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Fractal Fract
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Analysis and Applications of Fractional Calculus in Computational Physics
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Analysis and Applications of Fractional Calculus in Computational Physics

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

Deadline for manuscript submissions: 31 August 2026 | Viewed by 4149

Special Issue Editors

Dr. Slobodanka Galovic

E-Mail Website
Guest Editor
Vinca Institute of Nuclear Sciences, National Institute of the Republic of Serbia, University of Belgrade, Mike Petrovica Alasa 12-14, 11001 Belgrade, Serbia
Interests: fractional calculus; fractional operator; fractional-order and distributed order models; wave propagation; applied and computational mathematics; nonlinear dynamics; condensed matter physics; heat transfer; photothermal science; inverse problems; artificial intelligence; quantum transport
Special Issues, Collections and Topics in MDPI journals
Dr. Dalibor Chevizovich

E-Mail Website
Guest Editor
Vinca Institute of Nuclear Sciences, National Institute of the Republic of Serbia, University of Belgrade, Mike Petrovica Alasa 12-14, 11001 Belgrade, Serbia
Interests: quantum coherence phenomena; quantum biophysics; condensed matter physics;polaron theory; solitons; fractional-order models; fractional calculus

Special Issue Information

Dear Colleagues,

Fractional calculus is a powerful tool that enables more efficient modeling of physical processes in complex physical systems. It is used in the modeling of anomalous dissipative processes, describing physical systems with nonlinear behavior, modeling of electromagnetic field propagation in fractal and anisotropic media, consideration of memory effects in quantum mechanics, describing viscoelastic materials with fractional damping in classical mechanics, overcoming approximation of local equilibrium and locality in general in thermodynamics, etc.

The most commonly used definitions of fractional derivatives and integrals include the Riemann-Liouville and Caputo definitions. However, contemporary research and analysis of different physical models with complex initial and boundary conditions indicate the need to further develop and understand fractional operators. In addition, the analysis of fractional models often requires the development of specialized methods for solving fractional differential equations.

This Special Issue aims to present the advancement in the development of fractional operators and methods of solving fractional differential equations, as well as the novelty in applications of fractional calculus in various fields of physics.

Dr. Slobodanka Galovic
Dr. Dalibor Chevizovich
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 250 words) can be sent to the Editorial Office for assessment.

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 in physics
  • fractional operators
  • integral transformations of irrational functions
  • inverse Laplace transform of irrational functions
  • time-delayed fractional models
  • numerical methods in fractional problems

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Published Papers (4 papers)

