TY  - JOUR
AU  - Awad, Emad
AU  - El-Dhaba, A. R.
PY  - 2026
TI  - Fox H-function representations of temperature and displacement fields in unbounded domains with anomalous decelerating thermal conduction
JO  - AIMS Mathematics
SP  - 11489
EP  - 11519
VL  - 11
IS  - 4
AB  - Recent advances in fractional calculus have highlighted the role of distributed-order operators in modeling anomalous diffusion processes, particularly through bi-fractional diffusion equations of the natural type. In this work, we introduce a distributed-order fractional integral formulation that leads to a generalized bi-fractional Fourier law involving two Riemann–Liouville fractional integrals. The proposed constitutive relation captures a class of anomalous heat conduction characterized by decelerating thermal transport, wherein the effective thermal conductivity is relatively large in the short-time regime and diminishes in the long-time regime. For a quasi-static thermoelastic problem in an unbounded domain, exact analytical solutions for the temperature and displacement fields are derived and expressed in terms of the Fox H-function. Within the quasi-static framework, it is rigorously shown that the appropriate zero initial condition must be imposed on the normal stress rather than on the volumetric strain. A damped cosinusoidal boundary condition at infinity is incorporated and is shown to affect the elastic response due to the infinite propagation speed of mechanical disturbances under the quasi-static assumption. The coupled thermo-mechanical analysis reveals that thermal and mechanical fields exhibit analogous transitional behavior: Decelerating thermal conduction induces a corresponding retardation in the deformation of the medium.
UR  - https://doi.org/10.3934/math.2026473
DO  - 10.3934/math.2026473