The present study aims to determine the appropriate size of mesh or the number of the element (NoE) for flat- and curved plates, which is suggested to assess its safety subjected to axial compression based on the ultimate limit state (ULS) design and analysis concept. The unstiffened panel (= plate) and stiffened panel, considered primary members of ships and ship-shaped offshore structures, are subjected to repeated axial compression and tension caused by continued vertical bending moments applied to the hull girder. Plates are attached with stiffeners by welding, and 6, 8 or 10 elements are generally recommended to allocate in flat-plate's breadth direction in between stiffeners for finite-element (FE) modelling, which enables the presentation of the shape of initial deflection applied to the plate. In the case of the load-shorting curve for curved plate, it is reported that the nonlinear behaviour characteristics, i.e., snap-through, snap-back, secondary buckling and others, appear in typical flank angle. To take this into account, we investigated the preferred number of elements (6, 8 or 10) generally applied to the flat plate whether it is an appropriate or more fine-sized element (or mesh) that should be considered. A useful guide is documented based on obtained outcomes which may help structural engineers select optimised mesh-size to predict ultimate strength and understand its characteristic of the flat and curved plates.

Highlights

● The general shape of the empirical formulation in predicting the ultimate strength of the plate is proposed.

● An empirical formulation in predicting ULS of initially deflected and simply supported edged plate under longitudinal compression is developed based on general shape by determining four coefficients.

● The results of ULS by NLFEM, semi-analytical method, and direct calculation method by empirical formulations are compared, and its accuracy has been verified.

Turbulence is strongly associated with the vast majority of fluid flows in nature and industry. Traditionally, results given by the direct numerical simulation (DNS) of Navier-Stokes (NS) equations that relate to a famous millennium problem are widely regarded as ‘reliable' benchmark solutions of turbulence, as long as grid spacing is fine enough (i.e. less than the minimum Kolmogorov scale) and time-step is small enough, say, satisfying the Courant-Friedrichs-Lewy condition (Courant number < 1). Is this really true? In this paper a two-dimensional sustained turbulent Kolmogorov flow driven by an external body force governed by the NS equations under an initial condition with a spatial symmetry is investigated numerically by the two numerical methods with detailed comparisons: one is the traditional DNS, the other is the ‘clean numerical simulation' (CNS). In theory, the exact solution must have a kind of spatial symmetry since its initial condition is spatially symmetric. However, it is found that numerical noises of the DNS are quickly enlarged to the same level as the ‘true' physical solution, which finally destroy the spatial symmetry of the flow field. In other words, the DNS results of the turbulent Kolmogorov flow governed by the NS equations are badly polluted mostly. On the contrary, the numerical noise of the CNS is much smaller than the ‘true' physical solution of turbulence in a long enough interval of time so that the CNS result is very close to the ‘true' physical solution and thus can remain symmetric, which can be used as a benchmark solution for comparison. Besides, it is found that numerical noise as a kind of artificial tiny disturbances can lead to huge deviations at large scale on the two-dimensional Kolmogorov turbulence governed by the NS equations, not only quantitatively (even in statistics) but also qualitatively (such as spatial symmetry of flow). This highly suggests that fine enough spatial grid spacing with small enough time-step alone could not guarantee the validity of the DNS of the NS equations: it is only a necessary condition but not sufficient. All of these findings might challenge some of our general beliefs in turbulence: for example, it might be wrong in physics to neglect the influences of small disturbances to NS equations. Our results suggest that, from physical point of view, it should be better to use the Landau-Lifshitz-Navier-Stokes (LLNS) equations, which consider the influence of unavoidable thermal fluctuations, instead of the NS equations, to model turbulent flows.

Highlights

● A two-dimensional Kolmogorov flow is numerically solved by means of the direct numerical simulation (DNS) and clean numerical simulation (CNS), respectively.

● It is found that tiny numerical noises of the DNS result are quickly enlarged to a macroscopic level so that the DNS results are quickly polluted badly.

● Detailed comparisons between the CNS and DNS reveal that artificial numerical noises lead to large deviations of the turbulent flow even in long-term statistics.

In the fields of oceanography, hydrodynamics, and marine engineering, many mathematicians and physicists are interested in Burgers-type equations to show the different dynamics of nonlinear wave phenomena, one of which is a (3+1)-dimensional Burgers system that is currently being studied. In this paper, we apply two different analytical methods, namely the generalized Kudryashov (GK) method, and the generalized exponential rational function method, to derive abundant novel analytic exact solitary wave solutions, including multi-wave solitons, multi-wave peakon solitons, kink-wave profiles, stripe solitons, wave-wave interaction profiles, and periodic oscillating wave profiles for a (3+1)-dimensional Burgers system with the assistance of symbolic computation. By employing the generalized Kudryashov method, we obtain some new families of exact solitary wave solutions for the Burgers system. Further, we applied the generalized exponential rational function method to obtain a large number of soliton solutions in the forms of trigonometric and hyperbolic function solutions, exponential rational function solutions, periodic breather-wave soliton solutions, dark and bright solitons, singular periodic oscillating wave soliton solutions, and complex multi-wave solutions under various family cases. Based on soft computing via Wolfram Mathematica, all the newly established solutions are verified by back substituting them into the considered Burgers system. Eventually, the dynamical behaviors of some established results are exhibited graphically through three - and two-dimensional wave profiles via numerical simulation.

