Linear Damped Oscillations Underlying the Fractional Jeffreys Equation

Abstract

In this study, we consider a fractional-order extension of the Jeffreys equation (also known as the dual-phase-lag equation) by introducing the Reimann–Liouville fractional integral, of order $0<\nu <1$, to the Jeffreys constitutive law, where for $\nu =1$ it corresponds to the conventional Jeffreys equation. The kinetical behaviors of the fractional equation such as non-negativity of the propagator, mean-squared displacement, and the temporal amplitude are investigated. The fractional Langevin equation, or the fractional damped oscillator, is a special case of the considered integrodifferential equation governing the temporal amplitude. When $\nu =0$ and $\nu =1$, the fractional differential equation governing the temporal amplitude has the mathematical structure of the classical linear damped oscillator with different coefficients. The existence of a real solution for the new temporal amplitude is proven by deriving this solution using the complex integration method. Two forms of conditional closed-form solutions for the temporal amplitude are derived in terms of the Mittag–Leffler function. It is found that the proposed generalized fractional damped oscillator equation results in underdamped oscillations in the case of $0<\nu <1$, under certain constraints derived from the non-fractional case. Although the nonfractional case has the form of classical linear damped oscillator, it is not necessary for its solution to have the three common types of oscillations (overdamped, underdamped, and critical damped), unless a certain condition is met on the coefficients. The obtained results could be helpful for analyzing thermal wave behavior in fractals, heterogeneous materials, or porous media since the fractional-order derivatives are related to the porosity of media.

1. Backgrounds and Motivation

Heat transfer in fractals, heterogeneous materials, metals with nonuniformly distributed impurities, porous media, and nanoscale materials are only a few examples of the physical scenarios in which thermal conduction may experience odd behavior that differs from the standard diffusive “Fourier” behavior [1,2,3,4,5,6]. One of the useful mathematical tool for simulating such atypical behaviors is the fractional kinetic equation, see illustrative examples which introduced the fractional diffusion equation [7], the fractional Cattaneo equation [8], and the distributed-order fractional diffusion and relaxation equations [9,10]. Recently, a fractional version of the Jeffreys equation (also known as dual-phase-lag heat conduction equation) has been proposed in [11], and it can be related to the continuous-time random walk scheme [12] under some constraints. The fractional Jeffreys equation with three parameters has been recently shown to be a versatile mathematical tool for characterizing a variety of anomalous heat/mass transition behaviors, such as accelerated heat transfer mechanism, retarded heat transfer mechanism, and a crossover thermal conduction law that transits from ultrahigh thermal conductivity in the short-time domain to low thermal conductivity in the long time domain. It is worth noting the recent numerical results on the ordinary and the fractional dual-phase-lag heat conduction equations with different initial and boundary conditions [13,14]. Most recently, using a fractional kinetic approach, the turbulence spreading and anomalous diffusion on combs have been discussed in [15].

One of the early studies that focused on studying the temporal amplitude resulting from the space–time thermal wave equation governing temperature propagation with finite wave speed was that presented by Tzou [16,17,18]. The temporal amplitude corresponding to the classical thermal wave equation has the structure

$${\displaystyle \frac{d^{2}X\left(t\right)}{d{t}^2}}+\lambda {\displaystyle \frac{dX\left(t\right)}{dt}}+{\omega }^{2}X\left(t\right)=g\left(t\right),$$

(1)

where $\lambda$ and $\omega$ are positive constants, and $g\left(t\right)$ is the temporal profile of the external source. It was shown that the temporal amplitude $X\left(t\right)$ given in (1), with $\lambda$ and $\omega$ related to the classical thermal wave equation, displays “always” three types of oscillations: overdamped, critical damped, and underdamped. In addition, the case of underdamped oscillation leads to the thermal resonance phenomenon when the natural frequency exceeds a critical value [16,17,18]. In [19], Xu and Wang, found that the temporal amplitude $X\left(t\right)$ resulting from the dual-phase-lag “Jeffreys-type” equation has the same mathematical structure (1) but with different coefficients. However, $X\left(t\right)$ of the dual-phase-lag equation requires some restrictions on the constants to oscillate in any of the three ways: overdamped, critical damped, or underdamped. Otherwise, $X\left(t\right)$ does not exhibit oscillation.

In their examination of the motion of a rigid plate within a Newtonian fluid, Torvik and Bagley [20] formulated the fractional differential equation:

$${\displaystyle \frac{d^{2}X\left(t\right)}{d{t}^2}}+\lambda {\mathcal{D}}_t^{\nu }X\left(t\right)+{\omega }^{2}X\left(t\right)=g\left(t\right),$$

(2)

where $\nu =3/2$, and ${\mathcal{D}}_t^{\nu }$ is defined in their study as the Riemann–Liouville fractional derivative. Equation (2) was also studied in detail by Burov and Barkai [21,22] and Naber [23], where the fractional derivative used by them is defined in the Caputo sense with order $0<\nu <1$. The above studies agreed that the solution to Equation (2) has an oscillatory nature of underdamped type. Nonetheless, no existing research elucidates the characteristics of the space–time equation associated with such equation, especially when $0<\nu <1$. Equation (1) is referred to as the linear damped oscillator or damped-wave equation, but Equation (2) is known by several designations in the literature. When $1<\nu <2$, Equation (2) is referred to as the Torvik–Bagley equation, and when $0<\nu <1$, it is termed the linear fractional damped oscillator or fractional Langevin equation. Most recently, Equation (2) was generalized to a linear damped oscillator equation with two fractional parameters [24]:

$${\mathcal{D}}_t^{\mu }X\left(t\right)+\lambda {\mathcal{D}}_t^{\nu }X\left(t\right)+{\omega }^{2}X\left(t\right)=g\left(t\right),$$

(3)

where $1<\mu \le 2$ and $0<\nu \le 1$. We believe that equation (3) may represent the temporal amplitude of the fractional Cattaneo equation with two distinct fractional parameters [25]. Furthermore, Equation (2) encompasses some specific cases, such as the fractional undamped oscillator, and it can be extended to other cases; the reader may consult the current review [26].

In light of the aforementioned backgrounds, the aim of this study can be summarized in the following points:

• Revisiting the space–time equations that results in the temporal amplitude governed by (1), for example, the thermal wave equation and the dual-phase-lag equation.
• Finding the possible forms of the space–time equation that can produce the fractional differential Equation (2) governing the amplitude $X\left(t\right)$, in other words, what is the extended fractional Cattaneo equation which results in (2).
• Replacing the ordinary Jeffreys-type equation by an extended fractional version that yields a temporal amplitude that exclusively generalizes Equation (2).
• Deriving an exact solution in addition to the zeros of this solution.

The current study is organized as follows: in Section 2, we suggest a fractional version of the Jeffreys-type constitutive law including the Riemann–Liouville fractional integral. The properties of the resulting fractional Jeffreys-type equation and its special case, the fractional Cattaneo equation, are studied, specifically, the nonnegativity and the mean-squared displacement for the propagators are discussed. Moreover, the temporal amplitudes of the proposed equations are derived at the end of the second section, and some special cases are discussed. Section 3 is designated to discuss the nonfractional case of the temporal amplitude resulting from the ordinary Jeffreys equation by using the complex contour integral. The fractional differential equation generalizing Equation (2) is studied in detail in Section 4. Through a set of propositions, we show that the new temporal integrodifferential equation, including a new term, has complex poles in the left half complex plane. Using the complex contour integral, we study the existence of a real solution by deriving an analytical real form of the temporal amplitude, in addition to some corresponding exact solutions in terms of the Mittag–Leffler function. The numerical values of the zeros of the temporal amplitude are also calculated for different values of the fractional parameter. Lastly, we draw our conclusions and present the possible future generalizations of the model in Section 5.

2. Fractional Jeffreys Equation

In this section, we consider a fractional version of the Jeffreys equation, which has a temporal amplitude governed by a fractional differential equation generalizing the well-known one studied by Burov and Barkai [21,22] and Naber [23]. The non-fractional Jeffreys equation, known in the literature as the dual-phase-lag heat conduction [6], consists of the flux constitutive equation

$${\displaystyle \frac{1}{{\tau }_q}}q+{\displaystyle \frac{𝜕q}{𝜕t}}=-{\displaystyle \frac{\kappa }{{\tau }_q}}\left(1+{\tau }_T{\displaystyle \frac{𝜕}{𝜕t}}\right){\displaystyle \frac{𝜕T}{𝜕x}},$$

(4)

where $q\left(x,t\right)$ is the heat flux and $T\left(x,t\right)=\check{T}\left(x,t\right)-T_0$ is the temperature difference, $\check{T}\left(x,t\right)$ is the absolute temperature and $T_0$ is the room temperature. Also, the time constants ${\tau }_q$ and ${\tau }_T$ are defined as the flux phase lag and the temperature gradient phase lag, and $\kappa$ is the thermal conductivity. The spatial coordinate and the time variable are denoted by x and t, respectively. In addition, the thermal energy equation is given by

$$-{\displaystyle \frac{𝜕q}{𝜕x}}+Q=C{\displaystyle \frac{𝜕T}{𝜕t}},$$

(5)

where Q is the heat source term, and C is the heat capacity.

Let us consider the following generalization of the Jeffreys constitutive law (4):

$${\displaystyle \frac{1}{{\tau }_q^{2-\nu }}}{}_0{}^{RL}I_t^{1-\nu }q+{\displaystyle \frac{𝜕q}{𝜕t}}=-{\displaystyle \frac{\kappa }{{\tau }_q}}\left(1+{\tau }_T{\displaystyle \frac{𝜕}{𝜕t}}\right){\displaystyle \frac{𝜕T}{𝜕x}},$$

(6)

where ${}_0{}^{RL}I_t^{\nu }$ is the Riemann–Liouville fractional integral of order $\nu$, $0<\nu <1$, defined for any generic function as [27]

$${}_0{}^{RL}I_t^{\nu }f\left(t\right)≔\left\{\begin{array}{cc}{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}{\int }_0^t\frac{f\left(\tau \right)}{{\left(t-\tau \right)}^{1-\nu }}d\tau ,}\hfill & \nu >0,\hfill \\ {\displaystyle f\left(t\right),}\hfill & \nu =0.\hfill \end{array}\right.$$

(7)

It can be seen that when $\nu =1$, the fractional-order extended Equation (6) reduces to the Jeffreys Equation (4). Eliminating the flux q between Equations (5) and (6), we obtain a fractional-order Jeffreys-type equation as follows:

$${\displaystyle \frac{1}{{\tau }_q^{2-\nu }}}{}_0{}^C\mathcal{D}_t^{\nu }T+{\displaystyle \frac{{𝜕}^{2}T}{𝜕t^2}}=v^2\left(1+{\tau }_T{\displaystyle \frac{𝜕}{𝜕t}}\right){\displaystyle \frac{{𝜕}^{2}T}{𝜕x^2}}+{\displaystyle \frac{1}{C}}\left({\displaystyle \frac{1}{{\tau }_q^{2-\nu }}}{}_0{}^{RL}I_t^{1-\nu }+{\displaystyle \frac{𝜕}{𝜕t}}\right)Q,$$

