Generalized Boussinesq Equation Research Articles

An exactly divergence-free hybridized discontinuous Galerkin method for the generalized Boussinesq equations with singular heat source

The purpose of this work is to propose and analyze a hybridized discontinuous Galerkin (HDG) method for the generalized Boussinesq equations with singular heat source. We use polynomials of order k, k−1 and k to approximate the velocity, the pressure and the temperature. By introducing Lagrange multipliers for the pressure, the approximate velocity field obtained by the HDG method is shown to be exactly divergence-free and H(div)-conforming. Under a smallness assumption on the problem data and solutions, we prove by Brouwer’s fixed point theorem that the discrete system has a solution in two dimensions. If the viscosity and thermal conductivity are further assumed to be positive constants, a priori error estimates with convergence rate O(h) and efficient and reliable a posteriori error estimates are derived. Finally numerical examples illustrate the theoretical analysis and show the performance of the obtained a posteriori error estimator. 1991 Mathematics Subject Classification 65N12, 65N30, 65N50, 76N05.

Optimal error estimates of a second-order fully decoupled finite element method for the nonstationary generalized Boussinesq model

In this paper, we develop and analyze a fully decoupled finite element method for the non-stationary generalized Boussinesq equations, where the viscosity and thermal conductivity depend on the temperature. Based on some subtle implicit-explicit treatments for the nonlinear coupling terms, we develop a second-order in time, fully decoupled, linear and unconditionally energy stable scheme for solving this system. The unconditional stability of the fully discrete scheme with finite element approximation is proved. The optimal L2-error estimates are analyzed for temperature-dependent thermal conductivity system. Numerical experiments are presented to illustrate the convergence, accuracy and applicability of the proposed numerical scheme.

Global well-posedness of the 3D generalized Boussinesq equations

Global well-posedness of the 3D generalized Boussinesq equations

Existence and bifurcation of traveling wave solutions to a generalized Boussinesq equation with nonlinear dispersion

This paper is devoted to the analytical study of traveling wave solutions to a generalized Boussinesq equation with nonlinear dispersion. Utilizing the bifurcation method of dynamical systems, the existence of compacton and peakon solutions are established. Under certain parameter conditions, the compacton and peakon solutions can be bifurcated by smooth periodic wave solutions, smooth solitary wave solutions, and singular cusp solutions. Numerical simulations are supplied to corroborate the analytical results. Some previous results are extended.

Second order Lagrangians for (2 + 1)-dimensional generalized Boussinesq equations and an extension of the Krupka-Betounes equivalent

We determine second order Lagrangians for (2 + 1)-dimensional generalized Boussinesq equations and we discuss some aspects concerning conservation laws associated with invariance properties of their extended ‘full’ equivalents, in particular of Krupka–Betounes type. Such equivalents are constructed by means of a recursive formula involving geometric integration by parts formulae.

Global existence and time-decay rates of solutions to the generalized Boussinesq equation with weak damping

In this paper, we study the initial value problem for the generalized Boussineq equation with weak damping. The existence and time-decay rates of global solutions and its derivatives are established for all space dimensions d ≥ 1, provided that the norm of the initial data is suitably small. The negative Sobolev norms of the initial data in low frequency are shown to be preserved along time evolution and enhance the decay rates of global solutions. The proof is based on the energy method and flexible interpolation trick without investigating the corresponding linear equation.

Global well-posedness, scattering and blow-up for the generalized Boussinesq equation in high dimensions below the ground state energy

In this paper, we study the generalized Boussinesq equation in energy space. For spatial dimensions d≥3, nonlinear exponent α+1∈[1+4/d,1+4/(d−2)), initial data (u0,u1), we clarify the longtime behavior of the solution with energy E(u0,u1) below the ground state energy E(Qd,α,0). It depends on K(u0)=‖∇u0‖L2dα/2−2‖u0‖L22−(d−2)α/2. For K(u0)>K(Qd,α), the solution blows up in finite time. For K(u0)<K(Qd,α), the solution is global and scatters to a linear solution in energy space if u0,u1 are radial functions. To show the scattering, we use the concentration-compactness argument of Kenig-Merle [15] and the Morawetz-virial type estimate obtained in [7].

Influence of dissipation on solitary wave solution to generalized Boussinesq equation

Influence of dissipation on solitary wave solution to generalized Boussinesq equation

Solitons in Neurosciences by the Laplace–Adomian Decomposition Scheme

The paper concentrates on the solitary waves that are retrievable from the generalized Boussinesq equation. The numerical simulations are displayed in the paper that gives a visual perspective to the model studied in neurosciences. The Laplace–Adomian decomposition scheme makes this visualization of the solitons possible. The numerical simulations are being reported for the first time using an elegant approach. The results would be helpful for neuroscientists and clinical studies in Medicine. The novelty lies in the modeling that is successfully conducted with an impressively small error measure. In the past, the model was integrated analytically only to recover soliton solutions and its conserved quantities.

