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New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition

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

Deadline for manuscript submissions: closed (28 February 2026) | Viewed by 10649

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Special Issue Editors

Prof. Dr. Haci Mehmet Baskonus

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Guest Editor

Faculty of Education, Harran University, 63050 Sanliurfa, Turkey
Interests: applied science; partial differential equations; soliton theory; analytical methods; numerical methods
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Luis Manuel Sánchez Ruiz

E-Mail Website1 Website2
Guest Editor

ETSID-Department of Applied Mathematics, Universitat Politecnica de Valencia, 46022 Valencia, Spain
Interests: fractional calculus; analytical and computational methods; differential and difference equations; real and complex analysis; applied and computational mathematics; mathematical physics
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Dr. Armando Ciancio

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Guest Editor

Department of Biomedical and Dental Sciences and Morphofunctional Imaging, University of Messina, 98125 Messina, Italy
Interests: time series based on wavelets; analysis of solutions in the field of physical-mathematical models of rheological media; fractional calculus; mathematical models in economics and finance; physical-mathematical models for biological media and applications to biotechnological and medical sciences
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Recently, many new models have been developed to address real-world problems that represent serious threats to the future of humankind. These result from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research conducted on fractional ordinary or partial differential equations and other relevant topics relating to integer order has attracted the attention of experts from all over the world.

The focus of this Special Issue will be on reviewing new developments based on fractional differentiation and integration with respect to both theoretical and numerical aspects.

This Special Issue invites experts to share new ideas on theories, applications, numerical and analytical methods, and simulations of fractional calculus and fractional differential equations, as well as integer order. Topics of interest are defined below, and submissions relating to relevant fields are welcome.

New analytical and numerical methods to solve partial differential equations.
Computational methods for fractional differential equations.
The analysis, modeling, and control of phenomena in the following areas:

Electrical engineering;
Fluid dynamics and thermal engineering;
Mechanics;
Biology;
Physics;
Applied sciences;
Computer science.

Engineering problems.
Deterministic and stochastic fractional order models.

Prof. Dr. Haci Mehmet Baskonus
Prof. Dr. Luis Manuel Sánchez Ruiz
Dr. Armando Ciancio
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

new analytical and numerical methods to solve partial differential equations
computational methods for fractional differential equations
the analysis, modeling, and control of phenomena in the following areas
engineering problems
deterministic and stochastic fractional order models

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Published Papers (11 papers)

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Research

33 pages, 1626 KB

Open AccessArticle

Fractional Reaction–Diffusion Modelling of Immune-Mediated Demyelination in Multiple Sclerosis Under IFN-Beta and Glatiramer Acetate Therapy

by Aytekin Enver, Fatma Ayaz, Mehmet Yavuz and Fuat Usta

Fractal Fract. 2026, 10(5), 281; https://doi.org/10.3390/fractalfract10050281 - 23 Apr 2026

Viewed by 120

Abstract

We propose a dimensionally consistent fractional spatio-temporal PDE framework for modelling immune-mediated demyelination in multiple sclerosis (MS). The system couples effector and regulatory T cells, M1/M2 macrophage polarisation, pro- and anti-inflammatory cytokines, oligodendrocyte dynamics, and time-dependent therapeutic controls within a unified distributed-parameter structure. In contrast to ad hoc replacements of integerorder derivatives by Caputo fractional derivatives, the fractional extension proposed here is derived from an underlying continuous-time random walk (CTRW) process with Mittag–Leffler-distributed residence times. This stochastic derivation yields a governing system in which a single commensurate fractional order α ∈ (0, 1], together with a characteristic memory timescale τ0, ensures dimensional consistency and mass balance across all coupled components. The model is formulated as a system of nonlinear reaction–diffusion equations with cross-regulatory and multiplicative interaction terms governing immune amplification, cytokine feedback, and the demyelination–remyelination balance. Analytical interpretation shows how non-Markovian residence times induce Mittag–Leffler-type relaxation and thereby modify effective growth, decay, and stability properties. Numerical simulations compare classical and fractional dynamics, revealing that memory-driven kinetics prolong effector T-cell and M1-macrophage activity, attenuate reparative M2 and oligodendrocyte responses, and extend the effective action of bang–bang therapy inputs representing IFN-β and glatiramer acetate beyond their dosing windows. The results indicate that integer-order models may underestimate chronic inflammatory persistence and demyelination severity, while providing a mathematically and physically well-posed platform for memory-aware immune modelling and therapy evaluation in MS. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

27 pages, 1113 KB

Open AccessArticle

On the Investigation of Environmental Effects of ChatGPT Usage via the Newly Developed Mathematical Model in Caputo Sense

by Sherly K, Pundikala Veeresha and Haci Mehmet Baskonus

Fractal Fract. 2026, 10(3), 184; https://doi.org/10.3390/fractalfract10030184 - 11 Mar 2026

