Search Results (772)

This work introduces a new family of multivariate hybrid special polynomials, motivated by their growing relevance in mathematical modeling, physics, and engineering. We explore their core properties, including recurrence relations and shift operators, within a unified structural framework. By employing the factorization method, we derive various governing equations such as differential, partial differential, and integrodifferential equations. Additionally, we establish a related fractional Volterra integral equation, which broadens the theoretical foundation and potential applications of these polynomials. To support the theoretical development, we carry out computational simulations to approximate their roots and visualize the distribution of their zeros, offering practical insights into their analytical behavior. Full article

Asymmetric functional-order (variable-order) fractional diffusion–wave equations (FO-FDWEs) introduce considerable computational challenges, as the fractional order of the derivatives can vary spatially or temporally. To overcome these challenges, a novel spectral method employing generalized fractional-order Chelyshkov wavelets (FO-CWs) is developed to efficiently solve such equations. In this approach, the Riemann–Liouville fractional integral operator of variable order is evaluated in closed form via a regularized incomplete Beta function, enabling the transformation of the governing equation into a system of algebraic equations. This wavelet-based spectral scheme attains extremely high accuracy, yielding significantly lower errors than existing numerical techniques. In particular, numerical results show that the proposed method achieves notably improved accuracy compared to existing methods under the same number of basis functions. Its strong convergence properties allow high precision to be achieved with relatively few wavelet basis functions, leading to efficient computations. The method’s accuracy and efficiency are demonstrated on several practical diffusion–wave examples, indicating its suitability for real-world applications. Furthermore, it readily applies to a wide class of fractional partial differential equations (FPDEs) with spatially or temporally varying order, demonstrating versatility for diverse applications. Full article

In this paper, we study direct and inverse problems for a spatial-fractional Black–Scholes equation with space-dependent volatility. For the direct problem, we provide CN-WSGD (Crank–Nicholson and the weighted and shifted Grünwald difference) scheme to solve the initial boundary value problem. The latter aims to recover the implied volatility via observable option prices. Using a linearization technique, we rigorously derive a mathematical formulation of the inverse problem in terms of a Fredholm integral equation of the first kind. Based on an integral equation, an efficient numerical reconstruction algorithm is proposed to recover the coefficient. Numerical results for both problems are provided to illustrate the validity and effectiveness of proposed methods. Full article

The combined application of fibers and lightweight aggregates (LWAs) represents an effective approach to achieving high-strength, lightweight concrete. To enhance the predictability of the mechanical properties of fiber-reinforced lightweight aggregate concrete (LWAC), this study conducts an in-depth investigation into its hydration characteristics. In this study, high-strength LWAC was developed by incorporating low water absorption LWAs, various volume fractions of basalt fiber (BF) (0.1%, 0.2%, and 0.3%), and a ternary cementitious system consisting of 70% cement, 20% fly ash, and 10% silica fume. The hydration-related properties were evaluated through isothermal calorimetry test and high-temperature calcination test. The results indicate that incorporating 0.1–0.3% fibers into the cementitious system delays the early hydration process, with a reduced peak heat release rate and a delayed peak heat release time compared to the control group. However, fitting the cumulative heat release over a 72-h period using the Knudsen equation suggests that BF has a minor impact on the final degree of hydration, with the difference in maximum heat release not exceeding 3%. Additionally, the calculation model for the final degree of hydration in the ternary binding system was also revised based on the maximum heat release at different water-to-binder ratios. The results for chemically bound water content show that compared with the pre-wetted LWA group, under identical net water content conditions, the non-pre-wetted LWA group exhibits a significant reduction at three days, with a decrease of 28.8%; while under identical total water content conditions it shows maximum reduction at ninety days with a decrease of 5%. This indicates that pre-wetted LWAs help maintain an effective water-to-binder ratio and facilitate continuous advancement in long-term hydration reactions. Based on these results, influence coefficients related to LWAs for both final degree of hydration and hydration rate were integrated into calculation models for degrees of hydration. Ultimately, this study verified reliability of strength prediction models based on degrees of hydration. Full article

The primary objective of this work is to develop a comprehensive theory of weighted fractional Sobolev spaces within the framework of timescales. To this end, we first introduce a novel class of weighted fractional operators and rigorously define associated weighted integrable spaces on timescales, generalising classical notions to this non-uniform temporal domain. Building upon these foundations, we systematically investigate the fundamental functional-analytic properties of the resulting Sobolev spaces. Specifically, we establish their completeness under appropriate norms, prove reflexivity under appropriate duality pairings, and demonstrate separability under mild conditions on the weight functions. As a pivotal application of our theoretical framework, we derive two robust existence theorems for solutions to the proposed model. These results not only extend classical partial differential equation theory to timescales but also provide a versatile tool for analysing dynamic systems with heterogeneous temporal domains. Full article

In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing $\beta$-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations. Full article

