Symmetry

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33 pages, 1831 KB

Open AccessArticle

Observer-Based Stabilization of an Incommensurate Fractional-Order Discrete-Time SI Computer Virus Model

by Slim Dhahri, Essia Ben Alaia, Sahar Almashaan, Hatem Alwardi and Omar Naifar

Symmetry 2026, 18(6), 911; https://doi.org/10.3390/sym18060911 - 26 May 2026

Viewed by 96

This paper studies observer-based stabilization of a normalized incommensurate fractional-order discrete-time SI benchmark model for computer-virus propagation. The model is formulated with Caputo-like fractional-difference operators and allows the susceptible and infected compartments to have different memory orders. In contrast with a predictive malware-forecasting [...] Read more.

This paper studies observer-based stabilization of a normalized incommensurate fractional-order discrete-time SI benchmark model for computer-virus propagation. The model is formulated with Caputo-like fractional-difference operators and allows the susceptible and infected compartments to have different memory orders. In contrast with a predictive malware-forecasting model, the proposed system is explicitly treated as a dimensionless benchmark for qualitative analysis and control design. To clarify how the benchmark can be connected to empirical cybersecurity data, the revised formulation includes a calibration and fractional-order selection procedure based on normalized infection telemetry, admissible parameter sets, and loss minimization. The incommensurate orders are therefore interpreted as identifiable modeling parameters, not as arbitrary constants. The plant, observer, and control laws are formulated on the integer update grid, and the memory terms are implemented through the equivalent Volterra-type convolution representation. A nonlinear Luenberger-type observer is proposed under infected-state measurements, which is justified as a detectability-based cyber-monitoring configuration rather than a full observability assumption. The observer gain design, the full-state feedback design, and the observer-based output-feedback design are derived from first-order linearized incommensurate fractional-order models. The resulting criteria are expressed through characteristic-root conditions associated with linear incommensurate Caputo-type fractional-order difference systems. The scope of the theoretical claims is made explicit: the results provide local linearized-design guarantees and do not establish global or semi-global nonlinear stabilization. The nonlinear residuals, measurement-noise channel, incomplete-measurement formulation, and limitations of the linearized characteristic-root approach are stated explicitly so that the numerical section can assess robustness, sensitivity, and the effective region of attraction of the nonlinear closed loop. Full article

(This article belongs to the Special Issue Symmetry in Fractional Derivatives, Fractional Equations and Fractional Order Systems)

37 pages, 3174 KB

Open AccessArticle

Accountability-Aware Fractional Control for Embodied Intelligent Systems: Mittag-Leffler Stability and Conditional Proxemic Safety

by Slim Dhahri, Essia Ben Alaia, Sahar Almashaan, Hatem Alwardi and Omar Naifar

Symmetry 2026, 18(6), 889; https://doi.org/10.3390/sym18060889 - 24 May 2026

Viewed by 140

Abstract

This paper develops an accountability-aware fractional control framework for embodied intelligent systems in shared human environments. The approach combines a Caputo fractional-order stabilizing law, an intent-evidence realization with softmax belief reconstruction, and a conditional proxemic safety layer. Sufficient conditions are established for local [...] Read more.

This paper develops an accountability-aware fractional control framework for embodied intelligent systems in shared human environments. The approach combines a Caputo fractional-order stabilizing law, an intent-evidence realization with softmax belief reconstruction, and a conditional proxemic safety layer. Sufficient conditions are established for local Mittag-Leffler stability of the augmented error dynamics and forward invariance of the safe set. Numerical results are presented as a theorem-validation benchmark. For the base case with

α = 0.9

, the augmented error norm decays from

1.2359

to

9.90 \times 10^{- 3}

while the safety margin remains strictly positive, and the robustness condition is satisfied with a margin of

