KdV Burgers Research Articles

An Ultra-weak Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for the KdV–Burgers–Kuramoto Equation

An Ultra-weak Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for the KdV–Burgers–Kuramoto Equation

Passivity-based boundary control for Korteweg-de Vries-Burgers equations

This article considers the passivity-based boundary control for Korteweg-de Vries-Burgers (KdVB) equations. Both the input strict passivity (ISP) and the output strict passivity (OSP) are studied. By the Lyapunov functional method and Wirtinger's inequality, sufficient criteria are derived to establish ISP and OSP for KdVB equations with boundary disturbances. Moreover, when parameter uncertainties arise in KdVB equations, the robust ISP and OSP are also investigated and sufficient criteria are proposed to achieve the robust ISP and OSP. Simulation examples are also given to verify the effectiveness of theoretical results.

Why Stable Finite-Difference Schemes Can Converge to Different Solutions: Analysis for the Generalized Hopf Equation

The example of two families of finite-difference schemes shows that, in general, the numerical solution of the Riemann problem for the generalized Hopf equation depends on the finite-difference scheme. The numerical solution may differ both quantitatively and qualitatively. The reason for this is the nonuniqueness of the solution to the Riemann problem for the generalized Hopf equation. The numerical solution is unique in the case of a flow function with two inflection points if artificial dissipation and dispersion are introduced, i.e., the generalized Korteweg–de Vries-Burgers equation is considered. We propose a method for selecting coefficients of dissipation and dispersion. The method makes it possible to obtain a physically justified unique numerical solution. This solution is independent of the difference scheme.

Fractional Spectral Approaches for Some Fractional Partial Differential Equations

The multi-order linear and nonlinear partial differential equations of fractional order have been solved numerically. A computational strategy based on fractional spectral operational matrices (OM) and generalized fractional Lageurre (GFL), generalized fractional shifted Legendre (GFSL), generalized fractional modified Bernstein (GFMB) are provided. Our fractional spectral methods have been used to solve nonlinear fractional partial differential equations and fractional Korteweg-de Vries-Burgers (KdVB) equation. Making use of a variety of test and application examples, the effectiveness of the numerical solution have been satisfied.

On new computations of the time-fractional nonlinear KdV-Burgers equation with exponential memory

This paper examines the Korteweg–de Vries-Burgers (KdV-Burgers) equation with nonlocal operators using the exponential decay and Mittag-Leffler kernels. The Caputo-Fabrizio and Atangana-Baleanu operators are used in the natural transform decomposition method (NTDM). By coupling a decomposition technique with the natural transform methodology, the method provides an effective analytical solution. When the fractional order is equal to unity, the proposed approach computes a series form solution that converges to the exact values. By comparing the approximate solution to the precise values, the efficacy and trustworthiness of the proposed method are confirmed. Graphs are also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both operators outcome is examined using diagrams and numerical data. These graphs show how the approximated solution’s graph and the precise solution’s graph eventually converge as the non-integer order gets closer to 1. The outcomes demonstrate the method’s high degree of accuracy and its wide applicability to fractional nonlinear evolution equations. In order to further explain these concepts, simulations are run using a computationally packed software that helps interpret the implications of solutions. NTDM is considered the best analytical method for solving fractional-order phenomena, especially KdV-Burgers equations.

Passivity‐based boundary control for stochastic Korteweg–de Vries–Burgers equations

The passivity‐based boundary control is considered for stochastic Korteweg–de Vries–Burgers (SKdVB) equations. Both the stochastic input strictly passive (SISP) and stochastic output strictly passive (SOSP) are studied. By introducing Lyapunov functionals and Wirtinger's inequality, sufficient criteria are derived to establish SISP and SOSP for SKdVB equations with boundary disturbances. Moreover, when parameter uncertainties arise in SKdVB equations, the robust stochastic passivity is also investigated and sufficient criteria are presented to achieve the robust SISP and SOSP. Two numerical simulations are employed to show the effectiveness and advantages of our theoretical results.

How do various forces affect pressure waves in bubbly flows?

This study investigated the weakly nonlinear propagation of pressure waves in compressible, flowing water with spherical microbubbles, considering various forces. Previous theoretical studies on nonlinear pressure waves in bubbly flows did not consider the forces acting on the bubbles, although the validity of ignoring these forces has not been demonstrated. We focused on every possible force such as drag, gravity, buoyancy, and Bjerknes (acoustic radiation) forces acting on bubbles and studied their effects on pressure waves in a one-dimensional setting. Using a singular perturbation method, the Korteweg–de Vries–Burgers equation describing wave propagation was derived. The following results were obtained: (i) Bjerknes force on the bubbles enhanced the nonlinearity, dissipation, and dispersion of the waves; (ii) Drag, gravity, and buoyancy forces acting on the bubbles increased wave dissipation; (iii) Thermal conduction had the most substantial dissipation effect, followed by acoustic radiation, drag, buoyancy, and gravity. We confirmed that the dissipation due to forces on gas bubbles was quantitatively minor.

