Fractal Fract., Volume 9, Issue 8 (August 2025) – 14 articles

This study introduces a paradigm shift in permanent magnet synchronous motor (PMSM) control through the development of hybrid dual fractional-order sliding mode control (HDFOSMC) architecture integrated with evolutionary parameter learning (EPL). Conventional PMSM control frameworks face critical limitations in ultra-precision applications due to their inability to reconcile dynamic agility with steady-state precision under time-varying parameters and compound disturbances. The proposed HDFOSMC framework addresses these challenges via two synergistic innovations: (1) a dual fractional-order sliding manifold that fuses the rapid transient response of non-integer-order differentiation with the small steady-state error capability of dual-integral compensation, and (2) an EPL mechanism enabling real-time adaptation to thermal drift, load mutations, and unmodeled nonlinearities. Validation can be obtained through the comparison of the results on PMSM testbenches, which demonstrate superior performance over traditional fractional-order sliding mode control (FOSMC). By integrating fractional-order theory, sliding mode control theory, and parameter self-tuning theory, this study proposes a novel control framework for PMSM. The developed system achieves high-precision performance under extreme operational uncertainties through this innovative theoretical synthesis and comparative results. Full article

The present study addresses the European option pricing problem based on the Black–Scholes (B-S) model using a hybrid analytical approach known as the Sawi homotopy perturbation transform scheme (SHPTS). We formulate fractional derivatives in the Caputo sense to effectively capture the memory effects inherent in financial models. The competency and reliability of the SHPTS are demonstrated through two illustrative examples. This method produces a closed-form series solution that converges to the precise solution. We perform convergence and visual analyses to demonstrate the competency and reliability of the proposed scheme. The numerical findings further reveal that the strategy is straightforward to apply and very successful in resolving the fractional form of the B-S problem. Full article

In this work, Noether symmetries and conserved quantities of a non-standard Birkhoffian system based on the Caputo $\Delta$ Pfaff–Birkhoff principle on time scales are studied. Firstly, equations of motion for Caputo $\Delta$ non-standard Birkhoffian systems are set up from Caputo $\Delta$ variational principle. Secondly, invariance of Caputo non-standard Pfaff action on time scales is demonstrated, thus giving rise to Noether symmetry criterions which establish Noether’s theorems for the corresponding system. The validity of the methods and results presented in the paper is illustrated by means of examples provided at the end of the article. Full article

This paper is focused on developing a Galerkin finite element framework for the fractional Allen–Cahn equation with regularized logarithmic potential over the ${\mathbb{R}}^d$ ($d=1,2,3$) domain, where the regularization of the singular potential extends beyond the classical double-well formulation. A fully discrete finite element scheme is developed using a k-th-order finite element space for spatial approximation and a backward Euler scheme for the temporal discretization of a regularized system. The existence and uniqueness of numerical solutions are rigorously established by applying Brouwer’s fixed-point theorem. Moreover, the proposed numerical framework is shown to preserve the discrete energy dissipation law analytically, while a priori error estimates are derived. Finally, numerical experiments are conducted to verify the theoretical results and the inherent physical property, such as phase separation phenomenon and coarsening processes. The results show that the fractional Allen–Cahn model provides enhanced capability in capturing phase transition characteristics compared to its classical equation. Full article

In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing $\beta$-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations. Full article

This study proposes an L1 discretization scheme (an accurate second-order finite difference method) for time-fractional diffusion equations involving the Caputo-type Erdélyi–Kober operator, which models anomalous diffusion. Our key contributions include the following: (i) reformulation of the original problem into an equivalent fractional integral equation to facilitate analysis; (ii) development of a novel L1 scheme for temporal discretization, which is rigorously proven to realize second-order accuracy in time; (iii) derivation of positive definiteness properties for discrete kernel coefficients; (iv) discretization of the spatial derivative using the classical second-order centered difference scheme, for which its second-order spatial convergence is rigorously verified through numerical experiments (this results in a fully discrete scheme, enabling second-order accuracy in both temporal and spatial dimensions); (v) a fast algorithm leveraging sum-of-exponential approximation, reducing the computational complexity from $O\left(N^2\right)$ to $O(NlogN)$ and memory requirements from $O\left(N\right)$ to $O(logN)$, where N is the number of grid points on a time scale. Our numerical experiments demonstrate the stability of the scheme across diverse parameter regimes and quantify significant gains in computational efficiency. Compared to the direct method, the fast algorithm substantially reduces both memory requirements and CPU time for large-scale simulations. Although a rigorous stability analysis is deferred to subsequent research, the proven properties of the coefficients and numerical validation confirm the scheme’s reliability. Full article

