Fractional Dynamics of Laser-Induced Heat Transfer in Metallic Thin Films: Analytical Approach
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 < <i>p<...
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MDPI AG
2025-06-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/6/373 |
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| author | M. A. I. Essawy Reham A. Rezk Ayman M. Mostafa |
| author_facet | M. A. I. Essawy Reham A. Rezk Ayman M. Mostafa |
| author_sort | M. A. I. Essawy |
| collection | DOAJ |
| description | 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 < <i>p</i> ≤ 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. |
| format | Article |
| id | doaj-art-ab7d756964a045d5ae9dd0bdff6db166 |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-ab7d756964a045d5ae9dd0bdff6db1662025-08-20T03:24:37ZengMDPI AGFractal and Fractional2504-31102025-06-019637310.3390/fractalfract9060373Fractional Dynamics of Laser-Induced Heat Transfer in Metallic Thin Films: Analytical ApproachM. A. I. Essawy0Reham A. Rezk1Ayman M. Mostafa2Department of Basic Sciences, Higher Technological Institute, 6th of October City, Giza P.O. Box 4, EgyptDepartment of Basic Sciences, Higher Technological Institute, 6th of October City, Giza P.O. Box 4, EgyptDepartment of Physics, College of Science, Qassim University, Buraidah 51452, Saudi ArabiaThis 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 < <i>p</i> ≤ 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.https://www.mdpi.com/2504-3110/9/6/373time-fractional Cattaneo modelfinite Fourier cosine transform (FFCT)Laplace transformanalytical solutionthin filmsphotothermal energy |
| spellingShingle | M. A. I. Essawy Reham A. Rezk Ayman M. Mostafa Fractional Dynamics of Laser-Induced Heat Transfer in Metallic Thin Films: Analytical Approach Fractal and Fractional time-fractional Cattaneo model finite Fourier cosine transform (FFCT) Laplace transform analytical solution thin films photothermal energy |
| title | Fractional Dynamics of Laser-Induced Heat Transfer in Metallic Thin Films: Analytical Approach |
| title_full | Fractional Dynamics of Laser-Induced Heat Transfer in Metallic Thin Films: Analytical Approach |
| title_fullStr | Fractional Dynamics of Laser-Induced Heat Transfer in Metallic Thin Films: Analytical Approach |
| title_full_unstemmed | Fractional Dynamics of Laser-Induced Heat Transfer in Metallic Thin Films: Analytical Approach |
| title_short | Fractional Dynamics of Laser-Induced Heat Transfer in Metallic Thin Films: Analytical Approach |
| title_sort | fractional dynamics of laser induced heat transfer in metallic thin films analytical approach |
| topic | time-fractional Cattaneo model finite Fourier cosine transform (FFCT) Laplace transform analytical solution thin films photothermal energy |
| url | https://www.mdpi.com/2504-3110/9/6/373 |
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