Boussinesq Equations Research Articles

Comprehensive two-dimensional analytical modeling of groundwater levels in bi-directional sloping heterogeneous aquifers under variable recharge conditions

This paper develops a comprehensive two-dimensional (2D) groundwater model that surpasses traditional one-dimensional approaches by incorporating extensive hydrological and geological data typically acquired through well drilling. The primary objective of this research is to explore and characterize the complex dynamics of groundwater flow within sloping heterogeneous aquifers subject to rainfall-induced recharge events. To achieve this, the study utilizes the 2D Boussinesq equation, which is enhanced with meticulously defined boundary conditions to more accurately reflect real-world scenarios. The Integral Transform Technique (ITT) is employed to derive an analytical solution that integrates the effects of variable rainfall recharge, aquifer inclination angles, and inherent heterogeneity. This analytical model effectively captures varying recharge dynamics, both spatially and temporally, contributing to a better understanding and management of groundwater systems. The paper presents a new analytical framework, offering deeper insights into how natural factors impact porous medium behavior, thereby providing strategic references for sustainable water resource management.

Modeling of Filtration and Heat and Mass Transfer Processes in Groundwater

Purpose. The method of mathematical modeling is an important tool for solving complex problems involving the analysis of groundwater dynamics and heat and mass transfer processes in them when studying their contamination from various anthropogenic sources in the event of accidental spills of chemically hazardous substances, etc. The main purpose of the article is to develop a set of mathematical models for calculating the process of filtration of non-pressure groundwater, mass transfer of impurities and the process of heat transfer in groundwater. Methodology. The two-dimensional Boussinesq equation of filtration was used to predict the dynamics of groundwater. The two-dimensional equation of convective-diffusive transport of impurities was used to model the processes of mass transfer in groundwater. The process of freezing of individual sections of the groundwater flow is modeled using the Laplace equation for the velocity potential (calculation of the flow velocity field in a time-varying geometry) and the two-dimensional equation of heat transfer in groundwater. Finite difference schemes were used to solve the modeling equations of groundwater dynamics and heat and mass transfer. Findings. A set of mathematical models has been developed to calculate the process of filtration of non-pressure groundwater and its chemical contamination. The experiment has confirmed the adequacy of the constructed numerical model of filtration of a non-pressure groundwater flow. An effective mathematical model was developed that allows determining the temperature fields in groundwater during the operation of a well used to freeze certain sections of the flow. The results of computer modeling indicate the effectiveness of the developed mathematical models. Originality. Effective mathematical models for predicting the level of chemical contamination of groundwater, its dynamics and thermal regime are proposed. The constructed mathematical models make it possible to determine the dynamics of changes in the temperature regime of groundwater during the operation of wells through which refrigerant is supplied to freeze individual areas. A computer program has been developed that allows for a comprehensive assessment of groundwater conditions. Practical value. A set of computer programs has been developed to conduct a computational experiment to study the processes of filtration, chemical contamination of groundwater and heat transfer processes in them. This set of programs can be used for the scientific substantiation of engineering solutions aimed at protecting groundwater.

Proper orthogonal decomposition reduced-order model of the global oceans

AbstractA reduced-order model (ROM) of the global oceans is developed by projecting the hydrostatic Boussinesq equations of motion onto a proper orthogonal decomposition (POD) basis. Three-dimensional POD modes are calculated from the ocean fields of an ensemble climate reanalysis dataset. The coefficients in the POD ROM are calculated using a regression approach. The performance of various POD ROM configurations are assessed. Each configuration is derived from an alternate sea-water equation of state, linking the density and temperature fields. POD ROM variants incorporating an equation of state in which density is a quadratic function of temperature, are able to reproduce the statistics of the large-scale structures at a fraction of the computational cost required to numerically simulate this flow. Due to the speed and efficiency of calculation, such reduced-order models of the global geophysical system will enable researchers and policy makers to assess the physical risk for a broader range of potential future climate scenarios.

Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives

Abstract This research explores new soliton solutions to the Atangana–Baleanu derivative (ABD) fractional system of equations for ion sound and Langmuir waves (FISLW). We utilize the fractional ABD operator to convert our system into an ordinary differential equations. In recent years, machine learning (ML) evolves significantly in the context of data analysis and computing different solutions, which typically enables systems to operate wisely. Now, we are going to use numerous ML tools including matplotlib. pyplot as plt, scipy.integrate, mpl toolkits.mplot3d, and Axes3D to generate various types of optical solutions by using complete discriminant of the polynomial method. We will also analyze solutions for the hyperbolic function, trigonometric function, Jacobian elliptic function (JEF), and other solitary wave solutions. Solitons have extensive uses in pure and applied mathematics, including nonlinear partial differential equations: the Boussinesq equation, the nonlinear Schrödinger equation, and the sine-Gordon equation, Lie groups, Lie algebras, and differential and algebraic geometry. In addition, we study the chaotic behaviour, i.e., 2D, 3D, time series, Poincarè maps, and sensitivity analysis of our governing model. Sensitivity analysis explores how changes in a system’s variables affects its behaviour.

