KdV Burgers Kuramoto Equation Research Articles

Investigating the dynamics, synchronization of a novel chaotic system reduced from a modified KdV–Burgers–Kuramoto equation

Investigating the dynamics, synchronization of a novel chaotic system reduced from a modified KdV–Burgers–Kuramoto equation

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

A pseudo‐operational collocation method for variable‐order time‐space fractional KdV–Burgers–Kuramoto equation

The idea of this work is to provide a pseudo‐operational collocation scheme to deal with the solution of the variable‐order time‐space fractional KdV–Burgers–Kuramoto equation (VOSTFKBKE). Such the fractional partial differential equation (FPDE) has three characteristics of dissipation, dispersion, and instability, which make this equation is used to model many phenomena in diverse fields of physics. Numerical solutions are sought in a linear combination of two‐dimensional Jacobi polynomials as basis functions. In order to approximate unknown functions in terms of the basis vector, pseudo‐operational matrices are constructed to avoid integration. An error bound of the residual function is estimated in a Jacobi‐weighted space in the norms. Numerical results are compared with exact ones and those reported by other researchers to demonstrate the effectiveness of the recommended method.

Dynamics of Travelling Waves to KdV–Burgers–Kuramoto Equation with Marangoni Effect Perturbation

Dynamics of Travelling Waves to KdV–Burgers–Kuramoto Equation with Marangoni Effect Perturbation

SYMMETRY ANALYSIS, ANALYTICAL SOLUTIONS AND CONSERVATION LAWS OF A GENERALIZED KdV–BURGERS–KURAMOTO EQUATION AND ITS FRACTIONAL VERSION

Wave motions play an important role in fluid dynamics and engineering issues. In this paper, a systematic study to the generalized KdV–Burgers–Kuramoto equation is presented resort to symmetry method. First, based on the Lie point symmetries of the generalized KdV–Burgers–Kuramoto equation, we get invariants and invariant solutions. In particular, we obtain series solutions of generalized KdV–Burgers–Kuramoto equation. Meanwhile, we find that this equation just exists in Lie point symmetries. Then, we present a conservation law, and derive reciprocal Bäcklund transformations of conservation law for the first time. Furthermore, Bäcklund transformation of KdV–Burgers–Kuramoto equation associated with truncated Painlevé expansion is studied for the first time. Subsequently, mCK method is employed to study KdV–Burgers–Kuramoto equation. Lastly, we investigate symmetries of generalized time fractional KdV–Burgers–Kuramoto equation, and derive a conservation law. The obtained results illustrate that symmetry method is a very powerful method to deal with nonlinear partial differential equations. The results provide theoretical support for explaining complex nonlinear phenomena.

Hopf Bifurcation of KdV–Burgers–Kuramoto System with Delay Feedback

Chaotic phenomena may exist in nonlinear evolution equations. In many cases, they are undesirable but can be controlled. In this study, we deal with the chaos control of a three-dimensional chaotic system, reduced from a KdV–Burgers–Kuramoto equation. By adding a single delay feedback term into the chaotic system, we investigate the local stability and occurrence of Hopf bifurcation near the equilibrium point. Some dynamical properties including the direction and stability of bifurcated periodic solutions are presented by using the normal form theory and the center manifold theorem. Numerical simulations are illustrated which agree well with the theoretical results.

Numerical solution of variable-order space-time fractional KdV–Burgers–Kuramoto equation by using discrete Legendre polynomials

In this paper, a new version of the nonlinear space-time fractional KdV–Burgers–Kuramoto equation has been generated via the variable-order (VO) fractional derivatives defined in the Caputo type. A numerical method has been developed based on the discrete Legendre polynomials (LPs) and the collocation scheme for solving this equation. First, the solution of the problem is expanded in terms of the shifted discrete LPs. Then, this expansion and its derivatives, including the classical partial derivatives and the VO fractional partial derivatives are replaced in the equation. Eventually, the operational matrices of the shifted discrete LPs, including the classical derivatives and the VO fractional derivatives (which are derived in this study), and the collocation method are employed to convert the approximated problem into an algebraic system of equations. Some numerical results are given to illustrate the accuracy of the method.

Development of spreading symmetric two-waves motion for a family of two-mode nonlinear equations

In this work, a functional operator extracted from Korsunsky's technique is used to produce new two-mode nonlinear equations. These new equations describe the motion of two directional solitary-waves overlapping with an increasing phase-velocity and affected by two factors labeled as the dispersion and nonlinearity coefficients. To investigate the dynamics of this two-mode family, we construct the two-mode KdV–Burgers–Kuramoto equation (TMKBK) and two-mode Hirota–Satsuma model (TMHS). Two efficient schemes are used to assign the necessary constraints for existence of solutions and to extract them. The role of the phase-velocity on the motion of the obtained two-wave solutions is investigated graphically. Finally, all the obtained solutions are categorized according to their physical shapes.

PHYSICAL INSIGHT OF LOCAL FRACTIONAL CALCULUS AND ITS APPLICATION TO FRACTIONAL KDV–BURGERS–KURAMOTO EQUATION

Local fractional calculus is used to derive fractional partner of the KdV–Burgers–Kuramoto equation, its physical understanding is elucidated and its solution properties are revealed. Adomian’s decomposition method (ADM) and the variational iteration method (VIM) are applied to obtain its approximate analytical solution, the results have both theoretical importance and practical applications.