This study introduces a novel model reference adaptive control (MRAC) framework that incorporates fractional-order gradients (FGs) to regulate the displacement of an inverted pendulum-cart system. Fractional-order gradients have been shown to significantly improve convergence rates in domains such as machine learning and neural network optimization. Nevertheless, their integration with fractional-order error models within adaptive control paradigms remains unexplored and represents a promising avenue for research. The proposed control scheme extends the classical MRAC architecture by embedding Caputo fractional derivatives into the adaptive law governing parameter updates, thereby improving both convergence dynamics and control flexibility. To ensure optimal performance across multiple criteria, the controller parameters are systematically tuned using a multi-objective Particle Swarm Optimization (PSO) algorithm. Two fractional-order error models (FOEMs) incorporating fractional gradients (FOEM2-FG, FOEM3-FG) are investigated, with their stability formally analyzed via Lyapunov-based methods under conditions of sufficient excitation. Validation is conducted through both simulation and real-time experimentation on a physical pendulum-cart setup. The results demonstrate that the proposed fractional-order MRAC (FOMRAC) outperforms conventional MRAC, proportional-integral-derivative (PID), and fractional-order PID (FOPID) controllers. Specifically, FOMRAC-FG achieved superior tracking performance, attaining the lowest Integral of Squared Error (ISE) of $2.32\times {10}^{-5}$ and the lowest Integral of Squared Input (ISI) of 6.40 in simulation studies. In real-time experiments, FOMRAC-FG maintained the lowest ISE ($5.11\times {10}^{-6}$). Under real-time experiments with disturbances, it still achieved the lowest ISE ($1.06\times {10}^{-5}$), highlighting its practical effectiveness. Full article

This paper investigates the well-posedness of the inverse problem for the time-fractional Burgers equation, which aims to reconstruct initial conditions from terminal observations. Such equations are crucial for the modeling of hydrodynamic phenomena with memory effects. The inverse problem involves inferring initial conditions from terminal observation data, and such problems are typically ill-posed. A framework based on optimal control theory is proposed, addressing the ill-posedness via $H^1$ regularization. Three substantial results are achieved: (1) a rigorous mathematical framework transforming the ill-posed inverse problem into a well-posed optimization problem with proven existence of solutions; (2) theoretical guarantee of solution uniqueness when the regularization parameter is $\alpha >0$ and the stability is of order O($\delta$) with respect to observation noise ($\delta$); and (3) the discovery of a “super-stability” phenomenon in numerical experiments, where the actual stability index (0.046) significantly outperforms theoretical expectations (1.0). Finally, the theoretical framework is validated through comprehensive numerical experiments, demonstrating the accuracy and practical effectiveness of the proposed optimal control approach for the reconstruction of hydrodynamic initial conditions. Full article

The principles of fractal geometry have revolutionized the characterization of complex geometric objects since Benoit Mandelbrot’s groundbreaking work. Richardson’s method for determining the fractal dimension of boundaries laid the groundwork for Mandelbrot’s later developments in fractal theory. Despite extensive research, challenges remain in accurately calculating fractal dimensions, particularly when dealing with digital images and their inherent limitations. This study examines the numerical artifacts introduced by Richardson’s method when applied to the Von Koch Island, a classic fractal curve, and proposes a novel approach for computing fractal dimensions in image analysis. The Koch snowflake serves as a key example in this analysis; it serves to assess the algorithm of fractal dimension calculation as his theoretical one is known. However, there is a fundamental difference between the theoretical calculation of fractal dimension and the actual calculation of the fractal dimension from digital images with a given resolution undergoing discretization. We propose eight different calculation methods based on Richardson’s area–perimeter relationship: the Self-Convolution Patterns Research (SCPR) method accurately estimates the fractal dimension, as the 95% confidence interval includes the theoretical dimension. Full article

In this paper, we delve into the following nonlinear fractional Kirchhoff-type problem ${(a+b||(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\left|\right|}_2^2{)(-\Delta )}^{s}u+\lambda u=g\left(u\right)+{\left|u\right|}^{2_s^{*}-2}u$ in ${\mathbb{R}}^3$ with prescribed mass ${\int }_{{\mathbb{R}}^3}{\left|u\right|}^{2}dx=\rho >0,$ where $s\in ({\displaystyle \frac{3}{4}},1),\lambda \in \mathbb{R},2_s^{*}={\displaystyle \frac{6}{3-2s}}$. Under some general growth assumptions imposed on g, we employ minimization of the energy functional on the linear combination of Nehari and Poho$\breve{z}$aev constraints intersected with the closed ball in the $L^2\left({\mathbb{R}}^3\right)$ of radius $\rho$ to prove the existence of normalized ground state solutions to the equation. Moreover, we provide a detailed description for the asymptotic behavior of the ground state energy map. Full article

