 3c.

(**c**)

**Figure 3.** Temperature distributions from *t* = 0 to *t* = 0.1 (**a**), *t* = 0.0 to *t* = 1 (**b**), and *t* = 1 to *t* = 5 (**c**).
