A new Riccati equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation

In this paper, we present a new Riccati equation rational expansion method to uniformly construct a series of exact solutions for nonlinear evolution equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions. The solutions obtained in this paper include rational triangular periodic wave solutions, rational solitary wave solutions and rational wave solutions. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equation.     

广义Riccati展开法及其在foam drainage方程中的应用

应用双曲函数法结合Riccati方程,求得foam drainage方程的精确解.通过这种方法可以得到此方程的新的孤立波解与周期解,并且此方法可以用来求解其它许多的非线性演化方程.     

A generalized extended rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation

In this Letter, a generalized extended rational expansion method is used to construct exact solutions of the (1 + 1)-dimensional dispersive long wave equation. As a result, many new and more general exact solutions are obtained, the solutions obtained in this Letter include rational triangular periodic wave solutions, rational solitary wave solutions.     

The extended Riccati equation method for travelling wave solutions of ZK equation

In this article, the extended Riccati equation method is applied to seeking more general exact travelling wave solutions of the ZK equation. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. When the parameters are taken as special values, the solitary wave solutions are obtained from the hyperbolic function solutions. Similarly, the periodic wave solutions are also obtained from the trigonometric function solutions. The approach developed in this paper is effective and it may also be used for solving many other nonlinear evolution equations in mathematical physics.     

A new subspace iteration method for the algebraic Riccati equation

We consider the numerical solution of the continuous algebraic Riccati equation A*X + XA ? XFX + G = 0, with F = F*,G = G* of low rank and A large and sparse. We develop an algorithm for the low‐rank approximation of X by means of an invariant subspace iteration on a function of the associated Hamiltonian matrix. We show that the sought‐after approximation can be obtained by a low‐rank update, in the style of the well known Alternating Direction Implicit (ADI) iteration for the linear equation, from which the new method inherits many algebraic properties. Moreover, we establish new insightful matrix relations with emerging projection‐type methods, which will help increase our understanding of this latter class of solution strategies. Copyright © 2014 John Wiley & Sons, Ltd.     

A new Jacobi elliptic function rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation

With the aid of computerized symbolic computation, a new elliptic function rational expansion method is presented by means of a new general ansätz and is very powerful to uniformly construct more new exact doubly-periodic solutions in terms of rational formal Jacobi elliptic function of nonlinear evolution equations (NLEEs). As an application of the method, we choose a (1 + 1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by most existing Jacobi elliptic function methods and find other new and more general solutions at the same time. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.     

The extended Riccati equation mapping method for variable-coefficient diffusion-reaction and mKdV equations

In this paper, the extended Riccati equation mapping method is proposed to seek exact solutions of variable-coefficient nonlinear evolution equations. Being concise and straightforward, this method is applied to certain type of variable-coefficient diffusion-reaction equation and variable-coefficient mKdV equation. By means of this method, hyperbolic function solutions and trigonometric function solutions are obtained with the aid of symbolic computation. It is shown that the proposed method is effective, direct and can be used for many other variable-coefficient nonlinear evolution equations.     

A new inversion-free method for a rational matrix equation

Motivated by the classical Newton-Schulz method for finding the inverse of a nonsingular matrix, we develop a new inversion-free method for obtaining the minimal Hermitian positive definite solution of the matrix rational equation X+A^∗X^-1A=I, where I is the identity matrix and A is a given nonsingular matrix. We present convergence results and discuss stability properties when the method starts from the available matrix AA^∗. We also present numerical results to compare our proposal with some previously developed inversion-free techniques for solving the same rational matrix equation.     

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.     

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.     

A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials

In this paper, the quadratic Riccati differential equation is solved by He's variational iteration method with considering Adomian's polynomials. Comparisons were made between Adomian's decomposition method (ADM), homotopy perturbation method (HPM) and the exact solution. In this application, we do not have secular terms, and if λ$\lambda$, Lagrange multiplier, is equal -1$-1$ then the Adomian's decomposition method is obtained. The results reveal that the proposed method is very effective and simple and can be applied for other nonlinear problems.     

