Fractional Dynamics of Laser-Induced Heat Transfer in Metallic Thin Films: Analytical Approach

Abstract

This study introduces an innovative analytical solution to the time-fractional Cattaneo heat conduction equation, which models photothermal transport in metallic thin films subjected to short laser pulse irradiation. The model integrates the Caputo fractional derivative of order 0 < p ≤ 1, addressing non-Fourier heat conduction characterized by finite wave speed and memory effects. The equation is nondimensionalized through suitable scaling, incorporating essential elements such as a newly specified laser absorption coefficient and uniform initial and boundary conditions. A hybrid approach utilizing the finite Fourier cosine transform (FFCT) in spatial dimensions and the Laplace transform in temporal dimensions produces a closed-form solution, which is analytically inverted using the two-parameter Mittag–Leffler function. This function inherently emerges from fractional-order systems and generalizes traditional exponential relaxation, providing enhanced understanding of anomalous thermal dynamics. The resultant temperature distribution reflects the spatiotemporal progression of heat from a spatially Gaussian and temporally pulsed laser source. Parametric research indicates that elevating the fractional order and relaxation time amplifies temporal damping and diminishes thermal wave velocity. Dynamic profiles demonstrate the responsiveness of heat transfer to thermal and optical variables. The innovation resides in the meticulous analytical formulation utilizing a realistic laser source, the clear significance of the absorption parameter that enhances the temperature amplitude, the incorporation of the Mittag–Leffler function, and a comprehensive investigation of fractional photothermal effects in metallic nano-systems. This method offers a comprehensive framework for examining intricate thermal dynamics that exceed experimental capabilities, pertinent to ultrafast laser processing and nanoscale heat transfer.

1. Introduction

The study of heat transfer in materials subjected to intense short-duration laser pulses has garnered significant attention due to its relevance in a wide range of applications, such as laser material processing, nano-photonics, and ultrafast optics. Metallic thin films, which exhibit unique thermal properties compared to bulk materials, have become central to understanding photothermal phenomena at the nanoscale. The analysis of heat distribution inside a thin film due to photothermal short laser pulses involves understanding how the laser energy is absorbed by the material and how it propagates through the film [1,2]. The laser effect can vary depending on various factors such as pulse duration, energy density, and repetition rate [3]. Generally, in the case of short pulse mode, the heat distribution is more concentrated at the point of impact of the laser beam, leading to localized heating [4].

The traditional Fourier heat conduction model often fails to accurately describe the heat transport in such systems, especially when considering the ultrafast temporal dynamics associated with laser pulses [5,6]. To address these challenges, the Cattaneo equation, an extension of Fourier’s law that incorporates the finite speed of heat propagation, has been employed in the study of thermal transport. However, the classical Cattaneo model assumes instantaneous thermal propagation, which is inadequate for describing phenomena in materials with memory effects, or when the heat propagation occurs over time scales comparable to the laser pulse duration. In the case of fractional-order heat conduction, memory effects refer to a material’s ability to preserve a history of its thermal reaction, which means that the current state of heat flow is determined not only by the current temperature gradient but also by previous values. This is an important property in systems defined by fractional derivatives, which naturally include time-dependent memory kernels [7]. Similarly, anomalous thermal dynamics describe deviations from classical diffusion behavior, such as subdiffusive or wave-like thermal transport, often observed in heterogeneous or microscale systems [8]. These phenomena are effectively captured using fractional-order models, which generalize classical heat equations by incorporating nonlocal and history-dependent terms.

Advancements in fractional calculus have led to the incorporation of time-fractional derivatives into the Cattaneo model, thereby accounting for anomalous diffusion and more accurately modeling the heat transfer in systems with nonlocal behavior and temporal memory [9]. While short laser pulses produce immediate and highly localized thermal effects, the time-fractional Cattaneo model better describes the complex dynamics of photothermal propagation in metallic thin layers. The traditional method is enhanced by employing fractional derivatives to capture material memory effects and better characterize heat transfer, particularly under non-equilibrium conditions [10]. In recent years, the modeling of heat and mass transfer phenomena has progressively transitioned to generalized fractional frameworks to account for complex memory and spatial heterogeneity effects. In addition to standard Caputo derivatives, researchers have investigated distributed-order and variable-order formulations to tackle dynamic material behavior and multi-scale diffusion processes [11,12]. These methodologies offer enhanced modeling flexibility in systems where the diffusion dynamics are insufficiently characterized by a singular constant order. The applications of these generalized models encompass several domains, such as chemical and biological systems [13], glycolysis dynamics [14], and anomalous diffusion–reaction processes [15,16]. Concurrently, advancements in spectral and operator splitting techniques [17] have facilitated the efficient numerical resolution of these equations. The current study utilizes the Caputo derivative to obtain an exact analytical solution through Laplace and Fourier methods; these more generic operators present potential avenues for future research, especially in simulating nanoscale heat transfer that evolves spatially and temporally.