(8)

where $v^2=\kappa /\left(C{\tau }_q\right)$ is the speed of the thermal waves in the case of classical thermal wave equation, (i.e., the case ${\tau }_T=0$ and $\nu =1$), and ${}_0{}^C\mathcal{D}_t^{\nu }$ is the Caputo fractional derivative defined for any generic function $f\left(t\right)$ as follows

$${}_0{}^C\mathcal{D}_t^{\nu }f\left(t\right)\left\{\begin{array}{cc}{\displaystyle {}_0{}^{RL}I_t^{1-\nu }\left\{\frac{𝜕f\left(t\right)}{𝜕t}\right\},}\hfill & 0<\nu <1,\hfill \\ {\displaystyle \frac{𝜕f\left(t\right)}{𝜕t},}\hfill & \nu =1.\hfill \end{array}\right.$$

(9)

When ${\tau }_T=0$, Equation (8) reduces to an unusual version of the fractional Cattaneo equation, as seen in [8,25,28]

$${\displaystyle \frac{1}{{\tau }_q^{2-\nu }}}{}_0{}^C\mathcal{D}_t^{\nu }T+{\displaystyle \frac{{𝜕}^{2}T}{𝜕t^2}}=v^2{\displaystyle \frac{{𝜕}^{2}T}{𝜕x^2}}+{\displaystyle \frac{1}{C}}\left({\displaystyle \frac{1}{{\tau }_q^{2-\nu }}}{}_0{}^{RL}I_t^{1-\nu }+{\displaystyle \frac{𝜕}{𝜕t}}\right)Q,$$

(10)

whereas upon setting $\nu =1$, Equation (8) reduces to the well-known dual-phase-lag heat conduction Equation [6]

$${\displaystyle \frac{1}{{\alpha }_T}}{\displaystyle \frac{𝜕T}{𝜕t}}+{\displaystyle \frac{1}{v^2}}{\displaystyle \frac{{𝜕}^{2}T}{𝜕t^2}}=\left(1+{\tau }_T{\displaystyle \frac{𝜕}{𝜕t}}\right){\displaystyle \frac{{𝜕}^{2}T}{𝜕x^2}}+{\displaystyle \frac{1}{\kappa }}\left(1+{\tau }_q{\displaystyle \frac{𝜕}{𝜕t}}\right)Q,$$

(11)

and if further putting ${\tau }_T=0$ in (11), it becomes the classical thermal wave (Cattaneo, or Maxwell–Cattaneo–Vernotte) equation [16,17]:

$${\displaystyle \frac{1}{{\alpha }_T}}{\displaystyle \frac{𝜕T}{𝜕t}}+{\displaystyle \frac{1}{v^2}}{\displaystyle \frac{{𝜕}^{2}T}{𝜕t^2}}={\displaystyle \frac{{𝜕}^{2}T}{𝜕x^2}}+{\displaystyle \frac{1}{\kappa }}\left(1+{\tau }_q{\displaystyle \frac{𝜕}{𝜕t}}\right)Q,$$

(12)

where ${\alpha }_T=\kappa /C$ is the thermal diffusivity.

2.1. Properties

In this subsection, we would like to explore the properties of the above fractional Jeffreys Equation (8) and the fractional Cattaneo Equation (10). To this end, we consider an initial value problem governed by (8) with initial conditions:

$$T\left(x,0\right)=\delta \left(x\right),{\left.{\displaystyle \frac{𝜕T\left(x,t\right)}{𝜕t}}\right|}_{t=0}=v^2{\tau }_T{\displaystyle \frac{{𝜕}^2\delta \left(x\right)}{𝜕x^2}},$$

(13)

where $\delta \left(·\right)$ is the Dirac delta function. It is worthy mentioning that the second initial condition in (13) was used previously in [28]. The nth derivative of the Dirac delta function is called the nth differentiator function. In the case of $n=1$, the first-order derivative of the Dirac delta function is called the unit doublet function, refer to Equation (A.26) in [28]. The use of the derivative of the Dirac delta function should be imposed on the flux as stated in Equation (47) of [12]. In the current formulation, the flux was eliminated, so the condition (13)b compensates for the condition that should be imposed on the flux to calibrate the initial value problem on the Jeffreys equation with that on the Cattaneo equation. Similar initial conditions for the initial value problem governed by (10) can be attached by replacing the second initial condition in (13) by $𝜕T\left(x,0\right)/𝜕t=0$. Upon neglecting any effect of the heat sources, and applying the Laplace transform for time t, defined by $\tilde{f}\left(x,s\right)=\mathcal{L}\left\{f\left(x,t\right);t\right\}(x,s)={\int }_0^{\infty }f\left(x,t\right)exp(-st)dt$, $s\in \mathbb{C}$ is the Laplace parameter, Equation (8) with the initial conditions (13) can be written as

$${\displaystyle \frac{1}{{\tau }_q^{2-\nu }}}\left[s^{\nu }\tilde{T}\left(x,s\right)-s^{\nu -1}\delta \left(x\right)\right]+s^2\tilde{T}\left(x,s\right)-s\delta \left(x\right)=v^2\left(1+{\tau }_{T}s\right){\displaystyle \frac{{\mathrm{d}}^2\tilde{T}\left(x,s\right)}{\mathrm{d}{x}^2}}.$$

Next, applying the Fourier transform defined by $\widehat{f}\left(\xi ,t\right)=\mathcal{F}\left\{f\left(x,t\right);x\right\}(\xi ,t)={\int }_{-\infty }^{\infty }f\left(x,t\right)exp\left(-\mathsf{ı}\xi x\right)dx$, $\xi \in \mathbb{R}$ is the Fourier parameter for space x, the above equation gives the solution in the Laplace–Fourier space as follows:

$$\widehat{\tilde{T}}\left(\xi ,s\right)={\displaystyle \frac{s+{\tau }_q^{\nu -2}{s}^{\nu -1}}{s^2+{\tau }_q^{\nu -2}{s}^{\nu }+v^2\left(1+{\tau }_{T}s\right){\xi }^2}}.$$

(14)

Rewriting Equation (14) on the form

$$\widehat{\tilde{T}}\left(\xi ,s\right)={\displaystyle \frac{a^2\left(s\right)}{s}}{\displaystyle \frac{1}{a^2\left(s\right)+{\xi }^2}},a^2\left(s\right)={\displaystyle \frac{s^2+\frac{s^{\nu }}{{\tau }_q^{2-\nu }}}{v^2(1+{\tau }_{T}s)}},$$

and inverting the Fourier transform in (14) using the useful relation ${\mathcal{F}}^{-1}\left\{1/\left(a^2+{\xi }^2\right)\right\}=exp\left(-a\left|x\right|\right)/\left(2a\right)$, we obtain the solution in the Laplace domain as

$$\tilde{T}\left(x,s\right)={\displaystyle \frac{1}{2vs}}\sqrt{{\displaystyle \frac{s^2+{\tau }_q^{\nu -2}{s}^{\nu }}{1+{\tau }_{T}s}}}exp\left(-{\displaystyle \frac{\left|x\right|}{v}}\sqrt{{\displaystyle \frac{s^2+{\tau }_q^{\nu -2}{s}^{\nu }}{1+{\tau }_{T}s}}}\right).$$

(15)

The solution of the fractional Cattaneo Equation (10) in the Laplace domain can be deduced from (15) by setting ${\tau }_T=0$:

$$\tilde{T}\left(x,s\right)={\displaystyle \frac{1}{2vs}}\sqrt{s^2+{\tau }_q^{\nu -2}{s}^{\nu }}exp\left(-{\displaystyle \frac{\left|x\right|}{v}}\sqrt{s^2+{\tau }_q^{\nu -2}{s}^{\nu }}\right).$$

(16)

It should be pointed out that the analytical inversion of the Laplace transform of (15) and (16) seems to be troublesome. Therefore, we explore as far as possible the properties of the solution of Equations (8) and (10) based on Equations (15) and (16).

2.1.1. Non-Negativity at Stake

In numerous innovative physical methodologies, the acceptance of an analytical function $f\left(x,t\right)$ for the mathematical representation of a physical phenomenon necessitates that $f\left(x,t\right)$ qualifies as a probability density function (PDF) [29]. Before investigating the solution $T\left(x,t\right)$ of (8) or (10) subject to the initial condition (13), we commence with the following definition: The function $f\left(x,t\right)$, $t\ge 0$, is said to be a probability density function (PDF), if and only if the following conditions are satisfied: (a) The function $f\left(x,t\right)$ is normalized over the interval $-\infty <x<\infty$ and time $t\ge 0$; (b) The function $f\left(x,t\right)$ is non-negative for $-\infty <x<\infty$ if it initiates the process with nonnegative initial condition. The normalization condition can be checked by integrating both sides of (15) over the domain $-\infty <x<\infty$, we get upon inverting the Laplace transform

$${\int }_{-\infty }^{\infty }T\left(x,t\right)dx=1,t\ge 0.$$

(17)

Therefore, the solution to the initial value problem governed by (8) is normalized. Similar results can be deduced for the solution of (10).

For the case of ${\tau }_T=0$ and $\nu =1$, the non-negativity of the solution $T\left(x,t\right)$ for the classical Cattaneo equation was discussed by Awad [25]. For the fractional Cattaneo equation proposed in this work (i.e., ${\tau }_T=0$ and $0<\nu <1)$, the solution (15) in the Laplace domain defined on the positive real line, $\lambda =\mathfrak{R}\left\{s\right\}>0$, can be written as

$$\tilde{T}\left(x,\lambda \right)={\displaystyle \frac{1}{2v\lambda }}\sqrt{{\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }}exp\left(-{\displaystyle \frac{\left|x\right|}{v}}\sqrt{{\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }}\right).$$

(18)

We note that when $\nu =1,$ the function $\sqrt{{\lambda }^2+\lambda /{\tau }_q}=\sqrt{\lambda \left(\lambda +1/{\tau }_q\right)}$ is a complete Bernstein function being a squared of multiplication of two complete Bernstein functions, $\lambda$ and $\lambda +1/{\tau }_q$, refer to [30] and the discussion about Corollary 3.2 in [25]. Therefore, the solution $\tilde{T}\left(x,\lambda \right)$ is a completely monotone function being a product of two completely monotone functions $\sqrt{{\lambda }^2+\lambda /{\tau }_q}/\lambda$ and $exp\left(-\sqrt{{\lambda }^2+\lambda /{\tau }_q}\right)$. Since for any function $f\left(x,t\right)$, if its Laplace transform $\tilde{f}\left(x,\lambda \right)$, confined to the positive real axis, is a completely monotone function, then $f\left(x,t\right)$ is nonnegative. This proves the non-negativity of the solution $T\left(x,t\right)$ of the classical Cattaneo equation.

