Search Results (35)

This study investigates the influence of multiplicative noise—modeled by a Wiener process—and spatial-fractional derivatives on the dynamics of the space-fractional stochastic Regularized Long Wave equation. By employing a complete discriminant polynomial system, we derive novel classes of fractional stochastic solutions that capture the complex interplay between stochasticity and nonlocality. Additionally, the variational principle, derived by He’s semi-inverse method, is utilized, yielding additional exact solutions that are bright solitons, bright-like solitons, kinky bright solitons, and periodic structures. Graphical analyses are presented to clarify how variations in the fractional order and noise intensity affect essential solution features, such as amplitude, width, and smoothness, offering deeper insight into the behavior of such nonlinear stochastic systems. Full article

This study aims to explore the effect of spatial-fractional derivatives and the multiplicative standard Wiener process on the solutions of the stochastic fractional regularized long-wave equation (SFRLWE) and contribute to its analysis. We introduce a new systematic method that combines the auxiliary function method with the complete discriminant polynomial system. This method proves to be effective in discovering precise solutions for stochastic fractional partial differential equations (SFPDEs), including special cases. Applying this method to the SFRLWE yields new exact solutions, offering fresh insights. We investigated how noise affects stochastic solutions and discovered that more intense noise can result in flatter surfaces. We note that multiplicative noise can stabilize the solution, and we show how fractional derivatives influence the dynamics of noise. We found that the noise strength and fractional derivative affect the width, amplitude, and smoothness of the obtained solutions. Additionally, we conclude that multiplicative noise impacts and stabilizes the behavior of SFRLWE solutions. Full article

This paper proposes an innovative algorithm for forecasting the motion response of floating offshore wind turbines by employing force-to-motion transfer functions and state-space models. Traditional numerical integration techniques, such as the Newmark-β method, frequently struggle with inefficiencies due to the heavy computational demands of convolution integrals in the Cummins equation. Our new method tackles these challenges by converting the problem into a system output calculation, thereby eliminating convolutions and potentially enhancing computational efficiency. The procedure begins with the estimation of force-to-motion transfer functions derived from the hydrostatic and hydrodynamic characteristics of the wind turbine. These transfer functions are then utilized to construct state-space models, which compactly represent the system dynamics. Motion responses resulting from initial conditions and wave forces are calculated using these state-space models, leveraging their poles and residues. We validated the proposed method by comparing its calculated responses to those obtained via the Newmark-β method. Initial tests on a single-degree-of-freedom (SDOF) system demonstrated that our algorithm accurately predicts motion responses. Further validation involved a numerical model of a spar-type floating offshore wind turbine, showing high accuracy in predicting responses to both regular and irregular wave conditions, closely aligning with results from conventional methods. Additionally, we assessed the efficiency of our algorithm over various simulation durations, confirming its superior performance compared to traditional time-domain methods. This efficiency is particularly advantageous for long-duration simulations. The proposed approach provides a robust and efficient alternative for predicting motion responses in floating offshore wind turbines, combining high accuracy with improved computational performance. It represents a promising tool for enhancing the development and evaluation of offshore wind energy systems. Full article

This study presents numerical and experimental investigations on an oscillating water column (OWC) wave energy device integrated into a sloping breakwater. Regular waves were generated in a physical wave tank to investigate the hydrodynamic performance and extraction efficiency of the small-scale nested OWC device. Simultaneously, to complement various scenarios, numerical simulations were conducted using the open-source computational fluid dynamics platform OpenFOAM. The volume of fluid (VOF) method was employed to capture the complex evolution of the air–water interface, and an artificial source term (Forchheimer flow region) was introduced into the Navier–Stokes equations to replace the power take-off (PTO) system. By analyzing wave reflection properties, energy absorption efficiency, and wave run-up, the hydrodynamic characteristics of the inclined OWC device were explored. The comparison between the numerical and experimental results indicate a good consistence. A smaller front wall draft broadens the high-efficiency frequency bandwidth. For relatively long waves, increasing the air chamber width enhances energy conversion efficiency and reduces wave run-up. The optimal configuration was achieved with the following dimensionless parameters: front wall draft $a/h=1/3$, air chamber width $d_1/h=2/9$, and slope $i=2$. Due to the sloped structure, when compared with a vertical OWC, long waves can more easily enter the chamber. This causes the efficient frequency bandwidth to shift towards the low frequency range, allowing more wave energy to be converted into pneumatic energy. As a result, wave run-up is reduced, enhancing the protective function of the breakwater. Full article

The Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is a well-known regularized long-wave model that examines the propagation kinematics of water waves. The current work employs an effective approach, called the Riccati Modified Extended Simple Equation Method (RMESEM), to effectively and precisely derive the propagating soliton solutions to the (3+1)-dimensional WBBM equation. By using this upgraded approach, we are able to find a greater diversity of families of propagating soliton solutions for the WBBM model in the form of exponential, rational, hyperbolic, periodic, and rational hyperbolic functions. To further graphically represent the propagating behavior of acquired solitons, we additionally provide 3D, 2D, and contour graphics which clearly demonstrate the presence of kink solitons, including solitary kink, anti-kink, twinning kink, bright kink, bifurcated kink, lump-like kink, and other multiple kinks in the realm of WBBM. Furthermore, by producing new and precise propagating soliton solutions, our RMESEM demonstrates its significance in revealing important details about the model behavior and provides indications regarding possible applications in the field of water waves. Full article

Solving shallow water equations is crucial in science and engineering for understanding and predicting natural phenomena. To address the limitations of Physics-Informed Neural Network (PINN) in solving shallow water equations, we propose an improved PINN algorithm integrated with a deep learning framework. This algorithm introduces a regularization term as a penalty in the loss function, based on the PINN and Long Short-Term Memory (LSTM) models, and incorporates an attention mechanism to solve the original equation across the entire domain. Simulation experiments were conducted on one-dimensional and two-dimensional shallow water equations. The results indicate that, compared to the classical PINN algorithm, the improved algorithm shows significant advantages in handling discontinuities, such as sparse waves, in one-dimensional problems. It accurately captures sparse waves and avoids smoothing effects. In two-dimensional problems, the improved algorithm demonstrates good symmetry and effectively reduces non-physical oscillations. It also shows significant advantages in capturing details and handling complex phenomena, offering higher reliability and accuracy. The improved PINNs algorithm, which combines neural networks with physical mechanisms, can provide robust solutions and effectively avoid some of the shortcomings of classical PINNs methods. It also possesses high resolution and strong generalization capabilities, enabling accurate predictions at any given moment. Full article

The fractional regularized long wave equation and the fractional nonlinear shallow-water wave equation are the noteworthy models in the domains of fluid dynamics, ocean engineering, plasma physics, and microtubules in living cells. In this study, a reliable and efficient improved F-expansion technique, along with the fractional beta derivative, has been utilized to explore novel soliton solutions to the stated wave equations. Consequently, the study establishes a variety of reliable and novel soliton solutions involving trigonometric, hyperbolic, rational, and algebraic functions. By setting appropriate values for the parameters, we obtained peakons, anti-peakon, kink, bell, anti-bell, singular periodic, and flat kink solitons. The physical behavior of these solitons is demonstrated in detail through three-dimensional, two-dimensional, and contour representations. The impact of the fractional-order derivative on the wave profile is notable and is illustrated through two-dimensional graphs. It can be stated that the newly established solutions might be further useful for the aforementioned domains. Full article

The propagation of light beams in photovoltaic pyroelectric photorefractive crystals is modelled by a specific generalization of the nonlinear Schrödinger equation (GNLSE). We use a variational approximation (VA) to predict the propagation of solitary-wave inputs in the crystals, finding that the VA equations involve a dilogarithm special function. The VA predicts that solitons and breathers exist, and the Vakhitov–Kolokolov criterion predicts that the solitons are stable solutions. Direct simulations of the underlying GNLSE corroborates the existence of such stable modes. The numerical solutions produce both regular breathers and ones featuring beats (long-period modulations of fast oscillations). In the latter case, the Fourier transform of amplitude oscillations reveals a nearly discrete spectrum characterizing the beats dynamics. Numerical solutions of another type demonstrate the spontaneous splitting of the input pulse in two or several secondary ones. Full article

This study explores the effects of using space-fractional derivatives and adding multiplicative noise, modeled by a Wiener process, on the solutions of the space-fractional stochastic regularized long wave equation. New fractional stochastic solutions are constructed, and the consistency of the obtained solutions is examined using the transition between phase plane orbits. Their bifurcation and dependence on initial conditions are investigated. Some of these solutions are shown graphically, illustrating both the individual and combined influences of fractional order and noise on selected solutions. These effects appear as alterations in the amplitude and width of the solutions, and as variations in their smoothness. Full article

