Search Results (4,252)

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

This paper focuses on the problem, claimed in some works, of the non-existence of finite-time stable equilibria in nonlinear fractional differential equations. After dividing the equilibrium point into the initial equilibrium point and the finite-time equilibrium point, we provide sufficient conditions for the equilibrium point of a fractional stochastic differential equation. Then the finite-time stability of the equilibrium points of nonlinear fractional stochastic differential equations is presented. Finally, the correctness of the theoretical analysis is illustrated through an example. 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

This paper focuses on the numerical modeling of drug diffusion in biological tissues using fractional time-dependent parabolic equations with non-local boundary conditions. The model includes a Caputo fractional derivative to capture the non-local effects and memory inherent in biological processes, such as drug absorption and transport. The theoretical framework of the problem is based on the work of Alhazzani, et al.,which demonstrates the solution’s goodness, existence, and uniqueness. Building on this foundation, we present a robust numerical method designed to deal with the complexity of fractional derivatives and non-local interactions at the boundaries of biological tissues. Numerical simulations reveal how fractal order and non-local boundary conditions affect the drug concentration distribution over time, providing valuable insights into drug delivery dynamics in biological systems. The results underscore the potential of fractal models to accurately represent diffusion processes in heterogeneous and complex biological environments. Full article

Biochar has garnered considerable attention as a soil amendment due to its unique physicochemical properties. Its application not only enhances soil carbon sequestration but also improves nutrient availability. Incorporating biochar into soil is regarded as a promising strategy for mitigating global climate change while delivering substantial environmental and agricultural benefits. In this study, biochar was extracted from Siraitia grosvenorii and subsequently modified through alkali treatment. A laboratory incubation experiment was conducted to assess the effects of unmodified (JMC) and modified (GXC) biochar, applied at different rates (1%, 2%, and 4%), on organic carbon mineralization and soil nutrient dynamics. Results indicated that, at equivalent application rates, JMC-treated soils exhibited lower CO₂ emissions than those treated with GXC, with emissions increasing alongside biochar dosage. After the incubation, the 1% JMC treatment exhibited a mineralization rate of 17.3 mg·kg⁻¹·d⁻¹, which was lower than that of the control (CK, 18.8 mg·kg⁻¹·d⁻¹), suggesting that JMC effectively inhibited organic carbon mineralization and reduced CO₂ emissions, thereby contributing positively to carbon sequestration in Siraitia grosvenorii farmland. In contrast, GXC application significantly enhanced soil nutrient levels, particularly increasing available phosphorus (AP) by 14.33% to 157.99%. Furthermore, partial least squares structural equation modeling (PLS-SEM) identified application rate and pH as the key direct factors influencing soil nutrient availability. Full article

The aim of this work is to study a class of nabla fractional difference equations with anti-periodic conditions. First, we construct the related Green’s function. After deducing some of its useful properties, we obtain an upper bound for its sum. Then, using this bound, we are able to obtain three existence results based on the Banach contraction principle, Brouwer’s fixed point theorem, and Leray–Schauder’s nonlinear alternative, respectively. Then, we show some non-existence results for the studied problem, and existence results are also provided for a system of two equations of the considered type. Finally, we outline some particular examples in order to demonstrate the theoretical findings. Full article

This paper proposes a novel analytical method to address the Helmholtz fractional differential equation by combining the Aboodh transform with the Adomian Decomposition Method, resulting in the Aboodh–Adomian Decomposition Method (A-ADM). Fractional differential equations offer a comprehensive framework for describing intricate physical processes, including memory effects and anomalous diffusion. This work employs the Caputo–Fabrizio fractional derivative, defined by a non-singular exponential kernel, to more precisely capture these non-local effects. The classical Helmholtz equation, pivotal in acoustics, electromagnetics, and quantum physics, is extended to the fractional domain. Following the exposition of fundamental concepts and characteristics of fractional calculus and the Aboodh transform, the suggested A-ADM is employed to derive the analytical solution of the fractional Helmholtz equation. The method’s validity and efficiency are evidenced by comparisons of analytical and approximation solutions. The findings validate that A-ADM is a proficient and methodical approach for addressing fractional differential equations that incorporate Caputo–Fabrizio derivatives. 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

Liquid crystal elastomers (LCEs) have shown great potential in the field of soft robotics due to their unique actuation capabilities. Despite the growing number of experimental studies in the soft robotics field, theoretical research remains limited. In this paper, a dynamic model of a bionic arm using an LCE fiber as artificial muscle is established, which exhibits periodic oscillation controlled by periodic illumination. Based on the assumption of linear damping and angular momentum theorem, the dynamics equation of the model oscillation is derived. Then, based on the assumption of linear elasticity model, the periodic spring force of the fiber is given. Subsequently, the evolution equations for the cis number fraction within the fiber are developed, and consequently, the analytical solution for the light-excited strain is derived. Following that, the dynamics equation is numerically solved, and the mechanism of the controllable oscillation is elucidated. Numerical calculations show that the stable oscillation period of the bionic arm depends on the illumination period. When the illumination period aligns with the natural period of the bionic arm, the resonance is formed and the amplitude is the largest. Additionally, the effects of various parameters on forced oscillation are analyzed. The results of numerical studies on the bionic arm can provide theoretical support for the design of micro-machines, bionic devices, soft robots, biomedical devices, and energy harvesters. Full article

