Lie symmetry reductions and exact solutions of a coupled KdV–Burgers equation

Yang, Shaojie; Chen, Hua

doi:10.1016/j.amc.2014.01.044

Cited by 23 publications

(10 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The goal of the present paper is to investigate the steady double-diffusive convection of a viscous and electrically conductive fluid past a permeable vertical stretching/shrinking sheet using the Lie group symmetry method, which is very well described in the recent published papers by Das [30], Yang and Hua [31], and Stepanova [32]. Appropriate similarity variables are obtained from this group analysis, so that the partial differential equations are transformed into ordinary differential equations, which are then solved numerically.…”

Section: Nomenclaturementioning

confidence: 99%

Lie group symmetry method for MHD double-diffusive convection from a permeable vertical stretching/shrinking sheet

Roşca

Pop

2016

Computers & Mathematics with Applications

View full text Add to dashboard Cite

a b s t r a c tIn this paper, the problem of the steady magnetohydrodynamics (MHD) double-diffusive convection of a viscous and electrically conducting fluid from a permeable vertical stretching/shrinking sheet with the presence of buoyancy force is investigated numerically. The effect of the MHD on the flow, heat transfer and species concentration has been also discussed. The partial differential equations are reduced to ordinary (similarity) equations using the method of the Lie group symmetry. Numerical solutions of the resulting nonlinear boundary value problem are carried out using the function bvp4c from Matlab for different values of the governing parameters. It is found that the solutions of the ordinary (similarity) differential equations have two branches, upper and lower branch solutions, in a certain range of the stretching/shrinking, buoyancy, suction and magnetic parameters. It is shown that the reported results are in excellent agreement with those from the open literature for a forced convection flow when the concentration gradient is absent and the stretching sheet is impermeable. In order to establish which of upper and lower solutions are stable and which are not, a stability analysis has been performed. Thus, it is found that the upper branch solution is stable and, therefore, physically realizable in practice, while the lower branch solution is not stable and, thus, physically not realizable. The effects of the governing parameters on the reduced skin friction coefficient, reduced Nusselt number and reduced Sherwood number are shown in tables and figures. It results in that the governing parameters affect considerably the flow, heat transfer and concentration characteristics. In our opinion, the results are new and original.

show abstract

Section: Nomenclaturementioning

confidence: 99%

Lie group symmetry method for MHD double-diffusive convection from a permeable vertical stretching/shrinking sheet

Roşca

Pop

2016

Computers & Mathematics with Applications

View full text Add to dashboard Cite

show abstract

“…In spite of these problems, in recent years a variety of efficient and practical methods have been proposed by mathematicians and physicists. Some of these methods are the exp-function method [1], the Darboux transformation [2], the Lie group analysis [3], the modified simple equation method [4], the homogeneous balance scheme [5], the sine-cosine method, and the tanh-coth method. Some new and effective attempts at determining solutions of partial differential equations can be found in [6][7][8][9][10][11][12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

New Solutions of Gardner's Equation Using Two Analytical Methods

Ghanbari

Băleanu

2019

Front. Phys.

View full text Add to dashboard Cite

This article introduces and applies new methods to determine the exact solutions of partial differential equations that will increase our understanding of the capabilities of applied models in real-world problems. With these new solutions, we can achieve remarkable advances in science and technology. This is the basic idea in this article. To accurately describe this, some exact solutions to the Gardner's equation are obtained with the help of two new analytical methods including the generalized exponential rational function method and a Jacobi elliptical solution finder method. A set of new exact solutions containing four parameters is reported. The results obtained in this paper are new solutions to this equation that have not been introduced in previous literature. Another advantage of these methods is the determination of the varied solutions involving various classes of functions, such as exponential, trigonometric, and elliptic Jacobian. The three-dimensional diagrams of some of these solutions are plotted with specific values for their existing parameters. By examining these graphs, the behavior of the solution to this equation will be revealed. Mathematica software was used to perform the computations and simulations. The suggested techniques can be used in other real-world models in science and engineering.

show abstract

“…The investigation of the analytical solutions of NPDEs plays a prominent role in the study of nonlinear physical phenomena. In the recent past decades, there has been significant progress in the development of methods such as the inverse scattering method, 4 Hirota's bilinear method, 5 the similarity transformation method, [6][7][8][9] the homogeneous balance method, 10 the exp-function method, [11][12][13] the sine-cosine method, 14 the tanh function method, [15][16][17][18] the F-expansion method, 19 the Riccati equation rational expansion method, 20 the Weierstrass elliptic function method, 21 the Jacobi elliptic function expansion method, [22][23][24] the new coupled fractional reduced differential transform method, [25][26][27] the fractional subequation method, 28 and the Kudryashov and modified Kudryashov methods. 29 The main objective of the present work is to employ the improved Bernoulli subequation function method and the improved tan(Φ( )/2)-expansion method.…”

Section: Introductionmentioning

confidence: 99%

The new complex rational function prototype structures for the nonlinear Schrödinger‐inviscid burgers system

Ray

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, a system representing the coupling between the nonlinear Schrödinger equation and the inviscid burgers equation in modeling the interactions between short and long waves in fluids has been investigated. The new algorithms, like improved tan(Φ( )/2)-expansion method and improved Bernoulli subequation function method, have been proposed. The proposed methods have been discussed comprehensively in this article. Using these methods, some new prototype for exact solutions such as complex exponential, complex hyperbolic, and complex trigonometric function solutions have been obtained for nonlinear Schrödinger-inviscid burgers system. Through the present analysis, it has been established that, with the help of symbolic computation, these methods provide a straightforward and powerful mathematical tool for solving nonlinear evolution equations. KEYWORDS complex function solution, exponential function solution, hyperbolic function solution, improved Bernoulli subequation function method, improved tan(Φ( )/2)-expansion method, nonlinear Schrödinger equation, rational function solution, Schrödinger-inviscid burgers system, trigonometric function solution 6312

show abstract

Lie symmetry reductions and exact solutions of a coupled KdV–Burgers equation

Cited by 23 publications

References 5 publications

Lie group symmetry method for MHD double-diffusive convection from a permeable vertical stretching/shrinking sheet

Lie group symmetry method for MHD double-diffusive convection from a permeable vertical stretching/shrinking sheet

New Solutions of Gardner's Equation Using Two Analytical Methods

The new complex rational function prototype structures for the nonlinear Schrödinger‐inviscid burgers system

Contact Info

Product

Resources

About