An Analytical Solution for Regular Progressive Water Waves

The power series solution is a cheap and effective method to solve nonlinear problems, like the Duffing-van der Pol oscillator, the Volterra population model and the nonlinear boundary value problems. A novel power series method by considering the multiple scales $R_k$ in the power term $(t/R_k)^k$ is developed, which are derived explicitly to reduce the ill-conditioned behavior in the data interpolation. In the method a huge value times a tiny value is avoided, such that we can decrease the numerical instability and which is the main reason to cause the failure of the conventional power series method. The multiple scales derived from an integral can be used in the power series expansion, which provide very accurate numerical solutions of the problems considered in this paper.

In this paper, a fully nonlinear numerical wave tank model has been used to simulate the propagation of fully nonlinear waves in different water depths. In the numerical wave tank model, the fully nonlinear dynamic and kinematic free-surface boundary conditions have been applied and the boundary integral equation (BIE) solution to the Laplacian problem has been obtained using the Mixed Eulerian-Lagrangian (MEL) approach. The model solution has been verified through the comparison with the available experimental data. A convergence and accuracy study has been carried out to examine the time stepping scheme and the required mesh density. The nonlinearity effects were evident in the solution by the asymmetrical wave profile around both vertical and horizontal axis along with sharp high crests and broad flat troughs. Fully nonlinear wave propagation in deepwater, in transition zone and in shallow water has been simulated. The nonlinear solution has been compared to the linear solution for various waves. Shoaling coefficient and wave-number have been derived based on the nonlinear solution and compared to the linear theory solution for various wave characteristics.

Limiting conditions for cnoidal and Stokes waves : Laitone E. V., 1962. J. Geophys. Res., 67 (4): 1555–1564

We employ a fully nonlinear numerical model to generate and propagate long-crested gravity waves in a tank containing an incompressible invicid homogeneous fluid, initially at rest, with a horizontal free surface of finite extent and of finite depth. A non-orthogonal curvilinear coordinate system is constructed which follows the free surface and is “fitted” to the bottom topography of the tank and therefore tracks the entire fluid domain at all times. A waveform relaxation algorithm provides an efficient iterative method to solve the resulting discrete Laplace equation, and the full nonlinear kinematic and dynamic free surface boundary conditions are employed to propagate the solution. In addition, a monochromatic deterministic theoretical wave-maker, employing a Dirichlet type boundary condition, and a suitably tuned numerical beach are utilized in the numerical model. We first show that the model generates the appropriate dispersion relation for both the deep water and shallow water conditions. Subsequently, we calculate the energy and energy flux for small steepness waves and show that the model produces the expected linear results. We complete the paper by considering steeper waves where the linear results are suspect.

Problems connected with a shock interaction of a thin elastic cylindrical bar and an elastic sphere with an Uflyand-Mindlin plate of a finite size are considered. In enumerated problems, an impact process is accompanied by material local deformations and propagation of wave surfaces of a strong or weak discontinuity in elastic bodies coming in contact. The local deformations are taken into account in terms of the Hertz's contact theory, but the dynamic deformations behind the fronts of incident and reflected waves are determined by means of truncated power series with variable coefficients (what is known as ray series). These two types of deformations are mated on the contact region boundary. As the truncated ray series, as a rule, are not uniformly applicable in the whole region of the wave solution existence, then for their improvement the method of “forward-area” regularization is used which is based on a combination of the recurrent equations of the ray method with the method of multiple scales. The method of power series (ray method) allows one to find the main characteristics of the impact theory in an analytical form.

In this paper, a lattice Boltzmann model is developed for solving the combined sine-cosine-Gordon equation through selecting equilibrium distribution function properly. With the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. Some problems, which have exact solutions, are validated by the present model. From the simulations, we find that the numerical results agree well with the exact solutions or better than the numerical solutions reported in previous studies. The study indicates that the present method is very effective and accurate. The present model can be used to solve more other nonlinear wave problems.

