Bäcklund Transformation of Fractional Riccati Equation and Infinite Sequence Solutions of Nonlinear Fractional PDEs

Abstract

The Bäcklund transformation of fractional Riccati equation with nonlinear superposition principle of solutions is employed to establish the infinite sequence solutions of nonlinear fractional partial differential equations in the sense of modified Riemann-Liouville derivative. To illustrate the reliability of the method, some examples are provided.

1. Introduction

Recently, nonlinear fractional differential equations increasingly are used to describe nonlinear phenomena in fluid mechanics, biology, engineering, physics, and other areas of science [1–3]. Much efforts have been spent in recent years to develop various techniques to deal with fractional differential equations. However, for the nonlinear differential equations including fractional calculus, the analytical or numerical results are usually difficult to be obtained. It is therefore needed to find a proper method to solve the problem of nonlinear differential equations containing fractional calculus.

In the past, several methods have been formulated, such as Adomian decomposition method [4, 5], variational iteration method [6, 7], homotopy perturbation method [8, 9], differential transform method [10, 11], and fractional subequation method [12–14]. S. Zhang and H.-Q. Zhang [12] first proposed a new direct method called fractional subequation method in solving nonlinear time fractional biological population model and ()-dimensional space-time fractional Fokas equation, based on the homogeneous balance principle and Jumarie’s modified Riemann-Liouville derivative.

In this paper, based on the Bäcklund transformation technique and the known seed solutions, we will devise effective way for solving fractional partial differential equations. It will be shown that the use of the Bäcklund transformation allows us to obtain new exact solutions from the known seed solutions.

2. Bäcklund Transformation of the Fractional Riccati Equation and Nonlinear Superposition Principle

Firstly, we give some definitions and properties of the modified Riemann-Liouville derivative [15] which are used in this paper.

Assume that , denote a continuous (but not necessarily differentiable) function, and let denote a constant discretization span. Jumarie defined the fractional derivative in the limit form where

This definition is close to the standard definition of the derivative (calculus for beginners), and as a direct result, the th derivative of a constant, , is zero. An alternative, which is strictly equivalent to (1) is as follows:

Some properties of the fractional modified Riemann-Liouville derivative that were summarized in four useful formulas of them are which are direct consequences of the equality which holds for nondifferentiable functions. In the above formulas (5)-(6), is nondifferentiable function in (5) and (6) and differentiable in (7), is nondifferentiable, and is differentiable in (6) and nondifferentiable in (7).

Recall the fractional Riccati equation: S. Zhang and H.-Q. Zhang [12] derived some exact solutions to (8) as follows: where the generalized hyperbolic and trigonometric functions are defined as where is the Mittag-Leffler function.

Next, we introduce the Bäcklund transformation of fractional Riccati equation (8): that is, satisfies the fractional Riccati equation where are arbitrary parameters, , , , , and are the known solutions (9).

Specially, if we take in (11), the Bäcklund transformation of factional Riccati equation can be obtained as

By means of solutions (9), we can construct the following infinite sequence exact solutions of fractional Riccati equation (8). Here in the following cases we given several Bäcklund transformations of solutions.

Case 1. When , if is the solution of fractional Riccati equation (8), then the following are also the solutions of (8):

Case 2. When , if is the solution of fractional Riccati equation (8), then the following are also the solutions of (8):

Case 3. When , if is the solution of fractional Riccati equation (8), then the following are also the solutions of (8):

Case 4. When , if is the solution of fractional Riccati equation (8), then the following are also the solutions of (8): where are arbitrary parameters, , , , , and .

Case 5. When , if is the solution of fractional Riccati equation (8), then the following are also the solutions of (8): where , are arbitrary constant, and .

Nonlinear Superposition Principle.  (1) If , are the solutions of fractional Riccati equation (8), respectively, then the following are also the solutions of (8) which read where , are arbitrary nonzero constants.

(2) If , , and are the solutions of fractional Riccati equation (8), respectively, then the following are also the solutions of (8) which read where is an arbitrary nonzero constant.

Applying the nonlinear superposition formulas (19)-(20), we can obtain the following new infinite sequence exact solutions of fractional Riccati equation (8). For example, when , we can get the infinite sequence new solutions as follows: where , are arbitrary nonzero constants.