1.8641

. An

α

-sweep and a step-size convergence study further show that the fractional order induces a systematic safety–performance trade-off and that the reported behaviors are numerically stable. Additional simulations with four intent classes, bounded observation noise, and Monte Carlo uncertainty stress tests are included to strengthen the numerical evidence beyond the two-intent theorem-validation case. The manuscript also clarifies the quantitative interpretation of the accountability index, the conditional nature of the safety theorem, and an implementable sampled safety-filter realization for concrete robotic platforms. The results support the proposed framework as a mathematically consistent tool for shaping the balance between regulation and proxemic safety. Full article

(This article belongs to the Special Issue Symmetry in Fractional Derivatives, Fractional Equations and Fractional Order Systems)

26 pages, 2296 KB

Open AccessArticle

Insights into the Time-Fractional Nonlinear KdV-Type Equations Under Non-Singular Kernel Operators

by Mashael M. AlBaidani and Rabab Alzahrani

Symmetry 2026, 18(2), 391; https://doi.org/10.3390/sym18020391 - 23 Feb 2026

Cited by 2 | Viewed by 607

Abstract

In this study, nonlinear fractional Korteweg–de Vries (KdV) type equations with nonlocal operators are studied using Mittag–Leffler kernels and exponential decay. The KdV equations are well known for its use in modeling ion-acoustic waves in plasma, oceanic dynamics, and shallow-water waves. As a [...] Read more.

In this study, nonlinear fractional Korteweg–de Vries (KdV) type equations with nonlocal operators are studied using Mittag–Leffler kernels and exponential decay. The KdV equations are well known for its use in modeling ion-acoustic waves in plasma, oceanic dynamics, and shallow-water waves. As a result, mathematicians are working to examine modified and generalized versions of the basic KdV equation. In order to find the solutions of nonlinear fractional KdV equations, an extension of this concept is described in the current paper. The solution of fractional KdV equations is carried out using the well-known natural transform decomposition method (NTDM). To evaluate the problem, we employ the fractional operator in the Caputo–Fabrizio (CF) and the Atangana–Baleanu–Caputo sense (ABC) manner. Nonlinear terms can be handled with Adomian polynomials. The main advantage of this novel approach is that it might offer an approximate solution in the form of convergent series using easy calculations. The dynamical behavior of the resulting solutions have been demonstrated using graphs. Numerical data is represented visually in the tables. The solutions at various fractional orders are found and it is proved that they all tend to an integer-order solution. Additionally, we examine our findings with those of the iterative transform method (ITM) and the residual power series transform method (RPSTM). It is evident from the comparison that our approach offers better outcomes compared to other approaches. The results of the suggested method are very accurate and give helpful details on the real dynamics of each issue. The present technique can be expanded to address other significant fractional order problems due to its straightforward implementation. Full article

(This article belongs to the Special Issue Symmetry in Fractional Derivatives, Fractional Equations and Fractional Order Systems)

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20 pages, 16375 KB

Open AccessArticle

On Fractional Partial Differential Systems with Incommensurate Orders: Stability Analysis of Some Reaction–Diffusion Models

by Omar Kahouli, Amel Hioual, Adel Ouannas and Sulaiman Almohaimeed

Symmetry 2026, 18(1), 52; https://doi.org/10.3390/sym18010052 - 26 Dec 2025

Viewed by 593

Abstract

This work develops and analyzes an incommensurate fractional FitzHugh–Nagumo (FHN) reaction–diffusion system in which each state variable evolves with a distinct fractional order. The formulation extends the classical and commensurate fractional models by incorporating heterogeneous memory effects that break temporal symmetry between the [...] Read more.