A comparison of the spectral and the discrete singular convolution schemes for the KdV-type equations

The modified Equal-Width (MEW) equation is a fundamental model in the realm of partial differential equations that is used to simulate one-dimensional wave propagation in nonlinear media, incorporating dispersion processes. In this study, we have employed two contemporary analytical and numerical methods to obtain exact traveling wave solutions for this model. The derived solutions have been evaluated using a numerical technique, demonstrating excellent accuracy. Various graphical representations, including a contour plot, a two-dimensional graph, and a three-dimensional graph, have been utilized to illustrate and analyze the results. Our study offers significant contributions to the field by providing new insights into the dynamics of solitary wave solutions of the MEW model. Furthermore, our method proves to be a valuable and effective mathematical tool that can be utilized to solve various nonlinear wave problems, making it highly applicable in a range of physical and mathematical studies. By showcasing the properties of the investigated model and the primary advantages of the employed techniques, our study highlights the potential for further developments and advancements in this field.

ABSTRACTResearch on the response of random wave on offshore structures has received great deal of attention of many researchers and engineers in the design of marine structures. Most previous investigations have been limited to the regular waves. In this paper, based on Longuet–Higgins random wave theory and finite element method, a numerical model for random wave-induced seabed response is established. The seabed is treated as poroelastic medium and characterized by Biot’s partly dynamic equations (u–p model). The JONSWAP spectrum is adopted in Longuet–Higgins model, which is based on the cumulative superposition of linear diffraction solution. Based on the numerical results, the effects of random wave on seabed response are investigated by comparing with the corresponding Stokes wave and cnoidal wave. Then, a parametric study is conducted to examine the effect of wave and soil characteristic on the seabed.

The nonlinear partial differential equations for dispersive waves have special solutions representing uniform wavetrains. An expansion procedure is developed for slowly varying wavetrains, in which full nonlinearity is retained but the scale of the nonuniformity introduces a small parameter. The first-order results agree with those obtained previously by averaging techniques (Whitham 1965 a, b). The perturbation method provides a detailed description and deeper understanding, as well as a consistent development to higher approximations. This method for treating partial differential equations is analogous to the ‘multiple time scale' methods for ordinary differential equations in nonlinear vibration theory. It may also be regarded as a generalization of the geometrical optics approximation to nonlinear problems.

Parametric study and formulation of outer-brace geometric stress concentration factors in internally ring-stiffened tubular KT-joints of offshore structures

Decoupled State-Estimation in Energy Control Centres

Mathematical modeling using noninteger‐order derivatives has gained rapid attention among scientists and researchers in recent years. This manuscript investigates the dynamics of rabies transmission in a predator–prey framework by formulating the problem as a system of nonlinear fractional initial value problems (FIVPs). Rabies modeling is important for understanding and controlling the spread of this fatal disease in humans and animals. To solve these complex equations, we employ the Laplace Residual Power Series (LRPS) method, which combines the Laplace transform with the Residual Power Series to efficiently produce rapidly convergent fractional power series solutions. Our results demonstrate that LRPS provides accurate and computationally efficient solutions, offering valuable insights into the effects of fractional dynamics on rabies spread. These findings highlight the potential of LRPS for studying complex epidemiological systems and improving disease management strategies.

This paper is devoted to problem of mass transport of fluid for the surface progressive waves. Both Stokes and cnoidal waves are considered. New solutions for the transitional current are obtained. It is discovered that the mass transport of fluid in the direction of wave propagation exists only in the top layer. In the underlying layers a compensatory reverse flow is formed. The existence of a compensatory flow was verified experimentally. It is revealed that theoretical results duly conform to experimental data.

The standard method of multiple shooting for a system of n first order differential equations, with k unknown initial conditions requires the integration of k sets of variational equations on the first shot, and n sets of variational equations on every shot thereafter. This paper describes a variant of multiple shooting that requires the solution of k sets of variational equations on every shot. The technique applies to both linear and nonlinear boundary value problems. Techniques to deal with difficulties unique to the solution of nonlinear problems are suggested.