Investigation of soliton structures for dispersion, dissipation, and reaction time-fractional KdV–burgers–Fisher equation with the noise effect

ABSTRACT In this manuscript, the soliton structures for the time-fractional KdV–Burgers–Fisher equation with the effect of noise are investigated analytically. This is the dispersion–dissipation–reaction model. The third- and fifth-order time-fractional stochastic KdV–Burgers–Fisher equations are under consideration. These wave structures are constructed with the help of an extended generalized Riccati equation mapping method (EGREM). This method is a combined form of the G ′ / G expansion method with the generalized Riccati equation mapping method, and it will give the different forms of wave structures like shock, singular, combo, hyperbolic, trigonometric, mixed trigonometric, and rational solutions. These techniques are used symbolically with computational tools like Mathematica to demonstrate the efficiency and simplicity of the proposed strategy. Additionally, with the various relevant parameter values, the sketches of some solutions in the form of 3D and contour representations for the purpose of comprehending physical processes are drawn. These sketches clearly show the random behavior of these wave structures that are appearing in dispersion, dissipation, and reaction concentrations of these mathematical models.

Undular bore theory for the modified Korteweg-de Vries-Burgers equation.

We consider nonlinear wave structures described by the modified Korteweg-de Vries equation, taking into account a small Burgers viscosity for the case of steplike initial conditions. The Whitham modulation equationsare derived, which include the small viscosity as a perturbation. It is shown that for a long enough time of evolution, this small perturbation leads to the stabilization of cnoidal bores, and their main characteristics are obtained. The applicability conditions of this approach are discussed. Analytical theory is compared with numerical solutions and good agreement is found.

A mixed-flow model for heterogeneous vehicles enforcing a movement control protocol utilizing a vehicular size-based equilibrium speed function

This work addressed the effect of heterogeneous vehicle sizes on traffic flow fields by introducing a movement control protocol. Considering a continuum traffic model, a new equilibrium velocity function that is dependent on traffic density was introduced to account for the effect of vehicle size. The established model showed a quantitative comparison between the Optimal Velocity and Full Velocity Difference models. A neutral stability test was carried out to evaluate the model's capability of neutralizing flow fields. The density wave behavior near a critical point was portrayed by deducing the Korteweg–de Vries–Burgers equation through a nonlinear analysis. A series of numerical simulations, the outcomes of which agreed well with the analytical results, was performed to observe the overall flow field scenario.

$ H^1 $ solutions for a modified Korteweg-de Vries-Burgers type equation

<p>This paper modeled the dynamics of microbubbles coated with viscoelastic shells using the modified Korteweg-de Vries-Burgers equation, a nonlinear third-order partial differential equation. This study focused on the well-posedness of the Cauchy problem associated with this equation.</p>

Theoretical investigations on a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers model for a dilated artery, blood vessel or circulatory system with experimental support

Recent theoretical physics efforts have been focused on the probes for nonlinear pulse waves in, for example, variable-radius arteries. With respect to the nonlinear waves in an artery full of blood with certain aneurysm, pulses in a blood vessel, or features in a circulatory system, this paper symbolically computes out an auto-Bäcklund transformation via a noncharacteristic movable singular manifold, certain families of the solitonic solutions, as well as a family of the similarity reductions for a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers equation. Aiming, e.g., at the dynamical radial displacement superimposed on the original static deformation from an arterial wall, our results rely on the axial stretch of the injured artery, blood as an incompressible Newtonian fluid, radius variation along the axial direction or aneurysmal geometry, viscosity of the fluid, thickness of the artery, mass density of the membrane material, mass density of the fluid, strain energy density of the artery, shear modulus, stretch ratio, etc. We also highlight that the shock-wave structures from our solutions agree well with those dusty-plasma-experimentally reported.

Nonlinear propagation of dust-acoustic waves and its modulation instability

Investigation the nonlinear dust-acoustic waves in a complex plasma in Titan’s ionosphere sets to become a vital factor in understanding different wave profiles in such space plasma. As for dust on Titan, there is evidence that suggests the Moon has a dusty surface. The Cassini spacecraft, which explored Saturn and its moons, detected bright spots on Titan’s surface that could be the result of dust or ice particles reflecting sunlight. Based on the above finding, a convenient mathematical model is considered and an appropriate evolution equation (Korteweg–de Vries-Burgers KdVB equation) is derived. At low wavenumber, the KdVB equation is transformed to a complex Ginzburg-Landau (CGL) equation that describes the propagation of the wave packet in the system, which is an efficient tool for studying rogue waves in space plasmas. Different plasma parameters are examined on the propagating waves in Titan’s ionosphere.

