Lie symmetry classification and invariant solutions of time-fractional telegraph systems with variable coefficients

TL;DR

This paper classifies symmetries of time-fractional telegraph systems with variable coefficients, deriving exact solutions expressed via special functions, advancing understanding of fractional transport phenomena.

Contribution

It provides a complete Lie group classification for these systems, identifying symmetry-dependent cases and systematically deriving invariant solutions.

Findings

Identified three symmetry classes based on coefficient relationships.

Reduced fractional PDEs to ODEs using symmetry analysis.

Obtained explicit solutions involving Mittag-Leffler, Wright, and Fox H-functions.

Abstract

Time-fractional telegraph equations provide fundamental mathematical models for transport processes that exhibit memory and nonlocal effects in industrial and physical systems. These models arise naturally in heat transport in materials with thermal memory, wave propagation in viscoelastic media, and charge transport in spatially heterogeneous semiconductor devices. In this study, we investigate a class of time-fractional telegraph systems with spatially varying coefficients using Lie symmetry analysis and the Riemann--Liouville fractional derivative. We establish a complete Lie group classification for sufficiently differentiable coefficient functions and determine all functional forms that admit such symmetry extensions. The symmetry structure is shown to depend fundamentally on the relationship between the transport coefficient and the potential function, resulting in three distinct symmetry classes. For each case, optimal systems of one-dimensional Lie subalgebras are constructed, and the governing fractional partial differential equations are systematically reduced to fractional ordinary differential equations. Exact invariant solutions are obtained in closed form and expressed in terms of Mittag--Leffler functions, generalized Wright functions, and Fox $H$-functions. These analytical solutions provide valuable insights into fractional telegraph-type transport phenomena and serve as important benchmarks for validating numerical methods in industrial transport modeling and fractional evolution systems.