Well-posedness and Scattering for the Generalized Boussinesq Equation

In this paper, we show the local well-posedness of the generalized Boussinesq equation (gBQ) in and obtain the global well-posedness, finite-time blowup and small initial data scattering of gBQ in energy space . Moreover, we obtain the large radial initial data scattering of the defocusing case for by using the method of Dodson–Murphy [Proc. Amer. Math. Soc., 145 (2017), pp. 4859–4867].

Linear barycentric rational collocation method for solving a class of generalized Boussinesq equations

&lt;abstract&gt;&lt;p&gt;This paper is concerned with solving a class of generalized Boussinesq shallow-water wave (GBSWW) equations by the linear barycentric rational collocation method (LBRCM), which are nonlinear partial differential equations (PDEs). By using the method of direct linearization, those nonlinear PDEs are transformed into linear PDEs which can be easily solved, and the corresponding differentiation matrix equations of their discretization linear GBSWW equations are also given by a Kronecker product. Based on the error estimate of a barycentric interpolation, the rates of convergence for numerical solutions of GBSWW equations are obtained. Finally, three examples are presented to show theoretical results.&lt;/p&gt;&lt;/abstract&gt;

Higher-order rogue waves with controllable fission and asymmetry localized in a (3 + 1)-dimensional generalized Boussinesq equation

The purpose of this paper is to report the feasibility of constructing high-order rogue waves with controllable fission and asymmetry for high-dimensional nonlinear evolution equations. Such a nonlinear model considered in this paper as the concrete example is the (3 + 1)-dimensional generalized Boussinesq (gB) equation, and the corresponding method is Zhaqilao’s symbolic computation approach containing two embedded parameters. It is indicated by the (3 + 1)-dimensional gB equation that the embedded parameters can not only control the center of the first-order rogue wave, but also control the number of the wave peaks split from higher-order rogue waves and the asymmetry of higher-order rogue waves about the coordinate axes. The main novelty of this paper is that the obtained results and findings can provide useful supplements to the method used and the controllability of higher-order rogue waves.

Gevrey regularity and summability of the formal power series solutions of the inhomogeneous generalized Boussinesq equations

In this article, we investigate Gevrey and summability properties of the formal power series solutions of the inhomogeneous generalized Boussinesq equations. Even if the case that really matters physically is an analytic inhomogeneity, we systematically examine here the cases where the inhomogeneity is s-Gevrey for any s ⩾ 0, in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy: for any s ⩾ 1, the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s &lt; 1, the formal solutions are generically 1-Gevrey. In the latter case, we give in particular an explicit example in which the formal solution is s ′ -Gevrey for no s ′ &lt; 1, that is exactly 1-Gevrey. Then, we give a necessary and sufficient condition under which the formal solutions are 1-summable in a given direction arg ( t ) = θ. In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proofs of our various results.

A fractional-step DG-FE method for the time-dependent generalized Boussinesq equations

In this work a fractional-step DG-FE method for the time-dependent generalized Boussinesq equations is proposed and analysed. The scheme is composed of two steps. In the first step the original problem is reduced into several scalar elliptic equations. An intermediate velocity and temperature are solved simultaneously. Then in the second step, the incompressibility constraint is enforced and velocity is corrected to be discretely divergence free. Moreover, the introduced elliptic term in the correction step enables the imposition of correct Dirichlet boundary conditions at each temporal step, avoiding the artificial boundary layer introduced by classical pressure correction method. DG-FE discretization strategy is utilized, in which the discontinuous Galerkin spacial discretization for flow equations is employed to obtain local mass conservation and traditional finite element spacial discretization is adopted for heat equation to reduce degrees of freedom. By choosing different symmetry and penalty parameters, SIPG-FE and NIPG-FE methods can be utilized. The consistency and stability of both methods are proved. Preliminary error estimates proving the optimal spacial order and suboptimal temporal order are carried out. Based on a different error equation and the preliminary error estimates, the optimal temporal convergence order is obtained. Numerical tests including a benchmark simulating square cavity flow are then presented, to verify the theoretical analysis and validate the method.

