Short noteSoliton solution of fractional-order nonlinear differential equations based on the exp-function method
Introduction
In different disciplines of science and technology, fractional calculus has gained great importance. Cause fractional differential equations describe the natural phenomena better than integer order differential equations. These equations are one of the main tool in modelling and controlling of real-life problems [1], [2], [3]. But, in general, it is difficult to derive exact solutions to fractional differential equations. There are too many techniques such as adomian decomposition method, variational iteration method, homotopy analysis method, power series method, exp-function method and so on [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33]. In this paper, we intend to extend the exp-function method with modified Riemann–Liouville derivative. We use this method for time fractional differential equation and fractional advection–diffusion-reaction equation and obtain some new exact solutions. The Jumarie's modified Riemann–Liouville derivative of order α is defined by [34], [35]:
where denotes a continuous (but not necessarily first-order-differentiable) function. Some useful formulas:where a, b and c are constants.
Section snippets
Preliminaries and notation
We consider the following general nonlinear FDE of the typewhere u is an unknown function, and P is a polynomial of u and its partial fractional derivatives, in which the highest order derivatives and the nonlinear terms are involved. Li and He [36] proposed a fractional complex transformwhere τ and δ are non zero arbitrary constants. By using the fractional chain rulewhere is called the sigma index see [37], without loss
The time fractional modified nonlinear Kawahara equation
In this study, a new algorithm for solving the time fractional modified nonlinear Kawahara equation is [39]where α is a parameter and 0 < α ≤ 1.
Numerical solutions of time fractional modified nonlinear Kawahara equation have been introduced by Atangana and his friends. They have used the homotopy decomposition method and the Sumudu transform method which can be accepted relatively new analytical method.
For our goal, we present the following
The nonlinear fractional advection–diffusion-reaction equation
The second equation is called the nonlinear fractional advection–diffusion-reaction (ADR) equation which has the form [40]
In [40], exponential integrators approach (EIs) has been introduced this equation by Garrappa. Also the advantages of the proposed method are illustrated by means of some test problems.
For our purpose, we introduce the transformations (3.2) and same procedure we have the ODEwhere “U′” = . Same
Conclusion
In this work, the exp-function method and fractional complex transform are applied for exact solutions of fractional differential equations. Two examples are given to show that this method is powerful mathematical tool to find exact solutions of fractional differential equations. These new solutions verify that the method is reliable and effective method for fractional differential equations.
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