A simple method to construct the traveling wave solutions to nonlinear evolution equations

Liu, Chunping; Zhou, Ruguang; Zhou, Mi

doi:10.1016/s0375-9601(98)00497-6

Cited by 12 publications

(13 citation statements)

References 7 publications

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“…The first explicit form of a traveling wave solution for the Fisher equation was obtained by Ablowitz and Zeppetella [21] using the Painlevé analysis. The same result was obtained by Liu et al [29] using the method of undetermined coefficients. The problem of selection of appropriate speed has been discussed in [30,31].…”

Section: ∂ T U(x T) − D∂ 2 XX U(x T) = αU(x T)(1 − U(x T)/usupporting

confidence: 85%

Nonstandard perturbation approximation and travelling wave solutions of nonlinear reaction diffusion equations

Kaushik¹

2007

Numerical Methods Partial

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This paper deals with the construction of a nonstandard numerical method to compute the travelling wave solutions of nonlinear reaction diffusion equations at high wave speeds. Related general properties are studied using the perturbation approximation. At high wave speed the perturbation parameter approaches to zero and the problem exhibits a multiscale character. That is, there are thin layers where the solution varies rapidly, while away from these layers the solution behaves regularly and varies slowly. Most of the conventional methods fail to capture this layer behavior. Thus, the quest for some new numerical techniques that may handle the travelling wave solutions at high wave speeds earns relevance. In this paper, one such parameter robust nonstandard numerical scheme is constructed, in the sense that its numerical solution converges in the maximum norm to the exact solution uniformly well for all finite wave speeds. To overcome the difficulty due to the nonlinearity, the problem is linearized using the quasilinearization process followed by nonstandard finite difference discretization. An extensive amount of analysis is carried out which uses a suitable decomposition of the error into smooth and singular component and a comparison principle combined with appropriate barrier functions. The error estimates are obtained, which ensures uniform convergence of the method. A set of numerical experiment is carried out in support of the predicted theory that validates computationally the theoretical results.

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Section: ∂ T U(x T) − D∂ 2 XX U(x T) = αU(x T)(1 − U(x T)/usupporting

confidence: 85%

Nonstandard perturbation approximation and travelling wave solutions of nonlinear reaction diffusion equations

Kaushik¹

2007

Numerical Methods Partial

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“…Since Hopf's [6] and Cole's [7] independent proof that Eq. (1.1) can be reduced to the linear heat equation by a proper nonlinear transformation, numerous studies have approached its solution [8][9][10][11][12][13][14][15][16][17][18]. Formal generalizations of the Burgers equation are the Burgers-Huxley [19][20][21][22][23], Fisher [24][25][26], Korteweg-de Vries-Burgers [27][28][29] and Kuramoto-Sivashinsky [30][31][32] equations.…”

Section: Introductionmentioning

confidence: 99%

A fourth-order numerical scheme for solving the modified Burgers equation

Bratsos

2010

Computers & Mathematics with Applications

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a b s t r a c tA finite-difference scheme based on fourth-order rational approximants to the matrixexponential term in a two-time level recurrence relation is proposed for the numerical solution of the modified Burgers equation. The resulting nonlinear system, which is analyzed for stability, is solved using an already known modified predictor-corrector scheme. The results arising from the experiments are compared with the corresponding ones known from the available literature.

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“…Nowadays, the literature of traveling solitary wave solutions and the related methods to the Burgers-KdV equation is very comprehensive and quite many elegant methods have been proposed by mathematicians, engineers and physicists. Among them are McIntosh [31] by using the Painlevé analysis, Zhang and Liu et al [32,33] by using the method of undetermined coefficients, Jeffery and Mohamad et al [34,35] by using two different methods: an ansatz and a direct method, Halford and Vlieg-Hulstman [36,37] by using an algorithmic approach, Wang [38] by using the homogeneous balance method, Parkes et al [39−41] by using the automated tanh method, Feng [42,43] by introducing the first integral method and an analytical method [44] , Guo and Zhang [45] by using the Jacobi elliptic function method, and so on. However, all these bounded traveling solitary wave solutions mentioned above to the Burgers-KdV equation can be proved to be algebraically equivalent to each other [28,36] .…”

Section: The Burgers-kdv Equationmentioning

confidence: 99%

Burgers-Korteweg-de Vries equation and its traveling solitary waves

Feng¹,

Meng

2007

SCI CHINA SER A

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The Burgers-Korteweg-de Vries equation has wide applications in physics, engineering and fluid mechanics. The Poincaré phase plane analysis reveals that the Burgers-Korteweg-de Vries equation has neither nontrivial bell-profile traveling solitary waves, nor periodic waves. In the present paper, we show two approaches for the study of traveling solitary waves of the Burgers-Korteweg-de Vries equation: one is a direct method which involves a few coordinate transformations, and the other is the Lie group method. Our study indicates that the Burgers-Korteweg-de Vries equation indirectly admits one-parameter Lie groups of transformations with certain parametric conditions and a traveling solitary wave solution with an arbitrary velocity is obtained accordingly. Some incorrect statements in the recent literature are clarified.

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A simple method to construct the traveling wave solutions to nonlinear evolution equations

Cited by 12 publications

References 7 publications

Nonstandard perturbation approximation and travelling wave solutions of nonlinear reaction diffusion equations

Nonstandard perturbation approximation and travelling wave solutions of nonlinear reaction diffusion equations

A fourth-order numerical scheme for solving the modified Burgers equation

Burgers-Korteweg-de Vries equation and its traveling solitary waves

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