Ship-hull design is a complex process because the any slight local alteration in ship hull structure may significantly change the hydrostatic and hydrodynamic performances of a ship. To find the optimum hull shape under the design requirements, the state-of-art of ship hull design combines computational fluid dynamics computation with geometric modeling. However, this process is very computationally intensive, which is only suitable at the final stage of the design process. To narrow down the design parameter space, in this work, we have developed an AI-based deep learning neural network to realize a real-time prediction of the total resistance of the ship-hull structure in its initial design process. In this work, we have demonstrated how to use the developed DNN model to carry out the initial ship hull design. The validation results showed that the deep learning model could accurately predict the ship hull’s total resistance accurately after being trained, where the average error of all samples in the testing dataset is lower than 4%. Simultaneously, the trained deep learning model can predict the hip’s performances in real-time by inputting geometric modification parameters without tedious preprocessing and calculation processes. The machine learning approach in ship hull design proposed in this work is the first step towards the artificial intelligence-aided design in naval architectures.

The real-time prediction of a floating platform or a vessel is essential for motion-sensitive maritime activities. It can enhance the performance of motion compensation system and provide useful early-warning information. In this paper, we apply a machine learning technique to predict the surge, heave, and pitch motions of a moored rectangular barge excited by an irregular wave, which is purely based on the motion data. The dataset came from a model test performed in the deep-water ocean basin, at Shanghai Jiao Tong University, China. Using the trained machine learning model, the predictions of 3-DoF (degrees of freedom) motions can extend two to four wave cycles into the future with good accuracy. It shows great potential for applying the machine learning technique to forecast the motions of offshore platforms or vessels.

Highlights

● The real-time prediction of the motion of a floating platform in irregular wave.

● A machine learning technique is applied to predict the surge, heave, and pitch motions of a moored barge, which is purely based on the motion data.

● The dataset was generated from a model test performed in the deep-water ocean basin.

● The predictions of motions are validated by experimental measurements.

Nowadays seakeeping is mostly analyzed by means of model testing or numerical models. Both require a significant amount of time and the exact hull geometry, and therefore seakeeping is not taken into account at the early stages of ship design. Hence the main objective of this work is the development of a seakeeping prediction tool to be used in the early stages of ship design.

This tool must be fast, accurate, and not require the exact hull shape. To this end, an artificial intelligence (AI) algorithm has been developed. This algorithm is based on Artificial Neural Networks (ANNs) and only requires a number of ship coefficients of form.

The methodology developed to obtain the predictive algorithm is presented as well as the database of ships used for training the ANN. The data were generated using a frequency domain seakeeping code based on the boundary element method (BEM). Also, the AI predictions are compared to the BEM results using both, ship hulls included and not included in the database.

As a result of this work it has been obtained an AI tool for seakeeping prediction of conventional monohull vessels

Highlights

● Application of Artificial Neural Networks (ANNs) to predict seakeeping of monohulls in early ship design stages.

● High accuracy achieved by ANNs compared with traditional solvers.

● Methodology based on data augmentation, numerical computation, and ANN competition is applied.

● Fast computation method is developed achieving instant computation of seakeeping of monohulls.

● No need of hull shapes for computing seakeeping.

The simplified modified Camassa-Holm (SMCH) equation is an important nonlinear model equation for identifying various wave phenomena in ocean engineering and science. The new auxiliary equation (NAE) method has been applied to the SMCH equation. Base on the method, we have obtained some novel analytical solutions such as hyperbolic, trigonometric, exponential, and rational function solutions of the SMCH equation. For appropriate values of parameters, three dimensional (3D) and two dimensional (2D) graphs are designed by Mathematica. The stability of the model is also discussed in this manuscript. The dynamic and physical behaviors of the solutions derived from the SMCH equation have been extensively discussed by these plots. All our solutions are indispensable for understanding the nonlinear phenomena of dispersive waves that are important in ocean engineering and science. In addition, our results are essential to clarify the various oceanographic applications containing ocean gravity waves, offshore rig in water, energy associated with a moving ocean wave and numerous other related phenomena. Finally, the obtained solutions are helpful for studying wave interactions in many new structures and high-dimensional models.

The study of internal atmospheric waves, also known as gravity waves, which are detectable inside the fluid rather than at the fluid surface, is presented in this work. We have used the time-fractional and fuzzy-fractional techniques to solve the differential equation system representing the atmospheric internal waves model. The q-Homotopy analysis Shehu transform technique (q-HAShTM) is used to solve the model. The method helps find convergent solutions since it helps solve nonlinearity, and the fractional derivative can be easily computed using the Shehu transform. Finally, the obtained solution is compared for the particular case of α=1 with the HAM solution to explain the method's accuracy.

In this paper, an experimental investigation on the wave loads and structural motions of two semi-fixed semi-immersed horizontal cylinders type rafts in the free surface zone is conducted. The physical models are tested at the 1:4.5 scale and exposed to a range of regular and irregular waves in a wave flume at Queen's University Belfast. The physical models and experimental setup are discussed alongside an investigation of the hydrodynamic phenomena, surge forces, and dynamic responses that each structure exhibits in the coastal wave climates. Furthermore, an investigation into the wave attenuation by both models is carried out. The results show that the surge forces have a positive correlation with wave steepness for both models. Hydrodynamic phenomena such as wave runup and overtopping, radiative damping and reflected waves, constructive interference, diffraction and flow separation were identified during the experiments. A negative mean heave displacement is observed during the monochromatic sea states which could result in impact loading and submergence of the superstructure components and photovoltaic panels at full-scale. The results presented in this paper may be used to calibrate and verify numerical models that calculate the global responses and hydrodynamic forces. It may also benefit the design processes of geometrically similar floating solar technologies by providing data on surge loads, motion responses and hydrodynamic observations.