An extended simplest equation method and its application to several forms of the fifth-order KdV equation

In this paper, an extended simplest equation method is proposed to seek exact travelling wave solutions of nonlinear evolution equations. As applications, many new exact travelling wave solutions for several forms of the fifth-order KdV equation are obtained by using our method. The forms include the Lax, Sawada-Kotera, Sawada-Kotera-Parker-Dye, Caudrey-Dodd-Gibbon, Kaup-Kupershmidt, Kaup-Kupershmidt-Parker-Dye, and the Ito forms.     

An extended subequation rational expansion method with symbolic computation and solutions of the nonlinear Schrödinger equation model

To construct exact analytical solutions of nonlinear evolution equations, an extended subequation rational expansion method is presented and used to construct solutions of the nonlinear Schrödinger equation with varing dispersion, nonlinearity, and gain or absorption. As a result, many previous known results of the nonlinear Schrödinger equation can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. With computer simulation, the properties of a new non-travelling wave soliton-like solutions with coefficient functions and some elliptic function solutions are shown by some figures.     

A improved F‐expansion method and its application to Kudryashov–Sinelshchikov equation

On the basis of the F‐expansion method with a new sub‐equation and Exp‐function method, an improved F‐expansion method is introduced. As illustrative examples, the exact solutions expressed by exponential function, hyperbolic function of Kudryashov–Sinelshchikov equation for arbitrary α,β are derived. Some previous results are extended. The method is straightforward, concise and is a promising and powerful method for other nonlinear evolution equations in mathematical physics. Copyright © 2013 John Wiley & Sons, Ltd.     

A improved F‐expansion method and its application to the Zhiber–Shabat equation

Based on the F‐expansion method and Exp‐function method, an improved F‐expansion method is introduced. As illustrative examples, the exact solutions expressed by exponential function, hyperbolic functions, logarithmic function, and other type of functions for the Zhiber–Shabat equation are derived. Some previous results are extended. Copyright © 2012 John Wiley & Sons, Ltd.     

An extended rational interpolation method

To interpolate function, f(x), a ? x ? b, when we have some information about the values of f(x) and their derivatives in separate points on {x₀, x₁, … , x_n} ? [a, b], the Hermit interpolation method is usually used. Here, to solve this kind of problems, extended rational interpolation method is presented and it is shown that the suggested method is more efficient and suitable than the Hermit interpolation method, especially when the function f(x) has singular points in interval [a, b]. Also for implementing the extended rational interpolation method, the direct method and the inverse differences method are presented, and with some examples these arguments are examined numerically.     

Generalized extended tanh-function method and its application

In this paper, a new generalized extended tanh-function method is presented for constructing soliton-like, period-form solutions of nonlinear evolution equations (NEEs). Compared with most of the existing tanh-function method, extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By using this method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEEs. Make use of the method, we study the (3 + 1)-dimensional potential-YTSF equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions’ soliton-like solutions, singular soliton-like solutions, periodic form solutions.     

A new rational auxiliary equation method and exact solutions of a generalized Zakharov system

A new rational auxiliary equation method for obtaining exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. Its effectiveness is evinced by obtaining exact solutions of a generalized Zakharov system, some of which are new. It is shown that the G^′/G and the generalized projective Ricatti expansion methods are special cases of the auxiliary equation method. Further, due the solutions obtained, four other new and practicable rational methods are deduced.     

New exact solutions to MKDV-Burgers equation and (2 + 1)-dimensional dispersive long wave equation via extended Riccati equation method

In this paper, with the aid of symbolic computation and a general ansätz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansätz. The method can also be applied to other nonlinear partial differential equations.     

Homotopy analysis method for quadratic Riccati differential equation

In this paper, the quadratic Riccati differential equation is solved by means of an analytic technique, namely the homotopy analysis method (HAM). Comparisons are made between Adomian’s decomposition method (ADM), homotopy perturbation method (HPM) and the exact solution and the homotopy analysis method. The results reveal that the proposed method is very effective and simple.