KdV-Burgers Equation Research Articles

Analytical solutions of nonlinear time fractional evaluation equations via unified method with different derivatives and their comparison

This paper is devoted to addressings the fairly interesting soliton solutions for the time fractional combined Korteweg-de Vries-modified Korteweg-de Vries equation (KdV–mKdV equation) and modified Burgers-KdV equation. The unified method along with conformable, Beta and local M-derivative are used to construct the general structure of solitary wave soliton solutions. The method allows us to find solutions in both polynomial and rational forms. Further, the comparison of solutions are given out through 3D and 2D-plots to expose the impact of fractional parameter on the obtained solutions. The reported solutions are novel and have not been discussed in the literature.

Observer‐based H∞ control of a stochastic Korteweg–de Vries–Burgers equation

AbstractThis article mainly deals with observer‐based H∞ control problem for a stochastic Korteweg–de Vries–Burgers equation under point or averaged measurements. Due to the nonlinearity of the stochastic partial differential equations, special emphases are given to computation complexity. By constructing an appropriate Lyapunov functional, we derive sufficient conditions in terms of linear matrix inequalities to guarantee the internal exponential stability and H∞ performance of the perturbed closed‐loop system by means of the Lyapunov approach. Consistent simulation results that support the proposed theoretical statements are provided. Finally, we have made important instructions for future research directions.

Macro autonomous traffic flow model with traffic jerk and downstream vehicle information

PurposeIn the era of connected and autonomous vehicles, a large amount of surrounding vehicular information can be acquired by the focal vehicle in real time using vehicle-to-vehicle communication technology, such as the core variable of electronic throttle opening angle. Meanwhile, the traffic jerk, such as the non-compliance of drivers and pedestrians, worsens the chaos of the surrounding traffic environment. To reflect the future traffic environment, the authors simultaneously incorporate the electronic throttle (ET) and traffic jerk into the traditional continuum model. The authors derive the stability criterion of the enhanced continuum model via the perturbation method.Design/methodology/approachTo facilitate insight into the propagation and evolution mechanism of traffic jam near the stability condition, the authors use the nonlinear stability analysis method to derive the KdV-Burgers equation of proposed continuum model.FindingsThe new item of ET opening angle and traffic jerk have a positive impact on suppressing traffic congestion and improving road robustness.Originality/valueThe research on autonomous continuum models is rare. This model can better reflect the actual traffic, which can also provide a theoretical reference for the future traffic governance.

Stability analysis of traveling wave solutions of a generalized Korteweg–de Vries–Burgers equation with variable dissipation parameter

We consider traveling wave solutions of a generalized Korteweg–de Vries–Burgers equation. The dissipation coefficient depends on coordinate and time and has a smoothed step-like form at every instant of time. Small-scale processes of dissipation and dispersion determine the solution of the traveling wave problem in the high-gradient region. The flux function is non-convex and has two inflection points. It is shown that traveling wave solutions with the same wave speed can converge to the three different limiting values behind the wave. The Evans function technique is used to conduct linear stability analysis. We demonstrate that traveling wave solutions can be linearly stable for each of the three possible cases. Thus, we found linearly stable solutions in the form of a traveling wave, which correspond to admissible discontinuities in hyperbolic model. These discontinuities have the same speed but different states behind them.

Weakly nonlinear theory on ultrasound propagation in liquids containing many microbubbles encapsulated by visco-elastic shell

Aimed towards an application of ultrasound diagnosis using contrast agents, the dynamics of encapsulated bubbles has been theoretically investigated under the restriction of a single bubble. In this paper, we extend the theory for single bubble or some bubbles to that for many bubbles, and theoretically investigate weakly nonlinear propagation of ultrasound in an initially quiescent incompressible liquid, uniformly containing many microbubbles encapsulated by the shell as a viscoelastic body (Kelvin–Voigt model). As a result, we derived the Korteweg–de Vries–Burgers equation for a low-frequency long wave and clarified that the shell affects the advection, nonlinear, and dissipation (not dispersion) effects of ultrasound propagation. In particular, shell rigidity, surface tension, and shell viscosity increased the advection, nonlinear, and dissipation effects, respectively.