For the case of ${\tau }_T=0$ and $0<\nu <1,$ we note that the function $\sqrt{{\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }}=\sqrt{\lambda \left(\lambda +{\tau }_q^{\nu -2}{\lambda }^{\nu -1}\right)}$ cannot be verified as a complete Bernstein function or even a Bernstein function. Indeed, the presence of the function ${\lambda }^{\nu -1}$, which is a Stieltjes function [30], gives a trouble to prove that the function $\sqrt{{\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }}$ can be written as the squared root of a product of two complete Bernstein functions, implying the difficulty of verifying that the function $\sqrt{{\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }}$ is a complete Bernstein function. On the other hand, if we consider the function

$$\tilde{\eta }\left(\lambda \right)={\displaystyle \frac{1}{\sqrt{{\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }}}},$$

next, rewrite it on the form

$$\tilde{\eta }\left(\lambda \right)={\displaystyle \frac{{\lambda }^{(2-\nu )\left(\frac{1}{2}\right)-1}}{{\left({\lambda }^{2-\nu }+{\tau }_q^{\nu -2}\right)}^{\frac{1}{2}}}},$$

and invert the Laplace transform, we obtain [31]

$$\eta \left(t\right)=E_{2-\nu ,1}^{{\displaystyle \frac{1}{2}}}\left(-{\left({\displaystyle \frac{t}{{\tau }_q}}\right)}^{2-\nu }\right),$$

(19)

where $E_{\alpha ,\beta }^{\gamma }\left(·\right)$ is the Mittag–Leffler function with three parameters [32]. Using the property that $E_{\alpha ,\beta }^{\gamma }\left(·\right)$ is a completely monotone function if $0<\alpha \le 1$ and $\alpha \gamma \le \beta \le 1$, we can conclude that the function $\eta \left(t\right)$ is not a completely monotone function. Based on the definition of Stieltjes function which states that the function $\tilde{\eta }\left(\lambda \right)$ is a Stieltjes if there exists a completely monotone function $\eta \left(t\right)$ such that $\tilde{\eta }\left(\lambda \right)={\int }_0^{\infty }\eta \left(t\right)exp\left(-\lambda t\right)dt$, we have that the function $\tilde{\eta }\left(\lambda \right)$ is not a Stieltjes function. Since the reciprocal of a Stieltjes function is a complete Bernstein function and vice versa [29], we conclude that the function $\sqrt{{\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }}=1/\tilde{\eta }\left(\lambda \right)$ is not a complete Bernstein function. Additionally, if we conduct a simple numerical test, we find that the condition

$${\left(-1\right)}^{n-1}{\displaystyle \frac{d^n}{d{\lambda }^n}}\left\{\sqrt{{\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }}\right\}\ge 0,n\in \mathbb{N}$$

is not satisfied which proves that the function is not even a Bernstein function. Therefore, the nonnegativity of the solution $T\left(x,t\right)$ of the fractional Cattaneo Equation (10) is not guaranteed. We can check the above results by inverting numerically the Laplace transform of (16) using the following “Durbin” or “modified Dubner–Abate” formula [11,33]

$$\begin{array}{c}\hfill f\left(x,t\right)={\displaystyle \frac{2exp\left(at\right)}{T_1}}\left\{-{\displaystyle \frac{1}{2}}\tilde{f}\left(x,a\right)+\mathfrak{R}\left[\sum _{k=0}^{NSum}\tilde{f}\left(x,a+{\displaystyle \frac{2\pi \iota k}{T_1}}\right)cos\left({\displaystyle \frac{2\pi kt}{T_1}}\right)\right]\right.\\ \hfill -\left.\mathfrak{I}\left[\sum _{k=0}^{NSum}\tilde{f}\left(x,a+{\displaystyle \frac{2\pi \iota k}{T_1}}\right)sin\left({\displaystyle \frac{2\pi kt}{T_1}}\right)\right]\right\},\end{array}$$

(20)

where $at\cong 4.7\sim 10$ and the parameter $T_1$ satisfies the inequality $0<t\le 2T_1$. The convergence analysis for the series (20) was presented in many works when the series “Riemann sum approximation” or the series “Durbin formula” implemented with any symbolic mathematics program, see e.g., [11], or even approximated using any numerical technique, see e.g., [34]. The error of the Durbin series (20) is given by [33]

$${\mathrm{ERROR}}_{\mathrm{D}}\left({\displaystyle \frac{{\gamma }_0}{t}},t,T_1\right)=\sum _{j=1}^{\infty }exp\left(-{\displaystyle \frac{2{\gamma }_{0}j{T}_1}{t}}\right)f\left(x,2j{T}_1+t\right),$$

where

$$\left|{\mathrm{ERROR}}_{\mathrm{D}}\left({\displaystyle \frac{{\gamma }_0}{t}},t,T_1\right)\right|\le {\displaystyle \frac{E}{exp\left(2{\gamma }_0{T}_1/t\right)-1}},$$

and E is the bound $f(x,t)\le E$. Furthermore, we draw the reader’s attention to the mathematical stability of the inverse Laplace transform tested on the real inverse formula in [35] and the convergence and stability criteria for some methods of the complex inverse formula in [36].

The implementation of series (20) was done using an appropriate symbolic program where the parameter $NSum$ was set to be ${10}^6$. Figure 1 shows the solution (16) of the fractional Cattaneo Equation (10) without heat sources for different values of the fractional parameter $\nu$ and at the time $t=1$, where ${\tau }_q=5$ and $v=1$. The box over interval $\left[-1,1\right]$ is a hallmark of the classical Cataneo equation. The limits of this interval comes from choosing the velocity of the thermal waves as $v=1$. Indeed, $x=vt$ leads now to $x=t$ for the thermal wavefront. Because we choose the time $t=1$, the two wavefronts are at $x=-1$ and $x=1$. As $\nu$ decreases, it appears that the solution loses its nonnegativity as shown in Figure 1. This behavior confirms the theoretical prediction stating that the nonnegativity of $T\left(x,t\right)$ for the fractional Cattaneo equation is not guaranteed.

For the case of ${\tau }_T>0$ and $\nu =1$, the nonnegativity of solution of the non-fractional Jeffreys equation was discussed in details for one-, two-, and three-dimensional spaces in [11]. It has been proved that when ${\tau }_T>{\tau }_q$, the solution is always nonnegative, and when ${\tau }_q>{\tau }_T$ the two-dimensional and the three-dimensional propagators may be negative. In the special case of ${\tau }_T>0$ and $0<\nu <1$, the solution in the Laplace domain defined on the positive real line, $\lambda =\mathfrak{R}\left\{s\right\}>0$, becomes

$$\tilde{T}\left(x,\lambda \right)={\displaystyle \frac{1}{2v\lambda }}\sqrt{{\displaystyle \frac{{\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }}{1+{\tau }_T\lambda }}}exp\left(-{\displaystyle \frac{\left|x\right|}{v}}\sqrt{{\displaystyle \frac{{\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }}{1+{\tau }_T\lambda }}}\right).$$

(21)

Similar difficulties to those we encountered in trying to prove that the function $\sqrt{{\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }}$ is a complete Bernstein function still exist after replacing it with the function $\chi \left(\lambda \right)=\sqrt{\left({\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }\right)/\left(1+{\tau }_T\lambda \right)}$. Here, we invoke the numerical test by graphically representing the function

$${\zeta }_n\left(\lambda \right)={\left(-1\right)}^{n-1}{\displaystyle \frac{d^n}{d{\lambda }^n}}\left\{\chi \left(\lambda \right)\right\}={\left(-1\right)}^{n-1}{\displaystyle \frac{d^n}{d{\lambda }^n}}\left\{\sqrt{{\displaystyle \frac{{\lambda }^2+{\tau }_q^{\nu -2}{\lambda }^{\nu }}{1+{\tau }_T\lambda }}}\right\},n\in \mathbb{N}.$$

(22)

Bearing in mind that, if ${\zeta }_n\left(\lambda \right)\ge 0$, $\forall n\in \mathbb{N}$, then the positive function $\chi \left(\lambda \right)$ is said to be Bernstein, see [30], we can examine the cases in which such a function may be Bernstein. In Figure 2, we plot the curves of the function ${\zeta }_n\left(\lambda \right)$ for $n=1,\dots ,5$, where we choose three different values for the fractional parameter to be $\nu =0.2$, $0.4$, $0.6$, and ${\tau }_q=5$ and ${\tau }_T=0.1$ (i.e., ${\tau }_q>{\tau }_T)$. On the other hand, Figure 3 shows the curves of the function ${\zeta }_n\left(\lambda \right)$ for $n=1,\dots ,5$, where $\nu =0.2,0.4,0.6$, and ${\tau }_q=0.1$, ${\tau }_T=5$ (i.e., ${\tau }_q<{\tau }_T$). It is observed that the function ${\zeta }_n\left(\lambda \right)$ shows negative values in both cases, especially, when $\nu$ is close to zero. The range of the variable $\lambda =\mathfrak{R}\left\{s\right\}>0$, on which the function ${\zeta }_n\left(\lambda \right)$ is negative, varies according to the cases ${\tau }_q\lessgtr {\tau }_T$. It is also noted that when $\nu$ approaches one, the function ${\zeta }_n\left(\lambda \right)$ preserves nonnegativity for $n\in \mathbb{N}$. Generally speaking, the function is not nonnegative for $\nu \in \left(0,1\right)$, however, as $\nu \to 1$, ${\zeta }_n\left(\lambda \right)$ may be positive for all $n\in \mathbb{N}$. Therefore, the function $\chi \left(\lambda \right)$ is not generally a Bernstein function; however, it may be Bernstein as $\nu \to 1$. In other words, we cannot say that the solution of the fractional Jeffreys Equation (8) without the effects of external sources and subject to the initial conditions (13) is always positive, but we can conclude that it may be positive when $\nu$ approaches 1. To validate the above predictions for the solution of (8), we implement the series (20) by setting $f\left(x,t\right)=T\left(x,t\right)$, where $\tilde{T}\left(x,s\right)$ is given by (15). In Figure 4, we draw the solution $T\left(x,t\right)$ of the fractional Jeffreys equation for different values of time. Figure 4a is designated for the case ${\tau }_q=5$ and ${\tau }_T=0.1$ (${\tau }_q>{\tau }_T$), while Figure 4b is dedicated to the case ${\tau }_q=0.1$ and ${\tau }_T=5$ (${\tau }_q>{\tau }_T$). In both subplots, we set the velocity of thermal waves to one ($v=1$). We pick the time instant $t=10$ for Figure 4a, and the time instant $t=0.5$ for Figure 4b. It is salient that the solutions shown in Figure 4a,b record negative values in accordance with the above predictions. The wavy nature shown in Figure 4a (${\tau }_q>{\tau }_T$) is apparent, while the diffusive nature shown in Figure 4b (${\tau }_q<{\tau }_T)$ is clear. Also, we can see that as $\nu$ approaches one (e.g., $\nu =0.8)$, the solution becomes nonnegative in both cases.

