Search Results (4,366)

In this work, we numerically investigate the fractional clannish random walker’s parabolic equations (FCRWPEs) and the nonlinear fractional Cahn–Allen (NFCA) equation using the Hybrid Decomposition Method (HDM). The analysis uses the Atangana–Baleanu fractional derivative (ABFD) in the Caputo sense, which has a nonsingular and nonlocal Mittag–Leffler kernel (MLk) and provides a more accurate depiction of memory and heredity effects, to examine the dynamic behavior of the models. Using nonlinear analysis, the uniqueness of the suggested models is investigated, and distinct wave profiles are created for various fractional orders. The accuracy and effectiveness of the suggested approach are validated by a number of example cases, which also support the approximate solutions of the nonlinear FCRWPEs. This work provides significant insights into the modeling of anomalous diffusion and complex dynamic processes in fields such as phase transitions, biological transport, and population dynamics. The inclusion of the ABFD enhances the model’s ability to capture nonlocal effects and long-range temporal correlations, making it a powerful tool for simulating real-world systems where classical derivatives may be inadequate. Full article

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

This paper aims to discuss the positive mild solutions for nonlocal fractional variable exponent differential equations with concave and convex coefficients. Based on a specifically defined order cone, even under the influence of the $p(t)$-Laplacian operator and the fractional integral operator, we avoid making many assumptions on the nonlocal coefficient $\mathcal{A}$ and just require that $\mathcal{A}>0$ on a set of positive measures. Utilizing the fixed-point index theory on cones, some new results on the existence and multiplicity of positive mild solutions were obtained, which extend and enrich some previous research findings. Finally, numerical examples are used to verify the feasibility of our main results. Full article

Designing an effective electrical model for charged water bodies is of great significance in reducing the risk of electric shock in water and enhancing the safety and reliability of electrical equipment. Aiming to resolve the problems faced in using existing charged water body modeling methods, a practical circuit model of a charged water body is developed. The basic units of the model are simply constructed using fractional-order resistance–capacitance (RC) parallel circuits. The state variables of the model can be obtained by solving the circuit equations. In addition, a practical method for obtaining the circuit model parameters is also developed. This enables the estimation of the characteristics of charged water bodies under different conditions through model simulation. The effectiveness of the proposed method is verified by comparing the estimated voltage and leakage current of the model with the actual measured values. The comparison results show that the estimated value of the model is close to the actual characteristics of the charged water body. Full article

Concrete materials undergo a series of physical and chemical changes under high temperature, leading to the degradation of mechanical properties. This study investigates basalt fiber-reinforced concrete (BFRC) through high-temperature testing using the split Hopkinson pressure bar (SHPB) apparatus. Impact compression tests were conducted on specimens after exposure to elevated temperatures to analyze the effects of varying fiber content, temperature levels, and impact rates on the mechanical behaviors of BFRC. Based on fractional calculus theory, a dynamic constitutive equation was established to characterize the viscoelastic properties and high-temperature damage of BFRC. The results indicate that the dynamic compressive strength of BFRC decreases significantly with increasing temperature but increases gradually with higher impact rates, demonstrating fiber-toughening effects, thermal degradation effects, and strain rate strengthening effects. The proposed constitutive model aligns well with the experimental data, effectively capturing the dynamic mechanical behaviors of BFRC after high-temperature exposure, including its transitional mechanical characteristics across elastic, viscoelastic, and viscous states. The viscoelastic behaviors of BFRC are fundamentally attributed to the synergistic response of its multi-phase composite system across different scales. Basalt fibers enhance the material’s elastic properties by improving the stress transfer mechanism, while high-temperature exposure amplifies its viscous characteristics through microstructural deterioration, chemical transformations, and associated thermal damage. 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

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

This paper introduces a digital adaptive control framework for large-scale multivariable systems, integrating matrix linear Diophantine equations with block pole placement. The main innovation lies in adaptively relocating the full eigenstructure using matrix polynomial representations and a recursive identification algorithm for real-time parameter estimation. The proposed method achieves accurate eigenvalue placement, strong disturbance rejection, and fast regulation under model uncertainty. Its effectiveness is demonstrated through simulations on a large-scale winding process, showing precise tracking, low steady-state error, and robust decoupling. Compared with traditional non-adaptive designs, the approach ensures superior performance against parameter variations and noise, highlighting its potential for high-performance industrial applications. Full article

The current work studies difference problems including two different nabla operators coupled with general summation boundary conditions that depend on a parameter. After we deduce the Green’s function, we obtain an interval of the parameter, where it is strictly positive. Then, we establish a lower and upper bound of the related Green’s function and we impose suitable conditions of the nonlinear part, under which, using the classical Guo–Krasnoselskii fixed point theorem, we deduce the existence of at least one positive solution of the studied equation. After that, we impose more restricted conditions on the right-hand side and we obtain the existence of n positive solutions again using fixed point theory, which is the main novelty of this research. Finally, we give particular examples as an application of our theoretical findings. Full article

We investigate how simultaneous mass and radius measurements of massive neutron stars can help constrain the properties of dark matter possibly admixed in them. Within a fermionic dark matter model that interacts only through gravitation, along with a well-constrained nuclear matter equation of state, we show that the simultaneous mass and radius measurement of PSRJ0740+6620 reduces the uncertainty of dark matter central energy density by more than 50% compared to the results obtained from using the two observables independently, while other dark matter parameters remain unconstrained. Additionally, we find that the dark matter fraction $f_D$ should be smaller than 2% when constrained by the observed neutron star maximum mass alone, and it could be even smaller than 0.3% with the simultaneous measurement of mass and radius, supporting the conclusion that only a small amount of dark matter exists in dark matter admixed neutron stars (DANSs). Full article

