Search Results (29)

This paper investigates the chaotic and pattern dynamics of the time-fractional Ginzburg–Landau equation. First, we propose a high-precision numerical method that combines finite difference schemes with an improved Grünwald–Letnikov fractional derivative approximation. Subsequently, the effectiveness of the proposed method is validated through systematic comparisons with classical numerical approaches. Finally, numerical simulations based on this method reveal rich dynamical phenomena in the fractional Ginzburg–Landau equation: the system exhibits complex behaviors including chaotic oscillations and novel two- and three-dimensional pattern structures. This study not only advances the theoretical development of numerical solutions for fractional GLE but also provides a reliable computational tool for deeper understanding of its complex dynamical mechanisms. Full article

We present a unified framework for fuzzy statistical convergence of Grünwald–Letnikov (GL) fractional differences in Bag–Samanta fuzzy normed linear spaces, addressing memory effects and nonlocality inherent to fractional-order models. Theoretically, we establish the uniqueness, linearity, and invariance of fuzzy statistical limits and prove a Cauchy characterization: fuzzy statistical convergence implies fuzzy statistical Cauchyness, while the converse holds in fuzzy-complete spaces (and in the completion, otherwise). We further develop an inclusion theory linking fuzzy strong Cesàro summability—including weighted means—to fuzzy statistical convergence. Via the discrete Q-operator, all statements transfer verbatim between nabla-left and delta-right GL forms, clarifying the binomial GL↔discrete Riemann–Liouville correspondence. Beyond structure, we propose density-based residual diagnostics for GL discretizations of fractional initial-value problems: when GL residuals are fuzzy statistically negligible, trajectories exhibit Ulam–Hyers-type robustness in the fuzzy topology. We also formulate a fuzzy Korovkin-type approximation principle under GL smoothing: Cesàro control on the test set $\{1,x,x^2\}$ propagates to arbitrary targets, yielding fuzzy statistical convergence for positive-operator sequences. Worked examples and an engineering-style case study (thermal balance with memory and bursty disturbances) illustrate how the diagnostics certify robustness of GL numerical schemes under sparse spikes and imprecise data. Full article

Fractional-order models provide a powerful framework for capturing memory-dependent and viscoelastic dynamics in mechanical systems, which are often inadequately represented by classical integer-order characterizations. This study addresses the identification of dynamic parameters in both single-degree-of-freedom (1-DOF) and three-degree-of-freedom (3-DOF) Duffing oscillators with fractional damping, modeled using the Grünwald–Letnikov characterization. The 1-DOF system includes a cubic nonlinear restoring force and is excited by a harmonic input to induce steady-state oscillations. For both systems, time domain simulations are conducted to capture long-term responses, followed by Fourier decomposition to extract steady-state displacement, velocity, and acceleration signals. These components are combined with a GL-based fractional derivative approximation to construct structured regressor matrices. System parameters—including mass, stiffness, damping, and fractional-order effects—are then estimated using pseudoinverse techniques. The identified models are validated through a comparison of reconstructed and original trajectories in the phase space, demonstrating high accuracy in capturing the underlying dynamics. The proposed framework provides a consistent and interpretable approach for frequency domain system identification in fractional-order nonlinear systems, with relevance to applications such as mechanical vibration analysis, structural health monitoring, and smart material modeling. Full article

This paper presents a novel wireless control framework for electric drive systems by employing a fuzzy brain emotional learning neural network (FBELNN) controller in conjunction with a Disturbance Observer (DO). The communication scheme uses chaotic system dynamics to ensure data confidentiality and robustness against disturbance in wireless environments. To be applied to embedded microprocessors, the continuous-time chaotic system is discretized using the Grunwald–Letnikov approximation. To avoid the loss of generality of chaotic behavior, Lyapunov exponents are computed to validate the preservation of chaos in the discrete-time domain. The FBELNN controller is then developed to synchronize two non-identical chaotic systems under different initial conditions, enabling secure data encryption and decryption. Additionally, the DOB is introduced to estimate and mitigate the effects of bounded uncertainties and external disturbances, enhancing the system’s resilience to stealthy attacks. The proposed control structure is experimentally implemented on a wireless communication system utilizing ESP32 microcontrollers (Espressif Systems, Shanghai, China) based on the ESP-NOW protocol. Both control and feedback signals of the electric drive system are encrypted using chaotic states, and real-time decryption at the receiver confirms system integrity. Experimental results verify the effectiveness of the proposed method in achieving robust synchronization, accurate signal recovery, and a reliable wireless control system. The combination of FBELNN and DOB demonstrates significant potential for real-time, low-cost, and secure applications in smart electric drive systems and industrial automation. Full article

We propose a novel space-time discretization method for a time-fractional dissipative wave equation. The approach employs a structured framework in which a fully discrete formulation is produced by combining virtual elements for spatial discretization and the Newmark predictor–corrector method for the temporal domain. The virtual element technique is regarded as a generalization of the finite element method for polygonal and polyhedral meshes within the Galerkin approximation framework. To discretize the time-fractional dissipation term, we utilize the Grünwald-Letnikov approximation in conjunction with the predictor–corrector scheme. The existence and uniqueness of the discrete solution are theoretically proved, together with the optimal convergence order achieved and an error analysis associated with the $H^1$-seminorm and the $L^2$-norm. Numerical experiments are presented to support the theoretical findings and demonstrate the effectiveness of the proposed method with both convex and non-convex polygonal meshes. Full article