Highlights

● An experimental investigation of two scaled and simplified floating photovoltaic structures in a wave flume environment.

● The dynamic responses, surge forces, wave attenuation and hydrodynamic phenomena were recorded using various instruments.

● A negative mean heave displacement is observed during the monochromatic sea states which could result in impact loading.

The mathematical model of imbibition phenomenon through homogeneous as well as heterogeneous porous media is presented in this study. Various types of porous materials including Fragmented Mixture, Touchet silt loam, and Glass Beads are investigated and discussed in terms of the relative permeability, capillarity, and heterogeneity of the material on saturation rate of a reservoir. In the present model, the comparison of saturation level for different time and distance level have been discussed between homogeneous and heterogeneous porous medium for various types of sands. The reduced differential transform method (RDTM) is used to obtain approximate analytical solution of the proposed model. Numerical and graphical results are presented for a wide range of time and distance.

Highlights

● Here Purcell relative permeability model of imbibition phenomenon in double phase fluid flow in porous media is discussed

● The imbibition phenomenon is one of the useful phenomena occurs in secondary oil recovery process in Petroleum Engineering

● This study proposes the effect of heterogeneity on saturation rate for various types of porous materials

● The effect of capillary pressure and relative permeability on saturation rate are analysed for various porous materials Reduced differential transform method is used to obtain saturation rates for different distance and time levels

The Tension Leg Platform (TLP) is a hybrid, compliant platform designed to sustain springing and ringing responses that are correlated to short-period motion. Since the period of short-period motion is within the wave energy concentration region, TLPs may experience sensitive short-period motion, such as resonance and green water, that usually cause serious damage to TLPs. In this study, a precontrol methodology is presented as a solution to prevent TLP-sensitive short-period motion. By applying the precontrol methodology, the parameters of TLP can be predetermined, allowing TLP motion performance to meet the requirements of short-period motion before sensitive motions actually occur. For example, the damping coefficient should be less than 4.3, the tendons' stiffness should be larger than 0.91 × 10⁸, and the dimensionless draft should be less than 0.665. The development of a precontrol methodology is based on a solid theoretical foundation. First, a series of simple and high-fidelity numerical models are proposed to simulate the natural period of roll, natural period of heave, and green water height. Second, a constraint regime is generated based on the numerical models and the sensitive motion range of short-period motion. The constraint regime is divided into two parts: the control range (corresponding to sensitive short-period motion) and the feasible range (the complementary set of control ranges in the whole parameter constraint domain). Finally, TLP parameters are derived from the calculated feasible range. The precontrol methodology goes beyond the conventional approach of real-time control by changing the control from a remedial action to a preventive action.

Highlights

This work proposes a precontrol methodology to constrain the short-period motion of Tension Leg Platform (TLP) to prevent the occurrence of sensitive short-period motion in advance with simple and high-fidelity numerical models developed. The precontrol methodology goes beyond the conventional approach of real-time control by changing the control from a remedial action to a preventive action.

This work develops simple and high-fidelity numerical models for TLP's short-period motion: natural period of roll motion model based on stability theory and green water height model based on wave height spectral distribution in body coordinate system.

A linear electrohydrodynamic Kelvin-Helmholtz instability of the interface between two viscoelastic Rivlin-Ericksen fluids enclosed by two concentric horizontal cylinders has been studied via the viscoelastic potential flow theory. The dispersion equation of complex coefficients for asymmetric disturbance has been obtained by using normal mode technique. the stability criteria are analyzed theoretically and illustrated graphically. The imaginary part of growth rate is plotted versus the wave number. The influences of dynamic viscoelastic, uniform velocities, Reynolds number, electric field, dynamic viscosity, density fluids ratio, dielectric constant ratio and inner fluid fraction on the stability of the system are discussed. The study finds its significance in Ocean pipelines to transfer oil or gas such as Eastern Siberia-Pacific Ocean oil pipeline.

Highlights

● The linear electrohydrodynamic Kelvin-Helmholtz instability analysis of two viscoelastic Rivlin-Ericksen fluids has been studied via the viscoelastic potential flow theory.

● The dispersion equation of complex coefficients has been obtaind by using normal mode technique.

● The effect of various parameters on the stability of the system are discussed.

● some limiting cases are considered and recovered previous works.

In this article, the fractional diffusion-advection equation with resetting is introduced to promote the theory of anomalous transport. The fractional equation describes a particle's non-diffusive motion performing a random walk and is reset to its initial position. An analytical method is proposed to obtain the solution of the fractional equation with resetting via Fourier and Laplace transformations. We study the influence of the fractional-order and resetting rate on the probability distributions, and the mean square displacements are analyzed for different cases of anomalous regimes.

Highlights

● The fractional diffusion-advection equation has been solved in the case of resetting assumption.

● The probability distribution functions (non-Maxwellian distributions) of the analytical solutions have been illustrated.