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Research

23 pages, 3691 KB  
Article
High-Precision and Stability-Preserving Approximations to the Time-Fractional Harry Dym Model Using the Tantawy Technique
by Linda Alzaben, Wedad Albalawi, Rajaa T. Matoog and Samir A. El-Tantawy
Fractal Fract. 2026, 10(4), 217; https://doi.org/10.3390/fractalfract10040217 - 26 Mar 2026
Viewed by 345
Abstract
Fractional differential equations provide a flexible framework for describing evolutionary processes in complex media, where nonlocality and memory effects play central roles, and classical integer-order models are frequently inadequate to capture these behaviors. In this work, we revisit the time-fractional Harry Dym (HD) [...] Read more.
Fractional differential equations provide a flexible framework for describing evolutionary processes in complex media, where nonlocality and memory effects play central roles, and classical integer-order models are frequently inadequate to capture these behaviors. In this work, we revisit the time-fractional Harry Dym (HD) evolution equation in the Caputo sense and construct high-precision analytical approximations using the recently developed Tantawy technique (TT). The method generates a rapidly convergent fractional-power series in time without resorting to perturbative assumptions, auxiliary decomposition polynomials, linearization procedures, or integral transforms, and it remains computationally economical even at high approximation orders. Closed, compact expressions are derived up to the fifth-order approximation and can be systematically extended, yielding excellent agreement with the known exact solution of the classical/integer HD model and with approximations obtained via the new iterative method. A detailed error analysis is carried out by computing absolute and maximum residual errors over the entire computational domain, demonstrating the accuracy, stability, and robustness of the TT for the HD-type fractional nonlinear evolution equation. From a physical perspective, the proposed framework offers a reliable tool for modeling nonlinear wave structures in dispersive media with significant memory and, more generally, for treating a broad class of fractional nonlinear wave equations arising in physics and engineering. Full article
(This article belongs to the Special Issue Analysis and Applications of Fractional Calculus in Computational Physics)
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20 pages, 1691 KB  
Article
On the Tantawy Technique for Analyzing Fractional Kuramoto–Sivashinsky-Type Equations and Modeling Shock Waves in Plasmas and Fluids—Part (I), Planar Case
by Samir A. El-Tantawy, Alvaro H. Salas, Wedad Albalawi, Rania A. Alharbey and Ashwag A. Alharby
Fractal Fract. 2026, 10(2), 105; https://doi.org/10.3390/fractalfract10020105 - 3 Feb 2026
Cited by 1 | Viewed by 860
Abstract
The Kuramoto–Sivashinsky (KS) equation and its fractional generalizations (FKSs) arise as canonical models for a wide class of nonlinear dissipative–dispersive systems, including thin-film flows, combustion fronts, drift–wave turbulence in plasmas, and chemically reacting media, where shock-like and strongly localized structures play a central [...] Read more.
The Kuramoto–Sivashinsky (KS) equation and its fractional generalizations (FKSs) arise as canonical models for a wide class of nonlinear dissipative–dispersive systems, including thin-film flows, combustion fronts, drift–wave turbulence in plasmas, and chemically reacting media, where shock-like and strongly localized structures play a central role in the dynamics. Despite their apparent simplicity, KS-type models become analytically intractable once higher-order dissipation, geometric effects, and memory (fractional) operators are incorporated, and standard perturbative or transform-based schemes often lead to cumbersome recursive structures, slow convergence, or severe restrictions on the initial data. In this work, a novel direct approximation procedure, referred to as the Tantawy Technique (TT), is developed and implemented to solve and analyze planar fractional KS-type equations and their Burgers-type reductions in a systematic manner. The central difficulty is to construct, for a given physically motivated initial profile, a rapidly convergent series in fractional time that remains stable for a broad range of the fractional order and transport coefficients, while still retaining a clear link to the underlying shock-wave physics. To overcome this, the TT combines (i) a Tanh-based exact shock solution of the planar integer-order KS equation, obtained first as a reference via the standard Tanh method, with (ii) a carefully designed fractional-time ansatz in powers of tρ, where the spatial coefficients are determined recursively from the governing equation in the Caputo sense. This construction yields closed-form expressions for the first few terms in the approximation hierarchy and allows one to monitor convergence through residual and absolute error measures. Full article
(This article belongs to the Special Issue Analysis and Applications of Fractional Calculus in Computational Physics)
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20 pages, 1005 KB  
Article
A Note on Solutions of Fractional Third-Order Dispersive Partial Differential Equations Using the Natural Generalized Laplace Transform Decomposition Method
by Hassan Eltayeb, Shayea Aldossari and Said Mesloub
Fractal Fract. 2025, 9(12), 770; https://doi.org/10.3390/fractalfract9120770 - 25 Nov 2025
Cited by 1 | Viewed by 602
Abstract
The present research offers reliable analytical solutions for time-fractional linear and nonlinear dispersive Korteweg–de Vries (dKdV)-type equations by employing the Natural Generalized Laplace Transform Decomposition Method (NGLTDM). The nonlinear differential dispersive Korteweg–de Vries (dKdV) equation involves a nonlinear derivative term that depends on [...] Read more.
The present research offers reliable analytical solutions for time-fractional linear and nonlinear dispersive Korteweg–de Vries (dKdV)-type equations by employing the Natural Generalized Laplace Transform Decomposition Method (NGLTDM). The nonlinear differential dispersive Korteweg–de Vries (dKdV) equation involves a nonlinear derivative term that depends on ϕ and its partial derivative with respect to x. We employ Adomian polynomials to deal with this nonlinear part, and we utilize the Caputo derivative to illustrate the fractional part of the equation. The work provides exact theorems regarding the stability, convergence, and accuracy of the generated solutions. Illustrative examples demonstrate the effectiveness and precision of the method by delivering solutions for quickly converging series with easily calculable coefficients. We use Maple 2021 software to show graphical comparisons between the approximate and exact solutions to show how rapidly the method converges. Full article
(This article belongs to the Special Issue Analysis and Applications of Fractional Calculus in Computational Physics)
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24 pages, 774 KB  
Article
Electrical Analogy Approach to Fractional Heat Conduction Models
by Slobodanka Galovic, Marica N. Popovic and Dalibor Chevizovich
Fractal Fract. 2025, 9(10), 653; https://doi.org/10.3390/fractalfract9100653 - 9 Oct 2025
Cited by 1 | Viewed by 1437
Abstract
Fractional heat conduction models extend classical formulations by incorporating fractional differential operators that capture multiscale relaxation effects. In this work, we introduce an electrical analogy that represents the action of these operators via generalized longitudinal impedance and admittance elements, thereby clarifying their physical [...] Read more.
Fractional heat conduction models extend classical formulations by incorporating fractional differential operators that capture multiscale relaxation effects. In this work, we introduce an electrical analogy that represents the action of these operators via generalized longitudinal impedance and admittance elements, thereby clarifying their physical role in energy transfer: fractional derivatives account for the redistribution of heat accumulation and dissipation within micro-scale heterogeneous structures. This analogy unifies different classes of fractional models—diffusive, wave-like, and mixed—as well as distinct fractional operator types, including the Caputo and Atangana–Baleanu forms. It also provides a general computational methodology for solving heat conduction problems through the concept of thermal impedance, defined as the ratio of surface temperature variations (relative to ambient equilibrium) to the applied heat flux. The approach is illustrated for a semi-infinite sample, where different models and operators are shown to generate characteristic spectral patterns in thermal impedance. By linking these spectral signatures of microstructural relaxation to experimentally measurable quantities, the framework not only establishes a unified theoretical foundation but also offers a practical computational tool for identifying relaxation mechanisms through impedance analysis in microscale thermal transport. Full article
(This article belongs to the Special Issue Analysis and Applications of Fractional Calculus in Computational Physics)
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