3. Summary of the Method

In this section, we describe the main steps of the fractional subequation method for finding exact solutions of fractional differential equations.

Let us consider the fractional differential equation with independent variables and dependent variable , where , , , and are the modified Riemann-Liouville derivatives of with respect to , , , and , respectively.

Step 1. Using the variable transformation where and are constants to be determined later, the fractional differential equation (24) is reduced to a nonlinear fractional ordinary differential equation where “” .

Step 2. We suppose that (26) has the following solution: where are constants to be determined, positive integer can be determined by balancing the highest order derivatives and nonlinear terms in (24) or (26), and comes from the following Bäcklund transformation for the fractional Riccati equation: that is, satisfies the fractional Riccati equation where are the known solutions of (8).

Step 3. Substituting the explicit formal solution (27) into (26) and setting the coefficients of the powers of to be zero, we obtain an overdetermined nonlinear algebraic system in , , and .

Step 4. Solving the nonlinear algebraic system yields the explicit expressions of the parameters ,   , and . Then substituting these values into (27), we may obtain the exact solutions of the nonlinear fractional differential equation (24).

4. Applications of the Method

In this section, we present two examples to illustrate the applicability of the our method to solve nonlinear fractional partial differential equations.

Example 1. We first consider the space-time fractional bidirectional wave equations in the form [16] where represents the distance along the channel, is the elapsed time, the variable is the dimensionless deviation of the water surface from its undisturbed position, is the dimensionless horizontal velocity, , , are real constants. When , (30) is the generalization of bidirectional wave equations, which can be used as a model equation for the propagation of long waves on the surface of water with a small amplitude.
For our purpose, we introduce the following transformations: where is constant.
Substituting (32) into (30), we can know that (30) is reduced into a fractional ordinary differential equations:
We suppose that (33) has the solution in the form
Balancing the highest order derivative terms and nonlinear terms in (33), we have . Substituting (35) given the value of and along with (8) into (33) and then setting the coefficients of to zero, we can obtain a set of algebraic equations about , , , , , , and . Solving the algebraic equations by Maple, we have, where , are arbitrary constants.
Substituting the above result into (35), we obtain new types of exact solutions of (30) as follows: The expression appearing in these solutions is given by relations (14)–(18) and the nonlinear superposition formulas (19)-(20), where , , , and are real constants.

Example 2. We consider the following space-time fractional Sharma-Tasso-Olver (STO) equation [17] in the form where is an arbitrary constant and is a parameter describing the order of the fractional derivative. When , (38) is the generalization of classical nonlinear STO equation, which was first derived as an example of odd members of Burgers hierarchy by Tasso.
For our purpose, we introduce the following transformations: where is constant.
Substituting (39) into (38), we can know that (38) is reduced into a fractional ordinary differential equation:
We suppose that (40) has the solution in the form
Balancing and , we have . Substituting (41) given the value of along with (8) into (40) and then setting the coefficients of to zero, we can obtain a set of algebraic equations about , , and . Solving the algebraic equations by Maple, we have where is an arbitrary constant, and where , are arbitrary constants.
Substituting the above results into (41), we obtain new types of exact solutions of (38) as follows: The expression appearing in these solutions is given by relations (14)–(18) and the nonlinear superposition formulas (19)-(20), where , , and The expression appearing in these solutions is given by relations (14)–(18) and the nonlinear superposition formulas (19)-(20), where and .

5. Conclusion

Bäcklund transformation of the fractional Riccati equation with nonlinear superposition principle of known solutions is applied successfully for solving the system of nonlinear fractional differential equation. To the best of our knowledge, the solutions obtained in this paper have not been reported in the literature. It can be concluded that this method is very simple and reliable and proposes a variety of exact solutions to fractional differential equations.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to express sincere thanks to referees for their valuable suggestions and comments. The project are supported by the 211 Project and Start-up Research Funds for Doctors of Anhhui University (02303319-33190074), the Key Foundation of Anhui Education Bureau (KJ2013A028, KJ2012A019), the Research Fund for the Doctoral Program of Higher Education (20103401120002), and 211 Project of Anhui University (02303303-33030011).