This study proposes an L1 discretization scheme (an accurate second-order finite difference method) for time-fractional diffusion equations involving the Caputo-type Erdélyi–Kober operator, which models anomalous diffusion. Our key contributions include the following: (i) reformulation of the original problem into an equivalent fractional integral equation to facilitate analysis; (ii) development of a novel L1 scheme for temporal discretization, which is rigorously proven to realize second-order accuracy in time; (iii) derivation of positive definiteness properties for discrete kernel coefficients; (iv) discretization of the spatial derivative using the classical second-order centered difference scheme, for which its second-order spatial convergence is rigorously verified through numerical experiments (this results in a fully discrete scheme, enabling second-order accuracy in both temporal and spatial dimensions); (v) a fast algorithm leveraging sum-of-exponential approximation, reducing the computational complexity from $O\left(N^2\right)$ to $O(NlogN)$ and memory requirements from $O\left(N\right)$ to $O(logN)$, where N is the number of grid points on a time scale. Our numerical experiments demonstrate the stability of the scheme across diverse parameter regimes and quantify significant gains in computational efficiency. Compared to the direct method, the fast algorithm substantially reduces both memory requirements and CPU time for large-scale simulations. Although a rigorous stability analysis is deferred to subsequent research, the proven properties of the coefficients and numerical validation confirm the scheme’s reliability. Full article

In this paper, we primarily use the inverse operator method to find a unique series solution to a time–fractional wave equation with variable coefficients based on the Mittag–Leffler function. In addition, we also derive the series and integral convolution solutions to the Klein–Gordon equation using the Fourier transform and Green’s functions. Furthermore, our series solutions significantly simplify the process of finding solutions with several illustrative examples, avoiding the need for complicated integral computations. Full article

The time-dependent creep behavior of coal is essential for assessing long-term structural stability and operational safety in deep coal mining. Therefore, this work develops a full-stage creep constitutive model. By integrating fractional calculus theory with statistical damage mechanics, a nonlinear fractional-order (FO) damage creep model is constructed through serial connection of elastic, viscous, viscoelastic, and viscoelastic–plastic components. Based on this model, both one-dimensional and three-dimensional (3D) fractional creep damage constitutive equations are acquired. Model parameters are identified using experimental data from deep coal samples in the mining area. The result curves of the improved model coincide with experimental data points, accurately describing the deceleration creep stage (DCS), steady-state creep stage (SCS), and accelerated creep stage (ACS). Furthermore, a sensitivity analysis elucidates the impact of model parameters on coal creep behavior, thereby confirming the model’s robustness and applicability. Consequently, the proposed model offers a solid theoretical basis for evaluating the sustained stability of deep coal mining and has great application potential in deep underground engineering. Full article

In this paper, we investigate the regularization of the backward problem for a diffusion process with a time-fractional derivative. We propose a novel double-parameter regularization scheme that integrates the quasi-reversibility method for the governing equation with the quasi-boundary method. Theoretical analysis establishes the regularity and the convergence analysis of the regularized solution, along with a convergence rate under an a-priori regularization parameter choice rule in the general-dimensional case. Finally, numerical experiments validate the effectiveness of the proposed scheme. Full article

This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces $L^{\mathtt{p}(·)}$. To the best of our knowledge, the boundedness of such operators has not been addressed in any existing functional setting. We establish their boundedness under appropriate log-Hölder continuity and growth conditions on the exponent function $\mathtt{p}(·)$. To highlight the novelty and practical relevance of the proposed operator, we conduct a comparative analysis demonstrating its effectiveness in addressing convergence, regularity, and stability of solutions to partial differential equations. We also provide non-trivial examples that illustrate not only these properties but also show that, under this operator, a broader class of functions becomes locally integrable. The exponential decay factor notably broadens the domain of boundedness compared to classical Riesz and Bessel–Riesz potentials, making the operator more versatile and robust. Additionally, we generalize earlier results on Sobolev-type inequalities previously studied in constant exponent spaces by extending them to the variable exponent setting through our fractional operator, which reduces to the classical Riesz potential when the decay parameter $\lambda =0$. Applications to elliptic PDEs are provided to illustrate the functional impact of our results. Furthermore, we develop several new structural properties tailored to variable exponent frameworks, reinforcing the strength and applicability of the proposed theory. Full article

Understanding the rheological properties of fresh cement slurries is essential to maintain optimal pumpability, achieve dependable zonal isolation, and preserve long-term well integrity in oil and gas cementing operations and the 3D printing cement and concrete industry. However, accurately and efficiently modeling the rheological behavior of cement slurries remains challenging due to the complex fluid properties of fresh cement slurries, which exhibit non-Newtonian and thixotropic behavior. Traditional numerical solvers typically require mesh generation and intensive computation, making them less practical for data-scarce, high-dimensional problems. In this study, a physics-informed neural network (PINN)-based framework is developed to solve the governing equations of steady-state cement slurry flow in a tilted channel. The slurry is modeled as a non-Newtonian fluid with viscosity dependent on both the shear rate and particle volume fraction. The PINN-based approach incorporates physical laws into the loss function, offering mesh-free solutions with strong generalization ability. The results show that PINNs accurately capture the trend of velocity and volume fraction profiles under varying material and flow parameters. Compared to conventional solvers, the PINN solution offers a more efficient and flexible alternative for modeling complex rheological behavior in data-limited scenarios. These findings demonstrate the potential of PINNs as a robust tool for cement slurry rheological modeling, particularly in scenarios where traditional solvers are impractical. Future work will focus on enhancing model precision through hybrid learning strategies that incorporate labeled data, potentially enabling real-time predictive modeling for field applications. Full article