The present study expands upon previous analyses of non-Fourier heat conduction in laser-irradiated materials, including works by Xu and Wang [18], Talaee and Sarafrazi [19], and Al-Khairy and Al-Ofey [20], by presenting a more comprehensive framework that integrates fractional-order time derivatives, variability in laser absorption, and surface memory with relaxation time effects. Previous studies focused on classical or hyperbolic heat equations with time-dependent or spatially varying sources; however, our research enhances the modeling of photothermal propagation in metallic thin films through a time-fractional Cattaneo model, which more effectively accounts for thermal memory and nonlocal transport phenomena. The suggested model employs a physically realistic laser absorption profile, integrated with fractional boundary conditions, and is solved precisely using a mixed Laplace and finite Fourier cosine transform method. This hybrid method yields a closed-form analytical solution utilizing the Mittag–Leffler function, presenting novel insights into the dynamic thermal behavior of nanoscale systems subjected to pulsed laser activation. In contrast to previous analytical methods, our model integrates both the fractional temporal response and actual boundary interactions, establishing it as a more thorough instrument for examining ultrafast heat transfer in nanostructured materials, as a continuation of the experimentally validated study by Alkallas et al. [21]. While several previous studies have addressed time-fractional heat conduction using either Laplace or Fourier techniques individually, this work uniquely combines them in a targeted application to realistic, non-uniform laser-induced heating of metallic thin films, with boundary conditions reflecting surface-specific thermal interactions.

2. Mathematical Model

The primary heat transfer mechanism utilized to transfer the photothermal energy from the sample surface to its interior when a laser beam acts on the target is heat conduction. The time-fractional Cattaneo model, a modified Cattaneo–Vernotte (CV) model, accurately describes heat conduction in nonlocal and non-linear materials using fractional-order time derivatives [22]. This model is useful for researching heat distribution in memory effect and anomalous heat conduction materials [23]. It also accounts for heat propagation’s finite speed, unlike the Fourier heat conduction equation. Due to the quick and localized heat input, the time-fractional Cattaneo model is powerful for heat distribution in finite thin films under laser pulses [24]. The fractional Cattaneo model can describe the delayed material reaction to laser pulses that cause strong heat fluxes and non-equilibrium heat conduction [25].

The general time-fractional Cattaneo heat conduction equation can be expressed as [26]

$${\displaystyle \frac{\partial \theta }{\partial \zeta }}+{\delta }^p{\displaystyle \frac{{\partial }^{p+1}\theta }{\partial {\zeta }^{p+1}}}=\alpha {\nabla }^2\theta$$

(1)

where $0⩽p⩽1$ denotes the fractional order of differentiation and ${\displaystyle \frac{{\partial }^p}{\partial {\zeta }^p}}$ is the fractional differential operator with respect to time, according to the definition by Caputo [27]. It should be noted that the classical Fourier heat conduction equation is derived from Equation (1) when $\delta =0$, and the Cattaneo–Vernotte heat conduction equation is accounted for by $p=1$.

Herein, the time-fractional model represented by Equation (1) is constructed by generalizing the classical Cattaneo–Vernotte (CV) heat conduction equation to incorporate memory effects through fractional calculus. Specifically, it begins with the standard hyperbolic heat equation and replaces the classical time derivative in the constitutive relation with a Caputo fractional derivative of order $p\in \left(\mathrm{0,1}\right)$, a common approach for capturing nonlocal temporal behavior in materials with memory. This results in the appearance of a higher-order derivative term, where the derivative order becomes $f\left(p\right)=p+1$, reflecting the physical accumulation of memory in the system’s thermal response. This choice ensures consistency with the classical model when $p=1$, and provides a mathematically stable and physically interpretable formulation. However, other forms of the function $f\left(p\right)$ could also be considered, such as $f\left(p\right)=p^2+1,f\left(p\right)=2^p$, or other non-linear expressions, which would yield alternative models with different memory kernels and propagation characteristics. Similarly, the exponent to which the relaxation time $\tau$ is raised could be generalized. While these alternatives may initially appear to be of mathematical interest, they could also provide meaningful insights into heterogeneous materials, non-equilibrium transport, or scale-dependent behavior, particularly in nanostructured or functionally graded media.

2.1. Formulation of the Problem

In this section, the photothermal behavior within an isotropic thin film of thickness $L_0$ is theoretically modeled by the one-dimensional time-fractional Cattaneo heat conduction equation, which considers the variation in laser absorption as in Equation (2). The second term on the right-hand side of Equation (2) describes the internal heat generation due to absorption of laser energy inside the metallic thin film. In this term, $I\left(\zeta \right)$ highlights the time-dependent laser intensity profile, $\mathit{exp}\left(-\omega \xi \right)$ represents the exponential spatial attenuation of the laser beam due to absorption according to the Beer–Lambert law [28] that describes how light is absorbed as it passes through a medium, ω is the absorption coefficient of the material, and $\left(1-R_0\right)$ accounts for the fraction of incident laser energy absorbed (after reflection $R_0$). Within this setting, the film is initially in a state of thermal equilibrium $\theta ={\theta }_0$: Equation (3). The boundary conditions (Equations (3) and (4)) illustrate that as $\zeta >0$, the left side surface at $\xi =0$ is irradiated by a pulsed laser beam $q=q_0\left(U\left(\zeta \right)-U\left(\zeta -{\zeta }_0\right)\right)$, where $U\left(\zeta \right)$ is a unit step Heaviside function, and $q_0$ is the absorbed laser intensity. Meanwhile, the right-side surface at $\xi =L_0$ is adiabatic [18], which simplifies the analysis and isolates the front-side thermal response. This condition is reasonable for freestanding or thermally insulated thin films, especially over short time scales relevant to ultrafast laser heating. However, in practical experimental conditions where the film is supported by a substrate, heat transfer through the backside may occur. In such cases, convective or fixed-temperature boundary conditions may be incorporated to improve model accuracy.

