Search Results (1,863)

This paper presents closed-form solutions for a three-dimensional system of nonlinear difference equations with variable coefficients. The approach employs functional transformations and leverages generalized Fibonacci sequences to construct the solutions explicitly. These solutions reveal a profound connection to generalized Fibonacci recursions. The proposed method is based on sophisticated mathematical transformations that reduce the complex nonlinear system to a solvable linear form, followed by the derivation of general solutions through iterative techniques and harmonic analysis. Furthermore, we extend our results to a generalized class of systems by introducing flexible functional transformations, while rigorously maintaining the required regularity conditions. The findings demonstrate the effectiveness of this methodology in addressing a broad class of complex nonlinear systems and open new perspectives for modeling multivariate dynamical phenomena. The analysis further reveals two distinct dynamical regimes—an unbounded oscillatory growth phase and a bounded cyclic equilibrium—arising from the relative magnitude of the variable coefficients, thereby highlighting the method’s capacity to characterize both amplifying and self-regulating behaviors within a unified analytical framework. Full article

The paper investigates the oscillation, zero-convergence, and solutions of second-order neutral delay difference equations containing three nonlinear delayed terms with different growth rates. By using positivity and monotonicity conditions on an auxiliary function along with divergence-type conditions on the coefficient sequences of the neutral and delayed terms, the paper establishes new criteria that guarantee oscillation or convergence of all solutions. These novel findings extend and enhance several of the existing oscillation criteria established by the literature. Suggestions for further investigation are included with illustrative examples. Full article

In this paper, we derive a semi-discrete scheme using the central difference method, which perfectly preserves the conservation of mass and energy for the Hirota equation. By applying the Crank–Nicolson method for temporal discretization, we develop the fully discrete scheme that conserves mass and energy. It is shown that the accuracy of the fully discrete scheme is of the second order in space and time. Because the Crank–Nicolson discretization leads to a nonlinear algebraic system, an efficient iterative solver is proposed that linearizes and solves the resulting five-diagonal matrix at each iteration while treating high-order contributions iteratively to reduce computational cost. Numerical experiments are presented to demonstrate the accuracy and verify the conservation properties. Full article

Time-fractional interface problems arise in systems where interacting materials exhibit memory effects or anomalous diffusion. These models provide a more realistic description of physical processes than classical formulations and appear in heat conduction, fluid flow, porous media diffusion, and electromagnetic wave propagation. However, the presence of complex interfaces and the nonlocal nature of fractional derivatives makes their numerical treatment challenging. This article presents a numerical scheme that combines radial basis functions (RBFs) with the finite difference method (FDM) to solve time-fractional partial differential equations involving interfaces. The proposed approach applies to both linear and nonlinear models with constant or variable coefficients. Spatial derivatives are approximated using RBFs, while the Caputo definition is employed for the time-fractional term. First-order time derivatives are discretized using the FDM. Linear systems are solved via Gaussian elimination, and for nonlinear problems, two linearization strategies, a quasi-Newton method and a splitting technique, are implemented to improve efficiency and accuracy. The method’s performance is assessed using maximum absolute and root mean square errors across various grid resolutions. Numerical experiments demonstrate that the scheme effectively resolves sharp gradients and discontinuities while maintaining stability. Overall, the results confirm the robustness, accuracy, and broad applicability of the proposed technique. Full article

This paper analyzes the modified canonical Heisenberg commutation relations or GUP, from a standard Hamiltonian point of view. For a one-dimensional system, a such modified canonical Heisenberg commutation relation is defined by the commutator between a position $\widehat{x}$ and a momentum operator $\widehat{p}$ (called the deformed momentum), which becomes a function F of the same operators: $\left[\widehat{x},\widehat{p}\right]=F(\widehat{x},\widehat{p})$, that is, the Heisenberg algebra closes itself in general in a nonlinear way. The function F also depends on a parameter that controls the deformation of the Heisenberg algebra in such a way that for a null parameter value, one recovers the usual Heisenberg algebra $\left[\widehat{x},{\widehat{p}}_0\right]=i\hslash \mathbb{I}$. Thus, it naturally raises the following questions: What does a relation of this type mean in Hamiltonian theory from a standard point of view? Is the deformed momentum the canonical variable conjugate to the position in such a relation? Moreover, what are the canonical variables in this model? The answer to these questions comes from the existence of two different phase spaces: The first one, called the non-deformed phase (which is obtained for control parameter value equal to zero), is defined by the Cartesian $\widehat{x}$ coordinate and its non-deformed conjugate momentum ${\widehat{p}}_0$, which satisfies the standard quantum mechanical Heisenberg commutation relation. The second phase space, the deformed one, is given by the deformed momentum $\widehat{p}$ and a new position coordinate $\widehat{y}$, which is its canonical conjugate variable, so $\widehat{y}$ and $\widehat{p}$ also satisfy standard commutation relations. We construct a classical canonical transformation that maps the non-deformed phase space into the deformed one for a specific class of deformation functions F. Additionally, a quantum mechanical operator transformation is found between the two non-commutative phase spaces, which allows the Schrödinger equation to be written in both spaces. Thus, there are two equivalent quantum mechanical descriptions of the same physical process associated with a deformed commutation relation. Full article

