KdV-Burgers Equation Research Articles

Exponential input-to-state stability of delay Korteweg–de Vries–Burgers equations

The exponential input-to-state stability (EISS) is studied for delay Korteweg–de Vries–Burgers (DKdVB) equations. DKdVB equations are considered with distributed input and boundary input. Making use of the Lyapunov–Krasovskii functional method and inequality techniques, a sufficient condition is established to ensure the EISS for DKdVB equations. Besides the impact of the time delay on the EISS, this sufficient condition also shows the effect of the diffusion term on the EISS. In case of uncertainties of system’s coefficients, robust EISS of uncertain DKdVB equations is investigated. A criterion is obtained to guarantee the EISS for the uncertain DKdVB equation. Two numerical simulation examples are given to illustrate the validity of the derived results.

Non-linear wave propagation in a weakly compressible Kelvin-Voigt liquid containing bubbly clusters

The effect of bubble-bubble interaction on wave propagation in homogeneous weakly compressible viscoelastic bubbly flow is investigated using the reductive perturbation method. The bubble dynamics equation is derived using the kinetic energy conservation approach. The bubble dynamics and mixture equations are coupled with the equation of state for gas to investigate the shock wave propagation phenomenon in the mixture. A two-dimensional Korteweg-de VriesBurger (KdVB) equation in terms of a pressure profile is derived. It is found that the bubble-bubble interaction has no effect when using the parameters under our consideration.

Spatio-temporal fractional shock waves solution for fractional Korteweg-de Vries burgers equations

ABSTRACT We examine the nonlinear characteristics of localized spatial (temporal) fractional shock excitations in a viscous fluid. By applying the natural transform decomposition method and with the aid of standard Caputo's operators, the stationary shock solutions for the spatial (temporal) fractional Korteweg-de Vries Burgers equations are derived. The results reveal that the temporal (Spatial) fractional index significantly modifies the localized solutions for the shocks. Moreover, The Banach contraction analysis shows that the wave solutions converge at , where stands for the temporal (Spatial) variables. Numerical analysis confirms that the index α effect nonlinear waves coupling and therefore causes the steeping of the fractional shocks. Variations in γ on the other hand reduce the wave dispersion and in turn localized the shock potentials. This study is important to understand the nonlinear evolution of shock waves in collisional plasma where the spatial/temporal fractional shocks suffer modifications in the pulse amplitude as well as in the spatial extensions.

Nonlinear acoustic theory on flowing liquid containing multiple microbubbles coated by a compressible visco-elastic shell: Low and high frequency cases

Microbubbles coated by visco-elastic shells are important for ultrasound diagnosis using contrast agents, and the dynamics of single coated bubbles has been investigated in the literature. However, although a high number of contrast agents are used in practical situations, there has long been an absence of a nonlinear acoustic theory for multiple coated bubbles, except for our recent work by Kikuchi and Kanagawa [“Weakly nonlinear theory on ultrasound propagation in liquids containing many microbubbles encapsulated by visco-elastic shell,” Jpn. J. Appl. Phys. 60, SDDD14 (2021)], under several assumptions to be excluded. Aiming for generalization, in this study, we theoretically investigate weakly nonlinear propagation of ultrasound in liquid containing multiple bubbles coated by a visco-elastic shell with compressibility. Leveraging the method of multiple scales, both the Korteweg–de Vries–Burgers (KdVB) equation for a low-frequency long wave and nonlinear Schrödinger (NLS) equation for a high-frequency short wave are derived from the volumetric averaged equations for bubbly liquids based on a two-fluid model and the up-to-date model for single coated bubbles with shell compressibility. Neglected factors in our previous paper, i.e., compressibility of the shell and liquid, drag force acting on bubbles, bubble translation, and thermal conduction, are incorporated in the present KdVB and NLS equations; the proposed model will be regarded as a generic physico-mathematical model. The results show that shell compressibility attenuated ultrasound strongly and decreased nonlinearity of ultrasound. Finally, we compared the magnitudes of six dissipation factors (shell compressibility, shell viscosity, liquid compressibility, liquid viscosity, thermal effect, and drag force) for five typical ultrasound contrast agents, and a similar tendency between KdVB and NLS equations was revealed.

Dynamics and head-on collisions of multidimensional dust acoustic shock waves in a self-gravitating magnetized electron-depleted dusty plasma

The dynamics and collisions of dust acoustic (DA) shock excitations traveling in opposite directions are theoretically investigated in a three-dimensional self-gravitating magnetized electron-depleted dusty plasma whose ingredients are extremely warm positively and negatively charged massive dust grains as well as ions that follow the q-nonextensive distribution. A linear analysis and the extended Poincare–Lighthill–Kuo method are used to derive the dispersion relation, the two-sided Korteweg–de Vries Burgers equations, and the phase shift that occurs due to the wave interaction. It is found that gravitation introduces Jeans-like instability, reduces the wave damping rate, decays the aperiodic oscillatory structure of DA excitations, and strongly affects the amplitude, steepness, and occurrence of monotonic compressive and rarefactive shocks. Numerical simulations also highlighted the stabilizing role of the magnetic field and the singularities of the collision process of monotonic shock fronts as well as the undeniable influence of viscosity, ion nonextensivity, and obliqueness between counter-traveling waves on the phase shift and collision profiles. The present results may be useful to better understand interactions of dust acoustic shock waves in the laboratory and astrophysical scenarios, such as dust clouds in the galactic disk, photo-association regions separating H II regions from dense molecular clouds, Saturn's planetary ring, and Halley Comet.

