Search Results (3,531)

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

The shallow water equations (SWEs) model fluid flow in rivers, coasts, and tsunamis. Their nonlinearity challenges analytical solutions. We present a numerical algorithm combining the finite integration method with Chebyshev polynomial expansion (FIM-CPE) to solve one- and two-dimensional SWEs. The method transforms partial differential equations into integral equations, approximates spatial terms via Chebyshev polynomials, and uses forward differences for time discretization. Validated on stationary lakes, dam breaks, and Gaussian pulses, the scheme achieved errors below ${10}^{-12}$ for water height and velocity, while conserving mass with volume deviations under ${10}^{-5}$. Comparisons showed superior shock-capturing versus finite difference methods. For two-dimensional cases, it accurately resolved wave interactions over complex topographies. Though limited to wet beds and small-scale two-dimensional problems, the method provides a robust simulation tool. Full article

A rotating imperfect ring resonator of the gyroscope is modeled by a rotating thin ring with evenly distributed point masses. The free response of the rotating ring structure at constant speed is investigated, including the steady elastic deformation and wave response. The dynamic equations are formulated by using Hamilton’s principle in the ground-fixed coordinates. The coordinate transformation is applied to facilitate the solution of the steady deformation, and the displacements and tangential tension for the deformation are calculated by the perturbation method. Employing Galerkin’s method, the governing equation of the free vibration is casted in matrix differential operator form after the separation of the real and imaginary parts with the inextensional assumption. The natural frequencies are calculated through the eigenvalue analysis, and the numerical results are obtained. The effects of the point masses on the natural frequencies of the forward and backward traveling wave curves of different orders are discussed, especially on the measurement accuracy of gyroscopes for different cases. In the ground-fixed coordinates, the frequency splitting results in a crosspoint of the natural frequencies of the forward and backward traveling waves. The finite element method is applied to demonstrate the validity and accuracy of the model. Full article

In the present paper, a mathematical analysis of the Gardner equation with varying coefficients has been performed to give a more realistic model of physical phenomena, especially in regards to plasma physics. First, a Lie symmetry analysis was carried out, as a result of which a symmetry classification following the different representations of the variable coefficients was systematically derived. The reduced ordinary differential equation obtained is solved using the power-series method and solutions to the equation are represented graphically to give an idea of their dynamical behavior. Moreover, a fully connected neural network has been included as an efficient computation method to deal with the complexity of the reduced equation, by using traveling-wave transformation. The validity and correctness of the solutions provided by the neural networks have been rigorously tested and the solutions provided by the neural networks have been thoroughly compared with those generated by the Runge–Kutta method, which is a conventional and well-recognized numerical method. The impact of a variation in the loss function of different coefficients has also been discussed, and it has also been found that the dispersive coefficient affects the convergence rate of the loss contribution considerably compared to the other coefficients. The results of the current work can be used to improve knowledge on the nonlinear dynamics of waves in plasma physics. They also show how efficient it is to combine the approaches, which consists in the use of analytical and semi-analytical methods and methods based on neural networks, to solve nonlinear differential equations with variable coefficients of a complex nature. Full article

We establish a rigorous kinetic-theoretical framework to analyze causal propagation in thermal transport phenomena within relativistic ideal fluids, building a more rigorous framework based on the kinetic theory of gases. Specifically, we provide a refined derivation of the wave equation governing thermal and density fluctuations, clarifying its hyperbolic nature and the associated characteristic propagation speeds. The analysis confirms that thermal fluctuations in a simple non-degenerate relativistic fluid satisfy a causal wave equation in the Euler regime, and it recovers the classical expression for the speed of sound in the non-relativistic limit. This work offers enhanced mathematical and physical insights, reinforcing the validity of the hyperbolic description and suggesting a foundation for future studies in dissipative relativistic hydrodynamics. Full article

