Generalization of the simplest equation method for nonlinear non-autonomous differential equations

doi:10.1016/j.cnsns.2014.03.035

Communications in Nonlinear Science and Numerical Simulation

Volume 19, Issue 11, November 2014, Pages 4037-4041

https://doi.org/10.1016/j.cnsns.2014.03.035 Get rights and content

Highlights

•
Generalization of the simplest equation method for non-autonomous differential equations is suggested.
•
Using the method special solutions of three Painlevé equations are found.
•
The obtained special solutions are expressed in terms of the special functions defined by linear second order ODEs.

Abstract

It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order.

Introduction

Recently significant results were achieved in the development of methods to search for the exact solutions of nonlinear differential equations [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Such equations have been permanently appearing in different applied problems [22].

A lot of methods exist to look for exact solutions of nonlinear partially solvable differential equations, such as the singular manifold method [1], [2], the tanh-function method [3], [4], [5], the $G^{'} / G$ -expansion method [6], [7], the simplest equation method [11], [12], [13], [14], [15], [16], [17], [18], [19] and so on.

However, there is no systematic approach for finding the exact solutions of non-autonomous nonlinear differential equations. Meanwhile the non-autonomous DE are widely used in different applied problems and can appear when partial differential equations are solved. For instance, using the self-similar variables Korteweg-de Vries equation, modified Korteweg-de Vries equation, Burgers equation and others are reduced to non-autonomous ODEs. Moreover most of them can be converted to Painlevé equations.

For the first time the simplest equation method was presented in the work [11], [12] to solve autonomous nonlinear DEs. There are several advantages of this method, namely: generalization of a lot of methods applied before, simplicity of realization and taking into account all possible singularities of equation solved.

In this Letter we extend the application of the simplest equation method to non-autonomous nonlinear ODEs. Three Painlevé equations were solved to show the application of this approach. The Riccati equation was chosen as ODE of lower order than Painlevé equations. The non-autonomy of the considered nonlinear ODE is reflected in functional dependence on independent variable in all coefficients existing in the Riccati equation and in substitution (2.2).

Section snippets

Method applied

Let us consider a nonlinear non-autonomous differential equation: $E [w, w^{'}, \dots, w^{(n)}, z] = 0 .$

Assuming ODE (2.1) possesses the first order singularity we shall search for exact solutions in the form as: $w (z) = f_{1} (z) + f_{2} (z) Y (z),$ where $f_{1} (z), f_{2} (z)$ are functions which we have to find, $f_{2} (z) \neq 0$ identically. $Y (z)$ satisfies the Riccati equation: $Y_{z} (z) = a (z) Y^{2} (z) + b (z) Y (z) + c (z), a (z) \neq 0 .$

The general solution of the Riccati equation (2.3) has the pole of the first order with respect to movable points, and, hence, function

Exact solutions of the second Painlevé equation

Consider the second Painlevé equation: $w_{zz} = 2 w^{3} (z) + zw (z) + α .$

Let us apply the algorithm expounded in Section 1 to find the solution of Eq. (3.1). Completing step 1 we obtain equations for coefficients determination: $2 f_{2} (z) (f_{2} (z) - a (z)) (f_{2} (z) + a (z)) = 0;$ $3 f_{2} (z) a (z) b (z) + 2 f_{2_{z}} (z) a (z) - 6 f_{1} (z) f_{2}^{2} (z) + f_{2} (z) a_{z} (z) = 0;$ $f_{2} (z) b^{2} (z) + 2 f_{2_{z}} (z) b (z) + f_{2_{zz}} (z) - 6 f_{1}^{2} (z) f_{2} (z) + 2 f_{2} (z) a (z) c (z) - {zf}_{2} (z) + f_{2} (z) b_{z} (z) = 0;$ $f_{1_{zz}} (z) - 2 f_{1}^{3} (z) + f_{2} (z) b (z) c (z) + 2 f_{2_{z}} (z) c (z) - {zf}_{1} (z) + f_{2} (z) c_{z} (z) - α = 0 .$

Solving sequentially Eqs. (3.2), (3.3), (3.4) we find

Exact special solutions of the third Painlevé equation

Another equation studied is the third Painlevé equation: $w_{zz} = \frac{{(w_{z} (z))}^{2}}{w (z)} - \frac{w_{z} (z)}{z} + \frac{α w^{2} (z) + β}{z} + γ w^{3} (z) + \frac{δ}{w (z)} .$

In (4.1) only two constants, namely $α, β$ , are arbitrary. Meanings of others are determined from equations: $\begin{matrix} γ (γ - 1) = 0, \\ δ (δ - 1) = 0 . \end{matrix}$

If we assume $γ = 0$ , solutions of Eq. (4.1) do not exist in the form (2.2), (2.3) while the coefficient of the second power of $Y (z)$ turns out to be: $f_{2}^{2} (z) a^{2} (z) = 0,$

