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EXTENSION TO THE DIRECT METHODS AND APPLICATIONS TO THE GENERALIZED BURGERS EQUATION
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作者 ZhangQuanju FengFuye 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2002年第3期277-283,共7页
A generalization of the direct method of Clarkson and Kruskal for finding similarity reductions of partial differential equations with arbitrary functions is found and discussed for the generalized Burgers equation. T... A generalization of the direct method of Clarkson and Kruskal for finding similarity reductions of partial differential equations with arbitrary functions is found and discussed for the generalized Burgers equation. The corresponding reductions and the exact solutions due to the methods of the ordinary differential equations are then given by the methods. The results given here answer partially an open problem proposed by Clarkson, that is how to develop the direct method to seek symmetry reductions of nonlinear PDEs with arbitrary functions. 展开更多
关键词 direct method generalized burgers equation exact solution.
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Approximate homotopy similarity reduction for the generalized Kawahara equation via Lie symmetry method and direct method 被引量:1
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作者 刘希忠 《Chinese Physics B》 SCIE EI CAS CSCD 2010年第8期28-34,共7页
This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method. Using both methods it obtains the similarity reduction solutions an... This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method. Using both methods it obtains the similarity reduction solutions and similarity reduction equations of different orders, showing that the approximate homotopy direct method yields more general approximate similarity reductions than the approximate homotopy symmetry method. The homotopy series solutions to the generalized Kawahara equation are consequently derived. 展开更多
关键词 approximate homotopy symmetry method approximate homotopy direct method generalized Kawahara equation homotopy series solutions
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New Exact Travelling Wave Solutions for Generalized Zakharov-Kuzentsov EquationsUsing General Projective Riccati Equation Method 被引量:14
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作者 CHENYong LIBiao 《Communications in Theoretical Physics》 SCIE CAS CSCD 2004年第1期1-6,共6页
Applying the generalized method, which is a direct and unified algebraic method for constructing multipletravelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer alg... Applying the generalized method, which is a direct and unified algebraic method for constructing multipletravelling wave solutions of nonlinear partial differential equations (PDEs), and implementing in a computer algebraicsystem, we consider the generalized Zakharov-Kuzentsov equation with nonlinear terms of any order. As a result, wecan not only successfully recover the previously known travelling wave solutions found by existing various tanh methodsand other sophisticated methods, but also obtain some new formal solutions. The solutions obtained include kink-shapedsolitons, bell-shaped solitons, singular solitons, and periodic solutions. 展开更多
关键词 projective Riccati equation method generalized Zakharov-Kuzentsov equation exact solutions
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Trial function method and exact solutions to the generalized nonlinear Schrdinger equation with time-dependent coefficient 被引量:2
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作者 曹瑞 张健 《Chinese Physics B》 SCIE EI CAS CSCD 2013年第10期182-185,共4页
In this paper, the trial function method is extended to study the generalized nonlinear Schrodinger equation with time- dependent coefficients. On the basis of a generalized traveling wave transformation and a trial f... In this paper, the trial function method is extended to study the generalized nonlinear Schrodinger equation with time- dependent coefficients. On the basis of a generalized traveling wave transformation and a trial function, we investigate the exact envelope traveling wave solutions of the generalized nonlinear Schrodinger equation with time-dependent coefficients. Taking advantage of solutions to trial function, we successfully obtain exact solutions for the generalized nonlinear Schrodinger equation with time-dependent coefficients under constraint conditions. 展开更多
关键词 generalized nonlinear SchriSdinger equation exact solution trial function method
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Direct Reduction and Exact Solutions for Generalized Variable Coefficients 2D KdV Equation under Some Integrability Conditions 被引量:2
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作者 M.H.M.Moussa RehabM.El-Shiekh 《Communications in Theoretical Physics》 SCIE CAS CSCD 2011年第4期551-554,共4页