This paper aims to explore the numerical solution of non-linear fractional-order Burgers’-Huxley equation based on Caputo’s formulation of fractional derivatives. The equation serves as a versatile tool for analyzing a wide range of physical, biological, and engineering systems, facilitating valuable insights into nonlinear dynamic phenomena. The fractional operator provides a comprehensive mathematical framework that effectively captures the non-locality, hereditary characteristics, and memory effects of various complex systems. The approximation of temporal differential operator is carried out through finite difference based L1 scheme, while spatial discretization is performed using modified cubic B-spline basis functions. The stability as well as convergence analysis of the approach are also presented. Additionally, some numerical test experiments are conducted to evaluate the computational efficiency of a modified fourth-order cubic B-spline (M43BS) approach. Finally, the results presented in the form of tables and graphs highlight the applicability and robustness of M43BS technique in solving fractional-order differential equations. The proposed methodology is preferred for its flexible nature, high accuracy, ease of implementation and the fact that it does not require unnecessary integration of weight functions, unlike other numerical methods such as Galerkin and spectral methods. Full article

Accessibility and observability are two properties of dynamic models that provide insights into the structural relationships between their input, output, and state variables. They are closely related to controllability and structural local identifiability, respectively. Observability and identifiability determine, respectively, the possibility of inferring the unmeasured state variables and parameters of a model from output measurements; accessibility and controllability describe the possibility of driving its state by changing its input. Analysing these structural properties in nonlinear models of ordinary differential equations can be challenging, particularly when dealing with large systems. Two main approaches are currently used for their study: one based on differential geometry, which uses symbolic computation, and another one based on sensitivity calculations that uses numerical integration. These approaches are implemented in two MATLAB (R2024b) software tools: the differential geometry approach in STRIKE-GOLDD, and the sensitivity-based method in StrucID. These toolboxes differ significantly in their features and capabilities. Until now, their performance had not been thoroughly compared. In this paper we present a comprehensive comparative study of them, elucidating their differences in applicability, computational efficiency, and robustness against computational issues. Our core finding is that StrucID has a substantially lower computational cost than STRIKE-GOLDD; however, it may occasionally yield inconsistent results due to numerical issues. Full article

Loquat fruit is valued for its pleasant taste and favorable ripening period. However, its delicate texture and high perishability make it highly vulnerable to damage during packaging, so the fruit is usually packed by hand. Developing a fruit-sizing machine could increase commercial market opportunities. Automated mass detection reduces manual sorting errors and labor requirements. Overall, it enhances grading accuracy, speed, and uniformity in loquat processing. It also helps distinguish between ripe, underripe, and overripe fruits through subtle mass differences. Mass modeling has proven to be an effective baseline approach for the development and optimization of grading machines, and its efficiency has been demonstrated across different fruit types. Here, we present a comparative analysis of various models for mass modeling of six international and Italian loquat varieties (“Algerie,” “Peluche,” “Golden Nugget,” “Virticchiara,” “Nespolone di Trabia,” and “Claudia”) cultivated in southern Italy. On fifty fruits per variety, singular mass and spatial diameters [longitudinal (DL), maximum transverse (DT1), and minimum transverse (DT2) were measured. Linear and non-linear regression analyses, including quadratic, polynomial, and cubic models, were applied to both the complete dataset and individual varieties. A set of predictors was used, including DL (length), DT1 (width), and DT2 (thickness), ellipsoid and oblate spheroid volume. Model performance was evaluated based on higher R² values, and lower RMSE and MBE values. The best general model was obtained using an ellipsoidal volume (R² = 0.97, RMSE = 2.76). Both linear and cubic models demonstrated high suitability across all varieties, with ellipsoidal volume emerging as the most effective predictor. Conversely, (DL) based models were the least suitable, yielding the lowest (R² = 0.41) values in “Virticchiara.” The developed general and specific-variety models and equations provide a solid foundation for establishing high-performance systems for mass and size estimation, which can be effectively integrated into a fruit sizer machine. Full article