On trilinear and quadrilinear equations associated with the lattice Gel’fand–Dikii hierarchy

Introduced in Zhang et al. (2012), the trilinear Boussinesq equation is the natural form of the equation for the τ-function of the lattice Boussinesq system. In this paper we study various aspects of this equation: its highly nontrivial derivation from the bilinear lattice AKP equation under dimensional reduction, a quadrilinear dual lattice equation, conservation laws, and periodic reductions leading to higher-dimensional integrable maps and their Laurent property. Furthermore, we consider a higher Gel’fand–Dikii lattice system, its periodic reductions and Laurent property. As a special application, from both a trilinear Boussinesq recurrence as well as a higher Gel’fand–Dikii system of three bilinear recurrences, we establish Somos-like integer sequences.

Strong Ill-Posedness in \(\boldsymbol{L}^{\boldsymbol{\infty }}\) of the 2D Boussinesq Equations in Vorticity Form and Application to the 3D Axisymmetric Euler Equations

Strong Ill-Posedness in \(\boldsymbol{L}^{\boldsymbol{\infty }}\) of the 2D Boussinesq Equations in Vorticity Form and Application to the 3D Axisymmetric Euler Equations

Flow characteristics of the rip current system adjacent to a coastal vertical structure for irregular waves

Rip currents are common hazards found on many beaches around the world. The present research examines the rip current flow features adjacent to a coastal vertical structure with total reflection function to the incident waves. The laboratory experiments were conducted over the planar beach for both regular and irregular waves to explore the specific phenomenon for the irregular waves, and a higher-order Boussinesq equation model was adopted in the simulations to analyze the formation mechanism of the flow features. The study results show that, the standing wave pattern for the irregular wave is much weaker than that for the regular wave with inducing only one apparent node rip at the first node line in the reflection wave area. Moreover, the specific “single-vortex” flow pattern was formed in the crossing wave field for the irregular wave tests. This specific flow pattern is found essentially induced by the larger wave reflection rate of the vertical structure, which leads to the strong lateral flow against the downstream longshore current. This specific flow pattern prefers to present for the smaller wave incident angles and larger wave heights and periods, and leads to the feeding flow volume rates discharged more in the form of lateral flow. The C-shaped lateral flow is found mainly induced by the longshore gradient of the wave set-up instead of the non-uniform wave radiation stress nearshore.

Direct and Inverse Problems in the Generalized Logarithmic Boussinesq Equation

Direct and Inverse Problems in the Generalized Logarithmic Boussinesq Equation

Inverse Problem for the $$L$$-Operator in the Lax Pair of the Boussinesq Equation on the Circle

Inverse Problem for the $$L$$-Operator in the Lax Pair of the Boussinesq Equation on the Circle

Improved growth estimate for one-dimensional sixth-order Boussinesq equation with logarithmic nonlinearity

This paper provides an improved exponential growth estimate, surpassing the growth rate given in the previous work. This finding elucidates the impact of the power index k in the logarithmic nonlinearity uln|u|k on the growth behavior of the solution to the initial boundary value problem for the one-dimensional sixth-order nonlinear Boussinesq equation with logarithmic nonlinearity.

An Efficient Numerical Approach to Solve Fractional Coupled Boussinesq Equations

An Efficient Numerical Approach to Solve Fractional Coupled Boussinesq Equations

Analysis of the(3+1)-Dimensional Fractional Kadomtsev–Petviashvili–Boussinesq Equation: Solitary, Bright, Singular, and Dark Solitons

Analysis of the(3+1)-Dimensional Fractional Kadomtsev–Petviashvili–Boussinesq Equation: Solitary, Bright, Singular, and Dark Solitons

Boussinesq model for two-fluid system with surface- and interfacial tension

A Boussinesq model associated with surface and internal waves with surface tension and interfacial tension in a two-layer fluid is presented in the application range of weakly dispersive nonlinear. The nonlinear form of model definitions in terms of surface and interfacial tensions along with velocity potentials are provided. Based on the perturbation technique, the Boussinesq equations for both layers are derived. The present Boussinesq result is validated against existing published results and it has a good level of agreement. Further, the present model results of crests of surface and interface displacements are compared with existing published numerical OpenFOAM simulations. The dispersion relations for fast- and slow modes are derived in the quadratic form of wave frequencies in the presence of surface and interfacial tensions. The second-order surface and internal wave displacements along with velocity potentials for upper- and lower-layer fluids are presented. The effects of surface and interfacial tensions on dispersion characteristics, harmonic transfer functions, and shoaling gradients on different physical parameters associated with the present model are studied by analyzing several numerical results. In the end, a real physical problem is demonstrated in the application of surface tension with solvents in the gas-liquid interface.

