Travelling wave solutions to the twodimensional kdv. Asymptotic behavior of solutions to the generalized kdvburgers equation ikki fukuda department of mathematics, hokkaido university abstract we study the asymptotic behavior of global solutions to the initial value problem for the generalized kdvburgers equation. Lecture notes massachusetts institute of technology. The idea, suggested by gao ge in 1985, that kdvburgers equation can be regarded as the normal equation of turbulence is shown to be meaningful by the present paper using the travelling wave analytic solution of kdvburgers equation. These solutions may be degenerated into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function. The kortewegde vries equation kdv equation describes the theory of water waves in shallow channels, such as a canal. The existence and certain qualitative properties of travellingwave solutions to the kortewegde vriesburgers equation, are established. In this approach a number of unknown constants are. Lie symmetry analysis for the variable coefficients kdv. Solving the kdvburgerskuramoto equation by a combination. Travelling waves for a nonlocal kdvburgers equation. The local dynamics of the kdvburgers equation with periodic boundary conditions is studied. Kdv can be solved by means of the inverse scattering transform.
The first method is a numerical one, namely the finite differences with variable mesh. The kdvburgerskuramoto equation 18, 19 is where,, and are constants. Asymptotic behavior of solutions to the generalized. Karthigai selvam apg and research department of mathematics, thiagarajar college madurai 625 009, india email. In section 3, we obtain the exact solitary wave solutions of the variable co. Approximate analysis to the kdvburgers equation ut math. Burgers equation and kdv equation choices of 0 and p 2 lead 1 to the burgers equation 3. Solitons in the kortewegde vries equation kdv equation. The kdvburgers equation is a onedimension generalization of the model description of the density and velocity. Solution of the burgers equation with nonzero viscosity 1 2. Exact solutions of kdvburgers equation by expfunction method.
Equation 1 is an extension of the kdv burgers equation for the twodimensional case just as in the relationship between the kdv equation and the kp equation. Exact solutions to the kdvburgers equation sciencedirect. Pdf in this work we use a generalized tanh method to solve the korteweg devries equation and kortewegde vriesburgers equation. Lie group method for solving the generalized burgers. Asymptotic behavior of solutions to the generalized kdvburgers. A special nonlinear partial differential equation is derived that plays the role of a normal form, i. In section 3, we apply ham to construct approximate solutions for the general perturbed kdv burgers equation. Asymptotic behavior of solutions to the generalized kdvburgers equation. Travellingwave solutions to the kortewegde vriesburgers. The nonlinear advective terms are computed based on the classical constrained interpolation profile cip method, which is coupled with a highorder compact scheme for thirdorder derivatives in kortewegde vries burgers equation. Exact solutions of the rosenau hyman equation, coupled kdv. This equation is an important mathematical model arising in many di. In this paper, we use a computer algebra system to search for appropriate exact solutions. Homotopy perturbation method for fractional kdvburgers.
Application of the homotopy analysis method for solving the. One can expect that the solution to this equation converges to a selfsimilar solution to the burgers equation, due to earlier works related to this problem. Travelling wave solutions to the twodimensional kdvburgers. Analytical wave solutions for foam and kdvburgers equations. Pego oscillatory instability of traveling waves for a kdv burgers equation 2. Kdvburgers equation is transformed into the standard form of the kdvburgers equation under the generalized transformation. Chebyshev and legendre via galerkin method for solving kdv. Based on the analysis on the characteristics of the kdv equation, kuramotosivashinsky equation and kdvburgerskuramoto equation, a combination method is proposed to construct the explicit exact solutions for the kdvburgerskuramoto equation by combining with those of the kdv equation and kuramotosivashinsky equation. The proper analytical solution of the kortewegde vries. Examples of in nitedimensional case inverse scattering solutions. Approximate analysis to the kdvburgers equation zhaosheng feng department of mathematics university of texaspan american 1201 w. It may be a more flexible tool for physicists than the burgers equation. The kdvburgers equation is a nonlinear partid differential equation which arise in the study of many physical problems 1,2.
This equation is an important mathematical model arising in many different physical contexts to describe many phenomena which are simultaneously involved in nonlinearity, dissipation, dispersion, and instability, especially at the description of turbulence processes. Hokkaido university collection of scholarly and academic. At present, studies of the kdv equation v 0 and burgers equation 6 0. It has been used in several different fields to describe various physical phenomena of interest.