2.1.2. Mean-Squared Displacement

In this subsection, we calculate the mathematical form for the mean-squared displacement for the solution of the fractional Jeffreys Equation (8) without heat sources, in the case of occurrence of nonnegative solution, i.e., $\nu \to 1$. The mean-squared displacement can be obtained through the integral:

$$〈{\tilde{x}}^2\left(s\right)〉={\int }_{-\infty }^{\infty }{x}^2\tilde{T}\left(x,s\right)dx={\displaystyle \frac{2v^2}{s}}{\displaystyle \frac{1+{\tau }_{T}s}{s^2+{\tau }_q^{\nu -2}{s}^{\nu }}}.$$

(23)

By utilizing the Mittag–Leffler properties [31,32], we can invert the Laplace transform of Equation (23) to arrive at

$$〈x^2\left(t\right)〉=2v^2\left\{t^2{E}_{2-\nu ,3}\left(-{\left({\displaystyle \frac{t}{{\tau }_q}}\right)}^{2-\nu }\right)+{\tau }_{T}t{E}_{2-\nu ,2}\left(-{\left({\displaystyle \frac{t}{{\tau }_q}}\right)}^{2-\nu }\right)\right\}.$$

(24)

Using the asymptotic behavior of the Mittag–Leffler function, $E_{\alpha ,\beta }\left(-\lambda t^{\alpha }\right)\sim 1/\mathrm{\Gamma }\left(\beta \right),t\to 0$, and $E_{\alpha ,\beta }\left(-\lambda t^{\alpha }\right)\sim {\left(\lambda t^{\alpha }\right)}^{-1}/\mathrm{\Gamma }\left(\beta -\alpha \right),t\to \infty$, we obtain the short time behavior as follows:

$$〈x^2\left(t\right)〉\sim 2v^2\left\{{\displaystyle \frac{t^2}{2}}+{\tau }_{T}t\right\},t\to 0.$$

(25)

Thus, if ${\tau }_T$ is large enough, the mean-square displacement behaves in the short time as $〈x^2\left(t\right)〉\sim 2v^2{\tau }_{T}t$ (i.e., a diffusive behavior). On the contrary, if ${\tau }_T$ is small enough, the mean-squared displacement behaves in the short-time behavior as $〈x^2\left(t\right)〉\sim v^2{t}^2$ (i.e., a ballistic behavior). Moreover, the long-time behavior for the mean-squared displacement is given as

$$〈x^2\left(t\right)〉\sim 2v^2{\tau }_q^{2-\nu }\left\{{\displaystyle \frac{t^{\nu }}{\mathrm{\Gamma }\left(1+\nu \right)}}+{\tau }_T{\displaystyle \frac{t^{\nu -1}}{\mathrm{\Gamma }\left(\nu \right)}}\right\},t\to \infty .$$

(26)

Thus, for very large values of ${\tau }_T$, we have $〈x^2\left(t\right)〉\sim 2v^2{\tau }_q^{2-\nu }{\tau }_T{t}^{\nu -1}/\mathrm{\Gamma }\left(\nu \right)$, which represents unusual tail for the MSD. In addition, for small values of ${\tau }_T$, the mean squared displacement behaves as $〈x^2\left(t\right)〉\sim 2v^2{\tau }_q^{2-\nu }{t}^{\nu }/\mathrm{\Gamma }\left(1+\nu \right)$. When $\nu =1$ and ${\tau }_T>$${\tau }_q$, we have the fame behavior [12]:

$$〈x^2\left(t\right)〉\sim 2v^2\left\{\begin{array}{cc}{\displaystyle {\tau }_{T}t,}\hfill & t\to 0,\hfill \\ {\displaystyle {\tau }_{q}t,}\hfill & t\to \infty ,\hfill \end{array}\right.$$

(27)

namely, normal diffusion behavior in both the short and long time. A very special case is that in the case of very large values of ${\tau }_T\to \infty$, the mean-squared displacement behaves like

$$〈x^2\left(t\right)〉\sim 2v^2\left\{\begin{array}{cc}{\displaystyle {\tau }_{T}t,}\hfill & t\to 0,\hfill \\ {\displaystyle {\tau }_q{\tau }_T,}\hfill & t\to \infty ,\hfill \end{array}\right.$$

(28)

i.e., diffusive behavior in the short-time domain and immobilization behavior in the long-time domain.

2.2. Temporal Amplitude

In this subsection, we extract the temporal profile associated with the fractional Jeffreys Equation (8). Without the heat sources effect, we have the following integrodifferential equation

$${\displaystyle \frac{{\tau }_q^{\nu -1}}{{\alpha }_T}}{}_0{}^C\mathcal{D}_t^{\nu }T+{\displaystyle \frac{1}{v^2}}{\displaystyle \frac{{𝜕}^{2}T}{𝜕t^2}}=\left(1+{\tau }_T{\displaystyle \frac{𝜕}{𝜕t}}\right){\displaystyle \frac{{𝜕}^{2}T}{𝜕x^2}},$$

(29)

where ${\alpha }_T=v^2{\tau }_q.$ Supposing that the material parameters ${\tau }_q$ and ${\tau }_T$ are constants; therefore, there is a very short time interval in which we have that $\left[{\displaystyle \frac{1}{v^2{t}^2}}\right]=\left[{\displaystyle \frac{1}{{\alpha }_{T}t}}{\displaystyle \frac{{\tau }_q}{t}}\right]\gg \left[{\displaystyle \frac{1}{{\alpha }_T{\tau }_q}}{\left({\displaystyle \frac{{\tau }_q}{t}}\right)}^{\nu }\right]$ and $\left[{\displaystyle \frac{{\tau }_T}{t}}\right]\gg 1$ for ${\tau }_T\gg {\tau }_q>t$. In this temporal domain, the temperature is of diffusive nature. Indeed, using this approximation Equation (29) can be reduced to the diffusion equation as:

$${\displaystyle \frac{1}{v^2{\tau }_T}}{\displaystyle \frac{𝜕T}{𝜕t}}={\displaystyle \frac{{𝜕}^{2}T}{𝜕x^2}}.$$

(30)

Conversely, if the ${\tau }_T\ll t<{\tau }_q$, then $\left[{\displaystyle \frac{1}{v^2{t}^2}}\right]=\left[{\displaystyle \frac{1}{{\alpha }_{T}t}}{\displaystyle \frac{{\tau }_q}{t}}\right]\gg \left[{\displaystyle \frac{1}{{\alpha }_T{\tau }_q}}{\left({\displaystyle \frac{{\tau }_q}{t}}\right)}^{\nu }\right]$ and $\left[{\displaystyle \frac{{\tau }_T}{t}}\right]\ll 1$, and hence Equation (29) can be reduced to the wave equation as:

$${\displaystyle \frac{1}{v^2}}{\displaystyle \frac{{𝜕}^{2}T}{𝜕t^2}}={\displaystyle \frac{{𝜕}^{2}T}{𝜕x^2}}.$$

(31)

Let ${\omega }_n$ denote the natural frequency of the thermal wave in the short time-domain. Thus, the solution of Equation (31) can be expressed in the following form:

$$T\left(x,t\right)\sum _{n=1}^{\infty }{\phi }_n\left(x\right)exp\left(\mathsf{ı}{\omega }_{n}t\right),$$

(32)

where ${\phi }_n\left(x\right)$ are eigenfunctions to be ascertained. Additionally, we examine the boundary value problem of a thermal wave traveling in a one-dimensional thick plate of thickness L, guided by the wave Equation (31) and constrained by the boundary conditions as:

$$T\left(0,t\right)=T\left(L,t\right)=0.$$

(33)

Substituting the modal form (32) into the wave Equation (31), and using the boundary conditions (33), we obtain [37]

$${\omega }_n={\displaystyle \frac{n\pi v}{L}},n=1,2,\dots ,$$

(34)

and the eigenfunctions are one of the spectrum ${\phi }_n\left(x\right)\in \left\{sin\left({\displaystyle \frac{n\pi x}{L}}\right),n=1,2,\dots \right\}=\left\{sin\left({\displaystyle \frac{\pi x}{L}}\right),sin\left({\displaystyle \frac{2\pi x}{L}}\right),\dots \right\}.$ For the fractional Jeffreys Equation (29), the solution (32) is only a special case, and therefore, we generalize it to include the whole time domain as

$$T\left(x,t\right)≔\sum _{n=1}^{\infty }{\mathcal{E}}_n\left(t\right){\phi }_n\left(x\right),$$

(35)

where ${\mathcal{E}}_n\left(t\right)$ is the temporal amplitude of the solution of the fractional Jeffreys Equation (29) that needs to be determined. Substituting $T\left(x,t\right)$ in (35) into Equation (29), we obtain the following integrodifferential equation

$${\displaystyle \frac{d^2{\mathcal{E}}_n\left(t\right)}{d{t}^2}}+{\displaystyle \frac{1}{{\tau }_q^{2-\nu }}}{}_0{}^C\mathcal{D}_t^{\nu }{\mathcal{E}}_n\left(t\right)+{\omega }_n^2{\tau }_T{\displaystyle \frac{d{\mathcal{E}}_n\left(t\right)}{dt}}+{\omega }_n^2{\mathcal{E}}_n\left(t\right)=0.$$

(36)

It should be pointed out that Equation (36) is a generalized version of fractional damped wave equation which has not been discussed previously in the literature. It contains six familiar differential equations:

(a)

When ${\tau }_T=0$ and $0<\nu <1$, Equation (36) reduces to the fractional damped oscillator equation or fractional Langevin equation [21,22,23].

(b)

When ${\tau }_T=0$ and $1<\nu <2$, Equation (36) reduces to the Torvik–Bagley equation [20].

(c)

When ${\tau }_T=0$ and $\nu =1$, Equation (36) reduces to the linear damped wave equation studied by Tzou [16,17].

(d)

When ${\tau }_T=0$ and $\nu =0$, Equation (36) reduces to the linear undamped wave equation.

$${\displaystyle \frac{d^2{\mathcal{E}}_n\left(t\right)}{d{t}^2}}=-\left({\omega }_n^2+{\displaystyle \frac{1}{{\tau }_q^2}}\right){\mathcal{E}}_n\left(t\right).$$

(37)

(e)

When ${\tau }_T>0$ and $\nu =1$, Equation (36) reduces to the linear damped wave equation studied by [19,38].

(f)

When ${\tau }_T>0$ and $\nu =0$, Equation (36) reduces to a version of the linear damped oscillator:

$${\displaystyle \frac{d^2{\mathcal{E}}_n\left(t\right)}{d{t}^2}}+{\omega }_n^2{\tau }_T{\displaystyle \frac{d{\mathcal{E}}_n\left(t\right)}{dt}}+\left({\omega }_n^2+{\displaystyle \frac{1}{{\tau }_q^2}}\right){\mathcal{E}}_n\left(t\right)=0.$$

(38)

In the next two sections, we will investigate the wave behaviors (i.e., underdamped, critical damped, and overdamped oscillation) induced based on the nonfractional version of Equation (36).