Research on reliability control for enhancing power systems under random loads holds significant and undeniable importance in maintaining system stability, performance, and safety. The primary challenge lies in determining the reliability index while optimizing system parameters. To effectively address this challenge, we developed a novel intelligent algorithm and conducted an optimal reliability assessment for a Negative Stiffness Device (NSD) seismic isolation structure incorporating fractional-order damping. This algorithm combines the Gaussian Radial Basis Function Neural Network (GRBFNN) with the Particle Swarm Optimization (PSO) algorithm. It takes the reliability function with unknown parameters as the objective function, while using the Backward Kolmogorov (BK) equation, which governs the reliability function and is accompanied by boundary and initial conditions, as the constraint condition. During the operation of this algorithm, the neural network is employed to solve the BK equation, thereby deriving the fitness function in each iteration of the PSO algorithm. Then the PSO algorithm is utilized to obtain the optimal parameters. The unique advantage of this algorithm is its ability to simultaneously achieve the optimization of implicit objectives and the solution of time-dependent BK equations.To evaluate the performance of the proposed algorithm, this study compared it with the algorithm combines the GRBFNN with Genetic Algorithm (GA-GRBFNN)across multiple dimensions, including performance and operational efficiency. The effectiveness of the proposed algorithm has been validated through numerical comparisons and Monte Carlo simulations. The control strategy presented in this paper provides a solid theoretical foundation for improving the reliability performance of mechanical engineering systems and demonstrates significant potential for practical applications. Full article

In this study, we focus on solving the nonlinear time-fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) and modified Korteweg–de Vries (MKdV) equations, using the Yang transform iterative method (YTIM). This method combines the Yang transform with a new iterative scheme to construct reliable and efficient solutions. Readers can understand the procedures clearly, since the implementation of Yang transform directly transforms fractional derivative sections into algebraic terms in the given problems. The new iterative scheme is applied to generate series solutions for the provided problems. The fractional derivatives are considered in the Caputo sense. To validate the proposed approach, two numerical examples are analysed and compared with exact solutions, as well as with the results obtained from the fractional reduced differential transform method (FRDTM) and the q-homotopy analysis transform method (q-HATM). The comparisons, presented through both tables and graphical illustrations, confirm the enhanced accuracy and reliability of the proposed method. Moreover, the effect of varying the fractional order is explored, demonstrating convergence of the solution as the order approaches an integer value. Importantly, the time-fractional Hirota–Satsuma coupled KdV and modified Korteweg–de Vries (MKdV) equations investigated in this work are not only of theoretical and computational interest but also possess significant implications for achieving global sustainability goals. Specifically, these equations contribute to the Sustainable Development Goal (SDG) “Life Below Water” by offering advanced modelling capabilities for understanding wave propagation and ocean dynamics, thus supporting marine ecosystem research and management. It is also relevant to SDG “Climate Action” as it aids in the simulation of environmental phenomena crucial to climate change analysis and mitigation. Additionally, the development and application of innovative mathematical modelling techniques align with “Industry, Innovation, and Infrastructure” promoting advanced computational tools for use in ocean engineering, environmental monitoring, and other infrastructure-related domains. Therefore, the proposed method not only advances mathematical and numerical analysis but also fosters interdisciplinary contributions toward sustainable development. Full article

Normal and anomalous diffusion processes are characterized by the time evolution of the mean square displacement of a diffusing molecule ${\sigma }^2\left(t\right)$. When ${\sigma }^2\left(t\right)$ is a power function of time, the process is described by a fractional subdiffusion, fractional superdiffusion or normal diffusion equation. However, for other forms of ${\sigma }^2\left(t\right)$, diffusion equations are often not defined. We show that to describe diffusion characterized by ${\sigma }^2\left(t\right)$, the g-subdiffusion equation with the fractional Caputo derivative with respect to a function g can be used. Choosing an appropriate function g, we obtain Green’s function for this equation, which generates the assumed ${\sigma }^2\left(t\right)$. A method for solving such an equation, based on the Laplace transform with respect to the function g, is also described. 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

In this work, we study Caputo fractional boundary value problems and contribute to the theory of fractional differential equations by improving the results of Ferreira. Specifically, we establish sharper bounds for the Green’s functions associated with the problems and apply Rus’s fixed-point theorem. Our results hold under a less restrictive assumption, thereby extending the class of problems for which the existence and uniqueness of solutions can be ensured. This is demonstrated through numerical validation presented in the final stage of our analysis. An important aspect of this approach is that it avoids the need for strong contraction conditions, suggesting potential applicability to a broader range of differential equations. Full article