● The mean square displacement (MSD) analysis has been studied to determine the mode of displacement of particles followed over time (freely diffusing, transported, bound and limited in its movement).

In this article, we suggest a new form of modified Kudryashov's method (NMK) to study the Dual-mode Sawada Kotera model. We know very well that the more the solutions depend on many constants, the easier it is to study the model better by observing the change in the constants and what their impact is on the solutions. From this point of view, we developed the modified Kudryashov method and put it in a general form that contains more than one controllable constant. We have studied the model in this way and presented figures showing the correctness of what we hoped to reach from the proposed method. In addition to the results we reached, they were not sufficient, so we presented an extensive numerical study of this model using the finite differences method. We also came up with the local truncation error for the difference scheme is h6k2(1+k2). In addition, the analytical solutions we reached were compared with the numerical solutions, and we presented many forms that show that the results we reached are a clear contribution to this field.

Highlights

● We suggest a new form of modified Kudryashov's method (NKM) to study the Dual-mode Sawada Kotera model.

● We know very well that the more the solutions depend on many constants, the easier it is to study the model better by observing the change in the constants and what their impact is on the solutions.

● From this point of view, we developed the modified Kudryashov method and put it in a general form that contains more than one controllable constant.

● We have studied the model in this way and presented figures showing the correctness of what we hoped to reach from the proposed method.

● In addition to the results we reached, they were not sufficient, so we presented an extensive numerical study of this model using the finite differences method.

● We also came up with the local truncation error for the difference scheme is $h^6{k}^2(1+k^2)$ .

● In addition, the analytical solutions we reached were compared with the numerical solutions, and we presented many forms that show that the results we reached are a clear contribution to this field.

The dynamics of atmosphere and ocean can be examined under different circumstances of shallow water waves like shallow water gravity waves, Kelvin waves, Rossby waves and inertio-gravity waves. The influences of these waves describe the climate change adaptation on marine environment and planet. Therefore, the present work aims to derive symmetry reductions of Broer-Kaup-Kupershmidt equation in shallow water of uniform depth and then a variety of exact solutions are constructed. It represents the propagation of nonlinear and dispersive long gravity waves in two horizontal directions in shallow water. The invariance of test equations under one parameter transformation leads to reduction of independent variable. Therefore, twice implementations of symmetry method result into equivalent system of ordinary differential equations. Eventually, the exact solutions of these ODEs are computed under parametric constraints. The derive results entail several arbitrary constants and functions, which make the findings more admirable. Based on the appropriate choice of existing parameters, these solutions are supplemented numerically and show parabolic nature, intensive and non-intensive behavior of solitons.

Highlights

● Lie symmetry classification of Broer-Kaup-Kupershmidt equation in shallow water.

● The system describes the propagation of nonlinear and dispersive long gravity waves in two horizontal directions in shallow water.

● Symmetry reductions and invariant solutions are carried out.

● The obtained solutions are analyzed graphically.

● The solutions show parabolic nature, multi soliton, soliton fission phenomena.

Real-time predicting of stochastic waves is crucial in marine engineering. In this paper, a deep learning wave prediction (Deep-WP) model based on the ‘probabilistic' strategy is designed for the short-term prediction of stochastic waves. The Deep-WP model employs the long short-term memory (LSTM) unit to collect pertinent information from the wave elevation time series. Five irregular long-crested waves generated in the deepwater offshore basin at Shanghai Jiao Tong University are used to validate and optimize the Deep-WP model. When the prediction duration is 1.92s, 2.56s, and, 3.84s, respectively, the predicted results are almost identical with the ground truth. As the prediction duration is increased to 7.68s or 15.36s, the Deep-WP model's error increases, but it still maintains a high level of accuracy during the first few seconds. The introduction of covariates will improve the Deep-WP model's performance, with the absolute position and timestamp being particularly advantageous for wave prediction. Furthermore, the Deep-WP model is applicable to predict waves with different energy components. The proposed Deep-WP model shows a feasible ability to predict nonlinear stochastic waves in real-time.

Highlights

● A deep learning wave prediction (Deep-WP) model is proposed for stochastic waves.

● The model is based on an effective 'probabilistic' strategy.

● Three covariates are introduced, successfully improving the prediction accuracy.

● The model's performance is validated by experimental measurements.

This paper extracts some analytical solutions of simplified modified Camassa-Holm (SMCH) equations with various derivative operators, namely conformable and M-truncated derivatives that have been recently introduced. The SMCH equation is used to model the unidirectional propagation of shallow-water waves. The extended rational sine−cosine and sinh−cosh techniques have been successfully implemented to the considered equations and some kinds of the solitons such as kink and singular have been derived. We have checked that all obtained solutions satisfy the main equations by using a computer algebraic system. Furthermore, some 2D and 3D graphical illustrations of the obtained solutions have been presented. The effect of the parameters in the solutions on the wave propagation has been examined and all figures have been interpreted. The derived solutions may contribute to comprehending wave propagation in shallow water. So, the solutions might help further studies in the development of autonomous ships/underwater vehicles and coastal zone management, which are critical topics in the ocean and coastal engineering.

Highlights

● Simplified modified Camassa-Holm equation with conformable and M- fractional derivative order is investigated.

● The novel solutions of considered equations are obtained analytically.

● The solutions of the conformable and $M-$ truncated model are graphically compared in the figures for different values of $\alpha$ and $\beta$ that are in order of the derivative operator.