Based on the closed connections among the homogeneous balance (HB) method and Clarkson-KruSkal (CK) method, we study the similarity reductions of the generalized variable coefficients 2D KdV equation. In the meant... Based on the closed connections among the homogeneous balance (HB) method and Clarkson-KruSkal (CK) method, we study the similarity reductions of the generalized variable coefficients 2D KdV equation. In the meantime it is shown that this leads to a direct reduction in the form of ordinary differential equation under some integrability conditions between the variable coefficients. Two different cases have been discussed, the search for solutions of those ordinary differential equations yielded many exact travelling and solitonic wave solutions in the form of hyperbolic and trigonometric functions under some constraints between the variable coefficients. 展开更多
关键词 direct reduction method the generalized variable coefficients 2D KdV equation exact solutions
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Exact Solutions of Generalized Burgers-Fisher Equation with Variable Coefficients 被引量:1
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作者 陈博奎 闵松强 汪秉宏 《Communications in Theoretical Physics》 SCIE CAS CSCD 2010年第3期443-449,共7页
The generalized Riccati equation vational expansion method is extended in this paper. Several exact solutions for the generalized Burgers-Fisher equation with variable coefficients are obtained by this method, and som... The generalized Riccati equation vational expansion method is extended in this paper. Several exact solutions for the generalized Burgers-Fisher equation with variable coefficients are obtained by this method, and some of which are derived for the first time. It is concluded from the results that this approach is simple and efficient even in solving partial differential equations with variable coefficients. 展开更多
关键词 generalized Riccati equation rational expansion method generalized variable coefficient burgers-Fisher equation exact solution
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Some new exact solutions to the Burgers-Fisher equation and generalized Burgers-Fisher equation 被引量:1
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作者 姜璐 郭玉翠 徐淑奖 《Chinese Physics B》 SCIE EI CAS CSCD 2007年第9期2514-2522,共9页
Some new exact solutions of the Burgers-Fisher equation and generalized Burgers-Fisher equation have been obtained by using the first integral method. These solutions include exponential function solutions, singular s... Some new exact solutions of the Burgers-Fisher equation and generalized Burgers-Fisher equation have been obtained by using the first integral method. These solutions include exponential function solutions, singular solitary wave solutions and some more complex solutions whose figures are given in the article. The result shows that the first integral method is one of the most effective approaches to obtain the solutions of the nonlinear partial differential equations. 展开更多
关键词 the first integral method burgers-Fisher equation exact solutions
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New Exact Explicit Solutions of the Generalized Zakharov Equation via the First Integral Method 被引量:1
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作者 Yuhuai Sun Hanlei Hu Jian Zhang 《Open Journal of Applied Sciences》 2014年第5期249-257,共9页
The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the first integral meth... The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the first integral method, in the form of trigonometric and exponential functions. The results show the first integral method is an efficient way to solve the coupled nonlinear equations and get rich explicit analytical solutions. 展开更多
关键词 generalized ZAKHAROV equation First INTEGRAL method exact EXPLICIT solutions
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General Solution of Two Generalized Form of Burgers Equation by Using the (<i>G</i><sup>'</sup>/<i>G</i>)-Expansion Method 被引量:1
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作者 Abdollah Borhanifar Reza Abazari 《Applied Mathematics》 2012年第2期158-168,共11页
In this work, the (G'/G)-expansion method is proposed for constructing more general exact solutions of two general form of Burgers type equation arising in fluid mechanics namely, Burgers-Korteweg-de Vries (Burger... In this work, the (G'/G)-expansion method is proposed for constructing more general exact solutions of two general form of Burgers type equation arising in fluid mechanics namely, Burgers-Korteweg-de Vries (Burgers-KdV) and Burger-Fisher equations. Our work is motivated by the fact that the (G'/G)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system. 展开更多
关键词 (G'/G)-Expansion method generalized burgers-KdV equation generalized burgers-Fisher equation Hyperbolic FUNCTION solutionS Trigonometric FUNCTION solutionS
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Exact Solutions for the Generalized KdV Equation: Modified Homogeneous Balance Method
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作者 WANG Xiu-mei ZHU Yue-feng 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2008年第2期260-269,共10页
In this paper, we present a solution methodology to obtain exact solutions of some nonlinear evolution equation by modifying the homogeneous balance method. Based on the modified homogeneous balance method, several ki... In this paper, we present a solution methodology to obtain exact solutions of some nonlinear evolution equation by modifying the homogeneous balance method. Based on the modified homogeneous balance method, several kinds of exact(new) solutions of the generalized KdV equation are obtained. 展开更多