An exhaustive theoretical analysis of thermal effect inside bubbles for weakly nonlinear pressure waves in bubbly liquids

Weakly nonlinear propagation of pressure waves in initially quiescent compressible liquids uniformly containing many spherical microbubbles is theoretically studied based on the derivation of the Korteweg–de Vries–Burgers (KdVB) equation. In particular, the energy equation at the bubble–liquid interface [Prosperetti, J. Fluid Mech. 222, 587 (1991)] and the effective polytropic exponent are introduced into our model [Kanagawa et al., J. Fluid Sci. Technol. 6, 838 (2011)] to clarify the influence of thermal effect inside the bubbles on wave dissipation. Thermal conduction is investigated in detail using some temperature-gradient models. The main results are summarized as follows: (i) Two types of dissipation terms appeared; one was a well-known second-order derivative comprising the effect of viscosity and liquid compressibility (acoustic radiation) and the other was a newly discovered term without differentiation comprising the effect of thermal conduction. (ii) The coefficients of the KdVB equation depended more on the initial bubble radius rather than on the initial void fraction. (iii) The thermal effect contributed to not only the dissipation effect but also to the nonlinear effect, and nonlinearity increased compared with that observed by Kanagawa et al. (2011). (iv) There were no significant differences among the four temperature-gradient models for milliscale bubbles. However, thermal dissipation increased in the four models for microscale bubbles. (v) The thermal dissipation effect observed in this study was comparable with that in a KdVB equation derived by Prosperetti (1991), although the forms of dissipation terms describing the effect of thermal conduction differed. (vi) The thermal dissipation effect was significantly larger than the dissipation effect due to viscosity and compressibility.

Behaviour Analysis of the Padé Sumudu Adomian Decomposition Method Solution

Researchers in the past investigate the Sumudu Adomian Decomposition Method (SADM), the Laplace Adomian Decomposition Method (LADM), the Padé Sumudu Adomian Decomposition Methods (PSADM). In this paper we analyse the behaviour of the function P[L/M][.] called double Padé approximation using in the Padé Sumudu Adomian Decomposition Method (PSADM), and provide some criteriums for chosing L and M to obtain the best Padé approximation solution in the case of nonlinear Schrödinger equation and nonlinear KdV Burger's equation.

Weak dust acoustic shock wave in strongly coupled two-dimensional complex plasma

By using the equation of the state of P=αd+βdTd, we studied a shock wave propagation in a strongly coupled complex plasma. A Korteweg-de Vries–Burgers equation is obtained to describe a shock wave in strongly coupled complex plasma. Dependence of the shock wave speed on the piston velocity and the plasma parameters such as the screening parameter of the strongly coupled complex plasma are given analytically for a weak shock wave. It is in good agreement with previous results. It shows in the present paper that there are density oscillations in the postshock region which is similar to that found in the previous works. Dependence of the length between two first peaks of the density oscillation on the piston velocity and the plasma parameter are also given analytically which is also in agreement qualitatively with previous results.

Weakly nonlinear theory on pressure waves in bubbly liquids with a weak polydispersity

Weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid uniformly containing many spherical microbubbles is theoretically investigated, especially focusing on an initial small polydispersity of both the bubble radius and the number density of bubbles (i.e., void fraction), which appears in a field far from the sound source. Nonlinear waves in polydispersed bubbly liquids are classified into a form of three cases of nonlinear wave equation describing long-range propagation of waves. Using the method of multiple scales with perturbation expansions and the scaling relations of some nondimensional ratios, from the set of basic equations based on a two-fluid model, (i) for a low-frequency long wave, the Korteweg–de Vries–Burgers (KdVB) equation is derived and (ii) for a moderately high-frequency short wave, (ii-a) the NLS (nonlinear Schrödinger)-I (or LG (Landau–Ginzburg)-I) equation for a weak polydisperse medium and (ii-b) the NLS-II (or LG-II) equation in a strong polydisperse medium are derived in a unified manner. For all cases, polydispersity contributes to the advection effect of waves and induces variable coefficients into the KdVB, NLS-I, and NLS-II equations. Furthermore, the KdVB equation includes an inhomogeneous term owing to the polydispersity and the NLS-II equation includes second-order nonlinearity of polydispersity. The polydisperse effect is finally clarified quantitatively by focusing on the advection coefficients with a help of numerical examples.

Variational Principles for Two Compound Nonlinear Equations with Variable Coefficients

It is very important to seek explicit variational principles for nonlinear partial differential equations, which are theoretical bases for many methods to solve or analyze the nonlinear phenomena and problems. By designing the modified trial-Lagrange functional, different variational formulations are successfully and firstly established by the semi-inverse method for two kinds of compound nonlinear equation, i.e. the KdV-Burgers equation and the Burgers-BBM equation, respectively. Both of them contain the variable coefficients, which are time-dependent. Furthermore, the obtained variational principles are proved correct by minimizing the functionals with the calculus of variations.