Oscillatory instability of traveling waves for a kdv. Pdf a comparison between two different methods for solving kdv. Some travelling wave solutions of kdvburgers equation 1057 equation 14, we have m 2. At present, studies of the kdv equation v 0 and burgers equation 6 0 have been undertaken, but studies of the kdvburgers equation are just beginning. When we convert kbke into ordinary differential equation, and by integrating two times, and getting to zero the constants of. Some exact solutions of kdv burgers kuramoto equation to cite this article. Special attention is given to the correct computation of the. Some exact solutions of kdv burgers kuramoto kbk equation are derived by the anzas and tanh methods.
Exact traveling wave solutions of modified kdvzakharov. Also, the most general lie point symmetry group of the kbk equation are presented using the. Application of the homotopy analysis method for solving. Numerical methods for hyperbolic conservation laws 9 6. Pdf some exact solutions of kdvburgerskuramoto equation. Spatially periodic complexvalued solutions of the burgers and kdvburgers equations are studied in this paper. It arises from many physical contexts, for example, the propagation of undular bores in a shallow water 7, 8, the. Asymptotic behavior of solutions to the generalized kdv. After making a series of transformations, we convert the kortewegde vriesburgers equation into an emdenfowler equation. Section 2 derives the semidiscrete approximation to the system in a manner suggested in the paper of goodman and lax 11. It is a nonlinear equation which exhibits special solutions, known as solitons, which are stable and do not disperse with time. We study the asymptotic behavior of global solutions to the initial value problem for the generalized kdv burgers equation.
Shock waves due to thde kdvburgerss equation were observed in dusty plasma by nakamura et al. A generalized tanh function method based on riccati equation was derived to obtain multiple soliton solutions containing some trigonometric, hyperbolic and complex functions for the kdvb equation 15. Direct numerical simulations dns have substantially contributed to our understanding of the disordered. Numerical solution of kortewegde vriesburgers equation by. The basic idea is, using an algebraic map, to transform the whole real line into a bounded interval where we can apply a fourier expansion. Chapter 3 burgers equation one of the major challenges in the. The solution is illustrated to agree well with phase plane analysis.
Using this method, we have found some extra family of solutions, is the best thing about this method. Some new analytical solutions for the nonlinear time. Based on the analysis on the characteristics of the kdv equation, kuramotosivashinsky equation and kdv burgers kuramoto equation, a combination method is proposed to construct the explicit exact solutions for the kdv burgers kuramoto equation by combining with those of the kdv equation and kuramotosivashinsky equation. In mathematics, the kortewegde vries kdv equation is a mathematical model of waves on shallow water surfaces. In the present work, by introducing a new potential function and by using the hyperbolic tangent method and an exponential rational function approach, a travelling wave solution to the kdvburgers kdvb equation is presented. Some exact solutions of kdvburgerskuramoto equation to cite this article. The proper analytical solution of the kortewegde vriesburgers.
It is particularly notable as the prototypical example of an exactly solvable model, that is, a nonlinear partial differential equation whose solutions can be exactly and precisely specified. The equation and derivatives appears in applications including shallowwater waves and plasma physics. Putting 15 in 14 and equating the coe cients of powers of y to zero. The kdvburgers equation defines the waves on lower water surfaces.
In section 3, we apply ham to construct approximate solutions for the general perturbed kdvburgers equation. The kdvburgers kdvb equation which is derived by su and gardner appears in the study of the weak effects of dispersion, dissipation, and nonlinearity in waves propagating in a. Solitary wave solution of the variable coefficient kdv. New exact solutions to the kdvburgerskuramoto equation. The kortewegde vries equation is nonlinear, which makes numerical solution important. Pdf a comparison between two different methods for solving. For phase plane analysis the system is linearised around.
It is observed that both methods lead to the same type of solution. This equation is equal to the kdv equation if a viscous dissipation term vu, is added. Sl evolutionary vessels examples plan of the lecture. Yuanxi and jiashi 14 developed many solitary wave solutions for the kdvb equation by the superposition method based on the analysis on the features of the burgers, the kdv and the kdv burgers equations. Solitons in the kortewegde vries equation kdv equation introduction the kortewegde vries equation kdv equation describes the theory of water waves in shallow channels, such as a canal. Yuanxi and jiashi 14 developed many solitary wave solutions for the kdvb equation by the superposition method based on the analysis on the features of the burgers, the kdv and the kdvburgers equations. Soliton solutions of fractional order kdvburgers equation. In this paper, a hybrid compactcip scheme is proposed to solve kortewegde vries burgers equation. These solutions may be degenerated into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in. A travelling wave solution to the kdvburgers equation.
