Riccati Equation Rational Expansion Method Research Articles

New exact solutions of optical metamaterial model with the Kerr law nonlinearity

This paper deals with the investigation of the newly obtained solitary solutions of perturbed optical metamaterial with Kerr law nonlinearity. Two novel analytical techniques, namely, the extended Riccati equation rational expansion (ERERE) method and the extended rational [Formula: see text]-expansion (EREE) methods are applied to develop new solitary solutions. The resultant wave solutions are periodic wave solution, singular kink, anti-peakon, dark, singular anti-kink, anti-kink, gray and singular soliton solutions. The detailed dynamics of the retrieved solutions are the most potent influence of the proposed methods which shows the efficacy and fruitfulness of both the methods.

Computational wave solutions of generalized higher-order nonlinear Boussinesq dynamical wave equation

A Riccati equation rational expansion (RERE) method is implemented to attain solutions of nonlinear partial differential equations (NLPDEs). Subsequently, the technique is used to attain families or sets of nonlinear solutions for the third extended fifth-order nonlinear equation and the generalized (2 + 1)-dimensional Boussinesq equation. By the use of the RERE method, exciting and new forms of solution such as rational solitary wave solutions, solitary wave solutions and periodic solitary wave solutions are obtained. Some of the solutions are demonstrated graphically with specific values assigned to the parameters. It is worth noting that the application of this method is very effective and reliable.

Fractional Riccati Equation Rational Expansion Method For Fractional Differential Equations

In this paper, a new fractional Riccati equation rational expansion method is proposed to establish new exact solutions for fractional differential equations. For illustrating the validity of this metho d, we apply it to the nonlinear fractional Sharma-Tasso- Olever (STO) equation, the nonlinear time fractional biological population model and the nonlinear fractional foam drainage equation. Compared with the existing results in the literature, more exact solutions are o btained by the proposed method. We also illustrate the application of the established exact solutions.

New Exact Solutions of Rational Expansion Method for the Variable Coefficient Nonlinear Equation with Forced Term

With the aid of symbolic computation system Maple, several new kinds of generalized exact solutions for the variable coefficient combined KdV equation and Chaffee-Infante equation with forced term are obtained by using a new generalize Riccati equation rational expansion method. This approach can also be applied to other variable coefficient nonlinear evolution equations.

Application of rational expansion method for stochastic differential equations

By means of the Hermite transformation, a new general ansätz and the symbolic computation system Maple, we apply the Riccati equation rational expansion method [24] to uniformly construct a series of stochastic non-traveling wave solutions for stochastic differential equations. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions. The method can also be applied to solve other nonlinear stochastic differential equation or equations.

Applications of the New Compound Riccati Equations Rational Expansion Method and Fan's Subequation Method for the Davey-Stewartson Equations

We used what we called extended Fan's sub-equation method and a new compound Riccati equations rational expansion method to construct the exact travelling wave solutions of the Davey-Stewartson (DS) equations. The basic idea of the proposed extended Fan's subequation method is to take fulls advantage of the general elliptic equations, involving five parameters, which have many new solutions and whose degeneracies lead to special subequations involving three parameters like Riccati equation, first-kind elliptic equation, auxiliary ordinary equation and generalized Riccati equation. Many new exact solutions of the Davey-Stewartson (DS) equations including more general soliton solutions, triangular solutions, and double-periodic solutions are constructed by symbolic computation.

A new generalized Riccati equation rational expansion method and its application

With the aid of symbolic computation system Maple, several new kinds of generalized exact solutions for the compound KdV system and extended KdV-Burgers system with nonlinear terms of any order are obtained by using a new generalize Riccati equation rational expansion method. This approach can also be applied to other nonlinear evolution equations with nonlinear terms of any order.

The Generalized Multiple Riccati Equations Rational Expansion Method and Its Application to the (2+1)- dimensional Painlevé Integrable Burgers Equations

Article The Generalized Multiple Riccati Equations Rational Expansion Method and Its Application to the (2+1)- dimensional Painlevé Integrable Burgers Equations was published on July 1, 2009 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 10, issue 7).

A series of complexiton soliton solutions for non-linear Hirota–Satsuma equations using the generalized multiple Riccati equations rational expansion method

In this article, the generalized multiple Riccati equations rational expansion method has been used to construct a series of complexiton soliton solutions for the non-linear Hirota–Satsuma equations. With the help of symbolic computation software as Maple or Mathematica, we obtain many new types of complexiton soliton solutions, i.e. various combination of trigonometric periodic function and hyperbolic function solutions, various combination of trigonometric periodic function and rational function solutions, and various combination of hyperbolic function and rational function solutions.

Symbolic computation and non-travelling wave solutions of (2 + 1)-dimensional nonlinear evolution equations

In this paper, the multiple Riccati equations rational expansion method is further extended to construct non-travelling wave solutions of the (2 + 1)-dimensional Painlevé integrable Burgers equation and the (2 + 1)-dimensional Breaking soliton equation, as a result, some double solitary-like wave solutions and complexiton solutions of the two equations are obtained. The extended method can also be applied to solve some other nonlinear evolution equations.

