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Fractal Fract
Special Issues
Advanced Numerical Methods for Fractional Functional Models
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Advanced Numerical Methods for Fractional Functional Models

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

Deadline for manuscript submissions: closed (30 April 2026) | Viewed by 9703

Special Issue Editors

Dr. Kamyar Hosseini

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Guest Editor
Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
Interests: analytical methods; numerical methods; fractional differential equations; wave propagation; mathematical physics; nonlinear partial differential equations
Special Issues, Collections and Topics in MDPI journals
Dr. Khadijeh Sadri

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Guest Editor
Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
Interests: numerical analysis; fractional integral equations; fractional partial differential equation; mathematical models
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Evren Hınçal

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Guest Editor
Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
Interests: applied mathematics; statistics; epidemiology
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Soheil Salahshour

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Guest Editor
Faculty of Engineering and Natural Sciences, Istanbul Okan University, 34959 Istanbul, Turkey
Interests: fuzzy systems; fractional modelling; optical solitons; applied artificial intelligence
Special Issues, Collections and Topics in MDPI journals
Dr. Farzaneh Alizadeh

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Guest Editor
Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
Interests: lie groups; geometry; differential geometry; pure mathematics; Riemannian geometry; fractional differential equations

Special Issue Information

Dear Colleagues,

Many powerful techniques such as partial differential equations, integral equations, and integro-differential equations have been used to model a wide variety of nonlinear phenomena from nonlinear optics to plasma physics, circuit theory, and biology. Although the usefulness of such techniques in modeling nonlinear phenomena is undeniable, researchers have faced problems in achieving the necessary efficiency to do so. Today, these problems are mitigated when such techniques are used in combination with fractional operators, which is the subject of much research. Such problems can be handled with a wide range of useful methods including finite difference methods, radial basis function methods, and spectral methods (collocation, Galerkin, and Tau). The key goal of the current Special Issue is to present the latest research on the solutions to the above problems involving fractional operators using advanced numerical methods. Original research and review articles are highly welcomed. Potential topics include but are not limited to the following:

  • Advanced numerical methods for fractional partial differential equations;
  • Advanced numerical methods for fractional integral equations;
  • Advanced numerical methods for integro-differential equations involving fractional operators;
  • Advanced numerical methods for systems of fractional differential equations;
  • Advanced numerical methods for systems of fractional functional models.

Dr. Kamyar Hosseini
Dr. Khadijeh Sadri
Prof. Dr. Evren Hınçal
Prof. Dr. Soheil Salahshour
Dr. Farzaneh Alizadeh
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

  • numerical methods
  • approximation methods
  • fractional differential equations
  • fractional partial differential equations
  • fractional integral equations
  • fractional functional models

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Published Papers (7 papers)

