The transient response of an ice shelf to an incident wave packet from the open ocean is studied with a model that allows for extensional waves in the ice shelf, in addition to the standard flexural waves. Results are given for strains imposed on the ice shelf by the incident packet, over a range of peak periods in the swell regime and a range of packet widths. In spite of large differences in speeds of the extensional and flexural waves, it is shown that there is generally an interval of time during which they interact, and the coherent phases of the interactions generate the greatest ice shelf strain magnitudes. The findings indicate that incorporating extensional waves into models is potentially important for predicting the response of Antarctic ice shelves to swell, in support of previous findings based on frequency-domain analysis.

This study investigates the hydroelastic interaction of flexural gravity waves with multiple porous elastic plates of varying lengths in finite-depth water, employing an integral equation approach. The floating ice sheet is modelled as a flexible plate of uniform thickness, governed by the Euler–Bernoulli beam equation. The primary objective is to evaluate the effectiveness of porous elastic plates as wave barriers for shoreline protection in ice-covered regions. Within the framework of linearized theory, the problem is formulated as a boundary value problem (BVP) and solved using an eigenfunction expansion method with nonorthogonal eigenfunctions. The mode-coupling relation is utilized to transform the BVP into a system of Fredholm-type integral equations, which is subsequently solved using the multi-term Galerkin approximation technique with Chebyshev polynomials. The numerical analysis evaluates the reflection and transmission coefficients, hydrodynamic forces, and wave energy dissipation, with a particular focus on the influence of the permeability and flexibility of the submerged plates, along with other relevant parameters. Validation is conducted by comparing the results with those of previous studies under specific conditions. This research underscores the practical benefits of incorporating porosity and flexibility into the model, demonstrating improved wave reflection and energy dissipation. Additionally, the findings reveal that the thickness of the ice sheet plays a crucial role in optimizing breakwater performance. The research delivers key insights into mitigating wave-induced forces and offers a reliable framework for designing effective and sustainable coastal protection systems that safeguard shorelines from high waves.

This paper introduces a parallelizable lossless image compression algorithm designed for three-channel standard images and two-channel pathology images. The proposed algorithm builds on the Quite OK Image Format (QOI) by addressing its limitations in parallelizability and compression efficiency, thereby enhancing both the compression ratio and processing speed. By incorporating image context and optimizing pixel traversal sequences, the algorithm enables effective parallel processing, achieving rapid compression of million-pixel pathology images within milliseconds, and is scalable to larger whole-slide images. It also delivers exceptional performance in terms of both speed and compression ratio for standard images. Additionally, the low complexity lossless compression for images (LOCO-I) context prediction algorithm used in joint photographic experts group lossless standard (JPEG-LS) is parallelized to improve compression efficiency and speed. By implementing full-process parallelization across the entire compression workflow rather than confining parallelization to individual steps, this approach significantly enhances overall time performance.

Intense vortices have been observed within large-scale bushfires, and have been likened to “fire tornadoes”. This paper presents a simple mathematical model of such an event, and is based on a Boussinesq approximation relating temperature and density in the air. A linearized model is derived under the assumption that the temperature varies only slightly from ambient, and a solution to that model is presented in closed form. The nonlinear equations are solved in axisymmetric geometry, using a semi-numerical approach based on Fourier–Bessel series. The nonlinear and linearized results are in good agreement for small temperature excursions above ambient, but when larger deviations occur, nonlinear effects cause a type of flow reversion within the fire vortex. The cause of this effect is discussed in the paper.

Two fourth-order difference approximations for fractional derivatives based on Lubich-type second-order approximation with different shifts are derived. These approximations are applied to the space fractional diffusion equation with the Crank–Nicolson scheme. Here, we analyse the stability and convergence of these schemes and prove that they are unconditionally stable and convergent for a fractional order $\alpha $ ranging from $1$ to $2$. Numerical examples are presented to show that both schemes converge, and we obtain the correct convergence rates and unconditional stability.

This paper focuses on the Aw–Rascle model of traffic flow for the Born–Infeld equation of state with Coulomb-like friction, whose Riemann problem is solved with the variable substitution method. Four kinds of nonself-similar solutions are derived. The delta shock occurs in the solutions, although the system is strictly hyperbolic with a genuinely nonlinear characteristic field and a linearly degenerate characteristic field. The generalized Rankine–Hugoniot relation and entropy condition for the delta shock are clarified. The delta shock can be used to describe the serious traffic jam. Under the impact of the friction term, the rarefaction wave (R), shock wave (S), contact discontinuity (J) and delta shock ($\delta $) are bent into parabolic curves. Furthermore, it is proved that the $S+J$ solution and $\delta $ solution of the nonhomogeneous Aw–Rascle model tend to be the $\delta $ solution of the zero-pressure Euler system with friction; the $R+J$ solution and $R+\mbox {Vac}+J$ solution tend to be the vacuum solution of the zero-pressure Euler system with friction.