In the opposite case, i.e. $γ = 1$ , we can find solution. $f_{2}^{2} (f_{2} - a) (f_{2} + a) = 0;$ $f_{2}^{2} α - f_{2} a - 2 {za}^{2} f_{1} + 4 f_{2}^{2} {zf}_{1} - {zf}_{2} ab - f_{2} {za}_{z} = 0;$ ${af}_{1} f_{2} - 2 zaf$

Exact special solutions of the fourth Painlevé equation

One more equation we focus on is the fourth Painlevé equation: $w_{zz} = \frac{1}{2} \frac{{(w_{z} (z))}^{2}}{w (z)} + \frac{3}{2} w^{3} (z) + 4 {zw}^{2} (z) + 2 (z^{2} - α) w (z) + \frac{β}{w (z)} .$

Applying the generalized simplest equation method to solve (5.1) we obtain equations for coefficients definition: $f_{2}^{2} (f_{2} - a) (f_{2} + a) = 0;$ $6 f_{1} f_{2}^{2} - 2 {abf}_{2} + 4 {zf}_{2}^{2} - 2 a^{2} f_{1} - f_{2_{z}} a - a_{z} f_{2} = 0;$ $2 f_{1_{z}} f_{2} a - 6 f_{2} {abf}_{1} - 2 f_{2_{zz}} f_{2} + 24 {zf}_{1} f_{2}^{2} - 2 {acf}_{2}^{2} + 18 f_{1}^{2} f_{2}^{2} - b^{2} f_{2}^{2} - 2 f_{2} a_{z} f_{1} - 4 f_{2_{z}} {af}_{1} - 2 b_{z} f_{2}^{2} - 4 α f_{2}^{2} - 2 f_{2_{z}} {bf}_{2} + 4 z^{2} f_{2}^{2} + f_{2_{z}}^{2} = 0;$ $f_{1_{zz}} f_{2} + f_{2} b_{z} f_{1} + f_{2_{zz}} f_{1} - f_{1_{z}} f_{2_{z}} + f_{2_{z}} {cf}_{2} - f_{1_{z}} f_{2} b - 12 {zf}_{1}^{2} f_{2} + f_{2} b^{2} f_{1} + 4 α f_{1} f_{2} + 2 f_{2} ac + f_{1} c_{z} f_{2}^{2} - 4 z^{2} f_{1} f_{2} - 6 f_{1}^{3} f_{2} + 2 f_{2_{z}} {bf}_{1} = 0;$ $3 f_{1}^{4} + 2 β - 2 f_{2}$

Conclusion

In this letter we generalized the simplest equation method for non-autonomous nonlinear ordinary differential equations. To show the application of this approach we found exact special solutions of three Painlevé equations using it. The Riccati equation was chosen as the ordinary differential equation of lower order than Painlevé ODEs. This method allowed us to obtain exact special solutions expressed in terms of the special functions, namely, Airy functions in case of the second Painlevé

References (22)

N.A. Kudryashov
Exact soliton solutions of the generalized evolution equation of wave dynamics
PMM J Appl Math Mech
(1988)
E.J. Parkes et al.
An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations
Comput Phys Commun
(1996)
N.A. Kudryashov
A note on the $G^{'} / G$ -expansion method
Appl Math Comput
(2010)
A. Biswas
Solitary wave solution for the generalized Kawahara equation
Appl Math Lett
(2009)
N.A. Kudryashov
Meromorphic solutions of nonlinear ordinary differential equations
Commun Nonlinear Sci Numer Simul
(2010)
N.A. Kudryashov
Exact solutions of the generalized Kuramoto–Sivashinsky equation
Phys Lett A
(1990)
N.A. Kudryashov
Simplest equation method to look for exact solutions of nonlinear differential equations
Chaos Soliton Fract
(2005)
N.A. Kudryashov et al.
Extended simplest equation method for nonlinear differential equations
Appl Math Comput
(2008)
N.K. Vitanov
Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity
Commun Nonlinear Sci Numer Simul
(2010)
M.V. Demina et al.
From Laurent series to exact meromorphic solutions: the Kawahara equation
Phys Lett A
(2010)

N.K. Vitanov

Modified method of simplest equation: powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs

Commun Nonlinear Sci Numer Simul

(2011)

Cited by (0)

View full text

Letter to the EditorGeneralization of the simplest equation method for nonlinear non-autonomous differential equations

Highlights

Abstract

Introduction

Section snippets

Method applied

Exact solutions of the second Painlevé equation

Exact special solutions of the third Painlevé equation

Exact special solutions of the fourth Painlevé equation

Conclusion

PMM J Appl Math Mech

Comput Phys Commun

Appl Math Comput

Appl Math Lett

Commun Nonlinear Sci Numer Simul

Phys Lett A

Chaos Soliton Fract

Appl Math Comput

Commun Nonlinear Sci Numer Simul

Phys Lett A

Commun Nonlinear Sci Numer Simul

Letter to the Editor
Generalization of the simplest equation method for nonlinear non-autonomous differential equations