Search Results (2,895)

Lorenz-type systems are dissipative dynamical systems that are described by three nonlinear equations with derivatives of the first order and are capable of exhibiting chaotic behavior. The generalization of Lorenz-type equations by using general fractional derivatives (GFDs) and periodical kicks is proposed. GFDs allow us to use the general form of memory functions as operator kernels to describe nonlinear dynamics with memory. The exact analytical solutions of Lorenz-type equations with GFDs are derived in the general case for the wide class of nonlinearity and memory functions. Using the exact solutions, we obtain discrete maps with memory (DMMs) that describe kicked GF Lorenz-type systems with general forms of memory and nonlinearity. The proposed maps describe the exact solution of nonlinear equations with GFDs at discrete time points as the function of all past discrete moments of time. The proposed multi-dimensional DMMs are derived from kicked GF Lorenz-type equations with GFDs without any approximations. The proposed results and the method to derive multi-dimensional DMMs are derived for arbitrary dimensions. The importance and unusualness of the proposed results lies in the fact that obtained solutions for equations of the Lorenz-type system are exact analytical solutions. Full article

This paper presents a novel B-spline wavelet-based scheme for solving multi-term time–space variable-order fractional nonlinear diffusion-wave equations. By combining semi-orthogonal B-spline wavelets with a collocation approach and a quasilinearization technique, we transform the original problem into a system of algebraic equations. To enhance the computational efficiency, we derive the operational matrix formulation of the proposed scheme. We provide a rigorous convergence analysis of the method and demonstrate its accuracy and effectiveness through numerical experiments. The results confirm the robustness and computational advantages of our approach for solving this class of fractional differential equations. Full article

The meteo-tsunami as an atmospheric phenomenon is still being researched, and its effects beyond physical ones (destruction if they hit the shore) are yet to be fully classified. One such effect is an increase in false alarm occurrence in high frequency surface wave radar (HFSWR) systems used for vessel traffic surveillance and control. Unfortunately, this effect is characterized by high-dimensional and highly nonlinear functional dependencies that cannot be described by closed-form mathematical equations. Since an artificial neural network is a highly parallelized distributed architecture with a fast flow of signals from input to output designed not to execute a predefined set of commands, but to “learn” dependencies during the training process and to apply that knowledge to solve unknown, but similar problems, it is a natural solution to the presented problem. Hybrid empirical–neural model-based probabilistic neural networks (PNN) used in this research proved to be quite capable of recognizing when an increase in false alarms can be expected based on the monitoring of atmospheric conditions in the HFSWR network coverage area and to eliminate them from the system, thus increasing the safety and security of all the actors in maritime traffic. Full article

The coordinated scheduling of diesel generators, photovoltaic (PV) systems, and energy storage systems (ESS) is essential for improving the reliability and resilience of islanded microgrids in remote and mission-critical applications. This review systematically analyzes diesel–PV–ESSs from an “energy symbiosis” perspective, emphasizing the complementary roles of diesel power security, PV’s clean generation, and ESS’s spatiotemporal energy-shifting capability. A technology–time–performance framework is developed by screening advances over the past decade, revealing that coordinated operation can reduce the Levelized Cost of Energy (LCOE) by 12–18%, maintain voltage deviations within 5% under 30% PV fluctuations, and achieve nonlinear resilience gains. For example, when ESS compensates 120% of diesel start-up delay, the maximum disturbance tolerance time increases by 40%. To quantitatively assess symbiosis–resilience coupling, a dual-indicator framework is proposed, integrating the dynamic coordination degree (ζ ≥ 0.7) and the energy complementarity index (ECI > 0.75), supported by ten representative global cases (2010–2024). Advanced methods such as hybrid inertia emulation (200 ms response) and adaptive weight scheduling enhance the minimum time to sustain (MTTS) by over 30% and improve fault recovery rates to 94%. Key gaps are identified in dynamic weight allocation and topology-specific resilience design. To address them, this review introduces a “symbiosis–resilience threshold” co-design paradigm and derives a ζ–resilience coupling equation to guide optimal capacity ratios. Engineering validation confirms a 30% reduction in development cycles and an 8–12% decrease in lifecycle costs. Overall, this review bridges theoretical methodology and engineering practice, providing a roadmap for advancing high-renewable-penetration islanded microgrids. Full article