On the analytical and numerical solutions of the damped nonplanar Shamel Korteweg–de Vries Burgers equation for modeling nonlinear structures in strongly coupled dusty plasmas: Multistage homotopy perturbation method

The dissipative cylindrical and spherical (nonplanar) electrostatic low-frequency dust-acoustic waves (DAWs) including solitary and shock waves in a collisional and unmagnetized strongly coupled dusty plasma are investigated analytically and numerically. The present plasma model consists of inertialess particles including thermal elections and vortex-like positive ions distribution as well as inertial strongly coupled negatively charged dust grains. In the hydrodynamic regime, the fluid governed equations of the present model are reduced to the damped nonplanar Shamel Korteweg–de Vries Burgers (SKVB) equation using the reductive perturbation technique. In the absence of the dissipative effect, the damped nonplanar Shamel Korteweg–de Vries (SKdV) equation is recovered and solved analytically for the first time using a novel analytical approach in order to describe the dynamical mechanism of the dissipative nonplanar dust-acoustic solitary waves. Also, the damped nonplanar SKdV equation is solved numerically using the homotopy perturbation method (HPM) and the hybrid homotopy perturbation method with the moving boundary method which is called multistage HPM (MsHPM). Furthermore, in the presence of the dissipative effect, the damped nonplanar SKdVB equation is solved numerically via the HPM and MsHPM for studying the characteristics of the dissipative nonplanar dust-acoustic solitary and shock waves. For checking the accuracy of the obtained solutions, the maximum global residual error is estimated. Moreover, a comparison between the approximate analytical and numerical solutions is reported. Furthermore, the dependence of dissipative nonplanar structures (solitons and shocks) characteristics on various plasma parameters is examined.

Exact Solutions of the Two Dimensional KdV-Burger Equation by Generalized Kudryashov Method

In this article, the generalized Kudryashov method which provides the exact solutions is examined. It is possible to obtain new exact solutions of the nonlinear differential equations with this method. By implementing this developed method to the two-dimensional KdV-Burger equation, new exact solutions of this equation are found. This new exact solutions are the solutions that are not in the literature. In addition, two and three-dimensional graphics of these new exact solutions have been drawn and their physical behavior has been demonstrated.

The Effect of Weak Mutual Diffusion on Transport Processes in a Multiphase Medium

The Cauchy problem for a singularly perturbed system of equations describing a transport process with diffusion in a multiphase medium is considered. A formal asymptotic expansion of its solution is constructed in the case when exchange between the phases proceeds much more rapidly than the transport and diffusion processes. The case when the diffusion fluxes of the components have a mutual effect on each other is considered. Under the assumptions imposed on the data of the problem, the leading term of the asymptotics is described by a multidimensional generalized Burgers–Korteweg–de Vries equation. Under some additional conditions, the remainder is estimated by means of the residual.

Theoretical elucidation of effect of drag force and translation of bubble on weakly nonlinear pressure waves in bubbly flows

Theoretical investigation of the effects of a translation of bubbles and a drag force acting on bubbles on the wave propagation in bubbly flows has long been lacking. In this study, we theoretically and numerically investigate the weakly nonlinear (i.e., finite but small amplitude) propagation of plane progressive pressure waves in compressible water flows that contain uniformly distributed spherical gas bubbles with translation and drag forces. First, we assume that the gas and liquid phases flow at independent velocities. Then, the drag force and virtual mass force are introduced in an interfacial transport across the bubble–liquid interface in the momentum conservation equations. Furthermore, we consider the translation and spherically symmetric oscillations as bubble dynamics and deploy a two-fluid model to introduce the translation and drag forces. Bubbles do not coalesce, break up, extinct, or appear. For simplicity, the gas viscosity, thermal conductivities of the gas and liquid, and phase change and mass transport across the bubble–liquid interface are ignored. The following results are then obtained: (i) Using the method of multiple scales, two types of Korteweg–de Vries–Burgers equations with a correction term due to the drag force are derived. (ii) The translation of bubbles enhances the nonlinear effect of waves, and the drag force acting on bubbles contributes the nonlinear and dissipation effects of waves. (iii) The results of long-period numerical analysis verify that the temporal evolution of the wave (not flow) dissipation due to the drag force differs from that caused by the acoustic radiation.