The modulational instability of weakly nonlinear capillary-gravity waves (CGWs) on the surface of infinitely deep water with uniform vorticity background shear is examined. Assuming a narrow band of waves, the fourth-order nonlinear Schrödinger equation (NSE) is derived from Zakharov’s integral equation (ZIE). The analysis is restricted to one horizontal dimension, parallel to the direction along the wave propagation to take advantage of a formulation using potential flow theory. It is to be noted that the dominant new effect introduced to the fourth order is the wave-induced mean flow response. The key point of this paper is that the present fourth-order analysis shows considerable deviation in the stability properties of CGWs from the third-order analysis and gives better results consistent with the exact results. It is found that the growth rate of instability increases for negative vorticity and decreases for positive vorticity, and the effect of capillarity is to reduce the growth rate of instability. Additionally, the effect of vorticity on the Peregrine breather, which can be considered as a prototype for freak waves, is investigated.

We study the long time dynamic properties of the nonlocal Kuramoto–Sivashinsky (KS) equation with multiplicative white noise. First, we consider the dynamic properties of the stochastic nonlocal KS equation via a transformation into the associated conjugated random differential equation. Next, we prove the existence and uniqueness of solution for the conjugated random differential equation in the theory of random dynamical systems. We also establish the existence and uniqueness of a random attractor for the stochastic nonlocal equation.

This research introduces an adapted multidimensional fractional optimal control problem, developed from a newly established framework that combines first-order partial differential equations (PDEs) with inequality constraints. We methodically establish and demonstrate the optimality conditions relevant to this framework. Moreover, we illustrate that, under certain generalized convexity assumptions, there exists a correspondence between the optimal solution of the multidimensional fractional optimal control problem and a saddle point related to the Lagrange functional of the revised formulation. To emphasize the significance and practical implications of our findings, we present several illustrative examples.

Mathematical billiards is much like the real game: a point mass, representing the ball, rolls in a straight line on a (perfectly friction-less) table, striking the sides according to the law of reflection. A billiard trajectory is then completely characterized by the number of elastic collisions. The rules of mathematical billiards may be simple, but the possible behaviours of billiard trajectories are endless. In fact, several fundamental theory questions in mathematics can be recast as billiards problems. A billiard trajectory is called a periodic orbit if the number of distinct collisions in the trajectory is finite. We show that periodic orbits on such billiard tables cannot have an odd number of distinct collisions. We classify all possible equivalence classes of periodic orbits on square and rectangular tables. We also present a connection between the number of different equivalence classes and Euler’s totient function, which for any positive integer N, counts how many positive integers smaller than N share no common divisor with N other than $1$. We explore how to construct periodic orbits with a prescribed (even) number of distinct collisions and investigate properties of inadmissible (singular) trajectories, which are trajectories that eventually terminate at a vertex (a table corner).

We present a study of oblique-wave scattering by an H-shaped breakwater submerged in deep water. The H-shaped breakwater is designed using two thin vertical plates connected by a thin horizontal plate. The velocity potentials that describe the wave motion in different regions are expressed using the Havelock expansion. Two first-kind Fredholm-type integral equations are derived by applying the continuity of fluid velocity and pressure at the interface for the horizontal component of fluid velocity across the gap below and above the breakwater. The coefficients of wave reflection and transmission are derived in explicit forms that require the solution of integral equations. The solutions of integral equations are obtained by employing the Galerkin approximation that involves simple polynomials. In some limiting cases, wave scattering by submerged $\large \boldsymbol {\sqcap }$ and $\large \boldsymbol {\sqcup }$-shaped breakwaters is also studied. The correctness of the results is verified by comparing them with existing results reproduced in a limiting case and checking the wave energy balance equation. The results reveal that the H-, $\large \boldsymbol {\sqcap }$- and $\large \boldsymbol {\sqcup }$-shaped breakwaters show low wave transmission for most incident wavelengths. The horizontal plate between the vertical plates accounts for this low wave transmission. Wave forces and the overturning moment are also calculated. These breakwaters may be used in water regions that require low wave transmission.