Fractional differential equations are commonly employed to characterize the long-term interactions of nonlinear systems, although they complicate inverse problems and numerical treatment. This article extends projection methods in order to numerically solve fractional differential equations. We propose a new projection approach for solving initial fractional problems based on the Jacobi weight function, employing a variety of generalized Jacobi polynomials with the indices $\sigma ,\rho \in \mathbb{R}$. This symmetry yields some intriguingly novel outcomes. The order of convergence for this method is given in appropriately weighted Sobolev spaces. Full article

This study investigates the dynamics of optical solitons for the variable-coefficient coupled higher-order nonlinear Schrödinger equation (VCHNLSE) enriched with $\beta$-derivatives. By employing an extended direct algebraic method (EDAM), we successfully derive explicit soliton solutions that illustrate the intricate interplay between nonlinearities and variable coefficients. Our approach facilitates the transformation of the complex NLS into a more manageable form, allowing for the systematic exploration of diverse solitonic structures, including bright, dark, and singular solitons, as well as exponential, polynomial, hyperbolic, rational, and Jacobi elliptic solutions. This diverse family of solutions substantially expands beyond the limited soliton interactions studied in conventional approaches, demonstrating the superior capability of our method in unraveling new wave phenomena. Furthermore, we rigorously demonstrate the robustness of these soliton solutions against various perturbations through comprehensive stability analysis and numerical simulations under parameter variations. The practical significance of this work lies in its potential applications in advanced optical communication systems. The derived soliton solutions and the analysis of their dynamics provide crucial insights for designing robust signal carriers in nonlinear optical media. Specifically, the management of variable coefficients and fractional-order effects can be leveraged to model and engineer sophisticated dispersion-managed optical fibers, tunable photonic devices, and ultrafast laser systems, where controlling pulse propagation and stability is paramount. The presence of $\beta$-fractional derivatives introduces additional complexity to the wave propagation behaviors, leading to novel dynamics that we analyze through numerical simulations and graphical representations. The findings highlight the potential of the proposed methodology to uncover rich patterns in soliton dynamics, offering insights into their robustness and stability under varying conditions. This work not only contributes to the theoretical foundation of nonlinear optics but also provides a framework for practical applications in optical fiber communications and other fields involving nonlinear wave phenomena. Full article

In this article, we present an efficient and highly accurate numerical scheme that achieves exponential convergence for solving nonlinear Riesz distributed-order fractional differential equations (RDFDEs) in one- and two-dimensional initial–boundary value problems. The proposed method is based on a two-stage collocation framework. In the first stage, spatial discretization is performed using the shifted Legendre–Gauss–Lobatto (SL-G-L) collocation method, where the approximate solutions and spatial derivatives are expressed in terms of shifted Legendre polynomial expansions. This reduces the original problem to a system of fractional differential equations (FDEs) for the expansion coefficients. Then, the temporal discretization is achieved in the second stage via Romanovski–Gauss–Radau collocation approach, which converts the system into a system of algebraic equations that can be solved efficiently. The method is applied to one- and two-dimensional nonlinear RDFDEs, and numerical experiments confirm its spectral accuracy, computational efficiency, and reliability. Existing numerical approaches to distributed-order fractional models often suffer from poor accuracy, instability in nonlinear settings, and high computational costs. By combining the efficiency of Legendre polynomials for bounded spatial domains with the stability of Romanovski polynomials for temporal discretization, the proposed two-stage framework effectively overcomes these limitations and achieves superior accuracy and stability. Full article