$${\displaystyle \frac{\partial \theta }{\partial \zeta }}+{\delta }^p{\displaystyle \frac{{\partial }^{p+1}\theta }{\partial {\zeta }^{p+1}}}=\alpha {\displaystyle \frac{{\partial }^2\theta }{\partial {\xi }^2}}+{\displaystyle \frac{1}{\rho c}}\left(1-R_0\right)\omega I\left(\zeta \right)\mathit{exp}\left(-\omega \xi \right),0<\xi <L_0,\zeta >0.$$

(2)

$$\theta ={\theta }_0,{\displaystyle \frac{\partial \theta }{\partial \zeta }}=0,0⩽\xi ⩽L_0,\zeta =0.$$

(3)

$$-k{\displaystyle \frac{\partial \theta }{\partial \xi }}=q_0\left(f\left(\zeta \right)+{\delta }^p{\displaystyle \frac{{\partial }^{p}f}{\partial {\zeta }^p}}\right),\xi =0,\zeta >0$$

(4)

$$-k{\displaystyle \frac{\partial \theta }{\partial \xi }}=0,\xi =L_0,\zeta >0$$

(5)

where $f\left(\zeta \right)=U\left(\zeta \right)-U\left(\zeta -{\zeta }_0\right)$.

For further analysis, the following dimensionless quantities and parameters are introduced:

$$T={\displaystyle \frac{k\left(\theta -{\theta }_0\right)}{L_0{q}_0}},x={\displaystyle \frac{\xi }{L_0}},t={\displaystyle \frac{\alpha \zeta }{L_o^2}},\tau ={\displaystyle \frac{\alpha \delta }{L_o^2}},ϵ=\omega L_0,\phi \left(t\right)={\displaystyle \frac{I\left(t\right)}{q_0}}.$$

(6)

Substituting (6) into Equations (2)–(5) gives their analogous dimensionless form as follows:

$${\displaystyle \frac{\partial T}{\partial t}}+{\tau }^p{\displaystyle \frac{{\partial }^{p+1}T}{\partial t^{p+1}}}={\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+\left(1-R_0\right)ϵ\phi (t)\mathit{exp}\left(-\omega x\right),t>0,0⩽x⩽1$$

(7)

$$T(x,t)=0,{\displaystyle \frac{\partial }{\partial t}}T(x,t)=0,t=0,0⩽x⩽1$$

(8)

$$f(t)+{\tau }^p{\displaystyle \frac{{\partial }^{p}f\left(t\right)}{\partial t^p}}=-{\displaystyle \frac{\partial T}{\partial x}},t>0,x=0$$

(9)

$${\displaystyle \frac{\partial T}{\partial x}}=0,t>0,x=1$$

(10)

2.2. Analytical Solution

The solution methodology employs a hybrid analytical approach that integrates the Finite Fourier Cosine Transform (FFCT) in spatial dimensions with the Laplace transform in temporal dimensions to solve the time-fractional Cattaneo equation. The novelty of this approach lies in the precise formulation and integration of these transforms to handle fractional-order derivatives, non-uniform internal heat generation, and boundary-layer memory effects, resulting in an exact closed-form solution expressed via generalized Mittag–Leffler functions. The FFCT transforms the spatially dependent partial differential equation into a set of time-fractional ordinary differential equations by utilizing the orthogonality of cosine functions within a finite domain. The Laplace transform is subsequently employed to address the time-fractional Caputo derivative, transforming it into an algebraic expression in the Laplace domain. The resulting algebraic equation is explicitly solved, and the inverse Laplace transform is analytically executed using the two-parameter Mittag–Leffler function, which encapsulates the memory-dependent relaxation behavior typical of fractional-order systems. The inverse FFCT reconstructs the complete temperature field in both space and time, producing a closed-form expression that elucidates the influences of fractional order, relaxation time, and laser absorption on the photothermal response [29].

The finite Fourier Cosine transform (FFCT) of a function $T\left(x,t\right)$ is defined as [21,30]

$$F_c\left(T\left(x,t\right)\right)=T_c\left(n,t\right)={\int }_0^{L}T\left(x,t\right)\cos \left({\displaystyle \frac{n\pi x}{L}}\right)dx;0\le x\le L;n=1,2,3,\dots .$$

(11)

where the Kernel of the transformation is $\cos \left({\displaystyle \frac{n\pi x}{L}}\right)$.