The present paper proposes a three-dimensional (3D) spherical shell model for the magneto-electro-elastic (MEE) free vibration analysis of simply supported multilayered smart shells. A mixed curvilinear orthogonal reference system is used to write the unified 3D governing equations for cylinders, cylindrical panels and spherical shells. The closed-form solution of the problem is performed considering Navier harmonic forms in the in-plane directions and the exponential matrix method in the thickness direction. A layerwise approach is possible, considering the interlaminar continuity conditions for displacements, electric and magnetic potentials, transverse shear/normal stresses, transverse normal magnetic induction and transverse normal electric displacement. Some preliminary cases are proposed to validate the present 3D MEE free vibration model for several curvatures, materials, thickness values and vibration modes. Then, new benchmarks are proposed in order to discuss possible effects in multilayered MEE curved smart structures. In the new benchmarks, first, three circular frequencies for several half-wave number couples and for different thickness ratios are proposed. Thickness vibration modes are shown in terms of displacements, stresses, electric displacement and magnetic induction along the thickness direction. These new benchmarks are useful to understand the free vibration behavior of MEE curved smart structures, and they can be used as reference for researchers interested in the development of of 2D/3D MEE models. Full article

In 2071, the Hydraulic community will commemorate the second centenary of the Baré de Saint-Venant equations, also known as the Shallow Water Equations (SWE). These equations are fundamental to the study of open-channel flow. As non-linear partial differential equations, their solutions were largely unattainable until the development of computers and numerical methods. Following 1960, various numerical schemes emerged, with Preissmann’s scheme becoming the most widely employed in many software applications. In the 1990s, some researchers identified a significant limitation in existing software and codes: the inability to simulate transcritical flow. At that time, Preissmann’s scheme was the dominant method employed in hydraulics tools, leading the research community to conclude that this scheme could not handle transcritical flow due to suspected instability. In response to this concern, several researchers suggested modifications to Preissmann’s scheme to enable the simulation of transcritical flow. This paper will demonstrate that these accusations against the Preissmann scheme are unfounded and that the proposed improvements are unnecessary. The observed instability is not due to the numerical method itself, but rather a mathematical instability inherent to the SWE, which can lead to ill-posed conditions if a specific derived condition is not met. In the context of a friction slope formula based on Manning or Chézy types, the condition for ill-posedness of the 1D shallow water equations simplifies to the Vedernikov number condition, which is necessary for roll waves to develop in uniform flow. This derived condition is also relevant for the formation of roll waves in unsteady flow when the 1D shallow water equations become ill-posed. Full article

The Korteweg–de Vries (KdV) equation is a nonlinear evolution equation that reflects a wide variety of dispersive wave occurrences with limited amplitude. It has also been used to describe a range of major physical phenomena, such as shallow water waves that interact weakly and nonlinearly, acoustic waves on a crystal lattice, lengthy internal waves in density-graded oceans, and ion acoustic waves in plasma. The KdV equation is one of the most well-known soliton models, and it provides a good platform for further research into other equations. The KdV equation has several forms. The aim of this study is to introduce and investigate a (2+1)-dimensional generalized fifth-order KdV equation with power law nonlinearity (gFKdVp). The research methodology employed is the Lie group analysis. Using the point symmetries of the gFKdVp equation, we transform this equation into several nonlinear ordinary differential equations (ODEs), which we solve by employing different strategies that include Kudryashov’s method, the $(G^{\prime }/G)$ expansion method, and the power series expansion method. To demonstrate the physical behavior of the equation, 3D, density, and 2D graphs of the obtained solutions are presented. Finally, utilizing the multiplier technique and Ibragimov’s method, we derive conserved vectors of the gFKdVp equation. These include the conservation of energy and momentum. Thus, the major conclusion of the study is that analytic solutions and conservation laws of the gFKdVp equation are determined. 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