A new generalized compound Riccati equations rational expansion method to construct many new exact complexiton solutions of nonlinear evolution equations with symbolic computation

Based on symbolic computation and the idea of rational expansion method, a new generalized compound Riccati equations rational expansion method (GCRERE) is suggested to construct a series of exact complexiton solutions for nonlinear evolution equations. Compared with most existing rational expansion methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general complexiton solutions. The validity and reliability of the method is tested by its application to the (2+1)-dimensional Burgers equation. It is shown that more complexiton solutions can be found by this new method.

Generalized multiple Riccati equations rational expansion method with symbolic computation to construct exact complexiton solutions of nonlinear partial differential equations

In this paper, based on symbolic computation and the idea of rational expansion method, a generalized multiple Riccati equations rational expansion method (GMRERE) is proposed to uniformly construct a series of exact complexiton solutions for nonlinear partial differential equations. Compared with most existing tanh function methods and rational expansion methods, the proposed method not only recovers some known solutions, but also finds some new and general complexiton solutions. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equation.

Generalized Riccati equation rational expansion method and its application

By directly extending the Riccati equation rational expansion method, a new method is developed for constructing a series of exact analytical solutions of an averaged dispersion-managed fiber system equation (ADmE), including some new rational formal soliton-like solutions, triangle-like solutions and rational solutions.

Complexiton solutions of the (2+1)-dimensional dispersive long wave equation

In this pager a pure algebraic method implemented in a computer algebraic system, named multiple Riccati equations rational expansion method, is presented to construct a novel class of complexiton solutions to integrable equations and nonintegrable equations. By solving the (2+1)-dimensional dispersive long wave equation, it obtains many new types of complexiton solutions such as various combination of trigonometric periodic and hyperbolic function solutions, various combination of trigonometric periodic and rational function solutions, various combination of hyperbolic and rational function solutions, etc.

A New Generalized Riccati Equation Rational Expansion Method to GeneralizedBurgers–Fisher Equation with Nonlinear Terms of Any Order

In this paper, based on a new more generalansätz, a new algebraic method, named generalized Riccatiequation rational expansion method, is devised forconstructing travelling wave solutions for nonlinear evolutionequations with nonlinear terms of any order. Compared with mostexisting tanh methods for finding travelling wave solutions, theproposed method not only recovers the results by most known algebraicmethods, but also provides new and more general solutions. We choosethe generalized Burgers–Fisher equation with nonlinear terms of anyorder to illustrate our method. As a result, we obtain several newkinds of exact solutions for the equation. This approach can also beapplied to other nonlinear evolution equations with nonlinear termsof any order.

A new compound Riccati equations rational expansion method and its application to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Vesselov system

In this paper, based on a new general ansätze and symbolic computation, a new compound Riccati equations rational expansion method is proposed. Being concise and straightforward, it is applied to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Vesselov system. It is shown that more complexiton solutions can be found by this new method. The method can be applied to other nonlinear partial differential equations in mathematical physics.

New exact solutions for the Konopelchenko–Dubrovsky equation using an extended Riccati equation rational expansion method and symbolic computation

In this paper, based on computerized symbolic computation and a new general ansätz, an extended Riccati equation rational expansion method is presented to construct multiple exact solutions for nonlinear evolution equations and implemented in a computer algebraic system. The validity and reliability of the method are tested by its application to the Konopelchenko–Dubrovsky equation. The method can be applied to other nonlinear evolution equations in mathematical physics.

A new generalized Riccati equation rational expansion method to a class of nonlinear evolution equations with nonlinear terms of any order

With the aid of symbolic computation, a new algebraic method, named generalize Riccati equation rational expansion (GRERE) method, is devised for constructing travelling-wave solutions for nonlinear evolution equations (NEEs) with nonlinear terms of any order. Compared with most existing tanh methods for finding travelling-wave solutions, the proposed method not only recovers the results by most known algebraic methods, but also provides new and more general solutions. For illustration, we apply the generalized method to a class of nonlinear evolution equations with nonlinear term of any order. As a result, we obtain several new kinds of exact solutions. This approach can also be applied to other nonlinear evolution equations with nonlinear terms of any order.

A new general algebraic method with symbolic computation and its application to two nonlinear differential equations with nonlinear terms of any order

In this paper, with the help of symbolic computation system Maple, we present a new generalized Riccati equation rational expansion method and successfully obtain some new rational formal solutions of some nonlinear partial differential equation with nonlinear terms of any order. The presented method can also be applied to other nonlinear evolution equations with nonlinear terms of any order.

Extended Riccati Equation Rational Expansion Method and Its Applicationto Nonlinear Stochastic Evolution Equations

In this work, by means of a newmore general ansätz and the symbolic computation systemMaple, we extend the Riccati equation rational expansion method[Chaos, Solitons & Fractals 25 (2005) 1019] touniformly construct a series of stochastic nontravelling wavesolutions for nonlinear stochastic evolution equation. To illustratethe effectiveness of our method, we take the stochastic mKdVequation as an example, and successfully construct some new and moregeneral solutions including a series of rational formal nontravelingwave and coefficient functions' soliton-like solutions andtrigonometric-like function solutions. The method can also beapplied to solve other nonlinear stochastic evolution equation or equations.