The flow of a two-dimensional incompressible hybrid nanofluid over a stretching cylinder containing microorganisms with parallel effect of inclined magnetohydrodynamic was examined in the current study in relation to chemical reactions, heat source effect, nonlinear heat radiation, and multiple convective boundaries. The main objective of this research is the optimization of heat transfer with inclined MHD and variation in different physical parameters. The governing partial differential equations are transformed into a set of ordinary differential equations by applying the appropriate similarity transformations. The Runge–Kutta method is recognized for using shooting as a technique. Surface plots, graphs, and tables have been used to illustrate how various parameters affect the local Nusselt number, mass transfer, and heat transmission. It is demonstrated that when the chemical reaction parameter rises, the concentration and motile concentration profiles drop. The least responsive is the rate of heat transfer to changes in the inclined magnetic field and most associated with changes in the Biot number and radiation parameter shown in contour plot. The streamline graph illustrates the way fluid flow is affected simultaneously by the magnetic parameter M and an angled magnetic field. Local Nusselt number and local skin friction are improved by the curvature parameter and mixed convection parameter. The contours highlight the intricate interactions between restricted magnetic field, significant radiation, and substantial convective condition factors by displaying the best heat transfer. The three-dimensional surface, scattered graph, pie chart, and residual plotting demonstrate the statistical analysis of the heat transfer. The results support their use in sophisticated energy, healthcare, and industrial systems and enhance our theoretical knowledge of hybrid nanofluid dynamics. Full article

This paper investigates a coupled system of Hadamard fractional p-Laplacian differential equations defined on an unbounded interval, subject to infinitely many points boundary conditions and formulated under a resonance framework. Under suitable growth assumptions imposed on the nonlinear terms of the system, the existence of solutions is established by means of the Ge–Mawhin’s continuation theorem. Moreover, an example is constructed to demonstrate the applicability of the main results. Full article

This paper studies the exact and approximate relations between fluctuations in thermodynamic variables (pressure, density and temperature) that are imposed by the dilute-gas ($\mathcal{Z}=1$) equation-of-state ($\mathrm{EoS}$), which is a satisfactory approximation of air thermodynamics for a wide range of pressures and temperatures. It focuses on triple- and higher-order correlations, extending previous studies that concentrated on second-order moments, with emphasis on the mathematical relations, which are generally valid independently of the particular flow configuration. Exact equations are developed both involving only single-variable moments and relating the correlations between variables. These contain nonlinear terms generated by the density-temperature fluctuation product in the fluctuating $\mathrm{EoS}$. The importance of the nonlinear terms in the 6 exact equations between the 10 third-order moments is assessed using $\mathrm{DNS}$ (direct numerical simulation) data for compressible turbulent plane channel ($\mathrm{TPC}$) flows and analyzed using general statistical inequalities involving third-order and fourth-order moments. The corresponding linearized system between third-order moments is studied to determine approximate relations and 4-tuples of linearly independent moments. These mathematical tools are then used to analyze $\mathrm{TPC}$ flow $\mathrm{DNS}$ data on the triple correlations between the thermodynamic variables. Full article

A novel numerical scheme is developed in this work to approximate solutions (APPSs) for nonlinear fractional differential equations (FDEs) governed by Robin boundary conditions (RBCs). The methodology is founded on a spectral collocation method (SCM) that uses a set of basis functions derived from generalized shifted Jacobi (GSJ) polynomials. These basis functions are uniquely formulated to satisfy the homogeneous form of RBCs (HRBCs). Key to this approach is the establishment of operational matrices (OMs) for ordinary derivatives (Ods) and fractional derivatives (Fds) of the constructed polynomials. The application of this framework effectively reduces the given FDE and its RBC to a system of nonlinear algebraic equations that are solvable by standard numerical routines. We provide theoretical assurances of the algorithm’s efficacy by establishing its convergence and conducting an error analysis. Finally, the efficacy of the proposed algorithm is demonstrated through three problems, with our APPSs compared against exact solutions (ExaSs) and existing results by other methods. The results confirm the high accuracy and efficiency of the scheme. Full article

