Search Results (1,340)

This study introduces a new class of coupled differential systems described by fractal–fractional Caputo derivatives with both constant and state-dependent delays. In contrast to traditional delay differential equations, the proposed framework integrates memory effects and geometric complexity while capturing adaptive feedback delays that vary with the system’s state. Such a formulation provides a closer representation of biological and physical processes in which delays are not fixed but evolve dynamically. Sufficient conditions for the existence and uniqueness of solutions are established using fixed-point theory, while the stability of the solution is investigated via the Hyers–Ulam (HU) stability approach. To demonstrate applicability, the approach is applied to two illustrative examples, including a predator–prey interaction model. The findings advance the theory of fractional-order systems with mixed delays and offer a rigorous foundation for developing realistic, application-driven dynamical models. Full article

We address a fractional spatial Stefan problem derived from a non-Fourier heat flux model with a convective boundary condition at the fixed boundary. An explicit solution is obtained in terms of a three-parameter Mittag–Leffler function. A dimensionless formulation is used to derive a family of fractional spatial Stefan problems that depend on the Biot and Stefan numbers. Finally, a straightforward numerical method for approximating the solutions is presented, along with numerical experiments analyzing the influence of the physical parameters and the order of fractional differentiation. Full article

This research paper uses two-stage explicit fractional numerical schemes to solve fractional-order initial value problems of ODEs. The proposed methods exhibit structural symmetry in their formulation, contributing to enhanced numerical stability and balanced error behavior across computational steps. The schemes utilize constant and variable step sizes, allowing them to adapt efficiently to solve the considered fractional-order initial value problems. These schemes employ variable step-size control based on error estimation, aiming to minimize computational costs while maintaining good accuracy and stability. We discuss the linear stability of the proposed numerical schemes and observe that a higher-stability region is obtained when the fractional parameter value equals one. We also discuss consistency and convergence analysis of the proposed methods and observe that as the fractional parameter values rise from 0 to 1, the scheme’s convergence rate improves and achieves its maximum at 1. Several numerical test problems are used to demonstrate the efficiency of the proposed methods in solving fractional-order initial value problems with either constant or variable step sizes. The proposed numerical schemes’ results demonstrate better accuracy and convergence behavior than the existing methods used for comparison. Full article

Fractional-order chaotic systems have emerged as powerful tools in secure communications and multimedia protection owing to their memory-dependent dynamics, large key spaces, and high sensitivity to initial conditions. However, most existing fractional-order image encryption schemes rely on fixed-order chaos and conventional solvers, which limit their complexity and reduce unpredictability, while also neglecting the potential of variable fractional-order (VFO) dynamics. Although similar phenomena have been reported in some fractional-order systems, the coexistence of hidden attractors and stable equilibria has not been extensively investigated within VFO frameworks. To address these gaps, this paper introduces a novel discrete variable fractional-order dark matter–dark energy (VFODM-DE) chaotic system. The system is discretized using the piecewise constant argument discretization (PWCAD) method, enabling chaos to emerge at significantly lower fractional orders than previously reported. A comprehensive dynamic analysis is performed, revealing rich behaviors such as multistability, symmetry properties, and hidden attractors coexisting with stable equilibria. Leveraging these enhanced chaotic features, a pseudorandom number generator (PRNG) is constructed from the VFODM-DE system and applied to grayscale image encryption through permutation–diffusion operations. Security evaluations demonstrate that the proposed scheme offers a substantially large key space (approximately $2^{249}$) and exceptional key sensitivity. The scheme generates ciphertexts with nearly uniform histograms, extremely low pixel correlation coefficients (less than 0.04), and high information entropy values (close to 8 bits). Moreover, it demonstrates strong resilience against differential attacks, achieving average NPCR and UACI values of about 99.6% and 33.46%, respectively, while maintaining robustness under data loss conditions. In addition, the proposed framework achieves a high encryption throughput, reaching an average speed of 647.56 Mbps. These results confirm that combining VFO dynamics with PWCAD enriches the chaotic complexity and provides a powerful framework for developing efficient and robust chaos-based image encryption algorithms. Full article

