In this article,the(1/G')-expansion method,the Bernoulli sub-ordinary differential equation method and the modified Kudryashov method are implemented to construct a variety of novel analytical solutions to the(3+1...In this article,the(1/G')-expansion method,the Bernoulli sub-ordinary differential equation method and the modified Kudryashov method are implemented to construct a variety of novel analytical solutions to the(3+1)-dimensional Boiti-Leon-Manna-Pempinelli model representing the wave propagation through incompressible fluids.The linearization of the wave structure in shallow water necessitates more critical wave capacity conditions than it does in deep water,and the strong nonlinear properties are perceptible.Some novel travelling wave solutions have been observed including solitons,kink,periodic and rational solutions with the aid of the latest computing tools such as Mathematica or Maple.The physical and analytical properties of several families of closed-form solutions or exact solutions and rational form function solutions to the(3+1)-dimensional Boiti-Leon-Manna-Pempinelli model problem are examined using Mathematica.展开更多
This study presents a large family of the traveling wave solutions to the two fourth-order nonlinear partial differential equations utilizing the Riccati-Bernoulli sub-ODE method.In this method,utilizing a traveling w...This study presents a large family of the traveling wave solutions to the two fourth-order nonlinear partial differential equations utilizing the Riccati-Bernoulli sub-ODE method.In this method,utilizing a traveling wave transformation with the aid of the Riccati-Bernoulli equation,the fourth-order equation can be transformed into a set of algebraic equations.Solving the set of algebraic equations,we acquire the novel exact solutions of the integrable fourth-order equations presented in this research paper.The physical interpretation of the nonlinear models are also detailed through the exact solutions,which demonstrate the effectiveness of the presented method.The Bäcklund transformation can produce an infinite sequence of solutions of the given two fourth-order nonlinear partial differential equations.Finally,3D graphs of some derived solutions in this paper are depicted through suitable parameter values.展开更多
In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. By a fractional co...In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established.展开更多
The generalized sub-ODE method, the rational （G＇/G）-expansion method, the exp-function method and the sine-cosine method are applied for constructing many traveling wave solutions of nonlinear partial differential ...The generalized sub-ODE method, the rational （G＇/G）-expansion method, the exp-function method and the sine-cosine method are applied for constructing many traveling wave solutions of nonlinear partial differential equations （PDEs）. Some illustrative equations are investigated by these methods and many hyperbolic, trigonometric and rational function solutions are found. We apply these methods to obtain the exact solutions for the generalized KdV-mKdV （GKdV-mKdV） equation with higherorder nonlinear terms. The obtained results confirm that the proposed methods are efficient techniques for analytic treatment of a wide variety of nonlinear partial differential equations in mathematical physics. We compare between the results yielding from these methods. Also, a comparison between our new results in this paper and the well-known results are given.展开更多
The purpose of the present paper is twofold. First, the projective Riccati equations （PREs for short） are resolved by means of a linearized theorem, which was known in the literature. Based on the signs and values o...The purpose of the present paper is twofold. First, the projective Riccati equations （PREs for short） are resolved by means of a linearized theorem, which was known in the literature. Based on the signs and values of coeffcients of PREs, the solutions with two arbitrary parameters of PREs can be expressed by the hyperbolic functions, the trigonometric functions, and the rational functions respectively, at the same time the relation between the components of each solution to PREs is also implemented. Second, more new travelling wave solutions for some nonlinear PDEs, such as the Burgers equation, the mKdV equation, the NLS^＋ equation, new Hamilton amplitude equation, and so on, are obtained by using Sub-ODE method, in which PREs are taken as the Sub-ODEs. The key idea of this method is that the travelling wave solutions of nonlinear PDE can be expressed by a polynomial in two variables, which are the components of each solution to PREs, provided that the homogeneous balance between the higher order derivatives and nonlinear terms in the equation is considered.展开更多
In this article, we employ the perturbed Fokas-Lenells equation(FLE), which represents recent electronic communications. The Riccati-Bernoulli Sub-ODE method which does not depend on the balance rule is used for thefi...In this article, we employ the perturbed Fokas-Lenells equation(FLE), which represents recent electronic communications. The Riccati-Bernoulli Sub-ODE method which does not depend on the balance rule is used for thefirst time to obtain the new exact and solitary wave solutions of this equation. This technique is direct, effective and reduces the large volume of calculations.展开更多
文摘In this article,the(1/G')-expansion method,the Bernoulli sub-ordinary differential equation method and the modified Kudryashov method are implemented to construct a variety of novel analytical solutions to the(3+1)-dimensional Boiti-Leon-Manna-Pempinelli model representing the wave propagation through incompressible fluids.The linearization of the wave structure in shallow water necessitates more critical wave capacity conditions than it does in deep water,and the strong nonlinear properties are perceptible.Some novel travelling wave solutions have been observed including solitons,kink,periodic and rational solutions with the aid of the latest computing tools such as Mathematica or Maple.The physical and analytical properties of several families of closed-form solutions or exact solutions and rational form function solutions to the(3+1)-dimensional Boiti-Leon-Manna-Pempinelli model problem are examined using Mathematica.