In section 4, we discuss the accuracy of these solutions with the small perturbation term as illustrations. Abstract in this paper, we present a new pseudospectral method to solve the initial value problem associated to a nonlocal kdvburgers equation involving a caputotype fractional derivative. Some exact solutions of kdvburgerskuramoto kbk equation are derived by the anzas and tanh methods. In this paper, we discuss the liouville integrability of the burgerskortewegde vries equation under certain parametric condition. On new travelling wave solutions of the kdv and the kdv. We study the asymptotic behavior of global solutions to the initial value problem for the generalized kdvburgers equation. The strong stability preserving thirdorder rungekutta time.
Some travelling wave solutions of kdvburgers equation. For this the fractional complex transformation have been implemented to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations, in the sense of the jumaries modified riemannliouville derivative. Some exact solutions of kdvburgerskuramoto equation. It is shown that for any sufficiently large time t, there exists an explicit initial datum such that its corresponding solution of the burgers equation blows up at t. Notes on burgerss equation maria cameron contents 1. Extended bspline collocation method for kdvburgers equation. Pdf a pseudospectral method for a nonlocal kdvburgers. The above invariant condition 6 could be written into the equivalent form. The generalized kdvburgers equation ikki fukuda department of mathematics, hokkaido university abstract we study the asymptotic behavior of global solutions to the initial value problem for the generalized kdvburgers equation. Pdf a comparison between two different methods for. The strength of fractional kdv equation is the nonlocal property 11121415 16 1718192021. Pdf some exact solutions of kdvburgerskuramoto kbk equation are derived by the anzas and tanh methods. Pdf the kortewegde vriesburgers equation and its approximate. This work addresses the global regularity issue on solutions of the complex burgers and kortwegde vries kdv burgers equations ut.
With the aid of generalized elliptic method and fouriers transform method, the approximate solutions of double periodic form are obtained. Normal form for the kdvburgers equation springerlink. An approximate solution in series form is obtained by means of the adomian decomposition method. The first is a direct one based on a combination of solutions to the kdv equation and burgers equation. Attention will be focused on the spatially periodic solutions, namely x. Jul 15, 2004 in the present work, by introducing a new potential function and by using the hyperbolic tangent method and an exponential rational function approach, a travelling wave solution to the kdvburgers kdvb equation is presented. In this study, the lie group method for constructing exact and numerical solutions of the generalized timedependent variable coefficients burgers, burgerskdv, and kdv equations with initial and boundary conditions is presented. It allows us to represent, in section 3, the fast part of the system. One can expect that the solution to this equation converges to a. An approximate solution is obtained by means of the adomian decomposition method. In addition, the global convergence and regularity of series solutions is established for initial data satisfying.
Jul 16, 2016 the local dynamics of the kdvburgers equation with periodic boundary conditions is studied. Approximate solution of the burgerskortewegde vries equation. College, madurai 625 019, india abstractthe kdv burgers equation has wide applications in physics. The kdv burgers equation defines the waves on lower water surfaces.
The homotopy analysis method is applied to solve the variable coefficient kdv burgers equation. Now we will discuss some of the obtained results of viscous burgers equation and their graphical representations. Linear stability in order to consider the linear stability of the traveling wave, we seek solutions of 1. Also, the kdvburgers equation is a onedimension generalization of the model description of the density and velocity fields that takes into account pressure forces as well as the viscosity and the dispersion. In this article, we obtain a large number of exact traveling wave solutions including solitary wave solutions for modified kdvzk equation and viscous burgers equation through enhanced g gexpansion method. The kdvburgers kdvb equation which is derived by su and gardner appears in the study of the weak effects of dispersion, dissipation, and nonlinearity in waves propagating in a liquidfilled elastic tube.
Introduction qualitative analysis approximate solution conclusion acknowledgement. Lie group theory is applied to determine symmetry reductions which reduce the nonlinear partial differential equations to ordinary differential equations. Oscillatory instability of traveling waves for a kdvburgers. The homotopy analysis method is applied to solve the variable coefficient kdvburgers equation. In this article, the new exact travelling wave solutions of the timeand spacefractional kdvburgers equation has been found. The kdvburgers equation 2729 is a very common and important equation in studying solitary waves and acoustic waves in plasma physics 30. Let us consider the solution of kdvburgers equation is of the form.
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