A semi-analytical study of oblique wave interaction with two $\boldsymbol {\sqcap }$-shaped breakwater designs—floating and bottom-fixed structures—incorporating two thin porous plates is presented using linearized theory. Wave potential for both configurations is developed using the eigenfunction expansion method, considering both progressive and evanescent wave modes. The problem of oblique wave scattering by $\boldsymbol {\sqcap }$-shaped breakwaters is reduced to a set of coupled integral equations of first kind, based on horizontal velocity components. These equations are solved using the multi-term Galerkin approximation with appropriate basis functions to handle the square-root singularities at sharp edges of the porous barriers. The performance of the models is evaluated by examining reflection, transmission and energy dissipation coefficients, along with free surface elevation and horizontal drift force. We observe that increasing the plate length of the breakwaters attenuates the incident waves more effectively than increasing the width. Additionally, the floating $\boldsymbol {\sqcap }$-shaped breakwater significantly reduces the free surface elevation in the transmitted region. The results from the developed model can provide valuable insights for the design of wave–structure systems in shallow waters.

Electricity supply operators offer financial incentives to encourage large energy users to reduce their power demand during declared periods of increased demand from energy users such as residential homes. This demand flexibility enables electricity system operators to ensure adequate power supply and avoid the construction of peaking power plants.

Railway operators can sometimes reduce their power demand during specified peak demand periods without disrupting the train schedules. For trains with infrequent stops, such as intercity trains, it is possible to speed up trains prior to the peak demand period, slow down during the peak demand period, then speed up again after the peak demand period. We use simple train models to develop an optimal strategy that minimizes energy use for a fleet of trains subject to energy-use constraints during specified peak demand intervals. The strategy uses two sets of interacting parameters to find an optimal solution—a Lagrange multiplier for each energy-constrained time interval to control the speed of trains during each interval, and a Lagrange multiplier for each train to control the relative train speeds and ensure each train completes its journey on time.

We investigate a Leslie-type prey–predator system with an Allee effect to understand the dynamics of populations under stress. First, we determine stability conditions and conduct a Hopf bifurcation analysis using the Allee constant as a bifurcation parameter. At low densities, we observe that a weak Allee effect induces a supercritical Hopf bifurcation, while a strong effect leads to a subcritical one. Notably, a stability switch occurs, and the system exhibits multiple Hopf bifurcations as the Allee effect varies. Subsequently, we perform a sensitivity analysis to assess the robustness of the model to parameter variations. Additionally, together with the numerical examples, the FAST (Fourier amplitude sensitivity test) approach is employed to examine the sensitivity of the prey–predator system to all parameter values. This approach identifies the most influential factors among the input parameters on the output variable and evaluates the impact of single-parameter changes on the dynamics of the system. The combination of detailed bifurcation and sensitivity analysis bridges the gap between theoretical ecology and practical applications. Furthermore, the results underscore the importance of the Allee effect in maintaining the delicate balance between prey and predator populations and emphasize the necessity of considering complex ecological interactions to accurately model and understand these systems.

We study the behaviour of (resonant) dynamic B-tipping in a forced two-dimensional nonautonomous system, close to a nonsmooth saddle-focus (NSF) bifurcation. The NSF arises when a saddle-point and a focus meet at a border collision bifurcation. The emphasis is on the Stommel 2-box model, which is a piecewise-smooth continuous dynamical system, modelling thermohaline circulation. This model exhibits an NSF as parameters vary. By using techniques from the theory of nonsmooth dynamical systems, we are able to provide precise estimates for the general tipping behaviour close to the bifurcation as parameters vary. In particular, we consider the combination of both slow drift and also periodic changes in the parameters, corresponding, for example, to the effects of slow climate change and seasonal variations. The results are significantly different from the usual B-tipping point estimates close to a saddle-node bifurcation. In particular, we see a more rapid rate of tipping in the slow drift case, and an advancing of the tipping point under periodic changes. The latter is made much more pronounced when the periodic variation resonates with the natural frequency of the focus, leading both to much more complicated behaviour close to tipping and also significantly advanced tipping in this case.

A family of arbitrarily high-order energy-preserving methods are developed to solve the coupled Schrödinger–Boussinesq (S-B) system. The system is a nonlinear coupled system and satisfies a series of conservation laws. It is often difficult to construct a high-order decoupling numerical algorithm to solve the nonlinear system. In this paper, the original system is first reformulated into an equivalent Hamiltonian system by introducing multiple auxiliary variables. Next, the reformulated system is discretized by the Fourier pseudo-spectral method and the implicit midpoint scheme in the spatial and temporal directions, respectively, and a second-order conservative scheme is obtained. Finally, the scheme is extended to arbitrarily high-order accuracy by means of diagonally implicit symplectic Runge–Kutta methods or composition methods. Rigorous analyses show that the proposed methods are fully decoupled and can precisely conserve the discrete invariants. Numerical results show that the proposed schemes are effective and can be easily extended to other nonlinear partial differential equations.