Fractional-order analysis of voltage and current propagation in lossy transmission lines with thermal feedback

Saqib Murtaza; Lilia El Amraoui; Aceng Sambas; Ahmed Mir; Chemseddine Maatki; Muhammad N. Khan; Badr M. Alshammari; Lioua Kolsi; Saqib Murtaza; Lilia El Amraoui; Aceng Sambas; Ahmed Mir; Chemseddine Maatki; Muhammad N. Khan; Badr M. Alshammari; Lioua Kolsi

doi:10.3934/math.20251195

AIMS Mathematics

2025, Volume 10, Issue 11: 27191-27216. doi: 10.3934/math.20251195

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Fractional-order analysis of voltage and current propagation in lossy transmission lines with thermal feedback

1.
Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Campus Besut, Terengganu, 22200, Malaysia
2.
Artificial Intelligence for Sustainability and Islamic Research Center (AIRIS), Universiti Sultan Zainal Abidin, Gongbadak, Terengganu, 21300, Malaysia
3.
Department of Electrical Engineering, College of Engineering, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
4.
Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tamansari Gobras, Tasikmalaya, 46196, Indonesia
5.
Department of Chemical and Materials Engineering, College of Engineering, Northern Border University, Arar P.O. Box 1321, Saudi Arabia
6.
Department of Mechanical Engineering, College of Engineering, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
7.
Jadara University Research Center, Jadara University, 21110, Jordan
8.
Institute of Engineering Mathematics, University Malaysia Perlis (UniMAP), 02600, Malaysia
9.
Department of Electrical Engineering, College of Engineering, University of Ha'il, Ha'il City 81451, Saudi Arabia
10.
Department of Mechanical Engineering, College of Engineering, University of Ha'il, Ha'il City 81451, Saudi Arabia

Received: 25 September 2025 Revised: 05 November 2025 Accepted: 12 November 2025 Published: 21 November 2025
MSC : 34A08, 65M70, 94C05, 97Mxx

Transmission lines form the backbone of power and communication networks, yet their behavior cannot be predicted by classical transmission line theory, which neglects thermal feedback and fractional-order dynamics. To address these limitations, we develop an advanced fractal-fractional electro-thermal model of a lossy transmission line to analyze voltage and current propagation under thermal feedback. The model is based on the fractal-fractional derivative in the sense of Caputo and integrates the effects of series resistance, inductance, conductance, and capacitance together with temperature-dependent feedback. The coupled governing equations are derived as fractal-fractional order partial differential equations based on Kirchoff's current law (KCL) and Kirchoff's voltage law (KVL). The equations are solved numerically using the local radial basis functions (LRBF) scheme, a meshfree numerical technique, to investigate the spatio-temporal profiles of voltage, current, and temperature. The numerical results demonstrate that higher resistance and conductance increase both the attenuation and heating, while increasing capacitance reduces voltage propagation but enhances the current propagation. Furthermore, fractional and fractal orders enrich the analysis by introducing memory and dispersive effects. Overall, this study offers a more realistic and predictive framework for evaluating lossy transmission systems, with direct implications for improving reliability, thermal management, and performance in modern electrical and communication infrastructures.
- electro-thermal modeling,
- telegrapher's equations,
- fractional-order system,
- lossy transmission line,
- voltage and current propagation,
- thermal feedback
Citation: Saqib Murtaza, Lilia El Amraoui, Aceng Sambas, Ahmed Mir, Chemseddine Maatki, Muhammad N. Khan, Badr M. Alshammari, Lioua Kolsi. Fractional-order analysis of voltage and current propagation in lossy transmission lines with thermal feedback[J]. AIMS Mathematics, 2025, 10(11): 27191-27216. doi: 10.3934/math.20251195

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Abstract

Transmission lines form the backbone of power and communication networks, yet their behavior cannot be predicted by classical transmission line theory, which neglects thermal feedback and fractional-order dynamics. To address these limitations, we develop an advanced fractal-fractional electro-thermal model of a lossy transmission line to analyze voltage and current propagation under thermal feedback. The model is based on the fractal-fractional derivative in the sense of Caputo and integrates the effects of series resistance, inductance, conductance, and capacitance together with temperature-dependent feedback. The coupled governing equations are derived as fractal-fractional order partial differential equations based on Kirchoff's current law (KCL) and Kirchoff's voltage law (KVL). The equations are solved numerically using the local radial basis functions (LRBF) scheme, a meshfree numerical technique, to investigate the spatio-temporal profiles of voltage, current, and temperature. The numerical results demonstrate that higher resistance and conductance increase both the attenuation and heating, while increasing capacitance reduces voltage propagation but enhances the current propagation. Furthermore, fractional and fractal orders enrich the analysis by introducing memory and dispersive effects. Overall, this study offers a more realistic and predictive framework for evaluating lossy transmission systems, with direct implications for improving reliability, thermal management, and performance in modern electrical and communication infrastructures.

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