Applying FFCT to Equation (7) in line with the initial and boundary conditions (8–10), it yields

$${\tau }^p{\displaystyle \frac{{\mathrm{d}}^{p+1}}{\mathrm{d}{t}^{p+1}}}{T}_{\mathrm{c}}\left(n,t\right)+{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{T}_{\mathrm{c}}\left(n,t\right)+n^2{\pi }^2{T}_{\mathrm{c}}\left(n,t\right)=f\left(t\right)+{\tau }^p{\displaystyle \frac{{\mathrm{d}}^{p}f}{\mathrm{d}{t}^p}}+{\displaystyle \frac{\left(1-R\right){ϵ}^2\phi }{n^2{\pi }^2+{ϵ}^2}}\left[1-e^{-ϵ}\cos \left(n\pi \right)\right]$$

(12)

$$T_{\mathrm{c}}\left(n,t\right)=0,{\displaystyle \frac{\partial }{\partial t}}{T}_{\mathrm{c}}\left(n,t\right)=0\mathrm{a}\mathrm{t}t=0$$

(13)

where $T_{\mathrm{c}}(n,t)={\int }_0^{1}T(x,t)\cos (n\pi x)\mathrm{d}x$.

Therefore, the Laplace-transformed solution of Equation (12) is

$${\bar{T}}_{\mathrm{c}}(n,s)={\displaystyle \frac{\left(1+{\tau }^p{s}^p\right)F\left(s\right)+\frac{\frac{\left(1-R\right){ϵ}^2\phi }{n^2{\pi }^2+{ϵ}^2}\left[1-e^{-ϵ}\cos \left(n\pi \right)\right]}{S}}{{\tau }^p{s}^{1+p}+s+n^2{\pi }^2}}$$

(14)

where the Laplace transforms for the functions $T_{\mathrm{c}}(n,t)$ and $f\left(t\right)$ are defined, respectively, as follows:

$$L\left[T_{\mathrm{c}}\left(n,t\right)\right]={\bar{T}}_{\mathrm{c}}(n,s)={\int }_0^{\mathrm{\infty }}{T}_{\mathrm{c}}(n,t)\exp (-st)\mathrm{d}t$$

$$L\left[f\left(t\right)\right]=F(s)={\int }_0^{\mathrm{\infty }}f(t)\exp (-st)\mathrm{d}t.$$

When $(n\ne 0)$, Equation (14) is reformulated as follows:

$$\begin{array}{c}{\bar{T}}_{\mathrm{c}}(n,s)=F(s){\displaystyle \sum _{m=0}^{\mathrm{\infty }}}(-1{)}^m{\displaystyle \frac{n^{2m}{\pi }^{2m}}{{\left(s^p+{\tau }^{-p}\right)}^{m+1}}}\left[{\tau }^{-\left(m+1\right)p}{s}^{-\left(m+1\right)}+{\tau }^{-mp}{s}^{p-m-1}\right]\\ +{\displaystyle \frac{\left(1-R\right){ϵ}^2\phi }{n^2{\pi }^2+{ϵ}^2}}\left[1-e^{-ϵ}\cos \left(n\pi \right)\right]\sum _{m=0}^{\mathrm{\infty }}(-1{)}^m{\displaystyle \frac{n^{2m}{\pi }^{2m}{\tau }^{-(m+1)p}{s}^{-(m+2)}}{{\left(s^p+{\tau }^{-p}\right)}^{m+1}}}.\end{array}$$

(15)

Hence, $T_{\mathrm{c}}(n,t)$ can be obtained from Equation (15) by implementing the inverse Laplace transform in addition to the convolution theorem as below

$$\begin{array}{c}{T}_{\mathrm{c}}(n,t)={\int }_0^t\sum _{m=0}^{\mathrm{\infty }}{\displaystyle \frac{{(-1{)}^{m}n}^{2m}{\pi }^{2m}}{m!}}\left[{\tau }^{-\left(m+1\right)p}{\eta }^{m\left(p+1\right)+p}{E}_{p,1+p+m}^{\left(m\right)}\left(\Omega \right)+{{\tau }^{-mp}{\eta }^{m(p+1)}E}_{p,1+m}^{\left(m\right)}(\Omega )\right]f(t-\eta )\mathrm{d}\eta +\\ {\displaystyle \frac{\left(1-R\right){ϵ}^2\phi }{n^2{\pi }^2+{ϵ}^2}}\left[1-e^{-ϵ}\cos \left(n\pi \right)\right]\sum _{m=0}^{\mathrm{\infty }}{\displaystyle \frac{{(-1{)}^{m}n}^{2m}{\pi }^{2m}{\tau }^{-(m+1)p}{t}^{m\left(p+1\right)+p+1}}{m!}}{E}_{p,2+p+m}^{\left(m\right)}\left(\Omega \right)\end{array}$$

(16)

where $\Omega =-{\tau }^{-p}{t}^p$ and the double-parameter Mittag–Leffler function $E_{\beta ,\gamma }\left(z\right)$ and its Laplace transform are defined in [31]

$$E_{\beta ,\gamma }(z)=\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{z^n}{\mathsf{\Gamma }(n\beta +\gamma )}},\beta >0,\gamma >0,z\in \mathbf{C}.$$

$$E_{\beta ,\gamma }^{\left(k\right)}(z)={\displaystyle \frac{{\mathrm{d}}^k}{\mathrm{d}{z}^k}}{E}_{\beta ,\gamma }(z)=\sum _{j=0}^{\mathrm{\infty }}{\displaystyle \frac{(j+k)!z^j}{j!\mathsf{\Gamma }\left(\beta \right(j+k)+\gamma )}},$$

$$L\left(t^{\beta k+\gamma -1}{E}_{\beta ,\gamma }^{\left(k\right)}\left(\pm a{t}^{\beta }\right)\right)={\displaystyle \frac{k!s^{\beta -\gamma }}{{\left(s^{\beta }\mp a\right)}^{k+1}}},R(s)>|a{|}^{1/\beta },$$

where $L$ and $R$ represent the Laplace transform operator and the real part operator, respectively.