3. The Non-Fractional Case

When ${\tau }_T>0$ and $\nu =1$ which is a non-fractional case. Equation (36) becomes a linear damped oscillation equation as

$${\displaystyle \frac{d^2{\mathcal{E}}_n\left(t\right)}{d{t}^2}}+\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right){\displaystyle \frac{d{\mathcal{E}}_n\left(t\right)}{dt}}+{\omega }_n^2{\mathcal{E}}_n\left(t\right)=0.$$

(39)

It is worth mentioning that Equation (39) was investigated in [19] by using a direct approach based on the discriminant of the quadratic equation. In what follows, we re-examine this equation using a complex integration of the Laplace transform inversion. Such a complex integration approach is useful for understanding the procedures of the next section dedicated for the fractional case. Assume the following initial conditions:

$${\mathcal{E}}_n\left(0\right)={\gamma }_0,{\left.{\displaystyle \frac{d{\mathcal{E}}_n\left(t\right)}{dt}}\right|}_{t=0}={\gamma }_1.$$

(40)

The Laplace transform of Equation (39) is given by

$${\tilde{\mathcal{E}}}_n\left(s\right)={\displaystyle \frac{{\gamma }_{0}s}{s^2+\left({\omega }_n^2{\tau }_T+\frac{1}{{\tau }_q}\right)s+{\omega }_n^2}}+{\displaystyle \frac{\left({\omega }_n^2{\tau }_T+\frac{1}{{\tau }_q}\right){\gamma }_0+{\gamma }_1}{s^2+\left({\omega }_n^2{\tau }_T+\frac{1}{{\tau }_q}\right)s+{\omega }_n^2}}.$$

(41)

Despite the availability of inversing Laplace transform using tabulated formulas, we employ the complex inversion formula to focus attention on what will be followed for the fractional case. In the absence of branch points, complex integration can be implemented using the Bromwich contour $C=\left[\gamma -\mathsf{ı}\infty ,\gamma +\mathsf{ı}\infty \right]\cup \mathrm{\Omega }$, as shown in Figure 5. Therefore, the temporal profile is given by

$$\begin{array}{c}{\mathcal{E}}_n\left(t\right)={\displaystyle \frac{1}{2\pi \mathsf{ı}}}{\int }_{\gamma -\mathsf{ı}\infty }^{\gamma +\mathsf{ı}\infty }{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right)ds\hfill \\ =\sum _{i=1}^{2}Res\left\{{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right),s=s_i\right\}-{\displaystyle \frac{1}{2\pi \mathsf{ı}}}\underset{R\to \infty }{lim}{\int }_{\mathrm{\Omega }}{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right)ds,\hfill \end{array}$$

(42)

where the first term represents the sum of residues of ${\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right)$ inside the Bromwich contour C, and the integral of the second term vanishes as $R\to \infty$. The residues of ${\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right)$ come from the zeros of the denominator of ${\tilde{\mathcal{E}}}_n\left(s\right)$, namely,

$$s^2+\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)s+{\omega }_n^2=0.$$

(43)

The roots of (43) are given by

$$s_{1,2}={\displaystyle \frac{1}{2}}\left[-\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)\pm \sqrt{\mathrm{\Delta }}\right],$$

(44)

where $\mathrm{\Delta }$ is the discriminant given by

$$\mathrm{\Delta }={\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)}^2-4{\omega }_n^2.$$

(45)

According to the value of this discriminant, the locations of the poles are determined. We have three possibilities:

i.

If $\mathrm{\Delta }<0$, we have two complex conjugate roots as shown in Figure 5,

$$s_{1,2}={\displaystyle \frac{1}{2}}\left[-\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)\pm \mathsf{ı}\sqrt{{4{\omega }_n^2-\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)}^2}\right].$$

(46)

ii.

If $\mathrm{\Delta }>0$, we have two real roots $s_3$ and $s_4$ as shown in Figure 5,

$$s_{3,4}={\displaystyle \frac{1}{2}}\left[-\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)\pm \sqrt{{\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)}^2-4{\omega }_n^2}\right].$$

(47)

iii.

If $\mathrm{\Delta }=0$, we have two equal roots $s_5=s_6$ as shown in Figure 5,

$$s_5=s_6=-{\displaystyle \frac{1}{2}}\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right).$$

(48)

These three cases are potential provided that the discriminant $\mathrm{\Delta }⋚0$ is a real number, in the sense that ${\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)}^2⋚4{\omega }_n^2$, or alternatively,

$${\omega }_n^2-{\displaystyle \frac{2}{{\tau }_T}}{\omega }_n+{\displaystyle \frac{1}{{\tau }_q{\tau }_T}}⋚0,$$

(49)

which leads to

$${\omega }_n⋚{\displaystyle \frac{1}{{\tau }_T}}\pm {\displaystyle \frac{1}{{\tau }_T}}\sqrt{1-{\displaystyle \frac{{\tau }_T}{{\tau }_q}}}.$$

(50)

Equation (50) clearly indicates the existence of two frequency thresholds, or critical frequencies ${\omega }_{nc1}$ and ${\omega }_{nc2}$ given by:

$${\omega }_{nc1}={\displaystyle \frac{1}{{\tau }_T}}+{\displaystyle \frac{1}{{\tau }_T}}\sqrt{1-{\displaystyle \frac{{\tau }_T}{{\tau }_q}}},{\omega }_{nc2}={\displaystyle \frac{1}{{\tau }_T}}-{\displaystyle \frac{1}{{\tau }_T}}\sqrt{1-{\displaystyle \frac{{\tau }_T}{{\tau }_q}}},$$

(51)

where ${\omega }_{nc1}>{\omega }_{nc2}$. Being a frequency, ${\omega }_{nc1}$ and ${\omega }_{nc2}$ should be positive real numbers which requires

$${\tau }_T<{\tau }_q.$$

(52)

Therefore, condition (52) is mandatory for the occurrence of the three cases (46)–(48). In view of (49), the complex conjugate roots (46) occur if $\left({\omega }_n-{\omega }_{nc1}\right)\left({\omega }_n-{\omega }_{nc2}\right)<0$, which helps us to deduce the condition

$${\omega }_{nc2}<{\omega }_n<{\omega }_{nc1},$$

(53)

where we rejected the case ${\omega }_n<{\omega }_{nc2}$ and ${\omega }_{nc1}<{\omega }_n$ since it is impossible to find such a number. In addition, the two real roots $s_3$ and $s_4$ (47) occur if $\left({\omega }_n-{\omega }_{nc1}\right)\left({\omega }_n-{\omega }_{nc2}\right)>0$, namely

$${\omega }_n<{\omega }_{nc2}<{\omega }_{nc1},or{\omega }_n>{\omega }_{nc1}>{\omega }_{nc2}.$$

(54)

Finally, the two equal roots $s_5=s_6$ occur if $\left({\omega }_n-{\omega }_{nc1}\right)\left({\omega }_n-{\omega }_{nc2}\right)=0$, i.e., at least one of the following occurs:

$${\omega }_n={\omega }_{nc1},or{\omega }_n={\omega }_{nc2}.$$

(55)

The above three conditions can be summarized through the schematic diagram shown in Figure 6.

We now attempt to obtain the solution structure in each case of the roots (46)–(48) by calculating the residues of Equation (42). In the case of Equation (47), the poles are simple, thereby the solution is given as

$$\begin{array}{c}{\mathcal{E}}_n\left(t\right)=\underset{s\to s_3}{lim}\left(s-s_3\right){\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right)+\underset{s\to s_4}{lim}\left(s-s_4\right){\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right)\hfill \\ ={\displaystyle \frac{exp\left(s_{3}t\right)}{2s_3+\left({\omega }_n^2{\tau }_T+\frac{1}{{\tau }_q}\right)}}\left({\gamma }_0{s}_3+{\gamma }_0\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)+{\gamma }_1\right)\hfill \\ +{\displaystyle \frac{exp\left(s_{4}t\right)}{2s_4+\left({\omega }_n^2{\tau }_T+\frac{1}{{\tau }_q}\right)}}\left({\gamma }_0{s}_4+{\gamma }_0\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)+{\gamma }_1\right).\hfill \end{array}$$

(56)

Since $s_3$ and $s_4$ are negative real numbers, refer to Figure 5, the solution (56) is called overdamped temperature oscillation that occurs when the natural frequency satisfies (54), see Figure 6. Since the poles $s_1$ and $s_2$ in (46) are also simple, the temporal amplitude can be computed by replacing $s_3$ and $s_4$ in (56) with $s_1$ and $s_2$, we obtain

$$\begin{array}{c}{\mathcal{E}}_n\left(t\right)=exp\left(-{\displaystyle \frac{1}{2}}\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)t\right)\left\{{\gamma }_{0}cos\left(t\sqrt{{{\omega }_n^2-{\displaystyle \frac{1}{4}}\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)}^2}\right)\right.\hfill \\ \left.+\left({\displaystyle \frac{{\gamma }_0\left({\omega }_n^2{\tau }_T+\frac{1}{{\tau }_q}\right)+2{\gamma }_1}{\sqrt{{4{\omega }_n^2-\left({\omega }_n^2{\tau }_T+\frac{1}{{\tau }_q}\right)}^2}}}\right)sin\left(t\sqrt{{{\omega }_n^2-{\displaystyle \frac{1}{4}}\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)}^2}\right)\right\}.\hfill \end{array}$$

(57)

The solution (57) is called underdamped temperature oscillation that occurs when the natural frequency satisfies (53), refer to Figure 6. The repeated roots $s_5=s_6$ are then not simple, rather a second-order pole. Therefore, the residue of (42) is computed as

$$\begin{array}{c}{\mathcal{E}}_n\left(t\right)=Res\left\{{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right),s=s_5\right\}=\underset{s\to s_5}{lim}{\displaystyle \frac{d}{ds}}\left\{{\left(s-s_5\right)}^2{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right)\right\}\hfill \\ =exp\left(-{\displaystyle \frac{1}{2}}\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right)t\right)\left[\left({\displaystyle \frac{1}{2}}\left({\omega }_n^2{\tau }_T+{\displaystyle \frac{1}{{\tau }_q}}\right){\gamma }_0+{\gamma }_1\right)t+{\gamma }_0\right].\hfill \end{array}$$

(58)

Lastly, the solution (58) is called critically damped temperature oscillation which occurs when the natural frequency exactly equals any of the critical frequencies ${\omega }_{nc1}$ or ${\omega }_{nc2}$, refer to Equation (55) and Figure 6.