Viewed by 481

Abstract

This study explores the interconnection between the variables of ChatGPT usage, energy consumption, water consumption, and carbon dioxide CO₂ emissions. A new integer and fractional order model using the Caputo derivative is proposed to comprehend the long-term dependencies of these variables. Boundedness, and global and local stability are examined for the fractional order model. The equilibrium points of these variables are shown to determine the stability of the model. The Runge–Kutta 7 numerical method is employed for the integer order model, whereas the semi-implicit linear interpolation (L1) method is used for the fractional order model. The parameter sensitivity is conducted on the system’s parameters to understand the variables’ impact by varying the relevant parameters for the system. To increase the efficacy of our analysis, we used machine learning approaches to model and predict the dynamics of CO₂ emissions, energy and water consumption, and ChatGPT usage. The Prophet ML model stood out among the other methods because it is adept at identifying long-term growth trends, seasonal changes, and the impact of outside variables in intricate time-series data. It is extremely beneficial for research centered on sustainability, where accurate projections are essential for wellinformed decision-making, because it can produce robust, interpretable forecasts against missing values and outliers. Using the Prophet ML model, our research guarantees precise and expandable predictions and provides valuable information that can direct tactics to balance environmental sustainability and technological progress. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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15 pages, 641 KB

Open AccessArticle

Optical Solitons, Optimal Systems and Conserved Quantities of the Schrödinger Equation with Spatio-Temporal and Inter-Modal Dispersions

by Funda Turk

Fractal Fract. 2026, 10(2), 112; https://doi.org/10.3390/fractalfract10020112 - 5 Feb 2026

Cited by 1 | Viewed by 458

Abstract

In this study, we present a unified symmetry-conservation solution analysis of a well-posed resonant nonlinear Schrödinger (NLS)-type equation incorporating spatio-temporal dispersion and inter-modal dispersion. Working within the truncated M-fractional derivative framework, we first construct exact traveling-wave solution families via the Kudryashov expansion method, together with the corresponding parameter constraints and limiting cases. We then determine the admitted Lie point symmetries and establish the associated Lie algebra, including the commutator structure, adjoint representation, and an optimal system of one-dimensional subalgebras for classification. Using the conservation theorem, we derive conserved vectors associated with the fundamental invariances of the model; in the NLS setting and under suitable conditions, these quantities can be interpreted as generalized power (mass), momentum, and energy-type invariants. Overall, the results provide explicit wave profiles and structural invariants that enhance the interpretability of the model and offer benchmark expressions useful for further qualitative, numerical, and stability investigations in nonlinear dispersive wave dynamics. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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18 pages, 3128 KB

Open AccessArticle

Classification of Fractional-Order Chaos and Integer-Order Chaos Using a Multi-Branch Deep Learning Network Model

by Jingchan Lv, Hongcun Mao, Yu Wang and Zhihai Yao

Fractal Fract. 2025, 9(12), 822; https://doi.org/10.3390/fractalfract9120822 - 16 Dec 2025

Viewed by 574

Abstract

Fractional-order chaotic systems describe complex dynamic processes with memory effects and long-range correlations, while integer-order chaotic systems are generally viewed as a special case of fractional-order counterparts. This close relationship often renders the two difficult to distinguish in practice. However, existing studies mostly design analytical methods for integer-order or fractional-order chaotic systems separately, lacking a unified classification framework that does not rely on prior assumptions about the system order. In this paper, we propose a multi-branch deep learning model integrating a multi-scale convolutional neural network, time–frequency analysis, Transformer blocks, and dynamic memory network to classify integer-order chaos, fractional-order chaos, and steady states. Experiments are conducted on time series from canonical chaotic systems—including the Lorenz, Rössler, Lü, and Chen systems—in both integer- and fractional-order formulations, under two data generation protocols: varying initial conditions and varying system parameters. We evaluate the model under two scenarios: (1) assessing baseline classification performance on noise-free data and (2) testing robustness against increasing levels of Gaussian, salt-and-pepper and Rayleigh noise. The model achieves classification accuracy above 99% on noise-free data across all tested systems. Under noise interference, it demonstrates strong robustness: accuracy remains above 89.7% under high-intensity Gaussian noise. Moreover, we demonstrate that the model trained with fixed system parameters but varying initial conditions generalizes poorly to unseen parameter settings, whereas training with diverse system parameters—while fixing initial conditions—markedly improves generalization. This work offers a data-driven framework for distinguishing integer- and fractional-order chaos and highlights the critical role of training data diversity in building generalizable classifiers for dynamical systems. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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24 pages, 2666 KB