When $(n=0)$, the solution of Equation (14) can be obtained in the form

$$T_{\mathrm{c}}\left(0,t\right)={\int }_0^{t}f\left(\eta \right)\mathrm{d}\eta +\left(1-R\right)\left(1-e^{-ϵ}\right){\phi \tau }^{-p}{\int }_0^t\left(\mathrm{t}-\eta \right){\eta }^{p-1}{E}_{p,p}\left(-{\tau }^{-p}{\eta }^p\right)\mathrm{d}\eta ,$$

(17)

where $E_{p,p}\left(-{\tau }^{-p}{\eta }^p\right)=\sum _{j=0}^{\mathrm{\infty }}{\displaystyle \frac{{\left(\frac{-\eta }{\tau }\right)}^{pj}}{\mathsf{\Gamma }\left[p(j+1)\right]}}$.

Consequently, from Equations (16) and (17), the final solution of the time-fractional Cattaneo heat conduction can be derived using the finite cosine inverse Fourier transform as follows:

$$T(x,t)=T_{\mathrm{c}}(0,t)+2{\sum }_{n=1}^{\mathrm{\infty }}{T}_{\mathrm{c}}(n,t)\cos \left(n\pi x\right).$$

(18)

3. Results and Discussion

Figure 1 depicts how the temperature distribution T(x,t) inside the laser-heated thin film varies as the fractional parameter p in the time-fractional Cattaneo model is changed. The remaining parameters are kept constant (τ = 1, R = 0.1, ϵ = 103, φ = 1). This setting modifies the fractional derivative component in the Cattaneo model to regulate the material’s peculiar heat diffusion behavior and memory effects. That is, when it comes to laser heating, varying the value of p is a realistic technique for altering heat transmission qualities. The temperature profile in Figure 1a with p = 0.2 shows substantial oscillations and a high-amplitude wave-like structure, implying enhanced memory and nonlocal effects. A lower fractional order slows down the diffusion process, resulting in more prolonged thermal effects. Also, the oscillations are still visible but more damp in Figure 1b with p = 0.5 when compared to Figure 1a, showing a shift toward conventional diffusion behavior while keeping some aberrant traits. Meanwhile, in Figure 1c, T(x,t) becomes smoother with substantially fewer oscillations, indicating that thermal wave propagation is less impacted by memory effects, and becomes more diffusion-driven. Eventually, Figure 1d with p = 1 depicts the classical Cattaneo model (no fractional derivative), where the temperature distribution is the smoothest of all cases, with almost no oscillatory behavior, indicating a more traditional heat conduction profile with finite propagation speed and no anomalous diffusion.

A significant characteristic observed in Figure 1a–c is the existence of discontinuities or abrupt gradients in the thermal field, particularly evident at early pulse durations (notably for 0 < t < 1) at fractional orders of p. The presence of these discontinuities can be ascribed to the intrinsic memory effects and nonlocal transport mechanisms represented by the time-fractional Cattaneo model. At lower p values, the system demonstrates pronounced non-Fourier behavior, characterized by a significant dependence on thermal history. This results in fast, discontinuous temperature fluctuations, akin to phase transition phenomena or sudden energy redistribution events, especially during the early phases after laser pulse heating. The distinct profiles indicate a postponed yet vigorous release or absorption of energy, maybe associated with latent heat processes devoid of gradual temperature fluctuations. As p approaches 1, the thermal behavior progressively shifts towards Fourier-like diffusion, mitigating these discontinuities and resulting in more continuous and predictable temperature distributions. The observed discontinuities directly illustrate the interaction between fractional dynamics, early-time memory effects, and potential phase transition mechanisms in the film, highlighting the essential role of the fractional order in regulating the smoothness and continuity of the thermal field. Furthermore, the discontinuities in Figure 1a–c may be influenced by the Gibbs phenomenon, a common effect in Fourier series representations near sharp gradients as well. It should be noted that numerical artifacts can be obtained due to the series truncation and do not imply physical discontinuities. This must also be considered when interpreting such complicated behaviors.