In this paper, a linearized mixed finite volume element (MFVE) scheme is proposed to solve the nonlinear time fractional fourth-order reaction–diffusion models with the Riemann–Liouville time fractional derivative. By introducing an auxiliary variable $\sigma =\Delta u$, the original fourth-order model is reformulated into a lower-order coupled system. The first-order time derivative and the time fractional derivative are discretized by using the BDF2 formula and the weighted and shifted Grünwald difference (WSGD) formula, respectively. Then, a fully discrete MFVE scheme is constructed by using the primal and dual grids. The existence and uniqueness of a solution for the MFVE scheme are proven based on the matrix theories. The scheme’s unconditional stability is rigorously derived by using the Gronwall inequality in detail. Moreover, the optimal error estimates for u in the discrete $L^{\infty }\left(L^2(\Omega )\right)$ and $L^2\left(H^1(\Omega )\right)$ norms and for $\sigma$ in the discrete $L^2\left(L^2(\Omega )\right)$ norm are obtained. Finally, three numerical examples are given to confirm its feasibility and effectiveness. Full article

This article investigates modeling issues of fractional-order singular systems. The state estimation can be solved by using the particle filter. An improved Adaptive Moment Estimation (Adam) method—the Amsgrad algorithm can handle the optimization problem caused by parameter estimation. Thus, a hybrid approach that combines the particle filter and Amsgrad is proposed to estimate both parameters and states in fractional-order singular systems. This method leverages the strengths of the particle filter in handling nonlinear and high-dimensional problems, as well as the stability of the Amsgrad algorithm in optimizing parameters for dynamic systems. Then, the identification process is concluded to achieve a more accurate joint estimation. To validate the feasibility of the proposed hybrid algorithm, simulations involving three-order and four-order fractional-order singular systems are conducted. A comparative analysis with other algorithms demonstrates that the proposed method behaves better than the standard particle filter, Amsgrad and Gravitational search algorithm-Kalman filter algorithms. Full article

Focusing on the dielectric elastomer actuator (DEA), this paper proposes a backstepping implicit inverse adaptive control scheme with creep direct inverse compensation. Firstly, a novel fractional-order creep Krasnoselskii–Pokrovskii (FCKP) model is established, which effectively captures hysteresis behavior and creep dynamic characteristics. Significantly, this study pioneers the incorporation of the fractional-order method into a hysteresis-coupled creep model. Secondly, based on the FCKP model, the creep direct inverse compensation is developed to combine with the backstepping implicit inverse adaptive control scheme, where the implicit inverse algorithm avoids the construction of the direct inverse model to mitigate hysteresis. Finally, the proposed control scheme was validated on the DEA system control experimental platform. Under both single-frequency and composite-frequency conditions, it achieved mean absolute errors of 0.0035 and 0.0111, and root mean square errors of 0.0044 and 0.0133, respectively, demonstrating superior tracking performance compared to other control schemes. Full article

The palm-vein recognition system has garnered attention as a biometric technology due to its resilience to external environmental factors, protection of personal privacy, and low risk of external exposure. However, with recent advancements in deep learning-based generative models for image synthesis, the quality and sophistication of fake images have improved, leading to an increased security threat from counterfeit images. In particular, palm-vein images acquired through near-infrared illumination exhibit low resolution and blurred characteristics, making it even more challenging to detect fake images. Furthermore, spoof detection specifically targeting palm-vein images has not been studied in detail. To address these challenges, this study proposes the Fourier-gated feature-fusion network (FGFNet) as a novel spoof detector for palm-vein recognition systems. The proposed network integrates masked fast Fourier transform, a map-based gated feature fusion block, and a fast Fourier convolution (FFC) attention block with global contrastive loss to effectively detect distortion patterns caused by generative models. These components enable the efficient extraction of critical information required to determine the authenticity of palm-vein images. In addition, fractal dimension estimation (FDE) was employed for two purposes in this study. In the spoof attack procedure, FDE was used to evaluate how closely the generated fake images approximate the structural complexity of real palm-vein images, confirming that the generative model produced highly realistic spoof samples. In the spoof detection procedure, the FDE results further demonstrated that the proposed FGFNet effectively distinguishes between real and fake images, validating its capability to capture subtle structural differences induced by generative manipulation. To evaluate the spoof detection performance of FGFNet, experiments were conducted using real palm-vein images from two publicly available palm-vein datasets—VERA Spoofing PalmVein (VERA dataset) and PLUSVein-contactless (PLUS dataset)—as well as fake palm-vein images generated based on these datasets using a cycle-consistent generative adversarial network. The results showed that, based on the average classification error rate, FGFNet achieved 0.3% and 0.3% on the VERA and PLUS datasets, respectively, demonstrating superior performance compared to existing state-of-the-art spoof detection methods. Full article