Fox H-function representations of temperature and displacement fields in unbounded domains with anomalous decelerating thermal conduction

Emad Awad; A. R. El-Dhaba; Emad Awad; A. R. El-Dhaba

doi:10.3934/math.2026473

AIMS Mathematics

2026, Volume 11, Issue 4: 11489-11519. doi: 10.3934/math.2026473

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Fox H-function representations of temperature and displacement fields in unbounded domains with anomalous decelerating thermal conduction

Emad Awad ¹,
A. R. El-Dhaba ^{2
,
,}

1.
Department of Mathematics, Faculty of Education, Alexandria University, Souter St. El-Shatby, Alexandria P.O. Box 21526, Egypt
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia

Received: 09 February 2026 Revised: 21 March 2026 Accepted: 13 April 2026 Published: 24 April 2026
MSC : 34A08, 35B40, 74B05, 80A20

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.
- asymptotic solutions,
- Cauchy problem,
- Fox H-function,
- fractional calculus,
- quasistatic theory
Citation: Emad Awad, A. R. El-Dhaba. Fox H-function representations of temperature and displacement fields in unbounded domains with anomalous decelerating thermal conduction[J]. AIMS Mathematics, 2026, 11(4): 11489-11519. doi: 10.3934/math.2026473

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Abstract

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.

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