A collocation method for solving time fractional nonlinear Korteweg–de Vries–Burgers equation arising in shallow water waves

This paper focuses on numerical solutions of time fractional nonlinear Korteweg–de Vries–Burgers equation formulated with Caputo’s fractional derivative. For this purpose, a framework of combinations of collocation method with the finite-element method is provided using trigonometric quintic B-spline basis. The method consists of both spatial discretization and temporal discretization using approximate solution and Crank–Nicolson approach. Discretizing fractional derivative is made using [Formula: see text] algorithm which is derived from the definition of Caputo derivative using an approximate function. The stability analysis is established using von-Neumann stability technique. The numerical results obtained using the collocation method are presented via tables and graphics. The novel results demonstrate the efficiency and reliability of the method.

Inverse Optimal Control of Korteweg-de Vries-Burgers Equation*

We consider inverse optimal adaptive control for Korteweg-de Vries Burgers (KdVB) equation. First, we prove that the boundary control design in Balogh and Krstic (2000) is inverse optimal for a meaningful functional. Next, we achieve an adaptive boundary control design for KdVB equation with an unknown dissipation coefficient. Moreover, we prove that it is also inverse optimal for a meaningful functional. Two examples are given to show the validity of the proposed design.

The Burgers-KdV limit in one-dimensional plasma with viscous dissipation: A study of dispersion and dissipation effects

<abstract><p>The Burgers-KdV equation as a highly nonlinear model, is commonly used in weakly nonlinear analysis to describe small but finite amplitude ion-acoustic waves. In this study, we demonstrate that by considering viscous dissipation, we can derive the Burgers-KdV limit from a one-dimensional plasma system by using the Gardner-Morikawa transformation. This transformation allows us to obtain both homogeneous and inhomogeneous Burgers-KdV equations, which incorporate dissipative and dispersive terms, for the ionic acoustic system. To analyze the remaining system, we employ the energy method in Sobolev spaces to estimate its behavior. As a result, we are able to capture the Burgers-KdV dynamics over a time interval of order $ O(\varepsilon^{-1}) $, where $ \varepsilon $ represents a small parameter.</p></abstract>

Global asymptotic stability of the rarefaction waves to the Cauchy problem for the generalized Rosenau–Korteweg–de Vries–Burgers equation

Global asymptotic stability of the rarefaction waves to the Cauchy problem for the generalized Rosenau–Korteweg–de Vries–Burgers equation

A New Class of Difference Schemes With Intrinsic Parallelism for the KdV-Burgers Equation

A New Class of Difference Schemes With Intrinsic Parallelism for the KdV-Burgers Equation

Investigation of time fractional nonlinear KdV-Burgers equation under fractional operators with nonsingular kernels

<abstract><p>In this manuscript, the Korteweg-de Vries-Burgers (KdV-Burgers) partial differential equation (PDE) is investigated under nonlocal operators with the Mittag-Leffler kernel and the exponential decay kernel. For both fractional operators, the existence of the solution of the KdV-Burgers PDE is demonstrated through fixed point theorems of $ \alpha $-type $ \digamma $ contraction. The modified double Laplace transform is utilized to compute a series solution that leads to the exact values when fractional order equals unity. The effectiveness and reliability of the suggested approach are verified and confirmed by comparing the series outcomes to the exact values. Moreover, the series solution is demonstrated through graphs for a few fractional orders. Lastly, a comparison between the results of the two fractional operators is studied through numerical data and diagrams. The results show how consistently accurate the method is and how broadly applicable it is to fractional nonlinear evolution equations.</p></abstract>

Approximate analytical solutions for a class of generalized perturbed KdV-burgers equation

In this paper, we establish an efficient algorithm for solving a class of generalized perturbed KdV-Burgers equation with conformable time fractional derivative and He?s space fractal derivative. An illustrative example is presented.

A new semi-analytical solution of compound KdV-Burgers equation of fractional order

This article introduces and illustrates a novel approximation to the compound KdV-Burgers equation. For such a challenge, the q-homotopy analysis transform technique (q-HATM) is a potent approach. The suggested procedure avoids the complexity seen in many other methods and provides an approximation that is extremely near to the exact solution. The uniqueness theorem and convergence analysis of the expected problem are explored with the aid of Banach's fixed-point theory. Through a difference in the fractional derivative, the normal frequency for the fractional solution to this issue changes. All of the discovered solutions are illustrated in the figures and tables.

