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

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.