Fractional sub-equation method and its applications to nonlinear fractional PDEs
Introduction
Nonlinear complex physical phenomena are related to nonlinear partial differential equations (PDEs) which are involved in many fields from physics to biology, chemistry, mechanics, etc. Searching for exact solutions of nonlinear PDEs plays an important role in the study of these physical phenomena and gradually becomes one of the most important and significant tasks. In the past several decades, both mathematicians and physicists have made many significant work in this direction and presented some effective methods, such as the inverse scattering method [1], Hirota's bilinear method [2], Bäcklund transformation [3], Painlevé expansion [4], and homogeneous balance method [5]. In recent years, a class of methods [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18] called the sub-equation method for exact solutions of nonlinear PDEs has attracted much attention. The key ideas of the sub-equation method are that the exact solutions of the complicated nonlinear PDE can be expressed as a polynomial, the variable of which is one of the solutions of simple and solvable ordinary or partial differential equation (sub-equation for short), and the degree of the polynomial can be determined by balancing the highest-order derivative with nonlinear terms in the considered equation. The sub-equations which were often used are the Riccati equation, elliptic equation, projective Riccati equation, etc. For example, Fan et al. [6], [7], [8], [9], [10] proposed and developed a sub-equation method which greatly exceeds the applicability of the existing tanh, extended tanh methods and Jacobi elliptic function expansion method to construct in a uniform way a series of exact solutions of nonlinear PDEs including polynomial solutions, exponential solutions, rational solutions, trigonometric periodic wave solutions, hyperbolic and solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions.
However, as El-Sayed et al. [19] pointed out that to a large extent we have difficulty in finding exact analytical solutions of fractional differential equations appear more and more frequently in different research areas and engineering applications, and that an effective and easy-to-use method for such equations is needed. Fractional calculus is one of the generalizations of ordinary calculus. Generally speaking, there are two kinds of fractional derivatives. One is nonlocal fractional derivative, i.e., Caputo derivative and Riemann–Liouville derivative which have been used successfully in various fields of science and engineering. However, the Caputo derivative requires the function should be smooth and differentiable. Obviously, the nonlocal derivatives are not suitable for the investigation of the local behavior of fractional differentiable equations. The other one is the local fractional derivative, i.e., Kolwankar–Gangal (K–G) derivative [20], [21], [22], [23], [24], Chen's fractal derivative [25], [26] and Cresson's derivative [27], [28]. One of the famous examples is the devi-stair curve, which can be described by a continuous but nowhere differentiable function. Recently, there is new development of continuous but nowhere differentiable functions [29]. The present Letter is motivated by the desire to propose a fractional sub-equation method to construct exact analytical solutions of nonlinear fractional differential equations with the modified Riemann–Liouville derivative defined by Jumarie [29] which has merits over the original one, for example, the α-order derivative of a constant is zero. Owing to these merits, Jumarie's modified Riemann–Liouville derivative was successfully applied to the probability calculus [30], fractional Laplace problems [31] and fractional variational calculus [32]. Some useful formulas and results of Jumarie's modified Riemann–Liouville derivative were summarized in [29], three of them are which will be used in the following sections.
The rest of this Letter is organized as follows. In Section 2, we describe the fractional sub-equation method for solving fractional differential equations. In Section 3, we give two applications of the proposed method to nonlinear equations. In Section 4, some conclusions and discussions are given.
Section snippets
Basic idea of the fractional sub-equation method
In this section, we outline the main steps of the fractional sub-equation method for solving fractional differential equations. For a given fractional differential equation, say, in three variables x, y and t where , and are the modified Riemann–Liouville derivatives of u with respect to x, y and t, respectively.
To determine u explicitly, we take the following five steps:
Step 1. We take, if necessary, a suitable transformation
Applications
Let us employ in this section Eq. (9) as the sub-equation to solve two nonlinear fractional PDEs.
Example 1 We consider a time fractional biological population model of the form [19] where u denotes the population density and represents the population supply due to births and deaths. To begin with, we take the traveling wave transformation then Eq. (11) is reduced into a nonlinear fractional ODE easy to solve
Conclusions and discussions
We have proposed a fractional sub-equation method to solve fractional differential equations. Two nonlinear equations, namely, the time fractional biological population model and -dimensional space–time fractional Fokas equation are selected to test the effectiveness of the proposed method. As a result, some exact analytical solutions are obtained including the generalized hyperbolic function solutions, generalized trigonometric function solutions and rational solutions. To the best of our
Acknowledgements
We would like to express our sincere thanks to the editors and the referees for their valuable suggestions and comments. One of the authors Sheng Zhang would like to thank Dr. Guo-Cheng Wu for his helpful discussions. This work was supported by the Doctoral Academic Freshman Award of Ministry of Education of China and the Natural Science Foundation of Educational Committee of Liaoning Province of China, and the “Mathematics + X” Key Project of Dalian University of Technology.
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