Semi-analytical solutions to the bi-flux diffusion model using the generalised integral transform technique

Abstract

Abstract This study presents a high-precision analysis of the Bi-Flux Diffusion Model using the Generalised Integral Transform Technique (GITT). The Bevilacqua–Galeão–Costa (BGC) equation, which characterises anomalous diffusion via a dual-flux framework, is cast in dimensionless form, introducing the Bevilacqua number ( $$B_v$$ ) as the governing parameter for retention dynamics. The GITT framework is adapted to accommodate fourth-order spatial operators, incorporating a biharmonic eigenproblem and analytical filtering to rigorously enforce non-homogeneous boundary conditions. The method achieves spectral convergence in both one- and two-dimensional configurations, as verified through systematic numerical validation. Computational results confirm exponential convergence rates, even under high-gradient regimes and mixed boundary constraints, positioning GITT as a reliable benchmark for anomalous diffusion studies. The work furnishes semi-analytical modal solutions, spectral truncation criteria, and an extensible reference framework for future applications in reactive-diffusive systems and complex geometries.