Research on the supply–demand relationships of ecosystem services (ESs) in alpine pastoral regions remains relatively scarce, yet it is crucial for regional ecological management and sustainable development. This study focuses on the Sanjiangyuan Region, a typical alpine pastoral area and significant ecological barrier, to quantitatively assess the supply–demand dynamics of key ESs and their spatial heterogeneity from 2000 to 2020. It further aims to elucidate the underlying driving mechanisms, thereby providing a scientific basis for optimizing regional ecological management. Four key ES indicators were selected: water yield (WY), grass yield (GY), soil conservation (SC), and habitat quality (HQ). ES supply and demand were quantified using an integrated approach incorporating the InVEST model, the Revised Universal Soil Loss Equation (RUSLE), and spatial analysis techniques. Building on this, the spatial patterns and temporal evolution characteristics of ES supply–demand relationships were analyzed. Subsequently, the Geographic Detector Model (GDM) and Geographically and Temporally Weighted Regression (GTWR) model were employed to identify key drivers influencing changes in the comprehensive ES supply–demand ratio. The results revealed the following: (1) Spatial Patterns: Overall ES supply capacity exhibited a spatial differentiation characterized by “higher values in the southeast and lower values in the northwest.” Areas of high ES demand were primarily concentrated in the densely populated eastern region. WY, SC, and HQ generally exhibited a surplus state, whereas GY showed supply falling short of demand in the densely populated eastern areas. (2) Temporal Dynamics: Between 2000 and 2020, the supply–demand ratios of WY and SC displayed a fluctuating downward trend. The HQ ratio remained relatively stable, while the GY ratio showed a significant and continuous upward trend, indicating positive outcomes from regional grass–livestock balance policies. (3) Driving Mechanisms: Climate and natural factors were the dominant drivers of changes in the ES supply–demand ratio. Analysis using the Geographical Detector’s q-statistic identified fractional vegetation cover (FVC, q = 0.72), annual precipitation (PR, q = 0.63), and human disturbance intensity (HD, q = 0.38) as the top three most influential factors. This study systematically reveals the spatial heterogeneity characteristics, dynamic evolution patterns, and core driving mechanisms of ES supply and demand in an alpine pastoral region, addressing a significant research gap. The findings not only provide a reference for ES supply–demand assessment in similar regions regarding indicator selection and methodology but also offer direct scientific support for precisely identifying priority areas for ecological conservation and restoration, optimizing grass–livestock balance management, and enhancing ecosystem sustainability within the Sanjiangyuan Region. Full article

Integer-order models cannot characterize the dynamic behavior of the flexible two-link manipulator (FTLM) system accurately due to its viscoelastic characteristics and flexible oscillation. Hence, this paper proposes a fractional-order modeling method and identification algorithm for the FTLM system. Firstly, we exploit the memory and history-dependent properties of fractional calculus to describe the flexible link’s viscoelastic potential energy and viscous friction. Secondly, we establish a fractional-order differential equation for the flexible link based on the fractional-order Euler–Lagrange equation to characterize the flexible oscillation process accurately. Accordingly, we derive the fractional-order model of the FTLM system by analyzing the motor–link coupling as well as the symmetry of the system structure. Additionally, a system identification algorithm based on the multi-innovation integration operational matrix (MIOM) is proposed. The multi-innovation technique is combined with the least-squares algorithm to solve the operational matrix and achieve accurate system identification. Finally, experiments based on actual data are conducted to verify the effectiveness of the proposed modeling method and identification algorithm. The results show that the MIOM algorithm can improve system identification accuracy and that the fractional-order model can describe the dynamic behavior of the FTLM system more accurately than the integer-order model. Full article

To develop cement-based composite materials with exceptional impact resistance, this study investigates the impact resistance performance of steel fiber- and glass fiber-reinforced specimens, as well as steel fiber and glass fiber textile-reinforced specimens, through drop weight impact tests. The results showed that the impact resistance of specimens increases with the number of glass fiber textile layers, glass fiber volume fractions, and glass fiber lengths, with 36GF1.5SF1.0 exhibitinh ultra-high impact resistance with a failure impact energy of 114 kJ. Compared to the specimens reinforced with glass textiles, the specimens with glass fiber showed better impact resistance at the same volume fraction. The failure mode of unreinforced specimens is divided into several pieces, while fiber-reinforced specimens have local punching shear failure at the impact site, maintaining better integrity. An impact damage evolution equation and life prediction model based on a two-parameter Weibull distribution are developed. The research results will provide a reference for the selection of fibers for engineering applications. Full article