By employing the cubic B-spline functions, a collocation approach was devised in this study to address the Modified Regularized Long Wave (MRLW) equation. Then, we derived the corresponding nonlinear system and easily solved it using Newton’s iterative approach. It was established that the cubic B-spline collocation technique exhibits unconditional stability. The dynamics of solitary waves, including their pairwise and triadic interactions, were meticulously investigated utilizing the proposed numerical method. Additionally, the transformation of the Maxwellian initial condition into solitary wave formations is presented. To validate the current work, three distinct scenarios were compared against the analytical solution and outcomes from alternative methods under both $L_2$- and $L_{\infty }$-error norms. Primarily, the key strength of the suggested scheme lies in its capacity to yield enhanced numerical resolutions when employed to solve the MRLW equation, and these conservation laws show that the solitary waves have time and space translational symmetry in the propagation process. Finally, this paper concludes with a summary of our findings. Full article

In this paper, we apply two different methods, namely, the $\frac{G^{\prime }}{G}$-expansion method and the $\frac{G^{\prime }}{G^2}$-expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation in the conformable sense. The mentioned nonlinear partial differential equations (NPDEs) arise in diverse physical applications such as ion sound waves in plasma and waves on shallow water surfaces. There exist multiple wave solutions to many NPDEs and researchers are interested in analytical approaches to obtain these multiple wave solutions. The multi-exp-function method (MEFM) formulates a solution algorithm for calculating multiple wave solutions to NPDEs and at the end of paper, we apply the MEFM for calculating multiple wave solutions to the (2 + 1)-dimensional equation. Full article

In this study, a short-term prediction method for ship roll motion in waves based on convolutional neural network (CNN) is presented. Firstly, based on the ship roll motion equation, the data for free roll attenuation motion in still water, roll motion in regular waves, and roll motion excited by irregular waves are simulated, respectively. Secondly, the simulation data is normalized and preprocessed, and then the time-sliding window technique is applied to construct the training and testing sample sets. Thirdly, the CNN model is trained by learning from the constructed training sample sets, and the well-trained CNN model is applied to predict the roll motion. To validate the CNN model’s prediction accuracy and effectiveness, a comparison between the forecasted results and the simulation data is conducted. Meanwhile, the predicted results are also compared with that of the long-short-term memory (LSTM) neural network. The research results demonstrate that CNN can effectively achieve accurate prediction of ship roll motion in waves, and its prediction accuracy is the same as that of the LSTM neural network. Full article

The paper introduces a new two-level time-mesh difference scheme for solving the symmetric regularized long wave equation. The scheme consists of three steps. A coarse time-mesh and a fine time-mesh are defined, and the equation is solved using an existing nonlinear scheme on the coarse time-mesh. Lagrange’s linear interpolation formula is employed to obtain all preliminary solutions on the fine time-mesh. Based on the preliminary solutions, Taylor’s formula is utilized to construct a linear system for the equation on the fine time-mesh. The convergence and stability of the scheme is analyzed, providing the convergence rates of $O({\tau }_F^2+{\tau }_C^4+h^4)$ in the discrete $L_{\infty }$-norm for $u(x,t)$ and in the discrete $L_2$-norm for $\rho (x,t)$. Numerical simulation results show that the proposed scheme achieves equivalent error levels and convergence rates to the nonlinear scheme, while also reducing CPU time by over half, which indicates that the new method is more efficient. Furthermore, compared to the earlier time two-mesh method developed by the authors, the proposed scheme significantly reduces the error between the numerical and exact solutions, which means that the proposed scheme is more accurate. Additionally, the effectiveness of the new scheme is discussed in terms of the corresponding conservation laws and long-time simulations. Full article

This work focuses on the propagation of waves on the water’s surface, which can be described via different mathematical models. Here, we apply the generalized exponential rational function method (GERFM) to several nonlinear models of surface wave propagation to identify their multiple solitary wave structures. We provide stability analysis and graphical representations for the considered models. Additionally, this paper compares the results obtained in this work and existing solutions for the considered models in the literature. The effectiveness and potency of the utilized approach are demonstrated, indicating their applicability to a broad range of nonlinear partial differential equations in physical phenomena. Full article

The elastic wave equation with seismic tensorial force is solved in a homogeneous and isotropic medium (the Earth). Spherical-shell waves are obtained, which are associated to the primary P and S seismic waves. It is shown that these waves produce secondary waves with sources on the plane surface of a half-space, which have the form of abrupt walls with a long tail, propagating in the interior and on the surface of the half-space. These secondary waves are associated to the seismic mainshock. The results, previously reported, are re-derived using Fourier transformations and specific regularization procedures. The relevance of this seismic motion for the ground motion, the seismographs’ recordings and the effect of the inhomogeneities in the medium are discussed. Full article