Modulation theory for solitary waves generated by viscous flow over a step

The effect of dissipation on the flow of a fluid over a step or jump is examined. This flow has been described theoretically and numerically by using the framework of the forced Korteweg–de Vries–Burgers equation. This model includes the influence of the viscosity of the fluid beyond the Korteweg–de Vries approximation. Modulation theory solutions for the undular bores generated upstream and downstream of the forcing are found. Predictions show that the effect of viscosity plays a key role in determining the properties of waves such as the upstream and downstream solitary wave amplitudes and the widths of the bores. The numerical simulations have shown that the unsteady flow consists of upstream and downstream undular bores, which are connected by a locally steady solution over a step. These numerical solutions are compared with modulation theory results with the excellent agreement obtained. Data from the undular bores are all tabulated. Also, the numerical solutions of the Euler equations are compared with the results of the model, which gave good comparisons. Moreover, the comparisons with the experimental measurements of Lee et al. (1989) together with the Navier–Stokes equations show evidence of the significance of this model.

Stratonovich–Khasminskii averaging principle for multiscale random Korteweg–de Vries-Burgers equation

In this paper, we consider the multiscale random Korteweg–de Vries-Burgers (KdVB) equation, namely, the KdVB equation perturbated by a fast time oscillating external force and a random oscillating noise, the random oscillating noise is made up of a family of strong mixing stationary processes with singular small parameters. A Stratonovich–Khasminskii type averaging principle for multiscale random KdVB equation is established, in physics, this averaging principle can describe the asymptotic behavior for the propagation of small-amplitude long waves in nonlinear dispersive and dissipative media with singular structure and the soliton propagation in the random weakly viscous media or in the random field.

On the shock wave structures in anisotropy magnetoplasmas

In this work, the propagation of nonlinear electrostatic shock wave structures in an anisotropy pressure magnetoplasma composed of warm inertial ions and inertia-less Maxwellian electrons is reported. For this purpose, the technique of reductive perturbation is applied for reducing fluid equations of the current model to the Korteweg–de Vries Burgers (KdVB) equation with a second-order dissipative term and the KdVB–Kuramoto (KBK) equation with both second- and fourth-order dissipative terms. The impact of various plasma parameters, including the parallel ion pressure, perpendicular ion pressure, and dissipation parameter, on the significant characteristics of the shock wave profile is examined and discussed. In addition, a comparison between the profiles of KdVB shocks and KdVB–Kuramoto shocks is reported. We expect that KBK shock wave amplitudes become larger than the KdVB ones by taking the fourth-order dissipative into consideration. Thus, the results of the KBK equation may treat the difference between the theoretical and laboratory results or satellite observations.

Viscous Effect on Solitary Kelvin Wave in Open Cylindrical Channel under Precession

Viscous effect is introduced into the system of Navier–Stokes equations, that were derived to study the solitary Kelvin mode in an open cylindrical channel that precesses. Accordingly, three new weakly nonlinear models were derived: Korteweg–de Vries-Burgers, and two new Benjamin–Bona–Mahony-Burgers. The first was solved analytically by discussing the phase solution and numerically using an implicite finite difference method to track the solution with time under diffusion effect. The second two models were solved numerically only using the Quartic B-Spline collocation method. By manipulating the scaling the first model included only the gravity force effect, and the second included both gravity and Coriolis forces. The numerical method is tested experimentally by comparing the velocity solutions with ADV signal extracted from the ADV measurements under bore conditions, and the amplitude solution with the solitary kelvin mode.

Structures of Classical and Special Discontinuities for the Generalized Korteweg–de Vries–Burgers Equation in the Case of a Flux Function with Four Inflection Points

Structures of Classical and Special Discontinuities for the Generalized Korteweg–de Vries–Burgers Equation in the Case of a Flux Function with Four Inflection Points

Weakly nonlinear propagation of pressure waves in bubbly liquids with a polydispersity based on two-fluid model equations

Herein, the weak, nonlinear propagation of pressure waves in an initially quiescent liquid containing many small spherical gas bubbles is theoretically studied. We focus on the initial, polydispersity features of the bubble radius and bubble number density. Our analysis was not based on any assumptions about explicit polydispersity forms. Using equations based on a two-fluid model and the method of multiple scales with perturbation expansions, the Korteweg–de Vries–Burgers equation for a low-frequency long wave and nonlinear Schrödinger equation for a high-frequency short wave were derived. In both cases, the polydispersity contributes to the advection and dissipation effects of waves, and every coefficient in both equations includes the initial void fraction as one of the most important parameters, owing to the use of the two-fluid model.

Nonlinear density waves on graphene electron fluids

The hydrodynamic behavior of charged carriers leads to nonlinear phenomena such as solitary waves and shocks, among others. As an application, such waves might be exploited on plasmonic devices either for modulation or signal propagation along graphene waveguides. We study the nature of nonlinear perturbations following an approach similar to Sagdeev potential analysis and also by performing the reductive perturbation method on the hydrodynamic description of graphene electrons, taking into consideration the effect of Bohm quantum potential and odd viscosity. Thus, deriving a dissipative Kadomtsev-Petviashvili-Burgers (KPB) equation for the bidimensional flow as well as its unidimensional limit in the form of Korteweg--de Vries--Burgers (KdVB). The stability analysis of these equations unveils the existence of unstable modes that can be excited and launched through graphene plasmonic devices.