This main focus of this work is the fractional-order nonlinear Schrödinger equation with wave operators. First, a conservative difference scheme is constructed. Then, the discrete energy and mass conservation formulas are derived and maintained by the difference scheme constructed in this paper. Through rigorous theoretical analysis, it is proved that the constructed difference scheme is unconditionally stable and has second-order precision in both space and time. Due to the completely implicit property of the differential scheme proposed, a linearized iterative algorithm is proposed to implement the conservative differential scheme. Numerical experiments including one example with the fractional boundary conditions were studied. The results effectively demonstrate the long-term numerical behaviors of the fractional nonlinear Schrödinger equations with wave operators. Full article

Background/Objectives: Germ Cell Neoplasia In Situ (GCNIS) is considered the precursor lesion for the majority of testicular germ cell tumors (TGCTs). The aim of this study was to evaluate whether first-order radiomics features derived from volumetric diffusion tensor imaging (DTI) metrics—specifically apparent diffusion coefficient (ADC) and fractional anisotropy (FA) histogram parameters—can detect GCNIS. Methods: This study included 15 men with TGCTs and 10 controls. All participants underwent scrotal MRI, including DTI. Volumetric ADC and FA histogram metrics were calculated for the following tissues: group 1, TGCT; group 2: testicular parenchyma adjacent to tumor, histologically positive for GCNIS; and group 3, normal testis. Non-parametric statistics were used to assess differences in ADC and FA histogram parameters among the three groups. Pearson’s correlation analysis was followed by ordinal regression analysis to identify key predictive histogram parameters. Results: Widespread distributional differences (p < 0.05) were observed for many ADC and FA variables, with both TGCTs and GCNIS showing significant divergence from normal testes. Among the ADC statistics, the 10th percentile and skewness (p = 0.042), range (p = 0.023), interquartile range (p = 0.021), total energy (p = 0.033), entropy and kurtosis (p = 0.027) proved the most significant predictors for tissue classification. FA_energy (p = 0.039) was the most significant fingerprint of the carcinogenesis among the FA metrics. These parameters correctly characterized 88.8% of TGCTs, 87.5% of GCNIS tissues and 100% of normal testes. Conclusion: Radiomics features derived from volumetric ADC and FA histograms have promising potential to differentiate TGCTs, GCNIS, and normal testicular tissue, aiding early detection and characterization of pre-cancerous lesions. Full article

Fractional differential equations have emerged as a prominent focus of modern scientific research due to their advantages in describing the complexity and nonlinear behavior of many physical phenomena. In particular, when considering problems with initial-boundary value conditions, the solution of nonlinear fractional differential equations becomes particularly important. This paper aims to explore the fractional soliton solutions for the three-component fractional nonlinear Schrödinger (TFNLS) equation under the zero background. According to the Lax pair and fractional recursion operator, we obtain fractional nonlinear equations with Riesz fractional derivatives, which ensure the integrability of these equations. In particular, by the completeness relation of squared eigenfunctions, we derive the explicit form of the TFNLS equation. Subsequently, in the reflectionless case, we construct the fractional N-soliton solutions via the Riemann–Hilbert (RH) method. The analysis results indicate that as the order of the Riesz fractional derivative increases, the widths of both one-soliton and two-soliton solutions gradually decrease. However, the absolute values of wave velocity, phase velocity, and group velocity of one component of the vector soliton exhibit an increasing trend, and show power-law relationships with the amplitude. Full article

