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Fractal Fract
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Analysis of Heat Conduction and Anomalous Diffusion in Fractional Calculus
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Analysis of Heat Conduction and Anomalous Diffusion in Fractional Calculus

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 30 May 2025 | Viewed by 2936

Special Issue Editors

Prof. Dr. Aloisi Somer

E-Mail Website
Guest Editor
Departamento de Física, Universidade Estadual de Ponta Grossa, Av. Gen. Carlos Cavalcanti 4748, Ponta Grossa 84030-900, PR, Brazil
Interests: fractional differential operators; fractional dual-phase-lag heat conduction theory; spatial and time fractional derivative; fractional diffusion; anomalous heat diffusion; heat conduction
Prof. Dr. Ervin K. Lenzi

E-Mail Website
Guest Editor
Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil
Interests: anomalous diffusion; liquid crystals; impedance; fractional dynamics; nonextensive thermostatistics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus is a powerful tool for modeling physical phenomena in which classical integer-order calculus cannot capture the system's complexity. One such area where fractional calculus has been found to be particularly useful is the study of heat conduction and anomalous thermal diffusion. The classical Fourier law of heat conduction assumes that the heat flux is proportional to the temperature gradient, which leads to a linear heat conduction equation. However, this law can only sometimes accurately describe heat conduction in complex materials. The use of fractional differential operators in the heat conduction equation has been shown to be effective in modeling non-local and memory effects in heat conduction. This behavior has been observed in many physical systems, including biological systems and porous media. Thus, fractional calculus in thermal conduction and diffusion is an interesting research area that provides useful tools to investigate the anomalous thermodynamic process in several fields, such as physics, fluid dynamics, chemistry, and biology, among others. Its relevance lies in its ability to capture the complexity of these systems and provide a more accurate description of their behavior. We invite researchers to submit original research and review articles on the recent developments in fractional differential equations in anomalous diffusion and thermal conduction and their applications in science, technology, and engineering.

Prof. Dr. Aloisi Somer
Prof. Dr. Ervin K. Lenzi
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional calculus and fractal media
  • thermo-molecular physics
  • thermodynamics
  • heat and mass transfer
  • bio-heat transfer
  • fractional thermal conduction
  • anomalous thermal diffusion
  • nonequilibrium processes
  • nonequilibrium thermodynamics
  • kinetics theory

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Published Papers (4 papers)