● The considered method suggests trigonometric functions producing dark, singular, and trigonometric solitons etc.

In this paper, the q-deformed Sinh-Gordon equation is solved analytically using a new general form based on the extended tanh approach. The numerical solutions of the equation is obtained using a b-spline finite element method. Also, we present numerous figures to demonstrate the various solitons propagation patterns. This type of equation has not been previously dealt with in such ways, whether analytical or numerical. This study is very useful in studying several physical systems that have lost their symmetry.

Highlights

● The q-deformed Sinh-Gordon equation was investigated analytically using the new general form of the extended tanh approach and numerically using a b-spline finite element method in this paper.

● We also present numerous figures to demonstrate the various solitons propagation patterns.

● This type of equation has not been previously dealt with in such ways, whether analytical or numerical.

● We believe that this study will be very useful in studying physical systems that have lost their symmetry.

The sea surface reconstructed from radar images provides valuable information for marine operations and maritime transport. The standard reconstruction method relies on the three-dimensional fast Fourier transform (3D-FFT), which introduces empirical parameters and modulation transfer function (MTF) to correct the modulation effects that may cause errors. In light of the convolutional neural networks’ (CNN) success in computer vision tasks, this paper proposes a novel sea surface reconstruction method from marine radar images based on an end-to-end CNN model with the U-Net architecture. Synthetic radar images and sea surface elevation maps were used for training and testing. Compared to the standard reconstruction method, the CNN-based model achieved higher accuracy on the same data set, with an improved correlation coefficient between reconstructed and actual wave fields of up to 0.96-0.97, and a decreased non-dimensional root mean square error (NDRMSE) of around 0.06. The influence of training data on the deep learning model was also studied. Additionally, the impact of the significant wave height and peak period on the CNN model’s accuracy was investigated. It has been demonstrated that the accuracy will fluctuate as the wave steepness increases, but the correlation coefficient remains above 0.90, and the NDRMSE remains less than 0.11.

In this research work, we present proof of the existence and uniqueness of solution for a novel method called tempered fractional natural transforms (TFNT) and give error estimates. This efficient method is applied to models, such as the time-space tempered fractional convection-diffusion equation (FCDE) and tempered fractional Black-Scholes equation (FBSE). We obtain exact solutions for these models using our methodology, which is very important for knowing the wave behavior in ocean engineering models and for the studies related to marine science and engineering. Finding exact solutions to tempered fractional differential equations (TFDEs) is far from trivial. Therefore, the proposed method is an excellent addition to the myriad of techniques for solving TFDE problems.

In this paper, the higher dimensional generalized Korteweg-de-Varies-Zakharov-Kuznetsov (gKdV-ZK) equation is under investigation. This model is used in the field of plasma physics which describes the effects of magnetic field on the weak ion-acoustic wave. We have applied two techniques, called as ϕ⁶-model expansion method and the Hirota bilinear method (HBM) to explore the diversity of wave structures. The solutions are expressed in the form of hyperbolic, periodic and Jacobi elliptic function (JEF) solutions. Moreover, the solitary wave solutions are also extracted. A comparison of our results to well-known results is made, and the study concludes that the solutions achieved here are novel. Additionally, 3-dimensional and contour profiles of achieved outcomes are drawn in order to study their dynamics as a function of parameter selection. On the basis of the obtained results, we can assert that the proposed computational methods are straightforward, dynamic, and well-organized, and will be useful for solving more complicated nonlinear problems in a variety of fields, particularly in nonlinear sciences, in conjunction with symbolic computations. Additionally, our discoveries provide an important milestone in comprehending the structure and physical behavior of complex structures. We hope that our findings will contribute significantly to our understanding of ocean waves. This study, we hope, is appropriate and will be of significance to a broad range of experts involved in ocean engineering models.

In the frame of fully nonlinear potential flow theory, series solutions of interfacial periodic gravity waves in a two-layer fluid with free surface are obtained by the homotopy analysis method (HAM), and the related wave forces on a vertical cylinder are analyzed. The solution procedure of the HAM for the interfacial wave model with rigid upper surface is further developed to consider the free surface boundary. And forces of nonlinear interfacial periodic waves are estimated by both the classical and modified Morison equations. It is found that the estimated wave forces by the classical Morison equation are more conservative than those by the modified Morison’s formula, and the relative error between the total inertial forces calculated by these two kinds of Morison’s formulae remains over 25% for most cases unless the upper and lower layer depths are both large enough. It demonstrates that the convective acceleration neglected in the classical Morison equation is rather important for inertial force exerted by not only internal solitary waves but also interfacial periodic waves. All of these should further deepen our understanding of internal periodic wave forces on a vertical marine riser.

The nonlinear Kortewege-de Varies (KdV) equation is a functional description for modelling ion-acoustic waves in plasma, long internal waves in a density-stratified ocean, shallow-water waves and acoustic waves on a crystal lattice. This paper focuses on developing and analysing a resilient double parametric analytical approach for the nonlinear fuzzy fractional KdV equation (FFKdVE) under gH-differentiability of Caputo fractional order, namely the q-Homotopy analysis method with the Shehu transform (q-HASTM). A triangular fuzzy number describes the Caputo fractional derivative of order α, 0<α≤1, for modelling problem. The fuzzy velocity profiles with crisp and fuzzy conditions at different spatial positions are investigated using a robust double parametric form-based q-HASTM with its convergence analysis. The obtained results are compared with existing works in the literature to confirm the efficacy and effectiveness of the method.