4. The Fractional Case

In this section, we investigate the distinctive features of the new temporal profile (36), and see if it still have the same characteristics as that in the damped oscillator equation discussed previously [21,22,23]. The Laplace transform of Equation (36) subject to the initial conditions (40)

$${\tilde{\mathcal{E}}}_n\left(s\right)={\displaystyle \frac{{\gamma }_0\left(s+{\tau }_q^{\nu -2}{s}^{\nu -1}\right)}{s^2+{\tau }_q^{\nu -2}{s}^{\nu }+{\omega }_n^2{\tau }_{T}s+{\omega }_n^2}}+{\displaystyle \frac{{\gamma }_0{\omega }_n^2{\tau }_T+{\gamma }_1}{s^2+{\tau }_q^{\nu -2}{s}^{\nu }+{\omega }_n^2{\tau }_{T}s+{\omega }_n^2}}.$$

(59)

Because of the presence of the terms $s^{\nu -1}$ and $s^{\nu }$ in the numerator and the denominator of (59), respectively, a branch point at $s=0$ exists and a branch cut is needed in the contour integral to isolate the point $s=0$, as seen Figure 7. The poles of the function ${\tilde{\mathcal{E}}}_n\left(s\right)$ in (59) is again the zeros of its denominator, namely,

$$s^2+{\tau }_q^{\nu -2}{s}^{\nu }+{\omega }_n^2{\tau }_{T}s+{\omega }_n^2=0.$$

(60)

To find the zeros of (60), we firstly set $s=rexp\left(\mathsf{ı}\theta \right)$, which yields the following equations based on the real and imaginary parts are as:

$$r^{2}cos\left(2\theta \right)+{\tau }_q^{\nu -2}{r}^{\nu }cos\left(\nu \theta \right)+{\omega }_n^2{\tau }_{T}rcos\left(\theta \right)+{\omega }_n^2=0,$$

(61)

$$r^{2}sin\left(2\theta \right)+{\tau }_q^{\nu -2}{r}^{\nu }sin\left(\nu \theta \right)+{\omega }_n^2{\tau }_{T}rsin\left(\theta \right)=0.$$

(62)

The following propositions help to determine the possible locations of zeros:

Proposition 1.

There are no poles on the positive direction of the real axis $\mathrm{Re}\left\{s\right\}$, i.e., $s\ne r$.

Proof. 

Indeed, if we set $\theta =0$ in Equation (61), we get $r^2+{\tau }_q^{\nu -2}{r}^{\nu }+{\omega }_n^2{\tau }_{T}r+{\omega }_n^2=0$, which is impossible to be satisfied because all coefficients are positive numbers and $r\ge 0$. □

Proposition 2.

There are no poles on the negative direction of the real axis $\mathrm{Re}\left\{s\right\}$, i.e., $s\ne -r$.

Proof. 

Indeed, if we set $\theta =\pi$ in Equation (62), we get ${\tau }_q^{\nu -2}{r}^{\nu }sin\left(\nu \pi \right)=0$. Since $sin\left(\nu \pi \right)\ne 0$ for $0<\nu <1$, ${\tau }_q>0$, and $r\ge 0$, this equation cannot be satisfied for all values of r except $r=0$. □

Proposition 3.

There are no poles on the positive direction of the imaginary axis $\mathrm{Im}\left\{s\right\}$, i.e., $s\ne \mathsf{ı}r$.

Proof. 

Indeed, if we set $\theta =\pi /2$ in Equation (62), which results in ${\tau }_q^{\nu -2}{r}^{\nu }sin\left(\nu \pi /2\right)+{\omega }_n^2{\tau }_{T}r=0$. For $0<\nu <1$, we have $0<sin\left(\nu \pi /2\right)<1$, and therefore, this equation cannot be satisfied except at $r=0$. □

Proposition 4.

There are no poles on the negative direction of the imaginary axis $\mathrm{Im}\left\{s\right\}$, i.e., $s\ne \mathsf{ı}r$.

Proof. 

We can prove this proposition by setting $\theta =3\pi /2$ in Equation (62) and following a similar argument as in the above Proposition 3. □

Proposition 5.

There are no poles in the right half plane, $\mathrm{Re}\left\{s\right\}>0$.

Proof. 

We first set $0<\theta <\pi /2$, which represents possible poles that lie in the first quadrant of the complex plane. Since $sin\left(\theta \right)$, $sin\left(2\theta \right)$, and $sin\left(\nu \theta \right)$ are positive numbers for $0<\theta <\pi /2$, the only value that satisfies (62) is the zero solution, $r=0$. Similar results can be obtained from Equation (62) when setting $-\pi /2<\theta <0$, implying that the possible poles lie in the fourth quadrant. □

Proposition 6.

All poles are located on the left half plane. There are only two poles, due to the second-order Equation (60). In addition, if $s_1=r_{1}exp\left(\mathsf{ı}{\theta }_1\right)$ is a pole located in the second quadrant $\pi /2<{\theta }_1<\pi$, then the second pole $s_2$ is the complex conjugate of $s_1$, i.e., $s_2=r_{1}exp\left(\mathsf{ı}{\theta }_2\right)=r_{1}exp\left(-\mathsf{ı}{\theta }_1\right)$, $\pi <{\theta }_2<3\pi /2$, located in the third quadrant.

Proof. 

The proof of the last part can be checked by verifying that the two conjugates $\left(r_1,{\theta }_1\right)$ and $\left(r_1,-{\theta }_1\right)$ give similar forms in (61) and (62). □

Proposition 7.

One of the poles which is in the second quadrant is confined within the sector $\pi /2<\theta <\pi /\left(2-\nu \right)$, and the other pole is confined within the sector $-\pi /\left(2-\nu \right)<\theta <-\pi /2$.

Proof. 

First, we consider the pole lying within the second quadrant $\pi /2<\theta <\pi$, as Proposition 6 stated. Eliminating the term containing $r^{\nu }$ between (61) and (62), we get

$$r={\displaystyle \frac{1}{2}}\left\{-{\omega }_n^2{\tau }_T{\displaystyle \frac{sin\left(\left(1-\nu \right)\theta \right)}{sin\left(\left(2-\nu \right)\theta \right)}}+\sqrt{{\omega }_n^4{\tau }_T^2{\displaystyle \frac{{sin}^2\left(\left(1-\nu \right)\theta \right)}{{sin}^2\left(\left(2-\nu \right)\theta \right)}}+4{\omega }_n^2{\displaystyle \frac{sin\left(\nu \theta \right)}{sin\left(\left(2-\nu \right)\theta \right)}}}\right\},$$

(63)

where we reject the negative value since $r\ge 0$. Furthermore, the nonnegativity of r in Equation (63) leads to the necessary condition as:

$${\displaystyle \frac{sin\left(\nu \theta \right)}{sin\left(\left(2-\nu \right)\theta \right)}}>0.$$

(64)

For $0<\nu <1$ and $\pi /2<\theta <\pi$, we have that $sin\left(\nu \theta \right)>0$ and $sin\left(\left(2-\nu \right)\theta \right)⋚0$, namely, the condition (64) is not completely satisfied. To ensure (64) holding true, we have to confine the range of $\theta$ in the second quadrant to the sector

$${\displaystyle \frac{\pi }{2}}<\theta <{\displaystyle \frac{\pi }{2-\nu }}.$$

(65)

Secondly, we consider the pole lying within the third quadrant $\pi <\theta <3\pi /2$. Following the same procedures as the first part of the proof, we arrive at the second sector in the third quadrant:

$$-{\displaystyle \frac{\pi }{2-\nu }}<\theta <-{\displaystyle \frac{\pi }{2}}.$$

(66)

Hence, we have completed proof of Proposition 7. □

4.1. Existence of a Real Solution

Here, we raise a central question in this study as if the existence of two complex simple poles as described above leads to the existence of a real solution to Equation (36)? Using the complex integration, the solution of (36) can be implemented over the Hankel contour $\mathcal{C}=\left[\gamma -i\infty ,\gamma +i\infty \right]\cup {\mathrm{\Omega }}_1\cup \overline{AB}\cup {\mathrm{\Omega }}_2\cup \overline{CD}\cup {\mathrm{\Omega }}_3$ as illustrated graphically in Figure 7. The solution is then given by

$$\begin{array}{c}{\mathcal{E}}_n\left(t\right)={\displaystyle \frac{1}{2\pi \mathsf{ı}}}{\int }_{\gamma -\mathsf{ı}\infty }^{\gamma +\mathsf{ı}\infty }{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right)ds=\sum _{i=1}^{2}Res\left\{{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right),s=s_i\right\}\hfill \\ -{\displaystyle \frac{1}{2\pi \mathsf{ı}}}\left\{{\int }_{\overline{AB}}{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right)ds+{\int }_{\overline{CD}}{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right)ds\right\},\hfill \end{array}$$

(67)

where the integrations over ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_3$ eliminate themselves as $R\to \infty$, and the integration over the circle ${\mathrm{\Omega }}_2$ goes to zero as $\epsilon \to 0$. In accordance with the above propositions, we assume that there are two complex roots $s_1$ and $s_2$ for (60), such that $s_2$ is the complex conjugate of $s_1$, namely,

$$s_{1,2}=\alpha \pm \mathsf{ı}\beta =rexp\left(\pm \mathsf{ı}\theta \right),$$

(68)

where $r=\sqrt{{\alpha }^2+{\beta }^2}$, $tan\left(\theta \right)=\beta /\alpha$, r and $\theta$ are the solutions of nonlinear system (61) and (62). The residues in Equation (67) are determined through

$$\begin{array}{c}\sum _{i=1}^{2}Res\left\{{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right),s=s_i\right\}=\hfill \\ {\displaystyle \frac{exp\left(s_{1}t\right)}{2s_1+\nu {\tau }_q^{\nu -2}{s}_1^{\nu -1}+{\omega }_n^2{\tau }_T}}\left({\gamma }_0{s}_1+{\gamma }_0{\tau }_q^{\nu -2}{s}_1^{\nu -1}+{\gamma }_0{\omega }_n^2{\tau }_T+{\gamma }_1\right)\hfill \\ +{\displaystyle \frac{exp\left(s_{2}t\right)}{2s_2+\nu {\tau }_q^{\nu -2}{s}_2^{\nu -1}+{\omega }_n^2{\tau }_T}}\left({\gamma }_0{s}_2+{\gamma }_0{\tau }_q^{\nu -2}{s}_2^{\nu -1}+{\gamma }_0{\omega }_n^2{\tau }_T+{\gamma }_1\right).\hfill \end{array}$$

(69)

Substituting from (68) into (69) and implementing some algebraic simplifications, we obtain

$$\sum _{i=1}^{2}Res\left\{{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right),s=s_i\right\}=Res\left(t\right)=exp\left(\alpha t\right)\left[Acos\left(\beta t\right)+Bsin\left(\beta t\right)\right],$$

(70)

where A and B are real numbers given by

$$A={\displaystyle \frac{2}{D}}\left[{\gamma }_0\left(a_1+{\omega }_n^2{\tau }_T{a}_2\right)+{\gamma }_1{a}_2\right],B={\displaystyle \frac{2}{D}}\left[{\gamma }_0\left(b_1+{\omega }_n^2{\tau }_T{b}_2\right)+{\gamma }_1{b}_2\right],$$