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Research

21 pages, 3494 KiB  
Article
Fractional Heat Conduction with Variable Thermal Conductivity of Infinite Annular Cylinder Under Thermoelasticity Theorem of Moore–Gibson–Thompson
by Eman A. N. Al-Lehaibi
Fractal Fract. 2025, 9(5), 272; https://doi.org/10.3390/fractalfract9050272 - 23 Apr 2025
Viewed by 189
Abstract
To discuss the fractional order heat conduction based on Youssef’s model, a new mathematical model of a thermoelastic annular cylinder with variable thermal conductivity will be constructed in this work. The Moore–Gibson– Thompson theorem of generalized thermoelasticity will be considered and the governing [...] Read more.
To discuss the fractional order heat conduction based on Youssef’s model, a new mathematical model of a thermoelastic annular cylinder with variable thermal conductivity will be constructed in this work. The Moore–Gibson– Thompson theorem of generalized thermoelasticity will be considered and the governing equations will be derived in dimensionless forms. The Laplace transform technique will be used for a one-dimensional thermoelastic, isotropic, and homogeneous annular cylinder in which the interior surrounded surface is thermally shocked and there is an axial traction-free environment, while the outer surrounded surface has neither heat increment nor cubical deformation. The numerical results will be computed for the Laplace transform inversions by using Tzou’s iteration approach. The distributions of the cubical deformation, invariant average stress, axial stress, and temperature increment will be represented in figures to analyze and discuss. The results show that the fractional-order and variable thermal conductivity parameters have significant effects on all the studied functions. The physical behaviour of the thermal conductivity is closely aligned with the classification of thermal conductivity into weak, normal, and strong categories, which is essential. Full article
(This article belongs to the Special Issue Analysis of Heat Conduction and Anomalous Diffusion in Fractional Calculus)
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34 pages, 4698 KiB  
Article
Stat-Space Approach to Three-Dimensional Thermoelastic Half-Space Based on Fractional Order Heat Conduction and Variable Thermal Conductivity Under Moor–Gibson–Thompson Theorem
by Hamdy M. Youssef
Fractal Fract. 2025, 9(3), 145; https://doi.org/10.3390/fractalfract9030145 - 25 Feb 2025
Viewed by 324
Abstract
This study presents a mathematical model of a three-dimensional thermoelastic half-space with variable thermal conductivity under the definition of fractional order heat conduction based on the Moor–Gibson–Thompson theorem. The non-dimensional governing equations using Laplace and double Fourier transform methods have been applied to [...] Read more.
This study presents a mathematical model of a three-dimensional thermoelastic half-space with variable thermal conductivity under the definition of fractional order heat conduction based on the Moor–Gibson–Thompson theorem. The non-dimensional governing equations using Laplace and double Fourier transform methods have been applied to a three-dimensional thermoelastic, isotropic, and homogeneous half-space exposed to a rectangular thermal loading pulse with a traction-free surface. The double Fourier transforms and Laplace transform inversions have been computed numerically. The numerical distributions of temperature increment, invariant stress, and invariant strain have been shown and analysed. The fractional order parameter and the variability of thermal conductivity significantly influence all examined functions and the behaviours of the thermomechanical waves. Classifying thermal conductivity as weak, normal, and strong is crucial and closely corresponds to the actual behaviour of the thermal conductivity of thermoelastic materials. Full article
(This article belongs to the Special Issue Analysis of Heat Conduction and Anomalous Diffusion in Fractional Calculus)
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24 pages, 6135 KiB  
Article
A New Fractional Boundary Element Model for Anomalous Thermal Stress Effects on Cement-Based Materials
by Mohamed Abdelsabour Fahmy and Roqia Abdullah A. Jeli
Fractal Fract. 2024, 8(12), 753; https://doi.org/10.3390/fractalfract8120753 - 21 Dec 2024
Cited by 4 | Viewed by 523
Abstract
The novelty of this work is the development of a new fractional boundary element model based on the Caputo derivative to investigate anomalous thermal stress effects on cement-based materials. To obtain the BEM integral equations for the proposed formulation, we employ the weighted [...] Read more.
The novelty of this work is the development of a new fractional boundary element model based on the Caputo derivative to investigate anomalous thermal stress effects on cement-based materials. To obtain the BEM integral equations for the proposed formulation, we employ the weighted residuals technique, with the anisotropic fundamental solution serving as the weighting function in the anomalous heat governing equation. The Caputo fractional derivative was employed as an integrand for the domain integral of the proposed formulation. The time step selection is less dependent on the time derivative order. This allows the approach to overcome the non-locality of the fractional operators. The key benefit provided by the suggested formulation is the ability to analyze situations with tiny values of the fractional time derivative. The current BEM methodology proves that it is a useful tool for solving fractional calculus problems. Full article
(This article belongs to the Special Issue Analysis of Heat Conduction and Anomalous Diffusion in Fractional Calculus)
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16 pages, 1702 KiB  
Article
Influence of Local Thermodynamic Non-Equilibrium to Photothermally Induced Acoustic Response of Complex Systems
by Slobodanka Galovic, Aleksa I. Djordjevic, Bojan Z. Kovacevic, Katarina Lj. Djordjevic and Dalibor Chevizovich
Fractal Fract. 2024, 8(7), 399; https://doi.org/10.3390/fractalfract8070399 - 3 Jul 2024
Cited by 3 | Viewed by 1191
Abstract
In this paper, the time-resolved model of the photoacoustic signal for samples with a complex inner structure is derived including local non-equilibrium of structural elements with multiple degrees of freedom, i.e., structural entropy of the system. The local non-equilibrium is taken into account [...] Read more.
In this paper, the time-resolved model of the photoacoustic signal for samples with a complex inner structure is derived including local non-equilibrium of structural elements with multiple degrees of freedom, i.e., structural entropy of the system. The local non-equilibrium is taken into account through the fractional operator. By analyzing the model for two types of time-dependent excitation, a very short pulse and a very long pulse, it is shown that the rates of non-equilibrium relaxations in complex samples can be measured by applying the derived model and time-domain measurements. Limitations of the model and further directions of its development are discussed. Full article
(This article belongs to the Special Issue Analysis of Heat Conduction and Anomalous Diffusion in Fractional Calculus)
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