The aim of this paper is to study the unsteady korteweg-de vries equation that plays an important role in describing the shallow water. Two analytical techniques namely the Sardar-subequation method and the energy balance method are employed to seek the abundant traveling wave solutions for the first time. By these two methods, plenty of traveling wave solutions such as the bright solitary wave solutions, dark solitary wave solutions, singular periodic wave solutions and perfect periodic wave solution that expressed in terms of the generalized hyperbolic functions, generalized trigonometric functions and the cosine function are obtained. Finally, the dynamic behaviors of the solutions are described through the 3D plot and 2D curve. The results in this paper demonstrate that the proposed methods are powerful and effective to construct the traveling wave solutions of the nonlinear evolution equations in ocean engineering and science.

The main work of this paper is focused on construction the single traveling wave solution of the coupled Fokas-Lenells system, which is usually used to simulate the propagation of ultrashort optical pulses in birefringent fibers or crossing sea waves on the high seas. Firstly, the coupled Fokas-Lenells system is simplified into a nonlinear ordinary differential equation by traveling wave transformation and linear transformation. Then, using the well-known complete discriminant system of third-order polynomials, some single traveling wave solutions of the coupled Fokas-Lenells system are obtained including implicit solutions, rational function solutions and Jacobian function solutions.

In this work we considered bi-domain partial differential equations (PDEs) with two coupling interface conditions. The one domain is corresponding to the ocean and the second is to the atmosphere. The two coupling conditions are used to linked the interaction between these two regions. As we know that almost every engineering problem modeled via PDEs. The analytical solutions of these kind of problems are not easy, so we use numerical approximations. In this study we discuss the two essential properties, namely mass conservation and stability analysis of two types of coupling interface conditions for the ocean-atmosphere model. The coupling conditions arise in general circulation models used in climate simulations. The two coupling conditions are the Dirichlet-Neumann and bulk interface conditions. For the stability analysis, we use the Godunov-Ryabenkii theory of normal-mode analysis. The main emphasis of this work is to study the numerical properties of coupling conditions and an important point is to maintain conservativity of the overall scheme. Furthermore, for the numerical approximation we use two methods, an explicit and implicit couplings. The implicit coupling have further two algorithms, monolithic algorithm and partitioned iterative algorithm. The partitioned iterative approach is complex as compared to the monolithic approach. In addition, the comparison of the numerical results are exhibited through graphical illustration and simulation results are drafted in tabular form to validate our theoretical investigation. The novel characteristics of the findings from this paper can be of great importance in science and ocean engineering.

The current work aims to present abundant families of the exact solutions of Mikhailov-Novikov-Wang equation via three different techniques. The adopted methods are generalized Kudryashov method (GKM), exponential rational function method (ERFM), and modified extended tanh-function method (METFM). Some plots of some presented new solutions are represented to exhibit wave characteristics. All results in this work are essential to understand the physical meaning and behavior of the investigated equation that sheds light on the importance of investigating various nonlinear wave phenomena in ocean engineering and physics. This equation provides new insights to understand the relationship between the integrability and water waves’ phenomena.

Nonlinear evolution equations (NLEEs) are frequently employed to determine the fundamental principles of natural phenomena. Nonlinear equations are studied extensively in nonlinear sciences, ocean physics, fluid dynamics, plasma physics, scientific applications, and marine engineering. The generalized exponential rational function (GERF) technique is used in this article to seek several closed-form wave solutions and the evolving dynamics of different wave profiles to the generalized nonlinear wave equation in (3+1) dimensions, which explains several more nonlinear phenomena in liquids, including gas bubbles. A large number of closed-form wave solutions are generated, including trigonometric function solutions, hyperbolic trigonometric function solutions, and exponential rational functional solutions. In the dynamics of distinct solitary waves, a variety of soliton solutions are obtained, including single soliton, multi-wave structure soliton, kink-type soliton, combo singular soliton, and singularity-form wave profiles. These determined solutions have never previously been published. The dynamical wave structures of some analytical solutions are graphically demonstrated using three-dimensional graphics by providing suitable values to free parameters. This technique can also be used to obtain the soliton solutions of other well-known equations in engineering physics, fluid dynamics, and other fields of nonlinear sciences.

The application of the q-homotopy analysis Shehu transform method (q-HAShTM) to discover the estimated solution of fractional Zakharov-Kuznetsov equations is investigated in this study. In the presence of a uniform magnetic field, the Zakharov-Kuznetsov equations regulate the behaviour of nonlinear acoustic waves in a plasma containing cold ions and hot isothermal electrons. The q-HAShTM is a stable analytical method that combines homotopy analysis and the Shehu transform. This q-homotopy investigation Shehu transform is a constructive method that leads to the Zakharov-Kuznetsov equations, which regulate the propagation of nonlinear ion-acoustic waves in a plasma. It is a more semi-analytical method for adjusting and controlling the convergence region of the series solution and overcoming some of the homotopy analysis method’s limitations.