Figure 2 illustrates the temperature distribution T(x) within the metallic thin film under the time-fractional Cattaneo model for various values of the dimensionless absorption parameter ϵ, at adjusted values of p = 0.2, τ = 1, R = 0.1, t = 2.1, and φ = 1. Increasing ϵ (the amount of laser energy absorbed by the thin film) enhances the temperature magnitude throughout the thin film without altering the fundamental oscillatory nature of the distribution. A larger absorption value means that more of the laser energy is turned into heat within the material, which results in a higher temperature amplitude. The peak and trough values become more pronounced, indicating stronger thermal responses within the metallic thin film. The wave-like structure remains consistent in shape and frequency across the profiles, suggesting that elevating ϵ does not affect the spatial frequency of the thermal wave but instead amplifies the intensity of heat distribution. This results in higher thermal excitation, leading to more intense temperature peaks and a greater sensitivity to heat input, especially relevant under short-pulse laser heating. That the thin film absorption strength simplified in ϵ helps in controlling the thermal amplification in photothermal processes involving fractional-order models.

Figure 3 depicts the temperature distribution T(x) in a metallic thin film for different relaxation time τ values, employing the time-fractional Cattaneo model with parameters set at p = 0.2, ϵ = 10, R = 0.1, t = 2.1, and φ = 1. To accurately forecast and regulate the thermal dynamics of thin films subjected to laser heating, it is essential to comprehend the impact of relaxation time on temperature progression, particularly within the context of the time-fractional Cattaneo model, which integrates nonlocal effects and thermal memory. The relaxation time τ measures the lag in the material’s thermal response after energy is introduced by a laser pulse. A reduced τ leads to an expedited thermal response, generating steep temperature gradients and significant oscillations. A longer τ enables the system to adapt more gradually, resulting in a more uniform energy distribution and smoother temperature profiles. In Figure 3, an increase in τ markedly diminishes the amplitude of temperature oscillations. At reduced relaxation durations (τ = 0.25), the temperature profile displays high-frequency oscillations characterized by pronounced peaks and deep troughs, signifying a swift and vigorous thermal response. As τ increases, these oscillations increasingly diminish, indicating a shift from wave-like to diffusion-dominated heat transmission. This behavior results from the extended relaxation period, which allows the system ample opportunity to attain thermal equilibrium, facilitating more uniform energy distribution and diminishing temperature fluctuations. This pattern aligns with the Boltzmann distribution, which characterizes the statistical distribution of energy among particles in an equilibrium system. This theory posits that energy tends to reside in lower energy levels over time, resulting in more uniform and consistent thermal behavior [26].

The time-fractional Cattaneo model predicts a temporal alteration in the internal temperature distribution of the thin film when subjected to laser pulse heating. The temporal evolution influences the temperature distribution within the film due to heat conduction, thermal diffusion, and laser energy absorption. Figure 4 illustrates the temporal progression of the temperature distribution T in a metallic thin film according to the time-fractional Cattaneo model, with parameters p = 0.2, ϵ = 10³, R = 0.1, τ = 1, and φ = 1. As time advances from t = 2.1 to t = 2.5, the amplitude of the thermal oscillations significantly rises, indicating an ongoing buildup and dissemination of thermal energy within the film. Initially, the temperature profile displays significant oscillations with acute peaks, reflecting the wave-like characteristics of heat transfer in the Cattaneo model. The temperature fluctuations become more extreme with time, indicating heightened thermal responses from pulsed laser energy absorption and internal heat wave interactions. Notwithstanding the increase in amplitude, the spatial frequency of the oscillations remains mostly unchanged, indicating that although the intensity of the thermal field intensifies over time, the essential wave-like characteristics of heat transfer are maintained. This behavior underscores the interaction between fractional-order memory effects and finite propagation rates in regulating the nonlocal thermal dynamics of the system.

Figure 5 illustrates the temporal change in thermal behavior T for τ = 1, R = 0.5, ϵ = 10, and φ = 1. The temperature rises significantly over time. The initial response time for various fractional order parameters differs and increases with higher p values, each of which demonstrates distinct heat propagation velocities. Augmenting p in the fractional Cattaneo model often leads to a more gradual decline in temperature at the extremities of a finite metallic thin film subjected to laser pulse heating. A greater value of p results in more significant history-dependent behavior and extended memory effects in the heat conduction process. The parameter p substantially influences the thermal response; a lower p value (p = 0.4) results in a greater temperature increase due to the gradual temporal dissipation of heat, whereas a larger p value (p = 1) leads to a more subdued temperature escalation over time. This shows that an increase in p serves a damping or moderating function in the thermal profile. Furthermore, the temperature diminishes spatially from x = 0 to x = 1. In Figure 5a, the temperature increases more swiftly and is highest at the surface of the thin film (x = 0) due to boundary heating and proximity to the heat source. Also, for the adiabatic surface (x = 1) in Figure 5b, the overall temperature is lower, and the influence of fractional memory diminishes with distance from the heat source. At (x = 0), where the pulse is likely introduced, the memory effect is instantly observable and most pronounced; conversely, at (x = 1), the thermal wave has traveled, resulting in heat dispersion that diminishes the effects of thermal memory [32].