基金Supported by the National Natural Science Foundation of Education Depart ment of HenanProvince of China (2006110002) ,the Science Foundation of Henan University of Science and Technology(2004ZD002 ,2006ZY001)
文摘This study presents a large family of the traveling wave solutions to the two fourth-order nonlinear partial differential equations utilizing the Riccati-Bernoulli sub-ODE method.In this method,utilizing a traveling wave transformation with the aid of the Riccati-Bernoulli equation,the fourth-order equation can be transformed into a set of algebraic equations.Solving the set of algebraic equations,we acquire the novel exact solutions of the integrable fourth-order equations presented in this research paper.The physical interpretation of the nonlinear models are also detailed through the exact solutions,which demonstrate the effectiveness of the presented method.The Bäcklund transformation can produce an infinite sequence of solutions of the given two fourth-order nonlinear partial differential equations.Finally,3D graphs of some derived solutions in this paper are depicted through suitable parameter values.
文摘In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established.
文摘The generalized sub-ODE method, the rational （G＇/G）-expansion method, the exp-function method and the sine-cosine method are applied for constructing many traveling wave solutions of nonlinear partial differential equations （PDEs）. Some illustrative equations are investigated by these methods and many hyperbolic, trigonometric and rational function solutions are found. We apply these methods to obtain the exact solutions for the generalized KdV-mKdV （GKdV-mKdV） equation with higherorder nonlinear terms. The obtained results confirm that the proposed methods are efficient techniques for analytic treatment of a wide variety of nonlinear partial differential equations in mathematical physics. We compare between the results yielding from these methods. Also, a comparison between our new results in this paper and the well-known results are given.
基金The project supported in part by the Natural Science Foundation of Education Department of Henan Province of China under Grant No. 2006110002 and the Science Foundations of Henan University of Science and Technology under Grant Nos. 2004ZD002 and 2006ZY001
文摘The purpose of the present paper is twofold. First, the projective Riccati equations （PREs for short） are resolved by means of a linearized theorem, which was known in the literature. Based on the signs and values of coeffcients of PREs, the solutions with two arbitrary parameters of PREs can be expressed by the hyperbolic functions, the trigonometric functions, and the rational functions respectively, at the same time the relation between the components of each solution to PREs is also implemented. Second, more new travelling wave solutions for some nonlinear PDEs, such as the Burgers equation, the mKdV equation, the NLS^＋ equation, new Hamilton amplitude equation, and so on, are obtained by using Sub-ODE method, in which PREs are taken as the Sub-ODEs. The key idea of this method is that the travelling wave solutions of nonlinear PDE can be expressed by a polynomial in two variables, which are the components of each solution to PREs, provided that the homogeneous balance between the higher order derivatives and nonlinear terms in the equation is considered.
文摘In this article, we employ the perturbed Fokas-Lenells equation(FLE), which represents recent electronic communications. The Riccati-Bernoulli Sub-ODE method which does not depend on the balance rule is used for thefirst time to obtain the new exact and solitary wave solutions of this equation. This technique is direct, effective and reduces the large volume of calculations.