Current research on tower crane control lacks high-fidelity models and fails to account for the coupling effects between the tower crane structure and the hoisting and luffing systems. A new dynamic analysis platform based on the flexible multibody system theory is proposed in this investigation for the tower crane which contains a large-scale steel structure and hoisting mechanisms undergoing large displacements and large deformations. The Arbitrary Lagrangian–Eulerian–Absolute Nodal Coordinate Formulation (ALE–ANCF) cable element was employed to model the varying length of the steel rope in the hoisting mechanisms. Nonlinear kinetic equations were used to describe the motion of a luffing trolley. The solving strategy of the system’s dynamical equations are presented. Two different trajectories were tested. Simulation results demonstrate the feasibility and rationality of the proposed dynamic analysis platform. The primary conclusion is that this platform serves as a reliable and high-fidelity testbed for developing and evaluating advanced control algorithms under realistic dynamic conditions, thereby providing a dependable tool for both research and engineering applications. Full article

This paper presents a finite difference approach for solving the time-fractional Burgers’ equation, which is a model for nonlinear flow with memory effects. The method leverages the $L1-2$ formula for the fractional derivative and provides a novel linearization strategy to efficiently transform the system into a stable linear problem. Rigorous analysis establishes the existence, uniqueness, and pointwise-in-time convergence of the numerical solution in the $L_2$ norm. The proposed formulation achieves second-order time accuracy and fourth-order spatial accuracy under smooth initial conditions, with numerically verified temporal convergence rates of $O({\tau }^{1+\alpha }+{\tau }^2{t}_n^{\alpha -2})$ for solutions with weak singularities. Critically, numerical findings demonstrate that the method is robust and highly efficient, offering high-resolution solutions at a substantially lower computational cost than equivalent graded-mesh formulations. Full article

This study examines the nonlinear dynamical behavior of a Van der Waals system in the viscosity–capillarity regularization form. The solitary wave solutions of the proposed model are investigated using advanced analytical techniques, including the generalized Arnous method, the modified generalized Riccati equation mapping method, and the modified F-expansion approach. Additionally, we use mathematical simulations to enhance our comprehension of wave propagation. Moreover, a machine learning algorithm known as the multilayer perceptron regressor neural network was adopted to predict the performance results of our soliton solutions. Another important aspect of this study is the exploration of the chaos of the studied model by introducing a perturbed system. Chaotic analysis is supported by different techniques, such as return maps, power spectra, a bifurcation diagram, and a chaotic attractor. This multifaceted investigation not only emphasizes the rich dynamical pattern of the studied model but also presents a robust mathematical framework for studying nonlinear systems. The studied model also presents a robust mathematical framework for studying nonlinear systems. This study offers novel insights into nonlinear dynamics and wave phenomena by assessing the effectiveness of modern methodologies and clarifying the distinctive characteristics of a system’s nonlinear dynamics. Full article

The FitzHugh–Nagumo (FHN) equation in one dimension is solved in this paper using an improved physics-informed neural network (PINN) approach. Examining test problems with known analytical solutions and the explicit finite difference method (EFDM) allowed for the demonstration of the PINN’s effectiveness. Our study presents an improved PINN formulation tailored to the FitzHugh–Nagumo reaction–diffusion system. The proposed framework is efficiently designed, validated, and systematically optimized, demonstrating that a careful balance among model complexity, collocation density, and training strategy enables high accuracy within limited computational time. Despite the very strong agreement that both methods provide, we have demonstrated that the PINN results exhibit a closer agreement with the analytical solutions for Test Problem 1, whereas the EFDM yielded more accurate results for Test Problem 2. This study is crucial for evaluating the PINN’s performance in solving the FHN equation and its application to nonlinear processes like pulse propagation in optical fibers, drug delivery, neural behavior, geophysical fluid dynamics, and long-wave propagation in oceans, highlighting the potential of PINNs for complex systems. Numerical models for this class of nonlinear partial differential equations (PDEs) may be developed by existing and future model creators of a wide range of various nonlinear physical processes in the physical and engineering sectors using the concepts of the solution methods employed in this study. Full article