关键词 generalized KdV equation modified homogeneous balance method exact solutions
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Symmetry Groups and New Exact Solutions to (2+1)-Dimensional Variable Coefficient Canonical Generalized KP Equation 被引量:7
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作者 ZHANG Li-Hua LIU Xi-Qiang BAI Cheng-Lin 《Communications in Theoretical Physics》 SCIE CAS CSCD 2007年第3X期405-410,共6页
In this paper, the modified CK's direct method to find symmetry groups of nonlinear partial differential equation is extended to (2+1)-dimensional variable coeffficient canonical generalized KP (VCCGKP) equation... In this paper, the modified CK's direct method to find symmetry groups of nonlinear partial differential equation is extended to (2+1)-dimensional variable coeffficient canonical generalized KP (VCCGKP) equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation. 展开更多
关键词 (2+1)-dimensional variable coefficient canonical generalized KP (VCCGKP) equation modified CK's'direct method symmetry groups Lie symmetry similarity reductions new exact solutions
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Symmetry Transformations and Exact Solutions of a Generalized Hyperelastic Rod Equation 被引量:1
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作者 Ran Wang Xuegang Yuan +2 位作者 Hongwu Zhang Jing Zhang Na Lv 《Computers, Materials & Continua》 SCIE EI 2018年第5期345-357,共13页
In this paper,a nonlinear wave equation with variable coefficients is studied,interestingly,this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressi... In this paper,a nonlinear wave equation with variable coefficients is studied,interestingly,this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities.With the aid of Lou’s direct method1,the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained.The corresponding numerical examples of exact solutions are presented by using different coefficients.Particularly,while the variable coefficients are taken as some special constants,the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients,which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials.Using the same method to solve the nonlinear wave equation,the validity and rationality of this method are verified. 展开更多
关键词 generalized hyperelastic rod equation symmetry transformation Lou’s direct method exact solution
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Group Analysis, Fractional Explicit Solutions and Conservation Laws of Time Fractional Generalized Burgers Equation
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作者 Gang-Wei Wang A. H. Kara 《Communications in Theoretical Physics》 SCIE CAS CSCD 2018年第1期5-8,共4页
The generalized fractional Burgers equation is studied in this paper. Using the classical Lie symmetry method, all of the vector fields and symmetry reduction of the equation with nonlinearity are constructed. In part... The generalized fractional Burgers equation is studied in this paper. Using the classical Lie symmetry method, all of the vector fields and symmetry reduction of the equation with nonlinearity are constructed. In particular, an exact solution & provided by using the ansatz method. In addition, other types of exact solution are obtained via the invariant subspace method. Finally, conservation laws for this equation are derived. 展开更多
关键词 time fractional generalized burgers equation Lie symmetry method explicit solutions conservation laws
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A Generalized Variable-Coefficient Algebraic Method Exactly Solving (3+1)-Dimensional Kadomtsev-Petviashvilli Equation 被引量:3
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作者 BAI Cheng-Lin BAI Cheng-Jie ZHAO Hong 《Communications in Theoretical Physics》 SCIE CAS CSCD 2005年第5X期821-826,共6页
A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for (3+1)-dimensional Kadomtsev-Petviashvilli (KP) equation. Among them, th... A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for (3+1)-dimensional Kadomtsev-Petviashvilli (KP) equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions. 展开更多
关键词 generalized variable-coefficient algebraic method (3+1)-dimensional KP equation exact explicit solutions
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Travelling solitary wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order 被引量:2
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作者 邓习军 燕子宗 韩立波 《Chinese Physics B》 SCIE EI CAS CSCD 2009年第8期3169-3173,共5页
In this paper, the travelling wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some e... In this paper, the travelling wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some exact explicit travelling solitary wave solutions for the above equation are obtained. As a result, some minor errors and some known results in the previousl literature are clarified and improved. 展开更多
关键词 travelling wave solutions first integral method generalized burgers-Huxley equation with nonlinear terms of any order
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Direct perturbation method for perturbed complex Burgers equation
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作者 程雪苹 林机 姚建明 《Chinese Physics B》 SCIE EI CAS CSCD 2009年第2期391-394,共4页