This work develops and analyzes an incommensurate fractional FitzHugh–Nagumo (FHN) reaction–diffusion system in which each state variable evolves with a distinct fractional order. The formulation extends the classical and commensurate fractional models by incorporating heterogeneous memory effects that break temporal symmetry between the activator and inhibitor variables. After establishing the mathematical framework, the equilibrium states of the system are derived and subjected to a detailed local stability analysis in both diffusion-free and diffusion-driven regimes. Explicit stability criteria are obtained by examining the spectral properties of the linearized operator under incommensurate fractional dynamics. Numerical simulations based on a Caputo L1 discretization scheme corroborate the theoretical results and demonstrate how asymmetric memory orders influence transient behavior, convergence rates, and the qualitative structure of the solutions. The study provides the first systematic stability characterization of an incommensurate fractional FitzHugh–Nagumo reaction–diffusion model, highlighting the role of fractional-order asymmetry in shaping the system’s dynamical response. Full article

(This article belongs to the Special Issue Symmetry in Fractional Derivatives, Fractional Equations and Fractional Order Systems)

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22 pages, 2920 KB

Open AccessArticle

Delannoy Tau-Based Numerical Procedure for the Time-Fractional Cable Model

by Ahmed Gamal Atta, Mohamed A. Abdelkawy, Naher Mohammed A. Alsafri and Waleed Mohamed Abd-Elhameed

Symmetry 2025, 17(11), 1916; https://doi.org/10.3390/sym17111916 - 8 Nov 2025

Cited by 1 | Viewed by 599

Abstract

This study uses the spectral tau method to treat the time-fractional cable equation (TFCE). The proposed algorithm uses the shifted Delannoy polynomials, which are non-symmetric orthogonal. The orthogonality property of the non-symmetric shifted Delannoy polynomials and some representations facilitate obtaining accurate spectral approximations [...] Read more.

This study uses the spectral tau method to treat the time-fractional cable equation (TFCE). The proposed algorithm uses the shifted Delannoy polynomials, which are non-symmetric orthogonal. The orthogonality property of the non-symmetric shifted Delannoy polynomials and some representations facilitate obtaining accurate spectral approximations for the TFCE. Several numerical examples ensure the efficiency and accuracy of the method. We compare the suggested scheme to other algorithms and benchmark it against existing analytical solutions to demonstrate the high accuracy of our presented algorithm. Full article

(This article belongs to the Special Issue Symmetry in Fractional Derivatives, Fractional Equations and Fractional Order Systems)

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26 pages, 389 KB

Open AccessArticle

On Hilfer–Hadamard Tripled System with Symmetric Nonlocal Riemann–Liouville Integral Boundary Conditions

by Shorog Aljoudi, Hind Alamri and Alanoud Alotaibi

Symmetry 2025, 17(11), 1867; https://doi.org/10.3390/sym17111867 - 4 Nov 2025

Viewed by 595

Abstract

The objective of this manuscript is to investigate the existence, uniqueness criteria and Ulam–Hyers stability of solutions to tripled systems of the Hilfer–Hadamard type supplemented with symmetric nonlocal multi-point Riemann–Liouville integral boundary conditions. By converting the considered problem into an equivalent fixed-point problem, [...] Read more.

The objective of this manuscript is to investigate the existence, uniqueness criteria and Ulam–Hyers stability of solutions to tripled systems of the Hilfer–Hadamard type supplemented with symmetric nonlocal multi-point Riemann–Liouville integral boundary conditions. By converting the considered problem into an equivalent fixed-point problem, the existence and uniqueness are proven by application of the Leray–Schauder nonlinear alternative and Banach’s contraction principle, respectively. In addition, we discuss the Ulam–Hyers stability and generalized Ulam–Hyers stability of the results, and illustrative examples are also presented to demonstrate their correctness and effectiveness. Full article

(This article belongs to the Special Issue Symmetry in Fractional Derivatives, Fractional Equations and Fractional Order Systems)

27 pages, 665 KB

Open AccessArticle

Study of Stability and Simulation for Nonlinear (k, ψ)-Fractional Differential Coupled Laplacian Equations with Multi-Point Mixed (k, ψ)-Derivative and Symmetric Integral Boundary Conditions

by Xiaojun Lv and Kaihong Zhao

Symmetry 2025, 17(3), 472; https://doi.org/10.3390/sym17030472 - 20 Mar 2025

Cited by 3 | Viewed by 843

Abstract

The

(k, ψ)

-fractional derivative based on the k-gamma function is a more general version of the Hilfer fractional derivative. It is widely used in differential equations to describe physical phenomena, population dynamics, and biological genetic memory problems. In [...] Read more.