Lie Symmetries and Exact Solutions of KdV–Burgers Equation with Dissipation in Dusty Plasma

Lie Symmetries and Exact Solutions of KdV–Burgers Equation with Dissipation in Dusty Plasma

Application of Generalized Logistic Function to Travelling Wave Solutions for a Class of Nonlinear Evolution Equations

The generalized Logistic function that solves a first-order nonlinear ODE with an arbitrary positive power term of the dependent variable is introduced in this paper, by means of which the traveling wave solutions of a class of nonlinear evolution equations, including the generalized Fisher equation, the generalized Nagumo equation, the generalized Burgers-Fisher equation, the generalized Gardner equation, the generalized KdV-Burgers equation, and the generalized Benney equation, are obtained successfully. In these particular cases, traveling wave solutions of several important model PDEs in mathematical physics are also discovered.

Intermittent observer-based dissipative saturation control for Korteweg-de Vries-Burgers equation with stochastic noise and incomplete measurable information

In this paper, a (Q, S, R)-r˜-dissipative control rate based on intermittent observations is developed for the Korteweg-de Vries-Burgers equation (KdVB) with stochastic noise and incomplete measurable information. Different from the existing works, the measurement transmissions from the sensor to the controller and from the controller to the actuator are assumed to be imperfect (i.e., the phenomenon of data packet dropouts may occur intermittently), and an intermittent observer is proposed to track the part state of the equation rather than the existing traditional full-order Lebesgue observer. In addition, a (Q, S, R)-r˜-saturation dissipation index is selected to design a unified control for KdVB dynamics, which is more general than H∞ exponent. More importantly, the calculated (Q, S, R)-r˜-saturation dissipation performance is completed under the condition of partial state loss of KdVB, which has great research significance and value in practical engineering applications. Finally, an example is given to verify the effectiveness and superiority of the proposed method.

Time-fractional electron-acoustic shocks in magnetoplasma with superthermal electrons

The linear and nonlinear characteristics of electron-acoustic (EA) shocks are examined in an electron–ion (EI) magnetoplasma containing superthermal electrons. A linear dispersion relation is determined by using magneto hydrodynamic (MHD) equations, revealing the excitations of electron-acoustic and Langmuir modes. Employing the reduction perturbation (RP) method, a Korteweg-de Vries Burgers (KdVB) equation is derived that admits evaluation of the EA shocks. The latter has an effect on variations in the superthermality index (κe) and the streaming effect (vo). For the time fractional shock solutions, a time fractional KdVB equation is obtained by method of variation. The mathbbJ-transform decomposition method is also used to derive higher order shock wave solutions. The shock waves are significantly altered by the relevant plasma parameters. This study is important in understanding the coherent EA modes in superthermal magnetoplasmas, relevant to the magnetosphere, the ionosphere, as well as the laboratory facilities.

Contribution of initial bubble radius distribution to weakly nonlinear waves with a long wavelength in bubbly liquids

In this study, the weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid was theoretically investigated. This liquid contains several small uniformly distributed spherical polydisperse gas bubbles. The polydispersity considered here represents various types of initial bubble radii, and the liquid contains multiple bubbles, each with an initial radius. Using the method of multiple scales, we first derived the Korteweg–de Vries–Burgers (KdVB) equation with a correction term as a nonlinear wave equation. This equation describes the long-range wave propagation with weak nonlinearity, low frequency, and long wavelength in the polydisperse bubbly liquid using the basic equations in a two-fluid model. The utilization of the two-fluid model incorporates the dependence of an initial void fraction on each coefficient in the nonlinear, dissipation, and dispersion terms in the KdVB equation. Furthermore, unlike previous studies on waves in polydisperse bubbly liquids, we achieved the formulation without assuming an explicit form of the polydispersity function. Consequently, we discovered the contribution of polydispersity to the various effects of wave propagation, that is, the nonlinear, dissipation, and dispersion effects. In particular, the dispersion effect of the waves was found to be strongly influenced by polydispersity.

On the Instability of Monotone Traveling-Wave Solutions for a Generalized Korteweg–de Vries–Burgers Equation

On the Instability of Monotone Traveling-Wave Solutions for a Generalized Korteweg–de Vries–Burgers Equation

Finite-time boundary stabilization for Korteweg–de Vries–Burgers equations

The finite-time boundary stabilization is investigated for Korteweg–de Vries–Burgers (KdVB) equations. Firstly, a distributed controller is derived, and criteria are presented to guarantee the finite-time stability (FTS) of KdVB equations via the distributed controller. Then, considering the installation in real applications, a vogue control strategy, a boundary controller design, is adopted to ensure the FTS. In light of the FTS lemma, sufficient conditions are obtained for KdVB equations to ensure the FTS. When there exist uncertainties in the system’s coefficients, the robust finite-time stabilization is studied for uncertain KdVB equations. With the designed boundary controller, a sufficient condition is provided to ensure the FTS for the uncertain KdVB equation. Finally, the theoretical results are verified by numerical examples.