Search Results (1,133)

The purpose of this paper is to investigate a class of novel symmetric coupled fuzzy fractional partial differential equation system involving the Caputo–Katugampola (C-K) generalized Hukuhara (gH) derivative. Within the framework of C-K gH differentiability, two types of gH weak solutions are defined, and their existence is rigorously established through explicit constructions via employing Schauder fixed point theorem, overcoming the limitations of traditional Lipschitz conditions and thereby extending applicability to non-smooth and nonlinear systems commonly encountered in practice. A typical numerical example with potential applications is proposed to verify the existence results of the solutions for the symmetric coupled system. Furthermore, we introduce Ulam–Hyers stability (U-HS) theory into the analysis of such symmetric coupled systems and establish explicit stability criteria. U-HS ensures the existence of approximate solutions close to the exact solution under small perturbations, and thereby guarantees the reliability and robustness of the systems, while it prevents significant deviations in system dynamics caused by minor disturbances. We not only enrich the theoretical framework of fuzzy fractional calculus by extending the class of solvable systems and supplementing stability analysis, but also provide a practical mathematical tool for investigating complex interconnected systems characterized by uncertainty, memory effects, and spatial dynamics. Full article

The Chang 7 member of the Ordos Basin hosts abundant shale oil and gas resources and plays a vital role in the development of unconventional energy. This study investigates differences in damage evolution and underlying mechanisms between representative shale oil and shale gas reservoir cores from the Chang 7 member under fracturing fluid hydration. A combination of high-temperature expansion tests, nuclear magnetic resonance (NMR), and micro-computed tomography (Micro-CT) was used to systematically characterize macroscopic expansion behavior and microscopic pore structure evolution. Results indicate that shale gas cores undergo faster expansion and higher imbibition rates during hydration (reaching stability in 10 h vs. 23 h for shale oil cores), making them more vulnerable to water-lock damage, while shale oil cores exhibit slower hydration but more pronounced pore structure reconstruction. After 72 h of immersion in fracturing fluid, both core types experienced reduced pore volumes and structural reorganization; however, shale oil cores demonstrated greater capacity for pore reconstruction, with a newly formed pore volume fraction of 34.5% compared to 24.6% for shale gas cores. NMR and Micro-CT analyses reveal that hydration is not merely a destructive process but a dynamic “damage–reconstruction” evolution. Furthermore, the addition of clay stabilizers effectively mitigates water sensitivity and preserves pore structure, with 0.7% identified as the optimal concentration. The research results not only reveal the differential response law of fracturing fluid damage in the Chang 7 shale reservoir but also provide a theoretical basis and technical support for optimizing fracturing fluid systems and achieving differential production increases. Full article

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 introduce a class of triply coupled systems of differential equations with fractal–fractional Caputo derivatives and time-dependent delays. This framework captures long-memory effects and complex structural patterns while allowing delays to evolve over time, offering greater realism than constant-delay models. The existence and uniqueness of solutions are established using fixed point theory, and Hyers–Ulam stability is analyzed. A numerical scheme based on the Adams–Bashforth method is implemented to approximate solutions. The approach is illustrated through a numerical example and applied to a three-species food-chain model, comparing scenarios with and without time-dependent delays to demonstrate their impact on system dynamics. 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 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

To explore strategies for the safe utilization of farmland co-contaminated with cadmium (Cd) and lead (Pb), this field study systematically evaluated the impacts of humic acid (HA) and potassium fulvate (PF) at different application rates (0, 1500, 3000, and 4500 kg·ha⁻¹) on the growth, yield, and translocation of Cd and Pb within the soil–plant system of maize (Zea mays L.). The results showed that while HA and PF did not significantly alter total soil Cd and Pb concentrations, they markedly reduced their bioavailable fractions. This mitigation of heavy metal phytotoxicity significantly promoted maize growth and yield, with the high-dose HA treatment increasing yield by a maximum of 32.9%. Both amendments dose-dependently decreased Cd and Pb concentrations, bioconcentration factors (BCF), and translocation factors (TF) in all maize tissues, particularly in the grains. At equivalent application rates, PF was slightly more effective than HA in reducing heavy metal concentrations in the grains. Notably, a significant positive correlation was observed between Cd and Pb concentrations across all plant parts, confirming a synergistic accumulation and translocation mechanism. This synergy provides a physiological explanation for the broad-spectrum immobilization efficacy of these humic substances. In conclusion, applying HA and PF presents a dual-benefit strategy for increasing yield and reducing risks in Cd- and Pb-contaminated farmlands. This study proposes a differentiated application approach: PF is the preferred option when ensuring food-grade safety is the primary goal, whereas high-dose HA is more advantageous for maximizing yield in soils with low-to-moderate contamination risk. 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

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 study presents an innovative numerical method for solving linear fractional differential equations (LFDEs) using modified Bernstein polynomial bases. The proposed approach effectively addresses the challenges posed by the nonlocal nature of fractional derivatives, providing a robust framework for handling multiple initial and boundary value constraints. By integrating the LFDEs and approximating the solutions with modified fractional-order Bernstein polynomials, we derive operational matrices to solve the resulting system numerically. The method’s accuracy is validated through several examples, showing excellent agreement between numerical and exact solutions. Comparative analysis with existing data further confirms the reliability of the approach, with absolute errors ranging from 10⁻¹⁸ to 10⁻⁴. The results highlight the method’s efficiency and versatility in modeling complex systems governed by fractional dynamics. This work offers a computationally efficient and accurate tool for fractional calculus applications in science and engineering, helping to bridge existing gaps in numerical techniques. Full article

This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of memory effects without singularities. Unlike existing approaches, which are limited to either neutral or hybrid stochastic structures, the proposed framework unifies both features within a fractional setting, capturing the joint influence of randomness, history, and abrupt transitions in real-world processes. We establish the existence and uniqueness of mild solutions via the Picard approximation method under generalized Carathéodory-type conditions, allowing for non-Lipschitz nonlinearities. In addition, mean-square Mittag–Leffler stability is analyzed to characterize the boundedness and decay properties of solutions under stochastic fluctuations. Several illustrative examples are provided to validate the theoretical findings and demonstrate their applicability. Full article

Non-zero differential initial values hinder the application of fractional operator theory in practical systems. This paper proposes a differential initial values zeroing method, decomposing functions with non-zero differential initial values into a compensation function (Taylor polynomial) and a zeroing function (with all differential initial values being zero). A differential initial values “zeroing operator” is defined, with properties such as initial value annihilation and linearity, and operational rules compatible with unilateral Laplace transforms and Mikusinski calculus operators. Based on the zeroing operator, the “zeroing differential operator” is defined to extract the zero-initial-value differential intrinsic properties of the functions with non-zero differential initial values. Using the zeroing operator, fractional constitutive equations are reconstructed in both time and complex Laplace domains in the self-congruent physical space, introducing complex fractional operators and generalized fractional operators. Validated by the complex fractional constitutive model of bone, this method breaks the bottleneck of zero-initial-value assumption in fractional operator theory in the self-congruent physical space, providing a rigorous mathematical foundation and a standardized tool for modeling sophisticated fractional systems with non-zero differential initial values. Full article