# The modified Kudryashov method for solving some fractional-order nonlinear equations

- Serife Muge Ege
^{1}Email author and - Emine Misirli
^{1}

**2014**:135

https://doi.org/10.1186/1687-1847-2014-135

© Ege and Misirli; licensee Springer. 2014

**Received: **16 October 2013

**Accepted: **26 March 2014

**Published: **7 May 2014

## Abstract

In this paper, the modified Kudryashov method is proposed to solve fractional differential equations, and Jumarie’s modified Riemann-Liouville derivative is used to convert nonlinear partial fractional differential equation to nonlinear ordinary differential equations. The modified Kudryashov method is applied to compute an approximation to the solutions of the space-time fractional modified Benjamin-Bona-Mahony equation and the space-time fractional potential Kadomtsev-Petviashvili equation. As a result, many analytical exact solutions are obtained including symmetrical Fibonacci function solutions, hyperbolic function solutions, and rational solutions. This method is powerful, efficient, and it can be used as an alternative to establish new solutions of different types of fractional differential equations applied in mathematical physics.

## Keywords

## 1 Introduction

Nonlinear partial differential equations of integer order play an important role in describing many nonlinear phenomena such as mathematical biology, electromagnetic theory, fluid mechanics, signal processing, engineering, solid state physics, and other fields of science. With the help of computerized symbolic computations many researchers implemented various methods to establish the solutions to different nonlinear differential equations. For example, the Exp-function method [1–3], the Jacobi elliptic function expansion method [4, 5], the first integral method [6, 7], $({G}^{\prime}/G)$-expansion method [8, 9], the direct algebraic method [10], the Cole-Hopf transformation method [11], and others.

Nonlinear fractional differential equations (FDEs) are a generalization of classical differential equations of integer order. Recently, FDEs have attracted great interest, using the fractional derivatives. It is caused both by the development of the theory of fractional calculus itself and by the applications of such constructions in various real life problems. In the past decades the theory of fractional derivatives was represented principally as a pure theoretical field of mathematics effective only for mathematicians. However, in recent years many authors have noticed that derivatives of non-integer order are convenient for the description of the properties of various physical phenomena. It has been shown that fractional-order models are more sufficient than the formerly used integer-order models. Some physical considerations by means of the models based on derivatives of non-integer order are given in [12–15]. New exact solutions for fractional differential equations may help to understand better the corresponding wave phenomena they describe. In order to obtain the solutions for fractional differential equations, many numerical and analytical methods have been proposed so far (*e.g.* see [12–32]). But the application of a modified Kudryashov method to fractional differential equations has not been researched.

In this paper, we will apply the modified Kudryashov method for solving fractional partial differential equations in the sense of the modified Riemann-Liouville derivative as given by Jumarie [33, 34]. To illuminate the utility and validness of the method, we will apply it to the space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation and the space-time fractional potential Kadomstev-Petviashvili (PKP) equation.

## 2 Preliminaries and the modified Kudryashov method

which are direct results of the equality ${D}^{\alpha}x(t)=\mathrm{\Gamma}(1+\alpha )Dx(t)$, which holds for non-differentiable functions.

We present the main steps of the modified Kudryashov method as follows [35–39].

*u*of independent variables, $X=(x,y,z,\dots ,t)$:

where ${D}_{t}^{\alpha}u$, ${D}_{x}^{\alpha}u$, ${D}_{y}^{\alpha}u$, and ${D}_{z}^{\alpha}u$ are the modified Riemann-Liouville derivatives of *u* with respect to *t*, *x*, *y* and *z*. *P* is a polynomial in $u=u(x,y,z,\dots ,t)$ and its various partial derivatives, in which the highest-order derivatives and nonlinear terms are involved.

*k*,

*n*,

*m*and

*λ*are arbitrary constants. Then Eq. (2.6) reduces to a nonlinear ordinary differential equation of the form

*Q*is the solution of the equation

*N*in Eq. (2.11) we have the pole order for the general solution of Eq. (2.8). In order to determine the value of

*N*we balance the highest-order nonlinear terms in Eq. (2.8), analogously as in the classical Kudryashov method. Supposing ${u}^{l}(\xi ){u}^{(s)}(\xi )$ and ${({u}^{(p)}(\xi ))}^{r}$ are the highest-order nonlinear terms of Eq. (2.8) and balancing the highest-order nonlinear terms we have

## 3 Applications

### 3.1 Space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation

where $0<\alpha \le 1$, $t>0$, *u* is the function of $(x,t)$ and *v* is a nonzero positive constant.

This equation was first derived to describe an approximation for surface long waves in nonlinear dispersive media. It can also characterize the hydromagnetic waves in a cold plasma, acoustic waves in harmonic crystals, and acoustic gravity waves in compressible fluids.

where *k*, *c*, ${\xi}_{0}$ are constants.

*Q*is the solution of ${Q}_{\xi}=lna({Q}^{2}-Q)$. Balancing ${g}^{\u2033}$ and ${g}^{3}$ in Eq. (3.3), we compute

*ξ*in Eq. (3.4) we obtain

### 3.2 The space-time fractional potential Kadomstev-Petviashvili (PKP) equation

where $0<\alpha \le 1$, $t>0$.

where *k*, *l*, *c*, ${\xi}_{0}$ are constants.

where $Q=\frac{1}{1\pm {a}^{\xi}}$.

*Q*is the solution of ${Q}_{\xi}=lna({Q}^{2}-Q)$. Balancing the linear term of the highest order with the highest-order nonlinear term in Eq. (3.17), we compute

*ξ*in Eq. (3.18) we obtain

## 4 Conclusion

We have extended the modified Kudryashov method to solve fractional partial differential equations. As applications, for the space-time fractional potential Kadomstev-Petviashvili equation we found similar solutions to the ones previously obtained in [26, 32]. However, for the space-time fractional modified Benjamin-Bona-Mahony equation we have obtained new symmetrical hyperbolic Fibonacci function solutions with differences from the solutions obtained before [25]. If we take $a=e$, we can also find the other hyperbolic solutions similar to [25, 26, 32]. The method is based on the homogeneous balancing principle. Therefore, it can also be applied to other fractional partial differential equations where the homogeneous balancing principle is satisfied. We can easily conclude that symmetrical hyperbolic Fibonacci function solutions for the space-time fractional modified Benjamin-Bona-Mahony equation have not been reported in the previous studies.

## Declarations

### Acknowledgements

This research is supported by Ege University, Scientific Research Project (BAP), Project Number: 2012FEN037.

## Authors’ Affiliations

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