One method for finding exact solutions of nonlinear differential equations

Abstract

One of old methods for finding exact solutions of nonlinear differential equations is considered. Modifications of the method are discussed. Application of the method is illustrated for finding exact solutions of the Fisher equation and nonlinear ordinary differential equation of the seven order. It is shown that the method is one of the most effective approaches for finding exact solutions of nonlinear differential equations. Merits and demerits of the method are discussed.

Introduction

One of the first method for finding exact solutions of nonintegrable nonlinear partial differential equations was introduced in Ref. [1] and applied in Ref. [2]. However it seems to us that it was premature work. At that time investigators did not use the symbolic calculations and the method [1] did not came into notice.

Later we could observe appearance many other methods for finding exact solutions of nonlinear differential equations. Let us call here the following methods: the tanh – expansion method [3], [4], the simplest equation method [5], [6], the Jacobi elliptic – function method [7], the modified simplest equation method [8], [9], [10], the cosh – function method [11], [12], the Exp – function method [13], [14], [15], the G′/G – expansion method [17], [18], a transformed rational function method [19], application of the Hirota method for nonintegral nonlinear differential equations [16]. However some of the mentioned methods yield to the method by paper [1].

Currently we know a few of papers [20], [21], [22], [23] of application of our method for finding exact solutions of nonlinear ordinary differential equations. However our opinion is that authors of these papers did not take into consideration all special particularities of our modified approach.

The aim of this paper is to present our approach again and give some examples of application of our method.

In fact we look for exact solution taking into account the following expression

$$y(z)=\sum _{n=0}^N{a}_n{Q}^n\text{,}$$

where the function Q takes the form

$$Q=\frac{1}{1+{\mathrm{e}}^z}\text{.}$$

We note that the function Q is solution of equation

$$Q_z=Q^2-Q\text{.}$$

This equation allows us to find derivatives y_z, y_zz and so on. In the next section we present formulae for the six derivatives of solution y(z).

We demonstrate application of our method for finding exact solutions of two nonlinear evolution equations.

One of these equations is the famous Fisher equation

$$u_t=u_{\mathit{xx}}+u(1-u)$$

and another equation of the seven order in the form

$$u_t+u^2{u}_x+\alpha u_{\mathit{xxx}}+\beta u_{\mathit{xxxxx}}-\gamma u_{\mathit{xxxxxxx}}=0\text{.}$$

We use our method for obtaining exact solutions of these two equations using the symbolic calculations.