Modified Boussinesq Equation Research Articles

INITIAL BOUNDARY VALUE PROBLEM FOR THE NONLINEAR MODIFIED BOUSSINESQ EQUATION

The problem in Sobolev spaces is investigated for a modified Boussinesq equation with a homogeneous Neumann boundary condition and with classical initial conditions. Based on the compactness method, it is shown that the approximate analytical solution, constructed in the form of Galerkin’s sum over the system of eigenfunctions of the Neumann boundary value problem, *-weakly converges to the exact solution.

Breather Transitions and Their Mechanisms of a (2+1)-Dimensional Sine-Gordon Equation and a Modified Boussinesq Equation in Nonlinear Dynamics

Breather Transitions and Their Mechanisms of a (2+1)-Dimensional Sine-Gordon Equation and a Modified Boussinesq Equation in Nonlinear Dynamics

ENERGETIC POWER ESTIMATION OF SWELLS AND ORBITAL MARINE CURRENTS IN BENIN COASTAL ZONE (GULF OF GUINEA)

The announced depletion of fossil energy resources is fueling some speculation. It is therefore urgent to explore the different renewable energy sources available to Benin. In the ocean, in addition to swells, there are also orbital marine currents generated by them. An advantageous solution for the countries located in the Gulf of Guinea is the recovery and transformation of the energy of the swell and its orbital marine currents into electrical energy. Swells are adult waves that propagate freely at the atmosphere-ocean interface. In this work, based on measurements made by the Millenium Challenge Account (MCA-Benin) as part of the extension of the Autonomous Port of Cotonou, we have shown that the swells at the Autonomous Port of Cotonou are Airy swells or of Stokes. We have identified the modified Boussinesq equations, proposed by Peregrine in 1967, as the theory for characterizing the evolution of these swells during their formation in the coastal zone. The variations of the different wave parameters (height, wavelength, period, phase and group velocities) are characterized from the shoaling zone to the breaking point. With this theory the results obtained are in agreement with the experimental measurements made, they perfectly describe the amplification of the height of these waves and show that these swells become very energetic when they approach the coast.

Long-time asymptotics of the good Boussinesq equation with q xx-term and its modified version

Two modified Boussinesq equations along with their Lax pairs are proposed by introducing the Miura transformations. The modified good Boussinesq equation with initial condition is investigated by the Riemann–Hilbert method. Starting with the three-order Lax pair of this equation, the inverse scattering transform is formulated and the Riemann–Hilbert problem is established, and the properties of the reflection coefficients are presented. Then, the formulas of long-time asymptotics to the good Boussinesq equation and its modified version are given based on the Deift–Zhou approach of nonlinear steepest descent analysis. It is demonstrated that the results from the long-time asymptotic analysis are in excellent agreement with the numerical solutions. This is the first result on the long-time asymptotic behaviors of the good Boussinesq equation with qxx-term and its modified version.

Solitary and periodic wave solutions of the loaded Boussinesq and the loaded modified Boussinesq equation

In this article, we establish new traveling wave solutions for the loaded Boussinesq equation and the loaded modified Boussinesq equation by the functional variable method. The performance of this method is reliable and effective and gives the exact solitary wave solutions and periodic wave solutions of the loaded Boussinesq equation and its modifications. The traveling wave solutions obtained via this method are expressed by hyperbolic functions and the trigonometric functions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions and solitary wave solutions of kink type. This method presents a wider applicability for handling nonlinear wave equations.

Dynamic Effective Porosity Explains Laboratory Experiments on Watertable Fluctuations in Coastal Unconfined Aquifers

Watertable fluctuations are a characteristic feature of coastal unconfined aquifers. They interact with the vadose zone creating a dynamic effective porosity, for which a new (empirical) expression is proposed based on a dimensionless parameter related to the fluctuation frequency. After comparing with both experimental data and numerical simulations, the new expression is implemented into a modified Boussinesq equation, allowing for examination of the effects of the dynamic effective porosity on watertable fluctuations. The dispersion relation arising from the modified Boussinesq equation predicts laboratory experimental data accurately, highlighting the importance of the dynamic effective porosity in modeling watertable fluctuations in coastal unconfined aquifers. This in turn confirms the applicability of the real-valued expression of the dynamic effective porosity. An outcome is that the phase lag between the total moisture (above the watertable) and watertable height measured in laboratory experiments using vertical soil columns (1D systems) can be ignored when predicting watertable fluctuations in coastal unconfined aquifers (2D systems). The dynamic effective porosity is always smaller than the soil porosity, and so by comparison there is a reduction in vertical water exchange between the saturated and vadose zones. Consequently, watertable waves can propagate further landward without standing wave behaviors.