Long-term greenhouse operations face a critical challenge in the form of soil quality degradation, yet the key intervention periods and underlying mechanisms of this process remain unclear. This study aims to quantify the effects of greenhouse lifespan on the evolution of soil quality and to identify critical periods for intervention. We conducted a systematic survey of greenhouse operations in a representative area of Yunnan Province, Southwest China, and adopted a space-for-time substitution design. Using open-field cultivation (OF) as the control, we sampled and analyzed soils from vegetable greenhouses with greenhouse lifespans of 2 years (G2), 5 years (G5), and 10 years (G10). The results showed that early-stage greenhouse operation (G2) significantly increased soil temperature (ST) by 8.38–19.93% and soil porosity (SP) by 16.21–56.26%, promoted nutrient accumulation and enhanced aggregate stability compared to OF. However, as the greenhouse lifespan increased, the soil aggregates gradually disintegrated, particle-size distribution shifted toward finer clay fractions, and pH changed from neutral to slightly alkaline, exacerbating nutrient imbalances. Compared with G2, G10 exhibited reductions in mean weight diameter (MWD) and soil organic matter (SOM) of 2.41–5.93% and 24.78–30.93%, respectively. Among greenhouses with different lifespans, G2 had the highest soil quality index (SQI), which declined significantly with extended operation; at depths of 0–20 cm and 20–40 cm, the SQI of G10 was 32.59% and 38.97% lower than that of G2, respectively (p < 0.05). Structural equation modeling (SEM) and random forest analysis indicated that the improvement in SQI during the early stage of greenhouse use was primarily attributed to the optimization of soil hydrothermal characteristics and pore structure. Notably, the 2–5 years was the critical stage of rapid decline in SQI, during which intensive water and fertilizer inputs reduced the explanatory power of soil nutrients for SQI. Under long-term continuous cropping, the reduction in MWD and SOM was the main reason for the decline in SQI. This study contributes to targeted soil management during the critical period for sustainable production of protected vegetables in southern China. Full article

Mathematical optimization, in both continuous and discrete forms, is well established and widely applied. This work addresses a gap in the literature by focusing on large-number optimization, where integers or fractions with hundreds of digits occur in decision variables, objective functions, or constraints. Such problems challenge standard optimization tools, particularly when exact solutions are required. The suitability of computer algebra systems and high-precision arithmetic software for large-number optimization problems is discussed. Our first contribution is the development of Python implementations of an exact Simplex algorithm and a Branch-and-Bound algorithm for integer linear programming, capable of handling arbitrarily large integers. To test these implementations for correctness, analytic optimal solutions for nine specifically constructed linear, integer linear, and quadratic mixed-integer programming problems are derived. These examples are used to test and verify the developed software and can also serve as benchmarks for future research in large-number optimization. The second contribution concerns constructing partially increasing subsequences of the Collatz sequence. Motivated by this example, we quickly encountered the limits of commercial mixed-integer solvers and instead solved Diophantine equations or applied modular arithmetic techniques to obtain partial Collatz sequences. For any given number J, we obtain a sequence that begins at $2^J-1$ and repeats J times the pattern ud: multiply by $3x_j+1$ and then divide by 2. Further partially decreasing sequences are designed, which follow the pattern of multiplying by $3x_j+1$ and then dividing by $2^m$. The most general J-times increasing patterns (ududd, udududd, …, ududududddd) are constructed using analytic and semi-analytic methods that exploit modular arithmetic in combination with optimization techniques. 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

By using the Ekeland variational principle and Nehari manifold, we study the following fractional p-Laplacian Kirchhoff equations: $M\left({\left[u\right]}_{s,p}^p+{\int }_{{\mathbb{R}}^N}V\left(x\right){\left|u\right|}^{p}dx\right)[{(-\Delta )}_p^{s}u+$${V\left(x\right)\left|u\right|}^{p-2}u]={\lambda \left|u\right|}^{q-2}u\ln \left|u\right|,x\in {\mathbb{R}}^N,\left(\mathrm{P}\right)$. In these equations, $\lambda \in \mathbb{R}\setminus \left\{0\right\},$$p\in (1,+\infty )$, $s\in (0,1),sp<N,$$p_s^{*}={\displaystyle \frac{Np}{N-sp}}$, $M\left(\tau \right)=a+b{\tau }^{\theta -1}$, $a,b\in {\mathbb{R}}^{+},$$1<\theta <{\displaystyle \frac{p_s^{*}}{p}}$, $V\left(x\right)\in C({\mathbb{R}}^N,\mathbb{R})$ is a potential function and ${(-\Delta )}_p^s$ is the fractional p-Laplacian operator. The existence of solutions is deeply influenced by the positive and negative signs of $\lambda$. More precisely, (i) Equation (P) has one ground state solution for $\lambda >0$ and $p\theta <q<p_s^{*}$, with a positive corresponding energy value; and (ii) Equation (P) has at least two nontrivial solutions for $\lambda <0$ and $p<q<p_s^{*}$, with positive and negative corresponding energy values, respectively. Full article