Extreme surface winds and wave heights of tropical cyclones (TCs)—pose serious threats to coastal community, infrastructure and environments. In recent decades, progress in numerical wave modeling has significantly enhanced the ability to reconstruct and predict wave behavior. This review offers an in-depth overview of TC-related wave modeling utilizing different computational schemes, with a special attention to WAVEWATCH III (WW3) and Simulating Waves Nearshore (SWAN). Due to the complex air–sea interactions during TCs, it is challenging to obtain accurate wind input data and optimize the parameterizations. Substantial spatial and temporal variations in water levels and current patterns occurs when coastal circulation is modulated by varying underwater topography. To explore their influence on waves, this study employs a coupled SWAN and Finite-Volume Community Ocean Model (FVCOM) modeling approach. Additionally, the interplay between wave and sea surface temperature (SST) is investigated by incorporating four key wave-induced forcing through breaking and non-breaking waves, radiation stress, and Stokes drift from WW3 into the Stony Brook Parallel Ocean Model (sbPOM). 20 TC events were analyzed to evaluate the performance of the selected parameterizations of external forcings in WW3 and SWAN. Among different nonlinear wave interaction schemes, Generalized Multiple Discrete Interaction Approximation (GMD) Discrete Interaction Approximation (DIA) and the computationally expensive Wave-Ray Tracing (WRT) A refined drag coefficient (C_d) equation, applied within an upgraded ST6 configuration, reduce significant wave height (SWH) prediction errors and the root mean square error (RMSE) for both SWAN and WW3 wave models. Surface currents and sea level variations notably altered the wave energy and wave height distributions, especially in the area with strong TC-induced oceanic current. Finally, coupling four wave-induced forcings into sbPOM enhanced SST simulation by refining heat flux estimates and promoting vertical mixing. Validation against Argo data showed that the updated sbPOM model achieved an RMSE as low as 1.39 m, with correlation coefficients nearing 0.9881. Full article

Wave deformation and sediment transport nearest the shoreside are among the main reasons for sand erosion and beach profile changes. In particular, identifying the areas of incident-wave breaking and longshore current generation parallel to the shoreline is important for understanding the morphological changes of coastal beaches. In this study, a two-phase incompressible flow model along with a sandy sloping topography was employed to investigate the wave deformation and longshore current generation areas in a circular wave basin model. The finite volume method (FVM) was implemented to discretize the governing equations in cylindrical coordinates, the volume-of-fluid method (VOF) was adopted to differentiate the air–water interfaces in the control cells, and the zonal embedded grid technique was employed for grid generation in the cylindrical computational domain. The water surface elevations and velocity profiles were measured in different wave conditions, and the measurements showed that the maximum water levels per wave were high and varied between cases, as well as between cross-sections in a single case. Additionally, the mean water levels were lower in the adjacent positions of the approximated wave-breaking zones. The wave-breaking positions varied between cross-sections in a single case, with the incident-wave height, mean water level, and wave-breaking position measurements indicating the influence of downstream flow variation in each cross-section on the sloping topography. The cross-shore velocity profiles became relatively stable over time, while the longshore velocity profiles predominantly moved in the alongshore direction, with smaller fluctuations, particularly during the same time period and in measurement positions near the wave-breaking zone. The computed velocity profiles also varied between cross-sections, and for the velocity profiles along the cross-shore and longshore directions nearest the wave-breaking areas where the downstream flow had minimal influence, it was presumed that there was longshore-current generation in the sloping topography nearest the shoreside. The computed results were compared with the experimental results and we observed similar characteristics for wave profiles in the same wave period case in both models. In the future, further investigations can be conducted using the presented circular wave basin model to investigate the oblique wave deformation and longshore current generation in different sloping and wave conditions. Full article

This study explores the (1+1)-dimensional Klein–Fock–Gordon equation, a distinct third-order nonlinear differential equation of significant theoretical interest. The Klein–Fock–Gordon equation (KFGE) plays a pivotal role in theoretical physics, modeling high-energy particles and providing a fundamental framework for simulating relativistic wave phenomena. To find the exact solution of the proposed model, for this purpose, we utilized two effective techniques, including the sine-Gordon equation method and a new extended direct algebraic method. The novelty of these approaches lies in the form of different solutions such as hyperbolic, trigonometric, and rational functions, and their graphical representations demonstrate the different form of solitons like kink solitons, bright solitons, dark solitons, and periodic waves. To illustrate the characteristics of these solutions, we provide two-dimensional, three-dimensional, and contour plots that visualize the magnitude of the (1+1)-dimensional Klein–Fock–Gordon equation. By selecting suitable values for physical parameters, we demonstrate the diversity of soliton structures and their behaviors. The results highlighted the effectiveness and versatility of the sine-Gordon equation method and a new extended direct algebraic method, providing analytical solutions that deepen our insight into the dynamics of nonlinear models. These results contribute to the advancement of soliton theory in nonlinear optics and mathematical physics. Full article