The main objective of this study is to present a fundamental mathematical model for nerve impulse transport, based on the underlying physical phenomena, with a straightforward application in describing the functionality of prosthetic devices. The governing equation of the resultant model is a two-dimensional nonlinear partial differential equation with a time-fractional derivative of order $\alpha \in (0,1)$. novel and effective numerical approach for solving this fractional-order problem is constructed based on the virtual element method. Three basic technical building blocks form the basis of our methodology: the regularity theory related to nonlinearity, discrete maximal regularity, and a fractional variant of the Grünwald–Letnikov approximation. By utilizing these components, along with the energy projection operator, a fully discrete virtual element scheme is formulated in such a way that it ensures stability and consistency. We establish the uniqueness and existence of the approximate solution. Numerical findings confirm the convergence in the $L^2$–norm and $H^1$–norm on both uniform square and regular Voronoi meshes, confirming the effectiveness of the proposed model and method, and their potential to support the efficient design of sensory prosthetics. Full article

This paper investigates the strong convergence of a Milstein scheme for a stochastic semilinear subdiffusion equation driven by fractionally integrated multiplicative noise. The existence and uniqueness of the mild solution are established via the Banach fixed point theorem. Temporal and spatial regularity properties of the mild solution are derived using the semigroup approach. For spatial discretization, the standard Galerkin finite element method is employed, while the Grünwald–Letnikov method is used for time discretization. The Milstein scheme is utilized to approximate the multiplicative noise. For sufficiently smooth noise, the proposed scheme achieves the temporal strong convergence order of $O({\tau }^{\alpha })$, $\alpha \in (0,1)$. Numerical experiments are presented to verify that the computational results are consistent with the theoretical predictions. Full article

This paper presents a circuit design methodology for the analog realization of time-varying fractional-order chaotic systems. While most existing studies implement such systems by switching between two or more constant fractional orders, these approaches become impractical when the fractional order changes smoothly over time. To overcome this limitation, the proposed method introduces a transfer function approximation specifically designed for variable fractional-order integrators. The formulation relies on a linear and time-invariant definition of the fractional-order operator, ensuring compatibility with Laplace-domain analysis. Under the condition that the fractional-order function is Laplace-transformable and its Bode plot slope lies between $-20$ dB/decade and 0 dB/decade, the system is realized using op-amps and standard RC components. The Grünwald–Letnikov method is employed for numerical calculation of phase portraits, which are then compared with simulation and experimental results. The strong agreement among these results confirms the effectiveness of the proposed method. Full article

The accurate and efficient simulation of seismic wave energy dissipation and phase dispersion during propagation in subsurface media due to inelastic attenuation is critical for the hydrocarbon-bearing distinction and improving the quality of seismic imaging in strongly attenuating geological media. The fractional viscoelastic equation, which quantifies frequency-independent anelastic effects, has recently become a focal point in seismic exploration. We have developed a novel hybrid staggered-grid Grünwald–Letnikov (HSGGL) finite difference method for solving the fractional viscoelastic equation in the time domain. The proposed method achieves accurate and computationally efficient solutions by using a staggered grid to discretize the first-order partial derivatives of the velocity–stress equations, combined with Grünwald–Letnikov finite difference discretization for the fractional-order terms. To improve the computational efficiency, we employ a preset accuracy to truncate the difference stencil, resulting in a compact fractional-order difference scheme. A stability analysis using the eigenvalue method reveals that the proposed method confers a relaxed stability condition, providing greater flexibility in the selection of sampling intervals. The numerical experiments indicate that the HSGGL method achieves a maximum relative error of no more than 0.17% compared to the reference solution (on a finely meshed domain) while being significantly faster than the conventional global FD method (GFD). In a 500 × 500 computational domain, the computation times for the proposed methods, which meet the specified accuracy levels used, are only approximately 4.67%, 4.47%, 4.44%, and 4.42% of that of the GFD method. This indicates that the novel HSGGL method has the potential as an effective forward modeling tool for understanding complex subsurface structures by employing a fractional viscoelastic equation. Full article

Edge detection is an essential image processing act that is crucial for many computer vision applications such as object detection, image segmentation, face recognition, text recognition, medical imaging, and autonomous vehicles. Deep learning is the most advanced and widely used tool of them all. In this paper, we present a novel deep learning model and use image datasets to test it. Our model uses a fractional calculus tool, which could enhance gradient approaches’ performances. Specifically, we approximate the fractional-order derivative-order neural network (GLFNet) using a Grünwald–Letnikov fractional definition. First, the original dataset is subjected to a Grünwald–Letnikov fractional order. After that, the CNN model is updated with the new dataset, concluding the standard CNN procedure. The training rate, the improvement in the F-measure for identifying the effective edge while maintaining the CNN model’s memory consumption, and the values of the loss errors between the prediction and training processes were all tested using the MNIST dataset. Our experiments show that GLFNet considerably enhances edge detection. GLFNet outperformed CNN with an average loss error ratio of 15.40, suggesting fewer loss mistakes. The F1-measure ratio of 0.81 indicates that GLFNet can compete with CNN in terms of precision and recall. The training time for GLFNet was lowered by an average ratio of 1.14 when compared to CNN, while inference time was faster with a ratio of 1.14, indicating increased efficiency. These findings demonstrate the efficacy of introducing Grünwald–Letnikov fractional convolution into deep learning models, resulting in more precise and reliable edge detection while preserving comparable memory utilization. Full article