Effectively forecasting electricity generation, consumption, and pricing enhances power utilization efficiency, safeguards the stable operation of power systems, and assists power generation enterprises in formulating rational generation plans and dispatch schedules. The electricity generation, consumption, and pricing system exhibits complex chaotic dynamics. Establishing effective predictive models by leveraging the strong coupling and multi-scale uncertainty characteristics of nonlinear dynamical systems is a key challenge in grey modelling. This study leverages grey differential information to effectively transform differential equations into difference equations. Fractional-order cumulative generation operations enable more refined extraction of data characteristics. Based on the coupling and uncertainty features of electricity generation–consumption–pricing dynamics within complex power systems, three types of fractional-order multivariate grey models are established. These models both reflect the system’s dynamic relationships and expand the conceptual framework for grey prediction modelling. Simultaneously, the effectiveness of these three models is analyzed using data on generation, consumption, and prices from both new and traditional power sources within China’s electricity system. Employing identical annual data, the models are evaluated from two distinct perspectives: variations in the numbers of simulated and predicted variables. Experimental results demonstrate that all three novel models perform well. Finally, the most effective predictive application of the three models was selected to forecast electricity generation, consumption, and pricing in China. This provides a basis for China’s power system and supports national macro-level intelligent energy dispatch planning. Full article

The MHD (magnetic hydrodynamics) boundary problem in three-dimensional domains of certain types is considered within the framework of discrete potential theory. The discrete character of the information obtained from remote sensing of the Earth and planets of the Solar System can be taken into account when using the basic principles of this theory. This approach makes it possible to reconstruct the spatial distribution of magnetic fields and the velocity field with relatively high accuracy using the heterogeneous data in some network points. In order to restore the magnetic image of a planet with a so-called dynamo, the subsequent approximations approach is implemented. The unknown physical field is represented as a sum of terms of different magnitudes. Such an approach allows us to simplify the nonlinear partial differential equation system of magnetic hydrodynamics and extend it to discrete magnetic field and velocity vectors. The solution of the simplified MHD equation system is constructed for some classes of bounded domains in Cartesian coordinates in three-dimensional space. Full article

This study examines cosmic acceleration in the framework of $f(Q,T)$ gravity and compares it to the standard $\Lambda$CDM model. It considers a generalized nonlinear form of the nonmetricity, expressed as $f(Q,T)=-Q+{\alpha }_0{Q}^2/H_0^2+{\beta }_{0}T+{\eta }_0$, where ${\alpha }_0,{\beta }_0,$ and ${\eta }_0$ are constants, and $H_0$ is the current value of the Hubble constant. In the solution process, we did not rely on any additional conditions to solve the field equations; instead, the field equations were reduced to a time-dependent closed system of nonlinear first-order coupled differential equations for H and $\rho$. Subsequently, these differential equations were converted to the redshift space for numerical integration alongside the Runge–Kutta method. Furthermore, the study demonstrates that the deceleration parameter q changes sign from being positive in an early period of time at high redshift values to a negative value, passing through a transitional redshift $z_t\approx [0.766,0.769,0.771]$ and $z_t\approx [0.521,0.770,1.010]$, reaching their current values at $q_0=[-0.61,-0.60,-0.59]$ and $[-0.455,-0.595,-0.694]$ for different values of ${\beta }_0$ and ${\alpha }_0$, respectively. Similarly, the effective equation of state $w_{eff}$ shifted from the matter-dominated phase $w_{eff}=0$ at high redshift to a quintessence-like behavior at low redshift. Moreover, a super-accelerated or phantom-like regime with $q_0\simeq -1.59$ and $w_{\mathrm{eff},0}\simeq -1.40$ was obtained when ${\alpha }_0=0.55$ and ${\beta }_0=0.60$ were employed. The model analysis reveals that the universe is presently experiencing an accelerating expansion phase, propelled by a quintessence-type and phantom-like dark energy component, as corroborated by the Om(z) diagnostic test. The results obtained were strongly consistent with the concordance $\Lambda$CDM model. Full article