This paper investigates soliton solutions of the time-fractional Biswas–Arshed (BA) equation using the Extended Simplest Equation Method (ESEM). The model is analyzed under two distinct fractional derivative operators: the $\beta$-derivative and the M-truncated derivative. These approaches yield diverse solution types, including kink, singular, and periodic-singular forms. Also, in this work, a nonlinear second-order differential equation is reconstructed as a planar dynamical system in order to study its bifurcation structure. The stability and nature of equilibrium points are established using a conserved Hamiltonian and phase space analysis. A bifurcation parameter that determines the change from center to saddle-type behaviors is identified in the study. The findings provide insight into the fundamental dynamics of nonlinear wave propagation by showing how changes in model parameters induce qualitative changes in the phase portrait. The derived solutions are depicted via contour plots, along with two-dimensional (2D) and three-dimensional (3D) representations, utilizing Mathematica for computational validation and graphical illustration. This study is motivated by the growing role of fractional calculus in modeling nonlinear wave phenomena where memory and hereditary effects cannot be captured by classical integer-order approaches. The time-fractional Biswas–Arshed (BA) equation is investigated to obtain diverse soliton solutions using the Extended Simplest Equation Method (ESEM) under the $\beta$-derivative and M-truncated derivative operators. Beyond solution construction, a nonlinear second-order equation is reformulated as a planar dynamical system to analyze its bifurcation and stability properties. This dual approach highlights how parameter variations affect equilibrium structures and soliton behaviors, offering both theoretical insights and potential applications in physics and engineering. Full article

We consider first-order impulses for impulsive stochastic differential equations driven by fractional Brownian motion (fBm) with Hurst parameter $H\in ({\displaystyle \frac{1}{2}},1)$ involving a nonlinear $\varphi$-Laplacian operator. The system incorporates both state and derivative impulses at fixed time instants. First, we establish the existence of at least one mild solution under appropriate conditions in terms of nonlinearities, impulses, and diffusion coefficients. We achieve this by applying a nonlinear alternative of the Leray–Schauder fixed-point theorem in a generalized Banach space setting. The topological structure of the solution set is established, showing that the set of all solutions is compact, closed, and convex in the function space considered. Our results extend existing impulsive differential equation frameworks to include fractional stochastic perturbations (via fBm) and general $\varphi$-Laplacian dynamics, which have not been addressed previously in tandem. These contributions provide a new existence framework for impulsive systems with memory and hereditary properties, modeled in stochastic environments with long-range dependence. Full article

The solution of a nonhomogeneous linear Caputo fractional differential equation of order $nq,(n-1)<nq<n$ with Caputo fractional initial conditions can be expressed using suitable Mittag–Leffler functions. In order to extend this result to such a nonhomogeneous linear Caputo fractional differential equation of order $nq,(n-1)<nq<n,$ that also includes lower order fractional derivative terms, we can reduce such a problem to an n-system of Caputo fractional differential equations of order $q,0<q<1,$ with corresponding initial conditions. In this work, we use an approximation method to solve the resulting system of Caputo fractional differential equations of order q with initial conditions, using the fundamental matrix solutions involving the matrix Mittag–Leffler functions. Furthermore, we compute the fundamental matrix solution using the standard eigenvalue method. This fundamental matrix solution then allows us to express the component-wise solutions of the system using initial conditions, similar to the scalar case. As a consequence, we obtain solutions to linear nonhomogeneous Caputo fractional differential equations of order $nq,(n-1)<nq<n,$ with Caputo fractional initial conditions having lower-order Caputo derivative terms. We illustrate the method with several examples for two and three system, considering cases where the eigenvalues are real and distinct, real and repeated, or complex conjugates. Full article

In this study, we explore the soliton solutions of the truncated M-fractional fifth-order Korteweg–de Vries (KdV) equation by applying the improved modified extended tanh function method (IMETM). Novel analytical solutions are obtained for the proposed system, such as brigh soliton, dark soliton, hyperbolic, exponential, Weierstrass, singular periodic, and Jacobi elliptic periodic solutions. To validate these results, we present detailed graphical representations of selected solutions, demonstrating both their mathematical structure and physical behavior. Furthermore, we conduct a comprehensive linear stability analysis to investigate the stability of these solutions. Our findings reveal that the fractional derivative significantly affects the amplitude, width, and velocity of the solitons, offering new insights into the control and manipulation of soliton dynamics in fractional systems. The novelty of this work lies in extending the IMETM approach to the truncated M-fractional fifth-order KdV equation for the first time, yielding a wide spectrum of exact analytical soliton solutions together with a rigorous stability analysis. This research contributes to the broader understanding of fractional differential equations and their applications in various scientific fields. Full article