(71)

$$\begin{array}{c}{a}_1=2r^2+{\omega }_n^2{\tau }_{T}rcos\left(\theta \right)+\left(\nu +2\right){\tau }_q^{\nu -2}{r}^{\nu }cos\left(\left(2-\nu \right)\theta \right)\hfill \\ +{\omega }_n^2{\tau }_T{\tau }_q^{\nu -2}{r}^{\nu -1}cos\left(\left(1-\nu \right)\theta \right)+\nu {\tau }_q^{2\left(\nu -2\right)}{r}^{2\nu -2},\hfill \end{array}$$

(72)

$$a_2={\omega }_n^2{\tau }_T+2rcos\left(\theta \right)+\nu {\tau }_q^{\nu -2}{r}^{\nu -1}cos\left(\left(1-\nu \right)\theta \right),$$

(73)

$$b_1=\left(2-\nu \right){\tau }_q^{\nu -2}{r}^{\nu }sin\left(\left(2-\nu \right)\theta \right)+{\omega }_n^2{\tau }_T{\tau }_q^{\nu -2}{r}^{\nu -1}sin\left(\left(1-\nu \right)\theta \right)-{\omega }_n^2{\tau }_{T}rsin\left(\theta \right),$$

(74)

$$b_2=2rsin\left(\theta \right)-\nu {\tau }_q^{\nu -2}{r}^{\nu -1}sin\left(\left(1-\nu \right)\theta \right),$$

(75)

$$\begin{array}{c}D=4r^2+4{\omega }_n^2{\tau }_{T}rcos\left(\theta \right)+4\nu {\tau }_q^{\nu -2}{r}^{\nu }cos\left(\left(2-\nu \right)\theta \right)\hfill \\ +2\nu {\omega }_n^2{\tau }_T{\tau }_q^{\nu -2}{r}^{\nu -1}cos\left(\left(1-\nu \right)\theta \right)+{\nu }^2{\tau }_q^{2\left(\nu -2\right)}{r}^{2\nu -2}+{\omega }_n^4{\tau }_T^2.\hfill \end{array}$$

(76)

As shown through the above propositions, $\alpha$ is a negative real number and it is the decay rate responsible for the damping represented by the residue (70). On the other hand, $\beta$ is a positive real number and it plays as the frequency of oscillation in the case of fractional damped oscillator. Equation (70) indicates that the residue is a real function representing damped oscillations with damping factor $\alpha$ and frequency $\beta$.

Secondly, we compute the integrals in the second and third term on the right-hand-side of (67). Let us define the function

$$I\left(t\right)=-{\displaystyle \frac{1}{2\pi \mathsf{ı}}}\left\{{\int }_{\overline{AB}}{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right)ds+{\int }_{\overline{CD}}{\tilde{\mathcal{E}}}_n\left(s\right)exp\left(st\right)ds\right\}.$$

(77)

Using the substitution $s=-r=rexp\left(\mathsf{ı}\pi \right)$ along $\overline{AB}$, and the substitution $s=-r=rexp\left(-\mathsf{ı}\pi \right)$ along $\overline{CD}$, we obtain

$$\begin{array}{c}I\left(t\right)={\displaystyle \frac{{\tau }_q^{\nu -2}sin\left(\nu \pi \right)}{\pi }}\hfill \\ \times {\int }_0^{\infty }{\displaystyle \frac{\left({\gamma }_0{\omega }_n^2{r}^{\nu -1}+{\gamma }_1{r}^{\nu }\right)exp\left(-rt\right)}{{\left(r^2+{\omega }_n^2-{\omega }_n^2{\tau }_{T}r\right)}^2+2{\tau }_q^{\nu -2}\left(r^2+{\omega }_n^2-{\omega }_n^2{\tau }_{T}r\right)r^{\nu }cos\left(\nu \pi \right)+{\tau }_q^{2\left(\nu -2\right)}{r}^{2\nu }}}dr.\hfill \end{array}$$

(78)

It is worthy noting that Equation (78) reduces to Equation (3.15) in [23]. It is salient from (78) that the integral $I\left(t\right)$ is a real function and it shows a temporal exponential decay, i.e., a second damping but it will be proved later that the contribution from this integral is negligible compared to the damped oscillations of the residues (70). In view of Equations (67), (70) and (77), the solution of the fractional Equation (36) exists, and it is a real function of the time given by

$${\mathcal{E}}_n\left(t\right)=Res\left(t\right)+I\left(t\right)=exp\left(\alpha t\right)\left[Acos\left(\beta t\right)+Bsin\left(\beta t\right)\right]+I\left(t\right).$$

(79)

As $\nu =0,1$, the decaying function $I\left(t\right)$ vanishes.

4.2. Closed-Form Solution

The solution of (36) can be obtained in a closed form as we will show in this subsection. Equation (59) can be rewritten as follows

$${\tilde{\mathcal{E}}}_n\left(s\right)=\left[{\gamma }_0\left(s+{\tau }_q^{\nu -2}{s}^{\nu -1}\right)+{\gamma }_0{\omega }_n^2{\tau }_T+{\gamma }_1\right]\sum _{k=0}^{\infty }{\left(-1\right)}^k{\displaystyle \frac{{\left({\tau }_q^{\nu -2}{s}^{\nu }+{\omega }_n^2{\tau }_{T}s\right)}^k}{{\left(s^2+{\omega }_n^2\right)}^{k+1}}}.$$

(80)

Upon employing the binomial expansion to (80), we obtain

$${\tilde{\mathcal{E}}}_n\left(s\right)=\left[{\gamma }_0\left(s+{\tau }_q^{\nu -2}{s}^{\nu -1}\right)+{\gamma }_0{\omega }_n^2{\tau }_T+{\gamma }_1\right]\sum _{k=0}^{\infty }{\left(-{\tau }_q^{\nu -2}\right)}^k\sum _{l=0}^k\left({\displaystyle \genfrac{}{}{0.0pt}{}{k}{l}}\right){\left({\displaystyle \frac{{\omega }_n^2{\tau }_T}{{\tau }_q^{\nu -2}}}\right)}^l{\displaystyle \frac{s^{\nu k+\left(1-\nu \right)l}}{{\left(s^2+{\omega }_n^2\right)}^{k+1}}}.$$

(81)

Utilizing useful properties of Mittag–Leffler function [31,32], the Laplace transform in (81) can be inverted to the form

$$\begin{array}{c}{\mathcal{E}}_n\left(t\right)=\sum _{k=0}^{\infty }{\left(-{\left({\displaystyle \frac{t}{{\tau }_q}}\right)}^{2-\nu }\right)}^k\sum _{l=0}^k\left({\displaystyle \genfrac{}{}{0.0pt}{}{k}{l}}\right){\left({\omega }_n^2{\tau }_T{\tau }_q{\left({\displaystyle \frac{{\tau }_q}{t}}\right)}^{1-\nu }\right)}^l\hfill \\ \times \left[{\gamma }_0{E}_{2,\left(2-\nu \right)k-\left(1-\nu \right)l+1}^{k+1}\left(-{\omega }_n^2{t}^2\right)+{\gamma }_0{\left({\displaystyle \frac{t}{{\tau }_q}}\right)}^{2-\nu }{E}_{2,\left(2-\nu \right)k-\left(1-\nu \right)l-\nu +3}^{k+1}\left(-{\omega }_n^2{t}^2\right)\right.\hfill \\ +\left.\left({\gamma }_0{\omega }_n^2{\tau }_T+{\gamma }_1\right)t{E}_{2,\left(2-\nu \right)k-\left(1-\nu \right)l+2}^{k+1}\left(-{\omega }_n^2{t}^2\right)\right].\hfill \end{array}$$

(82)

Equation (82) is a closed-form solution of the fractional damped oscillator Equation (36), and it is supposed to be equivalent to the form (79), as we shall discuss this point in the next subsection. When ${\tau }_T=0$, Equation (36) reduces to the fractional damped oscillator equation, and hence the closed-form solution (82) reduces to

$$\begin{array}{c}{\mathcal{E}}_n\left(t\right)=\sum _{k=0}^{\infty }{\left(-{\left({\displaystyle \frac{t}{{\tau }_q}}\right)}^{2-\nu }\right)}^k\left[{\gamma }_0{E}_{2,\left(2-\nu \right)k+1}^{k+1}\left(-{\omega }_n^2{t}^2\right)\right.\hfill \\ \left.+{\gamma }_0{\left({\displaystyle \frac{t}{{\tau }_q}}\right)}^{2-\nu }{E}_{2,\left(2-\nu \right)k-\nu +3}^{k+1}\left(-{\omega }_n^2{t}^2\right)+t{E}_{2,\left(2-\nu \right)k+2}^{k+1}\left(-{\omega }_n^2{t}^2\right)\right],\hfill \end{array}$$

(83)

where all terms of the second series are zero except the first term ($l=0$). It is worth mentioning that the closed-form (83) was derived previously (refer to Equation (3.1) in [39], also in the recent review [26]). The form (82) can be derived in another form that may keep convergence in a larger time domain:

$$\begin{array}{c}{\mathcal{E}}_n\left(t\right)={\gamma }_0\left[1+\sum _{k=1}^{\infty }{\left(-{\omega }_n^2{t}^2\right)}^k\sum _{l=0}^k\left({\displaystyle \genfrac{}{}{0.0pt}{}{k}{l}}\right){\left({\displaystyle \frac{{\tau }_T}{t}}\right)}^l{E}_{2-\nu ,2k-l+1}^k\left(-{\left({\displaystyle \frac{t}{{\tau }_q}}\right)}^{2-\nu }\right)\right]\hfill \\ +\left({\gamma }_0{\omega }_n^2{\tau }_T+{\gamma }_1\right)t\sum _{k=0}^{\infty }{\left(-{\omega }_n^2{t}^2\right)}^k\sum _{l=0}^k\left({\displaystyle \genfrac{}{}{0.0pt}{}{k}{l}}\right){\left({\displaystyle \frac{{\tau }_T}{t}}\right)}^l{E}_{2-\nu ,2k-l+2}^{k+1}\left(-{\left({\displaystyle \frac{t}{{\tau }_q}}\right)}^{2-\nu }\right).\hfill \end{array}$$

(84)

By setting ${\tau }_T=0$, the solution (84) reduces to

$$\begin{array}{c}{\mathcal{E}}_n\left(t\right)={\gamma }_0\left[1+\sum _{k=1}^{\infty }{\left(-{\omega }_n^2{t}^2\right)}^k{E}_{2-\nu ,2k+1}^k\left(-{\left({\displaystyle \frac{t}{{\tau }_q}}\right)}^{2-\nu }\right)\right]\hfill \\ +{\gamma }_{1}t\sum _{k=0}^{\infty }{\left(-{\omega }_n^2{t}^2\right)}^k{E}_{2-\nu ,2k+2}^{k+1}\left(-{\left({\displaystyle \frac{t}{{\tau }_q}}\right)}^{2-\nu }\right),\hfill \end{array}$$

(85)

which is equivalent to the solution of the linear fractional damped oscillator (83).