An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation

In this paper, our purpose is to present a wavelet Galerkin method for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation, which describes nonlinear physical phenomena and involves instability, dissipation, and dispersion parameters. The presented computational method in this paper is based on Gegenbauer wavelets. Gegenbauer wavelets have useful properties. Gegenbauer wavelets and the operational matrix of integration, together with the Galerkin method, were used to transform the time-fractional KBK equation into the corresponding nonlinear system of algebraic equations, which can be solved numerically with Newton’s method. Our aim is to show that the Gegenbauer wavelets-based method is efficient and powerful tool for solving the KBK equation with time-fractional derivative. In order to compare the obtained numerical results of the wavelet Galerkin method with exact solutions, two test problems were chosen. The obtained results prove the performance and efficiency of the presented method.

Some exact solutions of KdV-Burgers-Kuramoto equation

Some exact solutions of KdV-Burgers-Kuramoto (KBK) equation are derived by the anzas and tanh methods. Also, the most general Lie point symmetry group of the KBK equation are presented using the basic Lie symmetry method. As well as, the non-classical and weak symmetries of this equation, as well as the corresponding similarity reductions, are investigated. Finally, the classical and non-classical symmetries of KBK and KdV-Burgers (KB) equations are compared.

Painlevé analysis and some solutions of variable coefficient Benny equation

In this paper, variable coefficient Benny equation (also called the KdV Burgers–Kuramoto equation) has been considered. By using the Painleve analysis and Lie group analysis methods, the Painleve properties and symmetries have been studied. Some solutions of the reduced ODEs are obtained.

Nonlinear stability of strong rarefaction waves for the generalized KdV–Burgers–Kuramoto equation with large initial perturbation

In was shown in Ruan et al. (2008) [3] that rarefaction waves for the generalized KdV–Burgers–Kuramoto equation are nonlinearly stable provided that both the strength of the rarefaction waves and the initial perturbation are sufficiently small. The main purpose of this work is concerned with nonlinear stability of strong rarefaction waves for the generalized KdV–Burgers–Kuramoto equation with large initial perturbation. In our results, we do not require the strength of the rarefaction waves to be small and when the smooth nonlinear flux function satisfies certain growth condition at infinity, the initial perturbation can be chosen arbitrarily in H 1 ( R ) , while for a general smooth nonlinear flux function, we need to ask for the L 2 -norm of the initial perturbation to be small but the L 2 -norm of the first derivative of the initial perturbation can be large and, consequently, the H 1 ( R ) -norm of the initial perturbation can also be large.

Persistence of travelling fronts of KdV–Burgers–Kuramoto equation

We study the travelling fronts of KdV–Burgers–Kuramoto equation from geometric singular perturbation point of view. Motivated by the analogue between travelling fronts and heteroclinic orbits of the corresponding ordinary differential equations, we prove the persistence of these waves for sufficiently small dissipation. The result of numerical investigation also establishes our analysis.

Application of He’s variational iteration method and Adomian’s decomposition method to the fractional KdV–Burgers–Kuramoto equation

In this work, the fractional KdV–Burgers–Kuramoto equation is studied. He’s variational iteration method (VIM) and Adomian’s decomposition method (ADM) are applied to obtain its solution. Comparison with HAM is made to highlight the significant features of the employed methods and their capability of handling completely integrable equations.

Application of homotopy analysis method to fractional KdV–Burgers–Kuramoto equation

Abstract In the Letter, homotopy analysis method that developed for integer-order differential equation is directly extended to derive explicit and numerical solutions of nonlinear fractional differential equation for the first time. The fractional derivatives are described in the Caputo sense. To our knowledge, the Letter represents the first available numerical solutions of the fractional KdV–Burgers–Kuramoto equation.

Asymptotic stability of the rarefaction wave for the generalized KdV–Burgers–Kuramoto equation

In this paper, we will study the Cauchy problem for the generalized KdV–Burgers–Kuramoto equation, which represents a dissipative, stroboscopic and unstable system in physics. When the initial data is a small disturbance of a rarefaction wave of the inviscid Burgers equation, we prove the global existence of the solution to the corresponding Cauchy problem and asymptotic stability of the rarefaction wave. The analysis is based on a priori estimates and the L 2 -energy method.

New exact solutions of the KdV–Burgers–Kuramoto equation

A generalized F-expansion method is proposed and applied to the KdV–Burgers–Kuramoto equation. As a result, many new and more general exact travelling wave solutions are obtained including combined non-degenerate Jacobi elliptic function solutions, solitary wave solutions and trigonometric function solutions. The method can be applied to other nonlinear partial differential equations in mathematical physics.

Periodic Solutions to KdV–Burgers–Kuramoto Equation

In this paper, a new special ansatz solution, where ellipticequation satisfied by elliptic functions is taken as anintermediate transformation, is applied to solve theKdV-Burgers-Kuramoto equation, and many more new periodicsolutions are obtained, including solutions expressed in terms ofJacobi elliptic functions, solution expressed in terms ofWeierstrass elliptic function.

Traveling wave solutions for nonlinear differential equations: A unified ansätze approach

Abstract The aim of the paper is to introduce a general transformation for constructing analytic solutions for nonlinear differential equations. The main thrust of this alternate approach is to manipulate a unified ansatze to obtain exact solutions that are general solutions of simpler integrable equations. The ansatze is based on either the choice of an integrable differential operator or on a basis set of functions. Trigonometric, hyperbolic, Weierstrass and Jacobi elliptic functions can be used as building blocks for obtaining the exact solutions. The technique is implemented to acquire traveling wave solutions for the KDV–Burgers–Kuramoto equation.