This study focuses on the analysis of soliton solutions within the framework of the time-fractional cubic–quintic nonlinear Schrödinger equation (TFCQ-NLSE), a powerful model with broad applications in complex physical phenomena such as fiber optic communications, nonlinear optics, optical signal processing, and laser–tissue interactions in medical science. The nonlinear effects exhibited by the model—such as self-focusing, self-phase modulation, and wave mixing—are influenced by the combined impact of the cubic and quintic nonlinear terms. To explore the dynamics of this model, we apply a robust analytical technique known as the sub-ODE method, which reveals a diverse range of soliton structures and offers deep insight into laser pulse interactions. The investigation yields a rich set of explicit soliton solutions, including hyperbolic, rational, singular, bright, Jacobian elliptic, Weierstrass elliptic, and periodic solutions. These waveforms have significant real-world relevance: bright solitons are employed in fiber optic communications for distortion-free long-distance data transmission, while both bright and dark solitons are used in nonlinear optics to study light behavior in media with intensity-dependent refractive indices. Solitons also contribute to advancements in quantum technologies, precision measurement, and fiber laser systems, where hyperbolic and periodic solitons facilitate stable, high-intensity pulse generation. Additionally, in nonlinear acoustics, solitons describe wave propagation in media where amplitude influences wave speed. Overall, this work highlights the theoretical depth and practical utility of soliton dynamics in fractional nonlinear systems. Full article

Flooding due to dam failures is a critical issue with significant impacts on human safety, infrastructure, and the environment. This study assessed the potential flood hazard that could be generated from breaching of the Alacranes dam in Villa Clara, Cuba. Thirteen reservoir breaching scenarios were simulated under several criteria for modeling the flood wave through the 2D Saint Venant equations using the Hydrologic Engineering Center’s River Analysis System (HEC-RAS). A sensitivity analysis was performed on Manning’s roughness coefficient, demonstrating a low variability of the model outputs for these events. The results show that, for all modeled scenarios, the terrain topography of the coastal plain expands the flood wave, reaching a maximum width of up to 105,057 km. The most critical scenario included a 350 m breach in just 0.67 h. Flood, velocity, and hazard maps were generated, identifying populated areas potentially affected by the flooding events. The reported depths, velocities, and maximum flows could pose extreme danger to infrastructure and populated areas downstream. These types of studies are crucial for both risk assessment and emergency planning in the event of a potential dam breach. Full article

The interaction between thermal fields and mechanical loads in thin-walled cylindrical shells introduces complex dynamic behaviors relevant to aerospace and mechanical engineering applications. This study investigates the axial stress wave propagation in a circular cylindrical shell subjected to combined thermal gradients and time-dependent harmonic compression. A semi-analytical model based on Donnell–Mushtari–Vlasov (DMV) shells theory is developed to derive the governing equations, incorporating elastic, inertial, and thermal expansion effects. Modal solutions are obtained to evaluate displacement and stress distributions across varying thermal and mechanical excitation conditions. Empirical Mode Decomposition (EMD) and Instantaneous Frequency (IF) analysis are employed to extract time–frequency characteristics of the dynamic response. Complementary Finite Element Analysis (FEA) is conducted to assess modal deformations, stress wave amplification, and the influence of thermal softening on resonance frequencies. Results reveal that increasing thermal gradients leads to significant reductions in natural frequencies and amplifies stress responses at critical excitation frequencies. The combination of analytical and numerical approaches captures the coupled thermomechanical effects on shell dynamics, providing an understanding of resonance amplification, modal energy distribution, and thermal-induced stiffness variation under axial harmonic excitation across thin-walled cylindrical structures. Full article