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Research

26 pages, 7629 KB  
Article
Dynamics and Efficient Numerical Simulation of a Fractional-Order T System
by Liping Yu and Hongyi Zhu
Fractal Fract. 2026, 10(5), 334; https://doi.org/10.3390/fractalfract10050334 (registering DOI) - 14 May 2026
Abstract
In this paper, we propose and numerically investigate a fractional T system. As a fractional generalization of the classical T model, the fractional order serves as a memory parameter governing the system dynamics. By employing the fractional stability criterion, the local stability of [...] Read more.
In this paper, we propose and numerically investigate a fractional T system. As a fractional generalization of the classical T model, the fractional order serves as a memory parameter governing the system dynamics. By employing the fractional stability criterion, the local stability of the equilibrium points is analyzed, and the existence of Hopf bifurcation is characterized. To efficiently simulate the long-time dynamics induced by fractional memory, a linear semi-implicit numerical scheme accelerated by a sum-of-exponentials approximation of the Caputo derivative is developed. The proposed scheme is shown to be stable and enables a significant reduction in computational cost compared with classical L1 and Grünwald–Letnikov methods. Numerical experiments, including time series, phase portraits, Lyapunov exponent computations, and bifurcation diagrams, demonstrate that varying the fractional order leads to transitions among stable, periodic, and chaotic regimes. In particular, pronounced transient dynamics are observed as the fractional order approaches its critical value, highlighting the memory-induced effects inherent in fractional-order systems. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)
24 pages, 1628 KB  
Article
A Fractional-Order Sliding Mode DTC–SVM Framework for Precision Control of Surgical Robot Actuators
by Fatma Ben Salem, Jaouhar Mouine and Nabil Derbel
Fractal Fract. 2026, 10(3), 193; https://doi.org/10.3390/fractalfract10030193 - 13 Mar 2026
Viewed by 328
Abstract
Precise and smooth actuation is a central requirement in surgical robotics, where small tracking errors or oscillations can directly affect task quality and safety. This paper studies the control of an induction-motor-driven surgical joint using a sliding-mode strategy enhanced by fractional-order operators and [...] Read more.
Precise and smooth actuation is a central requirement in surgical robotics, where small tracking errors or oscillations can directly affect task quality and safety. This paper studies the control of an induction-motor-driven surgical joint using a sliding-mode strategy enhanced by fractional-order operators and implemented within a DTC–SVM structure. The motivation is to improve motion smoothness and disturbance rejection without sacrificing the fast dynamic response offered by direct torque control. A dynamic model of the actuator is developed by combining the electrical equations of the induction motor with the mechanical dynamics of a robotic joint, including inertia, viscous friction, gravity-induced torque, and Coulomb friction. Fractional-order sliding surfaces are introduced for both position and flux regulation, and the closed-loop stability is examined through Lyapunov-based arguments. Simulation results show accurate trajectory tracking with limited overshoot and smooth transient responses. The motor speed remains well regulated, while stator flux and currents stay within admissible bounds. The electromagnetic torque adapts to load variations with reduced ripple, and the rotor pulsation remains bounded. Within the limits of numerical evaluation, these results indicate that the proposed fractional-order sliding-mode DTC–SVM scheme is suitable for precision-oriented surgical robotic actuation. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)
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27 pages, 375 KB  
Article
An Efficient and Accurate Numerical Approach for Fractional Bagley–Torvik Equations: Hermite Polynomials Combined with Least Squares
by Heba S. Osheba, Mohamed A. Ramadan and Taha Radwan
Fractal Fract. 2026, 10(1), 37; https://doi.org/10.3390/fractalfract10010037 - 7 Jan 2026
Viewed by 457
Abstract
This paper proposes an efficient and accurate numerical framework for solving fractional Bagley–Torvik equations, which model viscoelastic and memory-dependent dynamic systems. The method combines the Hermite polynomial approximation with a least-squares optimization scheme to achieve high-accuracy solutions. By leveraging the analytical properties of [...] Read more.
This paper proposes an efficient and accurate numerical framework for solving fractional Bagley–Torvik equations, which model viscoelastic and memory-dependent dynamic systems. The method combines the Hermite polynomial approximation with a least-squares optimization scheme to achieve high-accuracy solutions. By leveraging the analytical properties of Hermite polynomials, Caputo fractional derivatives were computed efficiently, avoiding the complexities of direct fractional differentiation. The resulting weighted least-squares formulation transforms the problem into a stable algebraic system. Numerical results confirm the method’s superior accuracy, rapid convergence, and robustness compared with existing techniques, demonstrating its potential for broader applications in fractional-order boundary value problems. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)
15 pages, 463 KB  
Article
On Extended Numerical Discretization Technique of Fractional Models with Caputo-Type Derivatives
by Reem Allogmany and S. S. Alzahrani
Fractal Fract. 2025, 9(5), 289; https://doi.org/10.3390/fractalfract9050289 - 28 Apr 2025
Cited by 2 | Viewed by 1409
Abstract
In this work, we investigate the extended numerical discretization technique for the solution of fractional Bernoulli equations and SIRD epidemic models under the Caputo fractional, which is accurate and versatile. We have demonstrated the method’s strength in examining complex systems; it is found [...] Read more.
In this work, we investigate the extended numerical discretization technique for the solution of fractional Bernoulli equations and SIRD epidemic models under the Caputo fractional, which is accurate and versatile. We have demonstrated the method’s strength in examining complex systems; it is found that the method produces solutions that are identical to the exact solution and approximate series solutions. The ENDT is its ability to proficiently handle complex systems governed by fractional differential equations while preserving memory and hereditary characteristics. Its simplicity, accuracy, and flexibility render it an effective instrument for replicating real-world phenomena in physics and biology. The ENDT method offers accuracy, stability, and efficiency compared to traditional methods. It effectively handles challenges in complex systems, supports any fractional order, is simple to implement, improves computing efficiency with sophisticated methodologies, and applies it to epidemic predictions and biological simulations. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)