Open AccessArticle

The Adaptive Backstepping Synchronization Control for a Kind of Variable-Order Fractional Uncertain Nonlinear Systems

by Xinyu Si, Kangkang Zhang, Jingfei Jiang, Juan L. G. Guirao and Hongkui Li

Fractal Fract. 2025, 9(12), 761; https://doi.org/10.3390/fractalfract9120761 - 24 Nov 2025

Viewed by 563

Abstract

This paper is concerned with the adaptive backstepping synchronization control for a class of variable-order (VO) fractional uncertain nonlinear system with external disturbances and a dead zone. A kind of VO fractional command filter is employed to cope with “explosion of complexity”. The unknown nonlinear term in considered system is decomposed into unknown parameters and error functions by the Szász–Mirakyan operator theory. A VO fractional disturbance observer derived in this paper is used to simultaneously deal with the difficulties brought by dead zone and external disturbances. Thus, a VO fractional backstepping synchronization controller with adaptive laws for the system handled is proposed; moreover, the stability of system controlled is established. Finally, numerical examples are given to validate the theoretical results. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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23 pages, 1089 KB

Open AccessArticle

On the Qualitative Stability Analysis of Fractional-Order Corruption Dynamics via Equilibrium Points

by Qiliang Chen, Kariyanna Naveen, Doddabhadrappla Gowda Prakasha and Haci Mehmet Baskonus

Fractal Fract. 2025, 9(10), 666; https://doi.org/10.3390/fractalfract9100666 - 16 Oct 2025

Cited by 1 | Viewed by 629

Abstract

The primary objective of this study is to provide a more precise and beneficial mathematical model for assessing corruption dynamics by utilizing non-local derivatives. This research aims to provide solutions that accurately capture the complexities and practical behaviors of corruption. To illustrate how corruption levels within a community change over time, a non-linear deterministic mathematical model has been developed. The authors present a non-integer order model that divides the population into five subgroups: susceptible, exposed, corrupted, recovered, and honest individuals. To study these corruption dynamics, we employ a new method for solving a time-fractional corruption model, which we term the q-homotopy analysis transform approach. This approach produces an effective approximation solution for the investigated equations, and data is shown as 3D plots and graphs, which give a clear physical representation. The stability and existence of the equilibrium points in the considered model are mathematically proven, and we examine the stability of the model and the equilibrium points, clarifying the conditions required for a stable solution. The resulting solutions, given in series form, show rapid convergence and accurately describe the model’s behaviour with minimal error. Furthermore, the solution’s uniqueness and convergence have been demonstrated using fixed-point theory. The proposed technique is better than a numerical approach, as it does not require much computational work, with minimal time consumed, and it removes the requirement for linearization, perturbations, and discretization. In comparison to previous approaches, the proposed technique is a competent tool for examining an analytical outcomes from the projected model, and the methodology used herein for the considered model is proved to be both efficient and reliable, indicating substantial progress in the field. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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34 pages, 505 KB

Open AccessArticle

Regularity and Qualitative Study of Parabolic Physical Ginzburg–Landau Equations in Variable Exponent Herz Spaces via Fractional Bessel–Riesz Operators

by Waqar Afzal, Mesfer H. Alqahtani, Mujahid Abbas and Daniel Breaz

Fractal Fract. 2025, 9(10), 644; https://doi.org/10.3390/fractalfract9100644 - 1 Oct 2025

Cited by 1 | Viewed by 942

Abstract

In this article, we investigate the regularization and qualitative properties of parabolic Ginzburg–Landau equations in variable exponent Herz spaces. These spaces capture both local and global behavior, providing a natural framework for our analysis. We establish boundedness results for fractional Bessel–Riesz operators, construct examples highlighting their advantage over classical Riesz potentials, and recover several known theorems as special cases. As an application, we analyze a parabolic Ginzburg–Landau operator with VMO coefficients, showing that our estimates ensure the boundedness and continuity of solutions. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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18 pages, 6736 KB

Open AccessArticle

Realization of Fractional-Order Current-Mode Multifunction Filter Based on MCFOA for Low-Frequency Applications

by Fadile Sen and Ali Kircay

Fractal Fract. 2025, 9(6), 377; https://doi.org/10.3390/fractalfract9060377 - 13 Jun 2025

Cited by 4 | Viewed by 1438

Abstract

The present work proposes a novel fractional-order multifunction filter topology in current-mode (CM), which is designed based on the Modified Current Feedback Operational Amplifier (MCFOA). The proposed design simultaneously generates fractional-order low-pass (FO-LPF), high-pass (FO-HPF), and band-pass (FO-BPF) outputs while utilizing an optimized set of essential active and passive elements, thereby ensuring simplicity, cost efficiency, and compatibility with integrated circuits (ICs). The fractional-order feature allows precise control over the transition slope between the passband and the stopband, enhancing design flexibility. PSpice simulations validated the filter’s theoretical performance, confirming a 1 kHz cut-off frequency, making it suitable for VLF applications such as military communication and submarine navigation. Monte Carlo analyses demonstrate robustness against parameter variations, while a low THD, a wide dynamic range, and low power consumption highlight its efficiency for high-precision, low-power applications. This work offers a practical and adaptable approach to fractional-order circuit design, with significant potential in communication, control, and biomedical systems. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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23 pages, 2348 KB