4. Validation and Physical Relevance

A verification of the obtained analytical solution has been provided in (Appendix A). It is worth emphasizing that, while the current study only presents an analytical treatment of the time-fractional Cattaneo model, recent experimental work supports the utilized mathematical framework. Alkallas et al. [21] found remarkable agreement between finite Fourier transform-based models and experimental findings of thermal diffusion in steel alloys using laser-induced breakdown spectroscopy (LIBS) under single- and double-pulse conditions. Although their model is based on traditional Fourier theory, the technique and experimental validation provide a sound framework for generalizing the analysis to fractional-order models. Building on this, our formulation incorporates memory effects and nonlocal thermal behavior via the time-fractional Caputo derivative, potentially providing more precise descriptions of heat transport under ultrafast laser excitation. Future research will seek to test the proposed fractional model using experimental settings like those described in [21], bridging the gap between theory and real-world thermal dynamics in thin metallic films.

It should be noted that the analytical results of this work have been corroborated by their alignment with recognized models in the literature. The behavior of the fractional Cattaneo model in our formulation substantially corresponds with the analytical findings presented by Xu and Wang [18], who examined analogous pulse heat flux situations in finite areas. Figure 6 presents a comparative analysis of the thermal behavior predicted by the present model and the benchmark solution (Xu & Wang, 2018) [18] for a fixed fractional order p = 0.8, relaxation time τ = 1, and zero-source terms (ϵ = φ = 0). At the early time t = 0.1 (Figure 6a), both models predict relatively flat temperature distributions, but the Xu & Wang solution exhibits slightly higher magnitudes near the surface, indicating a stronger thermal memory effect. By t = 0.5 (Figure 6b), the Xu & Wang model maintains higher temperatures across the domain, while the present model demonstrates a more dissipative response, consistent with smoother diffusion and reduced thermal inertia. These discrepancies can be attributed to differences in how each model treats boundary conditions and initial energy propagation, with the present model incorporating a more generalized framework that naturally leads to enhanced damping of thermal waves over time. Therefore, the comparison confirms that while both formulations align qualitatively, the present model provides a more physically realistic temperature profile in the absence of external sources, especially at later stages of the diffusion process. Moreover,

Table 1. Time development of the temperature T(x, t) at (a) x = 0 and (b) x = 1 in the case of a Fourier model with (τ = 0, φ = 0).

t	T(x, t)
	(a) x = 0		(b) x = 1
	Present	Ref. [18]	Present	Ref. [18]
0.1	0.372708	0.380162	0.1007231	0.1248162
0.3	0.517713	0.524029	0.2116723	0.2398079
0.5	0.719142	0.725336	0.4071926	0.4258017
0.7	0.934299	0.962019	0.6020973	0.6270351
1.0	1.195604	1.211253	0.8093021	0.8230743
1.5	1.821032	1.839273	1.3542706	1.3711425
2.5	1.998765	2.000000	1.9684253	1.9999785

presents the temporal evolution of temperature T (x, t) at the boundaries x = 0 and x = 1 under the classical Fourier model with (τ = φ = 0). The results show close agreement between the present model and Ref. [18] across all time steps, with minor deviations that diminish as time increases. At early times, slight underestimation is observed in the present model, particularly at x = 1, likely due to the cumulative effects of initial conditions and spatial discretization. However, by t = 2.5, the solutions converge almost perfectly, confirming the reliability and consistency of the present formulation in reproducing standard Fourier behavior.

The current application of Laplace and finite Fourier cosine transforms adheres to the mathematical framework established in previous research on hyperbolic and time-dependent laser heating, including works by Talaee and Sarafrazi [19] and Al-Khairy and Al-Ofey [20]. Moreover, the temperature profiles and thermal trends generated by the present model qualitatively align with the experimental findings of Alkallas et al. [21], who integrated analytical modeling with in situ laser spectroscopy measurements. The consistency among several sources validates the accuracy and physical significance of this work in light of prior numerical simulations or experimental challenges [33,34,35].

Table 2. Comparative Study: Transposed View of different studies based on overlay temperature profiles, peak temperature, time to peak, thermal wave speed, fractional order, and relaxation time.

	Overlay Temperature Profiles	Peak Temperature	Time to Peak	Thermal Wave Speed	Fractional Order (p or α)	Relaxation Time (τ)
Current Work (Essawy et al.)	Analytical via Mittag–Leffler; damped wavefronts depending on p and τ	Decreases with higher p and τ	Increases with p; delayed heat front	Slows down with increasing p and τ	p (0 < p ≤ 1); key modeling tool	Explicit τ; impacts damping and delay
Qiao et al. (2021) [36]	Numerical/semi-analytical; validated vs. experiments	Thickness-dependent; experimentally supported	Increases with film thickness	Inferred from simulations; thickness impact	Different α for heat flux and gradient	τ estimated via optimization
Li et al. (2019) [37]	Laplace-based for bi-layers; shows spatial delay	Affected by interface conditions	Memory and delay factor affect timing	Controlled by kernel function and delay	Fractional order affects thermoelastic coupling	Delay and memory functions involved
Dutta et al. (2020) [38]	Exact solutions for A6061/Cu3Zn2; deviates from Fourier	Material-specific; significantly lower in DPL vs. Fourier	Related to pulse duration; material impacts it	DPL gives finite speed vs. Fourier’s infinite	Not used explicitly; DPL model analog	τq and τT in DPL framework
Karakas et al. (2010) [39]	Two-temperature model; gold/chromium metal interactions	Influenced by multilayer dynamics	Two-step model shows delay	Hyperbolic wave propagation used	Not fractional	Electron-lattice relaxation time
Ji et al. (2019) [40]	FDM solutions; shows jump conditions	Lower than classical due to lagging terms	Delayed due to fractional operators	Defined via Knudsen number	α and α + 1 modeled via Caputo form	τq, τT in Caputo framework
Mozafarifard et al. (2020) [41]	Caputo-based numerical model; fits experimental profiles	Lower than classical; influenced by β and material	Slower rise for lower α	Wave-like front with non-Fourier features	β = 0.5–1 in Caputo model; fits data	Estimated from data; critical for model
Mozafarifard & Toghraie (2020) [42]	Subdiffusion profiles; validated; lower gradient	Lower than Fourier; matches experiments	Larger delays with smaller α	Slow subdiffusive waves	β ∈ (0.5, 1); Caputo subdiffusion	Parametrized with laser depth
Amoruso et al. (2014) [43]	MD simulation; thin film shows smoother, more uniform profile	Uniform for film; less variance than bulk material	Not directly given; inferred from heat propagation dynamics	Faster in bulk; slower, more confined in thin film	Not fractional; uses MD and thermodynamic simulations	τ not named but behavior inferred from two-step MD model