4.3. Numerical Validations

In the last part of this study, we verify the solutions (79), (82) and (84) derived above through numerical examples. For this goal, we choose the problem constants as

$${\gamma }_0=1,{\gamma }_1=1,{\omega }_n=1,{\tau }_T=0.1,{\tau }_q=5,$$

(86)

where we keep on the case ${\tau }_q>{\tau }_T$ which has been proven to be mandatory for oscillation as Section 3 described. Also, we choose the natural frequency ${\omega }_n$ obeying the inequality (53). In essence, the solution (79) depends mainly on finding explicit forms of the complex roots of the characteristic Equation (60). Therefore, determining such roots for $0<\nu <1$ is a crucial procedure to obtain the numerical results of (79). It appears that obtaining a symbolic form for the roots is impossible; thus, we determine the roots based on the choice in (86). Nevertheless, the presence of $s^{\nu }$ complicates the procedure, and we need further to compute the roots for every single value of $\nu$. For example, if $\nu =0.5$, we substitute $s=y^2$, and then Equation (60) yields

$$y^4+{\omega }_n^2{\tau }_T{y}^2+{\displaystyle \frac{1}{{\tau }_q^{1.5}}}y+{\omega }_n^2=0,$$

(87)

which has four roots can be obtained by any symbolic program, e.g., MATLAB R2019a. By squaring the roots of Equation (87) we get four possible roots for (60) which can be reduced to two by excluding the ones that do not satisfy the characteristic equation. We repeat this procedure and obtain complex roots as listed in

Table 1. Roots of Equation (60) subject to the choices (86) and different values for the fractional parameter $\nu$. The roots are $s_{1,2}=\alpha \pm \mathsf{ı}\beta$. We list here the real part $\alpha$ and the imaginary part $\beta$.

$\mathit{\nu }$	$\mathit{\alpha }=\mathbf{Re}\left\{{\mathit{s}}_{1,2}\right\}$	$\pm \mathit{\beta }=\mathbf{Im}\left\{{\mathit{s}}_{1,2}\right\}$
$0.9$	$-0.13572446142796015792$	$\pm 1.0055420464850605255$
$0.8$	$-0.12089430164463930864$	$\pm 1.0172050242476475366$
$0.7$	$-0.10666760367664177559$	$\pm 1.0245067144543005042$
$0.6$	$-0.09373551639167924395$	$\pm 1.0283727756330567944$
$0.5$	$-0.082435307949889785728$	$\pm 1.0296665716999075911$
$0.4$	$-0.072866628217380925004$	$\pm 1.0291229604172938558$
$0.3$	$-0.06498350054235819761$	$\pm 1.0273345005761487755$
$0.2$	$-0.058658536820465940149$	$\pm 1.0247624975190713669$
$0.1$	$-0.053724529570311836368$	$\pm 1.0217569374463880093$

.

In Figure 8, we display the solution (79) for different values of the fractional parameter $\nu$. The values $\alpha$ and $\beta$ are taken from Table 1, and they are converted to the polar form to obtain the corresponding r and $\theta$. The integral $I\left(t\right)$ in (78) is transformed using the Riemann definition to an infinite series. It is apparent that the fractional parameter $\nu$ affects the oscillation frequency and the temporal decay of the damped oscillation. This effect can be deduced easily from the dependence of $\alpha$ and $\beta$ on the choice of $\nu$ as the cases of Table 1. In view of Table 1, we can see that decreasing the fractional parameter $\nu$ decreases the decay of the damped oscillation ($\alpha$) and increases the oscillation frequency ($\beta$). It should be pointed that Naber [23], using a mathematical approach, proved that there are three distinct cases from the frequency near $\nu =0$. One of them is represented by Table 1, in which the frequency increases while moving away from $\nu =0$, then it falls as $\nu$ increases. The case $\nu =0$ in Figure 9 represents the solution of a version of the linear damped oscillation equation, refer to Case (f) of the Section 2.2. We further note that the contribution from the integral $I\left(t\right)$ is negligible compared to the contribution from the residue term $Res\left(t\right)$ in (70), as seen in Figure 9. In Figure 9a, the temporal decay is $A_{0}exp\left(\alpha t\right)$, where $A_0=\sqrt{A^2+B^2}$, A and B are given by (71)–(76).

The Mittag–Leffler function with three parameters is defined by [31,32,40]

$$E_{\alpha ,\beta }^{\gamma }\left(z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{\mathrm{\Gamma }\left(\gamma +n\right)}{\mathrm{\Gamma }\left(\gamma \right)\mathrm{\Gamma }\left(\alpha n+\beta \right)}}{\displaystyle \frac{z^n}{\mathrm{\Gamma }\left(n+1\right)}}.$$

(88)

Using the series representation (88), we compare the two solutions (82) and (84) with the solution (79) as shown in Figure 10. The solution to ${\mathcal{E}}_n\left(t\right)$ computed from (79) is denoted by ${\mathcal{E}}_n^{\left(1\right)}\left(t\right)$, the second computed from (82) is denoted by ${\mathcal{E}}_n^{\left(2\right)}\left(t\right)$ and the last computed from (84) is denoted by ${\mathcal{E}}_n^{\left(3\right)}\left(t\right)$. It is salient from Figure 10 that these three solutions agree on the interval $0<t<28$. In the case $t\ge 28$, the two solutions ${\mathcal{E}}_n^{\left(2\right)}\left(t\right)$ and ${\mathcal{E}}_n^{\left(3\right)}\left(t\right)$ fail to match the ${\mathcal{E}}_n^{\left(1\right)}\left(t\right)$; however, the solution ${\mathcal{E}}_n^{\left(3\right)}\left(t\right)$ consists of two Mittag–Leffler functions diverges slowly as compared with ${\mathcal{E}}_n^{\left(2\right)}\left(t\right)$, and it has computation time less than ${\mathcal{E}}_n^{\left(3\right)}\left(t\right)$ that consists of three Mittag–Leffler functions. General speaking, the divergence is familiar in such types of series [28,41]. In addition, this series varies according to the problem constants (86) and the upper sum limit that replaces ∞ in (88). To distinguish between the numerical values resulting from these three solutions, we list them in

Table 2. Numerical values of ${\mathcal{E}}_n^{\left(1\right)}\left(t\right)$, ${\mathcal{E}}_n^{\left(2\right)}\left(t\right)$, and ${\mathcal{E}}_n^{\left(3\right)}\left(t\right)$, for different values of the time.

t	${\mathcal{E}}_{\mathit{n}}^{\left(1\right)}\left(\mathit{t}\right)$	${\mathcal{E}}_{\mathit{n}}^{\left(2\right)}\left(\mathit{t}\right)$	${\mathcal{E}}_{\mathit{n}}^{\left(3\right)}\left(\mathit{t}\right)$
1	1.331236863580820	1.331236863435100	1.331236863435100
5	−0.364661370914947	−0.364661371209339	−0.364661371209332
10	−0.492324712316533	−0.492324712417995	−0.492324712422266
15	−0.097340084121303	−0.097340084010003	−0.097340090379925
20	0.129191929336087	0.129190923385198	0.129190492667021
25	0.098227985617070	0.097593075152690	0.098011078770671
28	−0.061841150514196	−0.086782407250272	−0.065655340247290
29	−0.051742658520400	−0.139447453779813	−0.061479491016212
30	0.005160433851042	−0.380474371633025	−0.019378272394193
31	0.050058958739673	−2.351444092768650	−0.012082346712144

.

5. Summary and Future Perspectives

In this work, we have suggested two fractional constitutive laws, using the Riemann–Liouville fractional integral, and generalizing the existing classical Jeffreys and the classical Cattaneo constitutive laws. The generalized constitutive laws yield fractional-order versions of the Jeffreys equation and the Cattaneo equation that have not been discussed in the literature to the best of our knowledge. The nonnegativity of the propagators for both fractional equations have been studied using the Bernstein function approach, and it is found that the propagators are in general not positive except some values of the fractional parameters $\nu$ which approaches to one. The mean-squared displacement is derived, in addition to their asymptotic behaviors.

Two fractional equations governing the temporal amplitudes are derived from the time fractional Jeffreys and Cattaneo equations, and it is found that the fractional equation resulting from the fractional Cattaneo equation coincides with the fractional Langevin equation or the fractional damped oscillator equation. Whilst the fractional differential equation governing the temporal amplitude resulting from the proposed fractional Jeffreys equations is exclusively discussed in this work. The temporal amplitude of the ordinary Jeffreys equation has been studied using the complex contour integral and it is found that although the differential equation governing this amplitude has the same mathematical structure of the linear damped oscillator equation, its solution should satisfy certain restriction to yield the three familiar types of oscillations (underdamped, overdamped, and critically damped). Further, we have studied the temporal amplitude of the suggested time-fractional Jeffreys equation using the same complex contour integral method. We show the existence of a real solution by deriving its analytical form. Utilizing the properties of the Mittag–Leffler function, we derive conditional closed-form solutions for the temporal amplitude, which are found to generalize existing formulas. The new term added to the fractional damped oscillator does not change its characteristics being underdamped oscillation, especially if the constraint of the nonfractional case is considered. The suggested fractional differential equation presented in this work is still in its early stages and needs more investigations by researchers, including experimental validation and numerical tests. In a future work, we may extend the present equation to include two or three fractional parameters.

To the best of our knowledge, Equation (29) has not yet been addressed in the literature. Consequently, it is anticipated that it will garner increased attention from researchers in the imminent future. Attention should be directed towards the uniqueness, existence, and stability of its solution. However, given the negative values that its propagator may produce, its suitability for simulating physical processes should be called into doubt. Nonetheless, our primary assertion was that the existence of a solution for the temporal profile of (29) indirectly demonstrates the existence of the solution for (29). In Equation (79), we demonstrated the existence of a real solution for the temporal profile described by the fractional differential Equation (36). Consequently, the solution (35) exists.

The intrinsic nonlocality of the fractional derivative is known to have a substantial relationship with the observed phenomenology. Therefore, the nature of damped oscillations represented by the current version may alter if different types of fractional derivatives are invoked instead of the Caputo-type used in this work. A comparative study on the use of different kernels of the fractional derivative employed in the generalized fractional damped oscillator is needed to enrich the fractional calculus applications.

A conditional phenomenon relevant to temperature oscillations and finite thermal wave speed, as well as the confirmation of underdamping for temperature oscillations and a relationship between the external applied frequency and the natural frequency of thermal waves, is the thermal resonance phenomenon, in which the temperature amplitude can be amplified at a specific frequency of the externally applied heat source. A recent experiment [42] has succeeded in observing the second sound in graphite at temperatures above 100 K by measuring the frequency of temperature oscillations. The fractional differential equation governing a certain phenomenon, as it has the underdamped oscillations, supports the use of fractional derivative in modeling various phenomena in physics and engineering.