So far, Lou's direct perturbation method has been applied successfully to solve the nonlinear SchrSdinger equation(NLSE) hierarchy, such as the NLSE, the coupled NLSE, the critical NLSE, and the derivative NLSE. Bu... So far, Lou's direct perturbation method has been applied successfully to solve the nonlinear SchrSdinger equation(NLSE) hierarchy, such as the NLSE, the coupled NLSE, the critical NLSE, and the derivative NLSE. But to our knowledge, this method for other types of perturbed nonlinear evolution equations has still been lacking. In this paper, Lou's direct perturbation method is applied to the study of perturbed complex Burgers equation. By this method, we calculate not only the zero-order adiabatic solution, but also the first order modification. 展开更多
关键词 direct perturbation method perturbed complex burgers equation asymptotical solutions
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Application of the Generalized Simplest Equation Method to the Burgers Equation
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作者 Yuexing Bai Sudao Bilige +1 位作者 Xiaoqing Gao Jianqing Lü 《Journal of Applied Mathematics and Physics》 2017年第1期101-109,共9页
We successfully constructed wide classes of exact solutions for the Burgers equation by using the generalized simplest equation method. This method yielded a B&aumlcklund transformation between the Burgers equatio... We successfully constructed wide classes of exact solutions for the Burgers equation by using the generalized simplest equation method. This method yielded a B&aumlcklund transformation between the Burgers equation and a related constraint equation. By dealing with the constraint equation, we obtained the traveling wave solutions and non-traveling wave solutions of the Burgers equation. 展开更多
关键词 The generalized Simplest equation method exact solution the burgers equation
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Explicit Solutions of (2+1)-Dimensional Canonical Generalized KP, KdV, and (2+1)-Dimensional Burgers Equations with Variable Coefficients
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作者 ZHANG Lin-Lin LIU Xi-Qiang 《Communications in Theoretical Physics》 SCIE CAS CSCD 2009年第11期784-790,共7页
In this paper, using the generalized (G1/G)-expansion method and the auxiliary differential equation method, we discuss the (2+1)-dimensional canonical generalized KP (CGKP), KdV, and (2+1)-dimensional Burge... In this paper, using the generalized (G1/G)-expansion method and the auxiliary differential equation method, we discuss the (2+1)-dimensional canonical generalized KP (CGKP), KdV, and (2+1)-dimensional Burgers equations with variable coetticients. Many exact solutions of the equations are obtained in terms of elliptic functions, hyperbolic functions, trigonometric functions, and rational functions. 展开更多
关键词 generalized (G1/G)-expansion method auxiliary equation exact solution
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DEGENERATE BOUNDARY LAYER SOLUTIONS TO THE GENERALIZED BENJAMIN-BONAMAHONY-BURGERS EQUATION
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作者 肖清华 陈正争 《Acta Mathematica Scientia》 SCIE CSCD 2012年第5期1743-1758,共16页
This paper is concerned with the convergence rates of the global solutions of the generalized Benjamin-Bona-Mahony-Burgers(BBM-Burgers) equation to the corresponding degenerate boundary layer solutions in the half-s... This paper is concerned with the convergence rates of the global solutions of the generalized Benjamin-Bona-Mahony-Burgers(BBM-Burgers) equation to the corresponding degenerate boundary layer solutions in the half-space.It is shown that the convergence rate is t-(α/4) as t →∞ provided that the initial perturbation lies in H α 1 for α 〈 α(q):= 3 +(2/q),where q is the degeneracy exponent of the flux function.Our analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant introduced in [1] 展开更多
关键词 generalized BBM-burgers equation degenerate boundary layer solutions convergence rates Hardy's inequality space-time weighted energy method
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Exact Solutions to the Boussinesq-Burgers Equations
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作者 Xiangzheng Li Baoan Li +1 位作者 Jinlan Chen Mingliang Wang 《Journal of Applied Mathematics and Physics》 2017年第9期1720-1724,共5页
A nonlinear transformation from the solution of a linear equation to the solution of the Boussinesq-Burgers equations is derived by using the simplified homogeneous balance method. Based on the nonlinear transformatio... A nonlinear transformation from the solution of a linear equation to the solution of the Boussinesq-Burgers equations is derived by using the simplified homogeneous balance method. Based on the nonlinear transformation and various given solutions of the linear equation, various exact solutions, including solitary wave solutions, rational solutions, the solutions containing hyperbolic functions and the solutions containing trigonometric functions, of the Boussinesq-Burgers equations are obtained. 展开更多
关键词 Boussinesq-burgers equations Nonlinear Transformation Simplified HOMOGENEOUS BALANCE method exact solutions
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