This paper focuses on the influence of the fractional-order (FO) resonant capacitor on the zero-voltage-switching quasi-resonant converter (ZVS QRC). The FO impedance model of the capacitor is introduced to the circuit model of the ZVS QRC; hence, a piecewise smooth FO model is developed for the converter. Numerical solutions of the converter are obtained by using both the fractional Adams–Bashforth–Moulton (F-ABM) method and Oustaloup’s rational approximation method. In addition, the analytical solution of the converter is obtained by the Grünwald–Letnikov (GL) definition, which reveals the influence of the FO resonant capacitor on the zero-crossing point (ZCP) and resonant state of the converter. An experimental platform was built to verify the results of the theoretical analysis and numerical calculation. Full article

Fractional derivative operators are finding applications in a wide variety of fields with their ability to better model certain phenomena exhibiting spatial and temporal nonlocality. One area in which these operators are applicable is in the field of electromagnetism, thereby modelling transient wave propagation in complex media. To apply fractional derivative operators to electromagnetic problems, the operator must adhere to certain principles, like the trigonometric functions invariance property. The Grünwald–Letnikov and Marchaud fractional derivative operators comply with these principles and therefore could be applied. The fractional derivative arises when modelling frequency-dispersive dielectric media. The time-domain convolution integral in the relation between the electric displacement and the polarisation density, containing an empirical extension of the Debye model, is approximated directly. A common approach is to recursively update the convolution integral by approximating the time series by a truncated sum of decaying exponentials, with the coefficients found through means of optimisation or fitting. The finite-difference time-domain schemes using this approach have shown to be more computationally efficient compared to other approaches using auxiliary differential equation methods. Full article

In this paper, the finite-difference time-domain (FDTD) method is derived for electromagnetic simulations in media described by the time-fractional (TF) constitutive relations. TF Maxwell’s equations are derived based on these constitutive relations and the Grünwald–Letnikov definition of a fractional derivative. Then the FDTD algorithm, which includes memory effects and energy dissipation of the considered media, is introduced. Finally, one-dimensional signal propagation in such electromagnetic media is considered. The proposed FDTD method is derived based on a discrete approximation of the Grünwald–Letnikov definition of the fractional derivative and evaluated in a code. The stability condition is derived for the proposed FDTD method based on a numerical-dispersion relation. The obtained numerical results are compared with the outcomes of reference frequency-domain simulations, proving the accuracy of the proposed approach. However, high spatial resolution is required in order to obtain accurate results. The developed FDTD method is, unfortunately, computation and memory demanding when compared to the ordinary FDTD algorithm. Full article

System identification is an important methodology used in control theory and constitutes the first step of control design. It is known that many real systems can be better characterized by fractional-order models. However, it is often quite complex and difficult to apply classical control theory methods analytically for fractional-order models. For this reason, integer-order models are generally considered in classical control theory. In this study, an alternative approximation method is proposed for fractional-order models. The proposed method converts a fractional-order transfer function directly into an integer-order transfer function. The proposed method is based on curve fitting that uses a quadratic system model and Equilibrium Optimizer (EO) algorithm. The curve fitting is implemented based on the unit step response signal. The EO algorithm aims to determine the optimal coefficients of integer-order transfer functions by minimizing the error between general parametric quadratic model and objective data. The objective data are unit step response of fractional-order transfer functions and obtained by using the Grünwald-Letnikov (GL) method in the Fractional-Order Modeling and Control (FOMCON) toolbox. Thus, the coefficients of an integer-order transfer function most properly can be determined. Some examples are provided based on different fractional-order transfer functions to evaluate the performance of the proposed method. The proposed method is compared with studies from the literature in terms of time and frequency responses. It is seen that the proposed method exhibits better model approximation performance and provides a lower order model. Full article

This study aims to validate the hypothesis that the pharmacokinetics of certain drug regimes are better captured using fractional order differential equations rather than ordinary differential equations. To support this research, two numerical methods, the Grunwald–Letnikov and the L1 approximation, were implemented for the two-compartment model with Michaelis–Menten clearance kinetics for oral and intravenous administration of the drug. The efficacy of the numerical methods is verified through the use of the method of manufactured solutions due to the absence of an analytic solution to the proposed model. The model is derived from a phenomenological process leading to a dimensionally consistent and physically meaningful model. Using clinical data, the model is validated, and it is shown that the optimal model parameters select a fractional order for the clearance dynamic for certain drug regimes. These findings support the hypothesis that fractional differential equations better describe some pharmacokinetics. Full article