Numerical simulation of coupled fractional‐order Whitham‐Broer‐Kaup equations arising in shallow water with Atangana‐Baleanu derivative

In this manuscript, fractional nonlinear coupled Whitham‐Broer‐Kaup equation associated with Atangana‐Baleanu fractional derivative is considered. General conditions under which a system solution exists and is unique are presented using the fixed‐point theorem method. For numerical simulations, coupled fractional modified Boussinesq equations and coupled fractional approximate long wave equations are investigated using the homotopy perturbation transform technique (HPTT). The suggested technique is an elegant compilation of the Laplace transform technique with the homotopy perturbation approach. The physical behavior of the obtained solutions has been presented graphically as well as in tables for diverse fractional order. Comparative simulations have been performed to validate the efficiency and accuracy of the suggested technique.

Convergence of an approximate solution of the Showalter-Sidorov-Dirichlet problem for the modified Boussinesq equation

Convergence of an approximate solution of the Showalter-Sidorov-Dirichlet problem for the modified Boussinesq equation

New Exact Solutions of Conformable Time-Fractional Bad and Good Modified Boussinesq Equations

The new exact solutions of the conformable time-fractional Bad and Good modified Boussinesq equations are obtained using the Exp-function method, which is different from previous literature works. These equations play a significant role in mathematical physics, engineering sciences and applied mathematics. Plentiful exact solutions with arbitrary parameters are effectively obtained by the method. The obtained solutions are shown graphically. It is shown that the Exp-function method provides a simpler but more effective mathematical tool for constructing exact solutions of non-linear evolution equations.

Traveling Wave Solution of Bad and Good Modified Boussinesq Equations with Conformable Fractional-Order Derivative

Traveling Wave Solution of Bad and Good Modified Boussinesq Equations with Conformable Fractional-Order Derivative

The Sine-Cosine Function Method for Exact Solutions of Nonlinear Partial Differential Equations

The Sine-Cosine function algorithm is applied for solving nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of nonlinear partial differential equations such as, The K(n + 1, n + 1) equation, Schrödinger-Hirota equation, Gardner equation, the modified KdV equation, perturbed Burgers equation, general Burger’s-Fisher equation, and Cubic modified Boussinesq equation which are the important Soliton equations.Keywords: Nonlinear PDEs, Exact Solutions, Nonlinear Waves, Gardner equation, Sine-Cosine function method, The Schrödinger-Hirota equation.

On the use of modified Boussinesq equation for studying double-layered hillslope recession characteristics

Recession flow analysis based on the Boussinesq equation has been widely used to estimate hydraulic parameters of hillslope aquifers. However, previous studies mainly focused on homogeneous hillslope aquifers and showed that the recession curves satisfy the power law, with two transition points and three power law regimes (b1, b2 and b3). Since the variation trends of b for homogeneous hillslopes are inconsistent and even opposite, it is necessary to investigate the law of recession curves of double-layered hillslope aquifers. In the current study, a modified Boussinesq equation applicable to double-layered hillslope aquifers was proposed. After validating against laboratory experiments and numerical simulations, this modified Boussinesq equation was used to examine the recession characteristics of double-layered hillslope aquifers. The results show that the recession curves of double-layered hillslope aquifers also follow three power law regimes. However, the variations of b values are more complex than those of homogeneous hillslopes. Specifically, while b monotonically changes with slope angle for homogeneous hillslopes, the variations of b are non-monotonic for double-layered hillslopes, depending on the soil configuration, e.g., the thickness of subsoil. These findings are of great significance for determining the hydraulic parameters of hillslopes.

Almost optimal local well-posedness for modified Boussinesq equations

In this article, we investigate a class of modified Boussinesq equations, for which we provide first an alternate proof of local well-posedness in the space \((H^s\cap L^\infty)\times (H^s\cap L^\infty)(\mathbb{R})\) (\(s\geq 0\)) to the one obtained by Constantin and Molinet [7]. Secondly, we show that the associated flow map is not smooth when considered from \(H^s\times H^s(\mathbb{R})\) into \(H^s(\mathbb{R})\) for s<0, thus providing a threshold for the regularity needed to perform a Picard iteration for these equations. For more information see https://ejde.math.txstate.edu/Volumes/2020/24/abstr.html

Quasi-periodic Solutions for Two Dimensional Modified Boussinesq Equation

In this paper, two dimensional modified Boussinesq equation $$\begin{aligned} u_{tt} +\Delta ^2u+ \Delta ( u^3 ) =0, \quad x\in {\mathbb {T}}^2,~t\in {\mathbb {R}} \end{aligned}$$ under periodic boundary conditions is considered. It is proved that the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form.