This paper addresses the issue of rotor swings in a high-power synchronous generator during stable operation with a stiff power grid. The analysis of electromechanical swings was conducted using a circuit model incorporating fractional-order derivatives. Assuming that variations in the load angle under small disturbances from a stable equilibrium are minor, a linearized differential equation describing the electrodynamic state of the synchronous machine was derived. Based on this linearized equation of motion and the identified parameters of the equivalent circuit, calculations were performed for a 200 MW turbogenerator. The results indicate that the electromechanical swings are characterized by a constant pulsation and a low damping factor. Calculations were also carried out using a lumped-parameter equivalent circuit model. Based on the obtained results, it can be stated that the fractional-order model provides a more accurate fit of the frequency characteristics compared with the classical model with the same number of rotor equivalent circuits. The relative approximation errors for the fractional-order model are, for the d-axis (one rotor equivalent circuit), relative magnitude error δ_m = 1.53% and relative phase error δ_φ = 6.32%, and for the q-axis (two rotor equivalent circuits), δ_m = 3.2% and δ_φ = 8.3%. To achieve comparable approximation accuracy for the classical model, the rotor electrical circuit must be replaced with two equivalent circuits in the d-axis and four equivalent circuits in the q-axis, yielding relative errors of δ_m = 2.85% and δ_φ = 6.51% for the d-axis, and δ_m = 1.86% and δ_φ = 5.49% for the q-axis. Full article

The accurate estimation of parameters in dynamical systems stands for an open key research issue in modeling, control, and fault diagnosis. The presence of noise in input and output signals poses a serious challenge for accurate real-time dynamical system parameter estimation. This paper proposes a new robust algebraic parameter estimation methodology for integer-order dynamical systems that explicitly incorporates the signal filtering dynamics within the estimator structure and enhances noise attenuation through fractional differentiation in frequency domain. The introduced estimation methodology is valid for Liouville-type fractional derivatives and can be applied to estimate online the parameters of differentially flat, oscillating or vibrating systems of multiple degrees of freedom. The parametric estimation can be thus implemented for a wide class of oscillating or vibrating, nth-order dynamical systems under noise influence in measurement and control signals. Positive values are considered for the inertia, stiffness, and viscous damping parameters of vibrating systems. Parameter identification can be also used for development of actuators and control technology. In this sense, validation of the algebraic parameter estimation is performed to identify parameters of a differentially flat, permanent-magnet direct-current motor actuator. Parameter estimation for both open-loop and closed-loop control scenarios using experimental data is examined. Experimental results demonstrate that the new parameter estimation methodology combining signal filtering dynamics and fractional calculus outperforms other conventional methods under presence of significant noise in measurements. Full article

We introduce the generalized fractional-order Dirac delta distribution ${\delta }_{\mathrm{GFODDF}}$, defined by applying the generalized fractional derivative (GFD) operator to the Heaviside function. This construction extends the classical Dirac delta to non-integer orders, allowing modeling of systems with memory and non-local effects. We establish fundamental properties—including shifting, scaling, evenness, derivative, and convolution—within a rigorous distributional framework and present explicit proofs. Applications are demonstrated by solving linear fractional differential equations and by modeling drug release with fractional kinetics, where the new delta captures impulse responses with long-term memory. Numerical illustrations confirm that ${\delta }_{\mathrm{GFODDF}}$ reduces to the classical delta when $\eta =1$, while providing additional flexibility for $0<\eta <1$. These results show that ${\delta }_{\mathrm{GFODDF}}$ is a powerful tool for fractional-order analysis in mathematics, physics, and biomedical engineering. Full article

This paper investigates linear two-sided fractional matrix delay differential equations (TSFMDDE). Firstly, the two-sided fractional delayed Mittag-Leffler matrix functions (TSFDMLMF) are constructed. Further, the representation of solutions of two-sided homogeneous and nonhomogeneous problems are studied, and Ulam–Hyers (UH) stability of a two-sided nonhomogeneous problem is discussed. Lastly, we provide a numerical example to demonstrate our results. In the numerical example, the fractional order $\beta =0.6$, delay $\varrho =2$, UH constant $u_h\approx 5.92479$, $n=2$, and $s\in [-2,4]$. Full article