The

(k, ψ)

-fractional derivative based on the k-gamma function is a more general version of the Hilfer fractional derivative. It is widely used in differential equations to describe physical phenomena, population dynamics, and biological genetic memory problems. In this article, we mainly study the

4 m + 2

-point symmetric integral boundary value problem of nonlinear

(k, ψ)

-fractional differential coupled Laplacian equations. The existence and uniqueness of solutions are obtained by the Krasnosel’skii fixed-point theorem and Banach’s contraction mapping principle. Furthermore, we also apply the calculus inequality techniques to discuss the stability of this system. Finally, three interesting examples and numerical simulations are given to further verify the correctness and effectiveness of the conclusions. Full article

(This article belongs to the Special Issue Symmetry in Fractional Derivatives, Fractional Equations and Fractional Order Systems)

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20 pages, 309 KB

Open AccessArticle

Fractional Optimal Control Problem for Symmetric System Involving Distributed-Order Atangana–Baleanu Derivatives with Non-Singular Kernel

by Bahaa Gaber Mohamed and Ahlam Hasan Qamlo

Symmetry 2025, 17(3), 417; https://doi.org/10.3390/sym17030417 - 10 Mar 2025

Cited by 11 | Viewed by 1486

Abstract

The objective of this work is to discuss and thoroughly analyze the fractional variational principles of symmetric systems involving distributed-order Atangana–Baleanu derivatives. A component of distributed order, the fractional Euler–Lagrange equations of fractional Lagrangians for constrained systems are studied concerning Atangana–Baleanu derivatives. We [...] Read more.

The objective of this work is to discuss and thoroughly analyze the fractional variational principles of symmetric systems involving distributed-order Atangana–Baleanu derivatives. A component of distributed order, the fractional Euler–Lagrange equations of fractional Lagrangians for constrained systems are studied concerning Atangana–Baleanu derivatives. We give a general formulation and a solution technique for a class of fractional optimal control problems (FOCPs) for such systems. The dynamic constraints are defined by a collection of FDEs, and the performance index of an FOCP is considered a function of the control variables and the state. The formula for fractional integration by parts, the Lagrange multiplier, and the calculus of variations are used to obtain the Euler–Lagrange equations for the FOCPs. Full article

(This article belongs to the Special Issue Symmetry in Fractional Derivatives, Fractional Equations and Fractional Order Systems)

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13 pages, 270 KB

Open AccessArticle

Existence of Positive Solutions for Singular Difference Equations with Nonlinear Boundary Conditions

by Hua Luo and Alhussein Mohamed

Symmetry 2024, 16(10), 1313; https://doi.org/10.3390/sym16101313 - 5 Oct 2024

Cited by 1 | Viewed by 1161

Abstract

In this paper, we delve into a discrete nonlinear singular semipositone problem, characterized by a nonlinear boundary condition. The nonlinearity, given by

f (u) - \frac{a}{u^{α}}

with

α > 0

, exhibits a singularity at

u = 0

and [...] Read more.

In this paper, we delve into a discrete nonlinear singular semipositone problem, characterized by a nonlinear boundary condition. The nonlinearity, given by

f (u) - \frac{a}{u^{α}}

with

α > 0

, exhibits a singularity at

u = 0

and tends towards

- \infty

as u approaches

0^{+}

. By constructing some suitable auxiliary problems, the difficulty that arises from the singularity and semipositone of nonlinearity and the lack of a maximum principle is overcome. Subsequently, employing the Krasnosel’skii fixed-point theorem, we determine the parameter range that ensures the existence of at least one positive solution and the emergence of at least two positive solutions. Furthermore, based on our existence results, one can obtain the symmetry of the solutions after adding some symmetric conditions on the given functions by using a standard argument. Full article

(This article belongs to the Special Issue Symmetry in Fractional Derivatives, Fractional Equations and Fractional Order Systems)

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