We present an analytical and numerical study of nonlinear wave localization in a one-dimensional rhombic (diamond) waveguide array that combines forward- and backward-propagating channels. This mixed-index configuration, realizable through Bragg-type couplers or corrugated waveguides, produces a tunable spectral gap and supports nonlinear self-localized states in both transmission and forbidden-band regimes. Starting from the full set of coupled-mode equations, we derive the effective evolution model, identify the role of coupling asymmetry and nonlinear coefficients, and obtain explicit soliton solutions using the method of multiple scales. The resulting envelopes satisfy a nonlinear Schrödinger equation with an effective nonlinear parameter $\theta$, which determines the conditions for soliton existence ($\theta >0$) for various combinations of focusing and defocusing nonlinearities. We distinguish solitons formed outside and inside the bandgap and analyze their dependence on the dispersion curvature and nonlinear response. Direct numerical simulations confirm the analytical predictions and reveal robust propagation and interactions of counter-propagating soliton modes. Order-of-magnitude estimates show that the predicted effects are accessible in realistic integrated photonic platforms. These results provide a unified theoretical framework for soliton formation in mixed-index lattices and suggest feasible routes for realizing controllable nonlinear localization in Bragg-type photonic structures. Full article

To address the difficulty in obtaining analytical solutions for the torsional vibration response of pile foundations in orthotropic layered soil foundations subjected to torsional excitation at the pile top, this study investigates a layered recursive algorithm based on the Hankel transform. An integral transformation method is employed to reduce the dimensionality of the coupled pile–soil torsional vibration equations, converting the three-dimensional system of partial differential equations into a set of ordinary differential equations. Combining the constitutive properties of transversely anisotropic strata with interlayer contact conditions, a transfer matrix model is established. Employing inverse transformation coupled with the Gauss–Kronrod integration method, an explicit frequency-domain solution for the torsional dynamic impedance at the pile top is derived. The research findings indicate that the anisotropy coefficient of the foundation significantly influences both the real and imaginary parts of the impedance magnitude. The sequence of soil layer distribution and the bonding state at interfaces jointly affect the nonlinear transmission characteristics of torque along the pile shaft. Full article

The Hertz equation is the most widely used equation for data processing in AFM nanoindentation experiments on soft samples when using spherical indenters. Although valid only for small indentation depths relative to the tip radius, it is usually preferred because it directly relates applied force to indentation depth. Sneddon derived accurate equations relating force and contact radius to indentation depth for shallow and deep indentations, but they are rarely used in practice. This paper presents analytical approaches to solving Sneddon’s nonlinear system. Using Taylor series expansions and a simple equation linking applied force, average contact radius, and indentation depth, we derive a two-term equation that directly relates force to indentation depth. This expression is accurate for h ≤ 1.5 R, where h is the indentation depth and R is the indenter radius, making it applicable to most practical AFM measurements on soft materials. It should be used instead of the Hertzian model for extracting Young’s modulus, thereby enhancing measurement accuracy without increasing the complexity of data processing. In addition, the results are generalized to produce a series solution that is valid for large indentation depths. The newly derived equations proposed in this paper are tested on both simulated and experimental data from cells, demonstrating excellent accuracy. Full article

In this article, we present an efficient numerical strategy for the two-dimensional nonlinear Schrödinger equation, focusing on its development and analysis. Our approach begins with proposing a nonlinear, energy-conservative, fourth-order, compact, alternating-direction, implicit (ADI) scheme. To boost efficiency when solving the associated nonlinear system, we then implement this scheme using a temporal two-mesh (TTM) algorithm. Under discretization with coarse time step ${\tau }_C$, fine time step ${\tau }_F$, and spatial mesh size h, the numerical scheme exhibits a convergence rate of order $\mathcal{O}({\tau }_C^4+{\tau }_F^2+h^4)$ in both the discrete $L_2$-norm and $H_1$-norm. To facilitate the convergence analysis under fine time discretization, we propose a novel technique along with several supporting lemmas that enable the estimation of the discrete $L_4$-norm error term over the temporal coarse mesh. Numerical experiments are then performed to validate the theoretical results and demonstrate the effectiveness of the proposed algorithm. The numerical results show that the new algorithm produces highly accurate results and preserves the conservation laws of mass and energy. Compared with the fully nonlinear compact ADI scheme, it reduces computational time while maintaining accuracy. Full article