presents a detailed comparison of nine published studies alongside the current analytical study, focusing on the modeling of laser-induced heat transfer in thin metallic films. The comparison is structured across six critical criteria: overlay temperature profiles, peak temperature, time to peak, thermal wave speed, fractional order, and relaxation time. These parameters provide insight into the physical accuracy, mathematical complexity, and practical relevance of each model.

5. Conclusions

In this study, an analytical solution was developed to examine the photothermal response of metallic thin films subjected to short laser pulses, employing the time-fractional Cattaneo model to account for nonlocality and memory effects in heat conduction. Exact solutions were derived using fractional calculus and Laplace transform methods, with inverse formulations involving Mittag–Leffler functions, capturing the complex, non-Fourier behavior of thermal waves. The model provides a detailed description of temperature distribution as a function of both space and time, incorporating finite propagation speeds and delayed thermal responses.

The findings reveal that introducing a fractional-order time derivative significantly alters heat transport compared to classical Fourier models. A lower fractional order leads to slower decay and extended thermal retention, particularly near the irradiated surface, where the memory effect is most pronounced and gradually weakens toward the adiabatic boundary. The relaxation time τ and laser absorption coefficient ϵ critically shape the thermal behavior; shorter τ values generate sharper gradients and high-frequency oscillations, while longer τ promotes smoother, diffusion-like temperature profiles, highlighting the shift from wave-like to diffusive transport mechanisms. An increase in ϵ intensifies the thermal amplitude without changing the spatial oscillation frequency, indicating stronger thermal responses under higher energy input while maintaining the system’s oscillatory nature. Temporal analysis further shows that as time progresses, the thermal wave amplitude increases, reflecting persistent internal thermal interactions and memory effects. At the illuminated surface, temperature rises sharply with sustained elevation at lower fractional orders, whereas at the insulated boundary, the heat wave arrives later, more diffused, but still influenced by fractional and relaxation dynamics. Enhanced absorption near the surface also magnifies temperature contrasts across the film thickness. Moreover, the thermal behavior at longer times aligns with the Boltzmann distribution, suggesting a statistical tendency toward uniform energy distribution as equilibrium is approached.

Ultimately, the presented model that incorporates fractional dynamics, relaxation mechanisms, and laser absorption effects offers profound insights into heat transport in metallic thin films. The results combine mathematical precision with physical significance, providing valuable guidance for applications in laser-based microfabrication, photothermal device optimization, biomedical therapies, and thermal management in nanoscale systems. Moreover, this theoretical consideration of the parameters driving heat diffusion throughout the thin film provides the sensitivity required for precise compositional analysis while causing less target damage owing to a directed laser beam. It focuses on the effects of the fractional order on the temporal and spatial distribution of the thermal field, offering valuable predictions for experimental investigations of ultrafast thermal phenomena in metallic nanostructures. As a result, laser parameter optimizations can be achieved for efficient ablation and plasma generation, allowing for better control over a variety of industrial applications.

6. Future Work

This study opens avenues for further research into the application of fractional models in other materials and geometries, expanding beyond metallic thin films to other nanostructured materials or layered systems. This approach may be extended to multi-dimensional configurations, variable fractional orders, and coupled thermoelastic or thermoelectric phenomena for an even richer understanding of fractional-order photothermal dynamics. Additionally, the integration of these models with experimental data could lead to deeper insights into the physical mechanisms of heat transfer in extreme conditions. Future research may address the experimental identification of key parameters such as the fractional order p, relaxation time τ, and absorption coefficient ϵ. These can be estimated using inverse methods, where model predictions are fitted to data from ultrafast thermo-reflectance, photo-thermal radiometry, or laser flash experiments. Optimization algorithms or Bayesian approaches may be employed to extract parameters from measured temperature profiles, enabling validation and practical application of the model. Also, upcoming studies could replace the adiabatic boundary with more realistic conditions (Robin or Dirichlet types) to simulate heat exchange with the substrate and better align with experimental setups.