This paper presents a study of nonlinear waves in shallow water. The Korteweg-de Vries (KdV) equation has a canonical version based on oceanography theory, the shallow water waves in the oceans, and the internal ion-acoustic waves in plasma. Indeed, the main goal of this investigation is to employ a semi-analytical method based on the homotopy perturbation transform method (HPTM) to obtain the numerical findings of nonlinear dispersive and fifth order KdV models for investigating the behaviour of magneto-acoustic waves in plasma via fuzziness. This approach is connected with the fuzzy generalized integral transform and HPTM. Besides that, two novel results for fuzzy generalized integral transformation concerning fuzzy partial g\mathcal{H}-derivatives are presented. Several illustrative examples are illustrated to show the effectiveness and supremacy of the proposed method. Furthermore, 2D and 3D simulations depict the comparison analysis between two fractional derivative operators (Caputo and Atangana-Baleanu fractional derivative operators in the Caputo sense) under generalized g\mathcal{H}-differentiability. The projected method (GHPTM) demonstrates a diverse spectrum of applications for dealing with nonlinear wave equations in scientific domains. The current work, as a novel use of GHPTM, demonstrates some key differences from existing similar methods.

In the present article, we have used the three-phase-lag model of thermoelasticity to formulate a two dimensional problem of non homogeneous, isotropic, double porous media with a gravitational field impact. Thermal shock of constant intensity is applied on the bounding surface. The normal mode procedure is employed to derive the exact expressions of the field quantities. These expressions are also calculated numerically and plotted graphically to demonstrate and compare theoretical results. The influences of non-homogeneity parameter, double porosity and gravity on the various physical quantities are also analyzed. A comparative study is done between three-phase-lag and GN-III models. Some limiting cases are also deduced from the current study.

The paper investigates the multiple rogue wave solutions associated with the generalized Hirota-Satsuma-Ito (HSI) equation and the newly proposed extended (3 + 1)-dimensional Jimbo-Miwa (JM) equation with the help of a symbolic computation technique. By incorporating a direct variable transformation and utilizing Hirota’s bilinear form, multiple rogue wave structures of different orders are obtained for both generalized HSI and JM equation. The obtained bilinear forms of the proposed equations successfully investigate the 1st, 2nd and 3rd-order rogue waves. The constructed solutions are verified by inserting them into original equations. The computations are assisted with 3D graphs to analyze the propagation dynamics of these rogue waves. Physical properties of these waves are governed by different parameters that are discussed in details.

The main idea of this article is the investigation of atmospheric internal waves, often known as gravity waves. This arises within the ocean rather than at the interface. A shallow fluid assumption is illustrated by a series of nonlinear partial differential equations in the framework. Because the waves are scattered over a wide geographical region, this system can precisely replicate atmospheric internal waves. In this research, the numerical solutions to the fuzzy fourth-order time-fractional Boussinesq equation (BSe) are determined for the case of the aquifer propagation of long waves having small amplitude on the surface of water from a channel. The novel scheme, namely the generalized integral transform (proposed by H. Jafari [35]) coupled with the Adomian decomposition method (GIADM), is used to extract the fuzzy fractional BSe in $\mathbb{R}, \mathbb{R}^{n}$ and (2nth)-order including $g \mathscr{H}$-differentiability. To have a clear understanding of the physical phenomena of the projected solutions, several algebraic aspects of the generalized integral transform in the fuzzy Caputo and Atangana-Baleanu fractional derivative operators are discussed. The confrontation between the findings by Caputo and ABC fractional derivatives under generalized Hukuhara differentiability are presented with appropriate values for the fractional order and uncertainty parameters $\wp \in[0,1] $ were depicted in diagrams. According to proposed findings, hydraulic engineers, being analysts in drainage or in water management, might access adequate storage volume quantity with an uncertainty level.

Pressure hulls play an important role in deep-sea underwater vehicles. However, in the ultra-high pressure environment, a highly destructive phenomenon could occur to them which is called implosion. To study the characteristics of the flow field of the underwater implosion of hollow ceramic pressure hulls, the compressible multiphase flow theory, direct numerical simulation, and adaptive mesh refinement are used to numerically simulate the underwater implosion of a single ceramic pressure hull and multiple linearly arranged ceramic pressure hulls. Firstly, the feasibility of the numerical simulation method is verified. Then, the results of the flow field of the underwater implosion of hollow ceramic pressure hulls in 11000 m depth is analyzed. There are the compression-rebound processes of the internal air cavity in the implosion. In the rebound stage, a shock wave that is several times the ambient pressure is generated outside the pressure hull, and the propagation speed is close to the speed of sound. The pressure peak of the shock wave has a negative exponential power function relationship with the distance to the center of the sphere. Finally, it is found that the obvious superimposed effect between spheres exists in the chain-reaction implosion which enhances the implosion shock wave.

With the acceleration of the investigation and development of marine resources, the detection and location of submarine pipelines have become a necessary part of modern marine engineering. Submarine pipelines are a typical weak magnetic anomaly target, and their magnetic anomaly detection can only be realized within a certain distance. At present, a towfish or an autonomous underwater vehicle (AUV) is mainly used as the platform to equip magnetometers close to the submarine pipelines for magnetic anomaly detection. However, the mother ship directly affects the towfish, thus causing control interference. The AUV cannot detect in real time, which affects the magnetic anomaly detection and creates problems regarding detection efficiency. Meanwhile, a two-part towed platform has convenient control, thus reducing the interference of the towed mother ship and real-time detection. If the platform can maintain constant altitude sailing through the controller, the data accuracy in the actual magnetic anomaly detection can be guaranteed. On the basis of a two-part towed platform, a magnetic detection system with constant altitude sailing ability for submarine pipelines was constructed in this study. In addition, experimental verification was conducted. The experimental verification research shows that the constant altitude sailing experiment of the two-part towed platform verifies that the platform has good constant altitude sailing ability in both a hydrostatic environment and the actual marine environment. Meanwhile, the offshore magnetic anomaly detection experiment of submarine pipelines verifies the stable measurement function of the magnetic field and the function of the system to detect magnetic anomaly of submarine pipelines.