On the Cauchy problem for the generalized Boussinesq equation with a damped term

On the Cauchy problem for the generalized Boussinesq equation with a damped term

Combined Compact Symplectic Schemes for the Solution of Good Boussinesq Equations

Good Boussinesq equations are considered in this work. First, we apply three combined compact schemes to approximate spatial derivatives of good Boussinesq equations. Then, three fully discrete schemes are developed based on a symplectic scheme in the time direction, which preserves the symplectic structure. Meanwhile, the convergence and conservation of the fully discrete schemes are analyzed. Finally, we present numerical experiments to confirm our theoretical analysis. Both our analysis and numerical tests indicate that the fully discrete schemes are efficient in solving the spatial derivative mixed equation.

Painlevé integrability and multiple soliton solutions for the extensions of the (modified) Korteweg-de Vries-type equations with second-order time-derivative

This work introduces two (3+1)-dimensional expansions of the Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. These extensions incorporate a second-order time-derivative term, similar to the Boussinesq equation. The Painlevé test is utilized to verify the integrability of each extended model. The bilinear form is employed to investigate the existence of multiple-soliton (MS) solutions for each system under consideration. Furthermore, we provide solutions in the form of lumps for the extended KdV equation. The multidimensional KdV-type equations surpass the standard KdV equation. However, they offer enhanced accuracy by representing a broader spectrum of nonlinear phenomena in plasma physics, fluid mechanics, tsunami phenomena, and other science disciplines. Furthermore, the aforementioned equations can be employed to analyze the characteristics of various acoustic waves (AW) in different plasma models, such as their amplitude, width, frequency, and dispersion, as well as the phase shifts after collisions.

Wronskian solutions and [formula omitted]–soliton solutions for the Hirota–Satsuma equation

By employing the Hirota method and Wronskian technique, we firstly give the bilinear form, N-soliton solutions and the Wronskian solutions of the Hirota–Satsuma equation. Explicit one- and two-soliton solutions are given for the Hirota–Satsuma equation. The solutions of good Boussinesq equation are obtained through the Miura transformation. The two solitons have the degenerated forms of antisoliton-antikink and W-shape type.

Pointwise convergence problem of the one‐dimensional Boussinesq‐type equation

In this paper, we consider the initial data problem of the one‐dimensional Boussinesq‐type equation. Inspired by the work of Compaan et al. (Int. Math. Res. Not. 2021, 599–650), by using the Fourier restriction norm method, we investigate the pointwise convergence of the one‐dimensional Boussinesq equation and cubic Boussinesq equation. We also only use the Strichartz estimates instead of Fourier restriction norm method to establish convergence conclusions of the one‐dimensional cubic Boussinesq equation. The key ingredients are some Strichartz estimates and high‐low frequency decomposition related to the nonlinear part of the integral form solution in the Duhamel form and the maximal function estimate related to inhomogeneous estimate.

Diverse variety of exact solutions for some nonlinear models via the [formula omitted]-expansion method

In this article, we explore several significant nonlinear physical models, including the Benjamin–Bona–Mahony–Peregrine–Burgers (BBMPB) equation, the Burgers–Korteweg–De Vries (BK) equation, the one-dimensional Oskolkov (OSK) equation, the Klein–Gordon (KG) equation with quadratic non-linearity, and the improved Boussinesq (IB) equation. Utilizing the (G′G)-expansion method ansatz, we derive new exact traveling wave solutions for these models. These solutions, expressed in the forms of rational, hyperbolic, and trigonometric functions, present a novel contribution distinct from existing literature. The physical dynamics of these solutions are elucidated through Mathematica simulations.

Dynamics of resonant soliton, novel hybrid interaction, complex N-soliton and the abundant wave solutions to the (2+1)-dimensional Boussinesq equation

The presented work concerns with some novel solutions of the (2+1)-dimensional Boussinesq equation (BE), which acts as an important model for shallow water wave. Some resonant soliton solutions such as the X-shape soliton (XSS) and Y-shape soliton (YSS) solutions are developed via imposing the resonant conditions on the N-soliton solutions (N-SSs) that developed by the Hirota bilinear approach(HBA). Based on the XSS and YSS solutions, the novel hybrid interactions including the interaction between the 1-soliton and the YSS, the interaction between two YSS solutions are extracted. In addition, the complex N-SSs are also explored and discussed. Finally, the travelling wave solutions including the bright solitary and kinky solitary wave solutions are studied by employing the Bernoulli sub-equation function method (BSEFM). The graphs of the attained solutions are drawn to show the physical properties. The findings of this study are expected to help us apprehend the dynamics of the (2+1)-dimensional BE better.