Open AccessArticle

Chaotic Analysis and Wave Photon Dynamics of Fractional Whitham–Broer–Kaup Model with β Derivative

by Muhammad Idrees Afridi, Theodoros E. Karakasidis and Abdullah Alhushaybari

Fractal Fract. 2025, 9(5), 287; https://doi.org/10.3390/fractalfract9050287 - 27 Apr 2025

Cited by 4 | Viewed by 981

Abstract

This study uses a conformable derivative of order

β

to investigate a fractional Whitham–Broer–Kaup (

FWBK

) model. This model has significant uses in several scientific domains, such as plasma physics and nonlinear optics. The enhanced modified Sardar sub-equation

EMSSE

approach is applied [...] Read more.

This study uses a conformable derivative of order

β

to investigate a fractional Whitham–Broer–Kaup (

FWBK

) model. This model has significant uses in several scientific domains, such as plasma physics and nonlinear optics. The enhanced modified Sardar sub-equation

EMSSE

approach is applied to achieve precise analytical solutions, demonstrating its effectiveness in resolving complex wave photons. Bright, solitary, trigonometric, dark, and plane waves are among the various wave dynamics that may be effectively and precisely determined using the

FWBK

model. Furthermore, the study explores the chaotic behaviour of both perturbed and unperturbed systems, revealing illumination on their dynamic characteristics. By demonstrating its validity in examining wave propagation in nonlinear fractional systems, the effectiveness and reliability of the suggested method in fractional modelling are confirmed through thorough investigation. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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18 pages, 3859 KB

Open AccessArticle

The Use of Artificial Intelligence in Data Analysis with Error Recognitions in Liver Transplantation in HIV-AIDS Patients Using Modified ABC Fractional Order Operators

by Hasib Khan, Jehad Alzabut, D. K. Almutairi and Wafa Khalaf Alqurashi

Fractal Fract. 2025, 9(1), 16; https://doi.org/10.3390/fractalfract9010016 - 30 Dec 2024

Cited by 32 | Viewed by 1583

Abstract

In this article, we focused on the fractional order modeling, simulations and neural networking to observe the correlation between severity of infection in HIV-AIDS patients and the role of treatments and control. The model is structured with eight classes and a modified Atangana–Baleanu derivative in Caputo’s sense. The model has several interlinking parameters which show the rates of transmission between classes. We assumed natural death and death on the disease severity in patients. The model was analyzed mathematically as well as computationally. In the mathematical aspects,

R_{0}

was plotted for different cases which play a vital role in the infection spread in the population. The model was passed through qualitative analysis for the existence of solutions and stability results. A computational scheme is developed for the model and is applied for the numerical results to analyze the intricate dynamics of the infection. It has been observed that there is a good resemblance in the results for the correlation between the hospitalization, vaccination and recovery rate of the patients. These are reaffirmed with the neural networking tools for the regression, probability, clustering, mean square error and fitting data. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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19 pages, 404 KB

Open AccessArticle

Modeling of (n,m)-Type Minkowski Pythagorean Hodograph Curves with Hopf Map and Applications

by Muhammed Talat Sariaydin and Aziz Yazla

Fractal Fract. 2024, 8(12), 705; https://doi.org/10.3390/fractalfract8120705 - 28 Nov 2024

Cited by 1 | Viewed by 1539

Abstract

In the present paper, regular spacelike spatial Minkowski Pythagorean hodograph (MPH) curves are characterized with rational rotation-minimizing frames (RRMFs). We define an Euler–Rodrigues frame (ERF) for MPH curves and by means of this concept, we reach the definition of MPH curves of type [...] Read more.

(n, m)

. Expressing the conditions provided by these curves in the form of a Minkowski–Hopf map that we define; it is aimed to establish a connection with the Lorentz force that occurs during the process of computer numerical control (CNC)-type sinker electronic discharge machines (EDMs). This approach is reinforced by split quaternion polynomials. We give conditions satisfied by MPH curves of low degree to be type

(n, m)

and construct illustrative examples. In five-axis CNC machines, rotation-minimizing frames are used for tool path planning, and in this way, unnecessary rotations in the tool frame are prevented and tool orientation is provided. Since we obtain MPH curves with RRMF using the ERF, finally we define the Fermi–Walker derivative and parallelism along MPH curves with respect to the ERF and give applications. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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