Traveling fronts in dissipative granular chains and nonlinear lattices

We consider an infinite chain of particles with nonlinear elastic and dissipative nearest neighbors interactions. Assuming the existence of a traveling front between uniformly compressed (or stretched) states, we obtain jump conditions relating the wave speed and limiting particle velocities to the relative displacements at infinity. Using this result, we characterise compression fronts in chains of touching beads, for viscoelastic contact laws that include a nonlinear elastic force (generalised Hertz contact) and viscous dissipation. We compute compression fronts numerically for the generalised Kuwabara–Kono model in which the viscous contact force is proportional to the derivative of the elastic force, without precompression of the chain. Steady fronts are obtained both as the end result of the compression of one end of the chain and using a shooting method which provides numerically exact traveling waves. Depending on the magnitudes of contact damping and strain applied downstream, we obtain either overdamped (monotonic) or underdamped (oscillatory) compression fronts. To explain this transition and approximate the front profiles, we consider a continuum limit valid when the exponent of the contact nonlinearity is close to (and above) unity. Using multiscale expansions, we formally derive two different amplitude equations for long waves, a Burgers equation with logarithmic nonlinearity, and a logarithmic Korteweg–de Vries (KdV)–Burgers equation for small contact damping. Both models possess traveling front solutions that are in good agreement with the front profiles computed numerically in the granular chain. The analysis of the logarithmic KdV–Burgers equation allows one to approximate the critical damping corresponding to the transition from underdamped to overdamped fronts.

Bifurcation analysis for a novel heterogeneous continuum model considering electronic throttle angle changes with memory

Heterogeneous traffic flow model such as mixes of trucks and cars or connected and regular vehicles is a central issue in traffic flow modeling. However, few researches focus on the bifurcation phenomena of heterogeneous traffic flow. Due to this point, a novel heterogeneous continuum model accounting for different ratios of multiple optimal velocity functions is proposed, in which electronic throttle angle changes with memory is also considered. A simple formula for the linear stability criterion of this improved model is theoretically deduced, and the Korteweg-de Vries-Burgers (KdV-Burgers) equation is derived to discuss the characteristics of density evolution through nonlinear analysis method. The conditions and stability of the Hopf bifurcation for the modified model are then proved minutely. Numerical simulations are executed to investigate how multiple maximum velocities, multiple safety spacings and electronic throttle angle changes with memory affect the density fluctuations, in which numerical experiments on fuel consumptions and emissions are also included. To address the problems mentioned above, we set the Hopf bifurcation as the initial point of the heterogeneous continuum traffic flow to analyze the evolution of bifurcation behavior in the heterogeneous traffic flow.

Evolution of nonlinear stationary formations in a quantum plasma at finite temperature

Abstract In this paper we have studied the gradual evolution of stationary formations in electron acoustic waves at a finite temperature quantum plasma. We have made use of Quantum hydrodynamics model equations and obtained the KdV-Burgers equation. From here we showed how the amplitude modulated solitons evolve from double layer structures through shock fronts and ultimately converging into solitary structures. We have studied the various parametric influences on such stationary structure and also showed how the gradual variations of these parameter affect the transition from one form to another. The results thus obtained will help in the generation and structure of the structures in their respective domain. Much of the experiments on dense plasma will benefit from the parametric study. Further we have studied amplitude modulation followed by a detailed study on chaos.

Numerical solution and characteristic study of time-fractional shocks collision

In this study, the interaction of counter-propagating ion acoustic shock waves in three-component unmagnetized plasmas with inertial warm ions, superthermal electrons and positrons are examined. By employing the extended Poincare-Lighthill-Kuo (PLK) method, two-sided Korteweg-deVries-Burgers (KdVB) equations and their corresponding phase shifts for the shock wave are also derived. The derived two-sided KdVB equations are converted to two-sided time-fractional KdVB (TFKdVB) equations by semi inverse method, which is then solved numerically by a local meshless method (LMM). The effects of the ratio of electron temperature to positron temperature, the spectral index and fractional concentration of positron component p on the phase shift are also examined. Figures are given to judge the performance of the LMM. Comparison is made with the exact solution for the classical case (i.e. ρ = 1) and with Petrov-Galerkin method (PGM) in case of fractional derivative. The influence of the fractional parameter on the behaviour of ion acoustic shock wave in e-p-i plasma is investigated.

Bifurcation analysis of a heterogeneous continuum traffic flow model

Motivated by the increasingly popular mixed traffic flow modeling, a novel heterogeneous continuum traffic flow model is proposed in this paper, which considers the differences of the driver's psychological headway and the driver's self-stabilizing effect on the optimal velocity related to the psychological headway. Leveraging the linear and nonlinear analysis approaches, the linear stability condition and the KdV-Burgers equation are yielded to describe the characteristic of traffic flow evolution. The bifurcation analysis on the proposed model is executed to discuss the existence and stability of Hopf bifurcation from a theoretical perspective. Numerical simulations are performed to verify the rationality of the abovementioned theory and explore the density evolution of heterogeneous traffic flow, including local cluster effect, fuel consumption and exhaust emissions and Hopf bifurcation, in which it is notable that the heterogeneous continuum traffic flow generates significantly different phenomena under different density conditions and the evolution of Hopf bifurcation in traffic flow is also involved.

Blowing-up solutions of the time-fractional dispersive equations

Abstract This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.