New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations

Abstract The main purpose of this article is to obtain the new solutions of fractional bad and good modified Boussinesq equations with the aid of auxiliary equation method, which can be considered as a model of shallow water waves. By using the conformable wave transform and chain rule, nonlinear fractional partial differential equations are converted into nonlinear ordinary differential equations. This is an important impact because both Caputo definition and Riemann–Liouville definition do not satisfy the chain rule. By using conformable fractional derivatives, reliable solutions can be achieved for conformable fractional partial differential equations.

Numerical simulation of oblique and multidirectional wave propagation and breaking on steep slope based on FEM model of Boussinesq equations

• A FEM model is established based on modified Boussinesq equations to simulate wave propagation and breaking. • The incident wave heights increase for different periods to catch different kinds of wave breaking. • Oblique regular and irregular incident waves and multidirectional waves are considered. • The initial breaking threshold parameter should take different values for oblique and multidirectional waves. Creating a representative numerical simulation of the propagation and breaking of waves along slopes is an important problem in engineering design. Most studies on wave breaking have focused on the propagation of normal incident waves on gentle slopes. In practice, however, waves on steep slopes are obliquely incident or multidirectional irregular waves. In this paper, the eddy viscosity term is introduced to the momentum equation of the improved Boussinesq equations to model wave dissipation caused by breaking and friction, and a numerical model based on an unstructured finite element method (FEM) is established based on the governing equations. It is applied to simulate wave propagation on a steep slope of 1:5. Parallel physical experiments are conducted for comparative analysis that considered a large number of cases, including those featuring of normal and oblique incident regular and irregular waves, and multidirectional waves. The heights of the incident wave increase for different periods to represent different kinds of waves breaking. Based on examination, the effectiveness and accuracy of the numerical model is verified through a comprehensive comparison between the numerical and the experimental results, including in terms of variation in wave height, wave spectrum, and nonlinear parameters. Satisfactory agreement between the numerical and experimental values shows that the proposed model is effective in representing the breaking of oblique incident regular waves, irregular waves, and multidirectional incident irregular waves. However, the initial threshold of the breaking parameter η t ( I ) takes different values for oblique and multidirectional waves. This needs to be paid attention when the breaking of waves is simulated using the Boussinesq equations.

Two Different Systematic Techniques to Seek Analytical Solutions of the Higher-Order Modified Boussinesq Equation

In this paper, we seek analytical solutions of the higher-order modified Boussinesq equation by two different systematic techniques. Employing the exp $(-\psi (z))$ -expansion method, exact solutions of the mentioned equation, including hyperbolic, exponential, trigonometric, and rational function solutions, have been obtained. Based on the work of Yuan et al., we proposed the extended complex method to seek exact solutions of the higher-order modified Boussinesq equation. It shows that the extended complex method can solve more differential equations in mathematical physics than the complex method. The idea of this paper can be used to the complex nonlinear systems of electrical and electronics engineering.

Boussinesq equations: M-fractional solitary wave solutions and convergence analysis

The studies of the dynamics behaviors of nonlinear models arising in ocean engineering play a significant role in our daily activities. This study investigates the nonlinear fractional modified Boussinesq equation and the fractional bad Boussinesq equation using the extended sinh-Gordon equation expansion method. Several travelling wave solutions are successfully constructed. By choosing suitable values of parameters, the 2D and 3D graphs of the reported solutions are successfully plotted. The convergence analysis of the applied method is also discussed.

Impact of fluid turbulent shear stress on failure surface of reservoir bank landslide

Based on geological investigation, in situ monitoring data analysis, and numerical simulation, a new approach to explain the effect of the reservoir water storage on activation of the sliding phenomenon is discussed in this paper. The paper proposed the method of groundwater level distribution function in case of different rate of water level rise in reservoir by using Boussinesq equation solved by Laplace change and Laplace inverse transformation. On comparing the in situ deformation monitoring data and fluid flow property near sliding surface, the mechanism of reservoir water-induced slope instability was understood. Fluid dynamic property affecting the slope deformation can be described as turbulent bottom shear stress at the failure surface for the phreatic water type landslide. The relationship between the stability coefficient, the deformation of the slope, and the fluid flow was established by using modified Boussinesq equation. Finally, with due consideration to the stability of the reservoir bank landslide, the method of calculation for rate of critical reservoir water storage was provided.

An application of the MEFM to the modified Boussinesq equation

In this paper, some travelling wave solutions of the Modified Boussinesq (MBQ) equation are obtained by using the modified expansion function method (MEFM). When the obtained solutions are commented, trigonometric functions including hyperbolic features are obtained. The 2D and 3D graphics of the solutions have been investigated by selecting appropriate parameters. All the obtained solutions provide the MBQ equation. In this work, all mathematical calculations are done with Wolfram Mathematica software.