Approximate solution to the generalized radial Boussinesq equation

This paper presents a method for constructing approximate semi-analytical solutions for the radially symmetric generalized Boussinesq equation, also known as the porous medium equation (PME). This work is a further extension of analysis done by Telyakovskiy et al. (2016), that studied water injection governed by the Boussinesq equation into an unconfined initially dry aquifer. The newly derived approximate solution also has two parts; one to approximate singular behavior near the well and a polynomial part to model far-field behavior. With the prescribed initial and boundary conditions, the analyzed problem can be rewritten in terms of similarity variables resulting in a boundary value problem for a nonlinear ordinary differential equation. Using higher-order physically significant moments of the solution, a closed-form expression for the position of the wetting front and the shape of the phreatic surface is obtained. The introduced solution is valid over a wide range of injection regimes: the time-independent, the power-law injection function with positive exponents, and exponential-law of injection. A highly accurate numerical solver similar to Telyakovskiy et al. (2016) is used to validate the presented approximate solutions. Plus, comparison is made with the experimental data on prediction of the wetting front position for the special case of the Boussinesq equation.

Dynamics and bidirectional lumps of the generalized Boussinesq equation with time-space dispersion term: Application of surface gravity waves

In this paper, we revisit the generalized Boussinesq equation that include a new time-space dispersion term. The Kudryashov expansion method is used to obtain singular and nonsingular bidirectional bell-shaped solutions. Also, a modified version of the rational sine-cosine method is proposed to extract bidirectional concave-periodic solutions. Then, the Hirota bilinear method and the Cole-Hopf transformation are implemented to extract bidirectional lump-solitons. On the other hand, the dynamics of Boussinesq’s parameters, nonlinearity and time-space dispersion, are investigated by using graphical analysis. Finally, the obtained solutions to the proposed Boussinesq model are verified by direct substitution in the governing equation.

Blow-Up and Global Solvability of the Cauchy Problem for a Pseudohyperbolic Equation Related to the Generalized Boussinesq Equation

Blow-Up and Global Solvability of the Cauchy Problem for a Pseudohyperbolic Equation Related to the Generalized Boussinesq Equation

A decoupled and iterative finite element method for generalized Boussinesq equations

In this work, a decoupled and iterative finite element method is developed and analyzed for steady-state generalized Boussinesq equations, in which both the viscosity and thermal conductivity depend on the temperature. By utilizing the solutions obtained in the previous iteration step, the coupled system is reduced to Navier-Stokes equations with temperature-dependent viscosity and a linearized convection-diffusion equation, which can be solved in parallel. The well-posedness and stability of the scheme are proved. Finite element errors and iterative errors are analyzed for the cases of both temperature-dependent and temperature-independent thermal conductivity. Numerical examples are provided to demonstrate the convergence, accuracy and applicability of the proposed method.

Bilinear form and nonlinear waves of a (1+1)-dimensional generalized Boussinesq equation for the gravity waves over water surface

Gravity waves are studied in, e.g., the oceanic and atmospheric sciences. In this paper, for the head-on interaction of oblique gravity wave profiles over the water surface, we investigate a (1+1)-dimensional generalized Boussinesq equation. Bilinear form is derived, and the higher-order rogue waves, breather and hybrid solutions in the determinant form are constructed via the Hirota bilinear method and Kadomtsev–Petviashvili hierarchy reduction. Influence of α, β and γ on the solutions is discussed, where α, β and γ are the nonzero coefficients in that equation. For the first-order rogue-wave solutions, we find that the wave widths and amplitudes are related to α and β, and the locations of the extreme points depend on α, β and γ. Structures of the higher-order rogue waves are also given. We construct three types of the first-order breathers, and the second-order breathers are obtained through the combinations of the first-order ones. α, β and γ are found to influence the amplitudes and locations of the first-order breathers. Hybrid solutions comprising the one rogue wave, the higher-order breathers and solitons are discussed graphically. Those results reveal the nonlinear properties of oblique gravity waves over the water surface, which can help the study on nonlinear waves in fluid mechanics, plasma physics and fiber optics.

Extension of the sine-Gordon expansion scheme and parametric effect analysis for higher-dimensional nonlinear evolution equations

Different wave solutions assist to interpret phenomena in different aspects of optics, physics, plasma physics, engineering, and other related subjects. The higher dimensional generalized Boussinesq equation (gBE) and the Klein-Gordon (KG) equation have remarkable applications in the field of quantum mechanics, recession flow analysis, fluid mechanics etc. In this article, the soliton solutions of the higher-dimensional nonlinear evolution equations (NLEEs) have been extracted through extending the sine-Gordon expansion method and we analyze the effect of the associated parameters and the phenomena establishing the lump, kink, rogue, bright-dark, spiked, periodic wave, anti-bell wave, singular soliton etc. Formerly, the sine-Gordon expansion (sGE) method was used only to search for lower-dimensional NLEEs. In order to illustrate the latency, we have portrayed diagrams for different values of parameters and it is noteworthy that the properties of the features change as the parameters change.