This paper presents analytical studies carried out explicitly on a higher-dimensional generalized Zakharov-Kuznetsov equation with dual power-law nonlinearity arising in engineering and nonlinear science. We obtain analytic solutions for the underlying equation via Lie group approach as well as direct integration method. Moreover, we engage the extended Jacobi elliptic cosine and sine amplitude functions expansion technique to seek more exact travelling wave solutions of the equation for some particular cases. Consequently, we secure, singular and nonsingular (periodic) soliton solutions, cnoidal, snoidal as well as dnoidal wave solutions. Besides, we depict the dynamics of the solutions using suitable graphs. The application of obtained results in various fields of sciences and engineering are presented. In conclusion, we construct conserved currents of the aforementioned equation via Noether’s theorem (with Helmholtz criteria) and standard multiplier technique through the homotopy formula.

The (2+1)-dimensional Chaffee-Infante has a wide range of applications in science and engineering, including nonlinear fiber optics, electromagnetic field waves, signal processing through optical fibers, plasma physics, coastal engineering, fluid dynamics and is particularly useful for modeling ion-acoustic waves in plasma and sound waves. In this paper, this equation is investigated and analyzed using two effective schemes. The well-known tanh-coth and sine-cosine function schemes are employed to establish analytical solutions for the equation under consideration. The breather wave solutions are derived using the Cole-Hopf transformation. In addition, by means of new conservation theorem, we construct conservation laws (CLs) for the governing equation by means of Lie-Bäcklund symmetries. The novel characteristics for the (2+1)-dimensional Chaffee-Infante equation obtained in this work can be of great importance in nonlinear sciences and ocean engineering.

In the case of negligible viscosity and surface tension, the B-KP equation shows the evolution of quasi-one-dimensional shallow-water waves, and it is growingly used in ocean physics, marine engineering, plasma physics, optical fibers, surface and internal oceanic waves, Bose-Einstein condensation, ferromagnetics, and string theory. Due to their importance and applications, many features and characteristics have been investigated. In this work, we attempt to perform Lie symmetry reductions and closed-form solutions to the weakly coupled B-Type Kadomtsev-Petviashvili equation using the Lie classical method. First, an optimal system based on one-dimensional subalgebras is constructed, and then all possible geometric vector yields are achieved. We can reduce system order by employing the one-dimensional optimal system. Furthermore, similarity reductions and exact solutions of the reduced equations, which include arbitrary independent functional parameters, have been derived. These newly established solutions can enhance our understanding of different nonlinear wave phenomena and dynamics. Several three-dimensional and two-dimensional graphical representations are used to determine the visual impact of the produced solutions with determined parameters to demonstrate their dynamical wave profiles for various examples of Lie symmetries. Various new solitary waves, kink waves, multiple solitons, stripe soliton, and singular waveforms, as well as their propagation, have been demonstrated for the weakly coupled B-Type Kadomtsev-Petviashvili equation. Lie classical method is thus a powerful, robust, and fundamental scientific tool for dealing with NPDEs. Computational simulations are also used to prove the effectiveness of the proposed approach.

This study presents a large family of the traveling wave solutions to the two fourth-order nonlinear partial differential equations utilizing the Riccati-Bernoulli sub-ODE method. In this method, utilizing a traveling wave transformation with the aid of the Riccati-Bernoulli equation, the fourth-order equation can be transformed into a set of algebraic equations. Solving the set of algebraic equations, we acquire the novel exact solutions of the integrable fourth-order equations presented in this research paper. The physical interpretation of the nonlinear models are also detailed through the exact solutions, which demonstrate the effectiveness of the presented method.The Bäcklund transformation can produce an infinite sequence of solutions of the given two fourth-order nonlinear partial differential equations. Finally, 3D graphs of some derived solutions in this paper are depicted through suitable parameter values.

This work is devoted to get a new family of analytical solutions of the (2+1)-coupled dispersive long wave equations propagating in an infinitely long channel with constant depth, and can be observed in an open sea or in wide channels. The solutions are obtained by using the invariance property of the similarity transformations method via one-parameter Lie group theory. The repeated use of the similarity transformations method can transform the system of PDEs into system of ODEs. Under adequate restrictions, the reduced system of ODEs is solved. Numerical simulation is performed to describe the solutions in a physically meaningful way. The profiles of the solutions are simulated by taking an appropriate choice of functions and constants involved therein. In each animation, a frame for dominated behavior is captured. They exhibit elastic multisolitons, single soliton, doubly solitons, stationary, kink and parabolic nature. The results are significant since these have confirmed some of the established results of S. Kumar et al. (2020) and K. Sharma et al. (2020). Some of their solutions can be deduced from the results derived in this work. Other results in the existing literature are different from those in this work.