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30 pages, 5977 KB  
Article
Enhanced Numerical Solutions for Fractional PDEs Using Monte Carlo PINNs Coupled with Cuckoo Search Optimization
by Tauqeer Ahmad, Muhammad Sulaiman, David Bassir, Fahad Sameer Alshammari and Ghaylen Laouini
Fractal Fract. 2025, 9(4), 225; https://doi.org/10.3390/fractalfract9040225 - 2 Apr 2025
Cited by 6 | Viewed by 3082
Abstract
In this study, we introduce an innovative approach for addressing fractional partial differential equations (fPDEs) by combining Monte Carlo-based physics-informed neural networks (PINNs) with the cuckoo search (CS) optimization algorithm, termed PINN-CS. There is a further enhancement in the application of quasi-Monte Carlo [...] Read more.
In this study, we introduce an innovative approach for addressing fractional partial differential equations (fPDEs) by combining Monte Carlo-based physics-informed neural networks (PINNs) with the cuckoo search (CS) optimization algorithm, termed PINN-CS. There is a further enhancement in the application of quasi-Monte Carlo assessment that comes with high efficiency and computational solutions to estimates of fractional derivatives. By employing structured sampling nodes comparable to techniques used in finite difference approaches on staggered or irregular grids, the proposed PINN-CS minimizes storage and computation costs while maintaining high precision in estimating solutions. This is supported by numerous numerical simulations to analyze various high-dimensional phenomena in various environments, comprising two-dimensional space-fractional Poisson equations, two-dimensional time-space fractional diffusion equations, and three-dimensional fractional Bloch–Torrey equations. The results demonstrate that PINN-CS achieves superior numerical accuracy and computational efficiency compared to traditional fPINN and Monte Carlo fPINN methods. Furthermore, the extended use to problem areas with irregular geometries and difficult-to-define boundary conditions makes the method immensely practical. This research thus lays a foundation for more adaptive and accurate use of hybrid techniques in the development of the fractional differential equations and in computing science and engineering. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)
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31 pages, 817 KB  
Article
Qualitative Analysis of Generalized Power Nonlocal Fractional System with p-Laplacian Operator, Including Symmetric Cases: Application to a Hepatitis B Virus Model
by Mohamed S. Algolam, Mohammed A. Almalahi, Muntasir Suhail, Blgys Muflh, Khaled Aldwoah, Mohammed Hassan and Saeed Islam
Fractal Fract. 2025, 9(2), 92; https://doi.org/10.3390/fractalfract9020092 - 1 Feb 2025
Cited by 1 | Viewed by 1307
Abstract
This paper introduces a novel framework for modeling nonlocal fractional system with a p-Laplacian operator under power nonlocal fractional derivatives (PFDs), a generalization encompassing established derivatives like Caputo–Fabrizio, Atangana–Baleanu, weighted Atangana–Baleanu, and weighted Hattaf. The core methodology involves employing a PFD with a [...] Read more.
This paper introduces a novel framework for modeling nonlocal fractional system with a p-Laplacian operator under power nonlocal fractional derivatives (PFDs), a generalization encompassing established derivatives like Caputo–Fabrizio, Atangana–Baleanu, weighted Atangana–Baleanu, and weighted Hattaf. The core methodology involves employing a PFD with a tunable power parameter within a non-singular kernel, enabling a nuanced representation of memory effects not achievable with traditional fixed-kernel derivatives. This flexible framework is analyzed using fixed-point theory, rigorously establishing the existence and uniqueness of solutions for four symmetric cases under specific conditions. Furthermore, we demonstrate the Hyers–Ulam stability, confirming the robustness of these solutions against small perturbations. The versatility and generalizability of this framework is underscored by its application to an epidemiological model of transmission of Hepatitis B Virus (HBV) and numerical simulations for all four symmetric cases. This study presents findings in both theoretical and applied aspects of fractional calculus, introducing an alternative framework for modeling complex systems with memory processes, offering opportunities for more sophisticated and accurate models and new avenues for research in fractional calculus and its applications. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)
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30 pages, 2587 KB  
Article
A Local Radial Basis Function Method for Numerical Approximation of Multidimensional Multi-Term Time-Fractional Mixed Wave-Diffusion and Subdiffusion Equation Arising in Fluid Mechanics
by Kamran, Ujala Gul, Zareen A. Khan, Salma Haque and Nabil Mlaiki
Fractal Fract. 2024, 8(11), 639; https://doi.org/10.3390/fractalfract8110639 - 29 Oct 2024
Cited by 3 | Viewed by 2119
Abstract
This article develops a simple hybrid localized mesh-free method (LMM) for the numerical modeling of new mixed subdiffusion and wave-diffusion equation with multi-term time-fractional derivatives. Unlike conventional multi-term fractional wave-diffusion or subdiffusion equations, this equation features a unique time–space coupled derivative while simultaneously [...] Read more.
This article develops a simple hybrid localized mesh-free method (LMM) for the numerical modeling of new mixed subdiffusion and wave-diffusion equation with multi-term time-fractional derivatives. Unlike conventional multi-term fractional wave-diffusion or subdiffusion equations, this equation features a unique time–space coupled derivative while simultaneously incorporating both wave-diffusion and subdiffusion terms. Our proposed method follows three basic steps: (i) The given equation is transformed into a time-independent form using the Laplace transform (LT); (ii) the LMM is then used to solve the transformed equation in the LT domain; (iii) finally, the time domain solution is obtained by inverting the LT. We use the improved Talbot method and the Stehfest method to invert the LT. The LMM is used to circumvent the shape parameter sensitivity and ill-conditioning of interpolation matrices that commonly arise in global mesh-free methods. Traditional time-stepping methods achieve accuracy only with very small time steps, significantly increasing the computational time. To overcome these shortcomings, the LT is used to provide a more powerful alternative by removing the need for fine temporal discretization. Additionally, the Ulam–Hyers stability of the considered model is analyzed. Four numerical examples are presented to illustrate the effectiveness and practical applicability of the method. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)
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