Fractional calculus first started in 1600s, with the question of G.W. Leibnitz, can integer-order derivative be generalized for non-integer derivatives?, to L’Hospital . In the last few decades, fractional differential equation (FDE) has been rediscovered by applied scientists, proving to be very useful in various fields: physics (classic and quantum mechanics, thermodynamics, fluid mechanics, optics, plasma etc.), chemistry, biology, economics, engineering, signal and image processing, and control theory and so on. In recent years, many important definitions of FDEs, Riemann-Liouville, Caputo, Grunwald-Letnikov and conformable derivative etc., are found to solve the nonlinear FDEs [2, 3, 4, 5]. Among them, the conformable derivative definition is newly defined by Khalil and et al. , which is very closer to definition of general calculus. So it is more aplicable and practical than the other definitions. Many effective methods and techniques for obtaining the exact solutions of nonlinear FDEs, such as, Adomian decomposition method , the homotopy perturbation method , the ansatz method [9, 10], the sub-equation method [11, 12], the exp-function method [13, 14], the first integral method [15, 16], the functional variable method [17, 18], (G’/G) expansion method [19, 20], the extended trial equation method , the modified simple equation method , the modified Kudryashov method , the exp(phi(ε))-expansion method , the modified trial equation method , the finite difference metod  and so on., have been used and improved various group of mathematicians and scientist.
In this study, first we examine the exact solutions of the space-time nonlinear fractional Klein-Gordon equation, which is well known, linear and nonlinear Klein-Gordon equations model many problems in classical and quantum mechanics, solitons, and condensed matter physics [23,27,28]. Then we observe the (2+1) dimensional time-fractional Zoomeron Equation, which is a convenient model to display the novel phenomena associated with boomerons and trappons is studied [29,30]. These two equations have been solved using different methods before and we have found new exact solutions by using the auxiliary equation method.
The rest of this paper is arranged as follows: In Section 2, we describe the key ideas of the conformable fractional derivative. In Section 3, we consume auxiliary equation method to search the exact solutions of the space-time fractional Klein-Gordon equation and (2+1)-dimensional time-fractional Zoomeron Equation. Finally, Section 4 consists of the conclusion.
In this section, we reviewed some identity and characteristics of the conformable fractional derivative that are exposed further in this article.
Given a function f: [0, ∞) → R, then the CFD of f of order α is defined by
- Tα (af + bg) = aTα (f) + bTα (g) , a, b ∈ R,
- Tα (tp) = ptp-α, p ∈ R,
- f(t) = λ ∈ R, for all constant functions Tα (λ) = 0,
- Tα (fg) = fTα (g) + gTα (f),
- (6)If, in addition to f differentiable, then
(Chain Rule) Assume functions f,g : [0, ∞) → R be α-differentiable, then the following rule is obtained
Let 0 < α ≤ 1 and 0 ≤ a < b. A function f : [a, b] → R is α-fractional integrable on [a, b] if the integral
Let f ∈ C [a, b] and 0 < α ≤ 1. Then ,
3 Description of the Method
A brief description of the method is presented in this section. For this purpose, consider the following nonlinear CFPDE,
By virtue of the extended tanh-function method we assume that the solution of the Eq. (7) is of the form
In this section, we consider two nonlinear CFDEs as an application of the auxiliary equation method.
4.1 Space-Time Fractional Klein-Gordon Equation
Let us consider nonlinear space-time fractional Klein-Gordon equation as follow :
The balancing number is found 2 by balancing the highest order derivative term with the highest power nonlinear term. Then the solution of the Eq. (12) becomes
4.2 (2+1)-Dimensional Time-Fractional Zoomeron Equation
Taking the conformable Integration of the Eq.(51) twice with respect to ξ, then we have,
In this article, new exact solutions of CFPDEs, namely, the nonlinear space-time fractional Klein-Gordon equation and the (2+1)-dimensional time-fractional Zoomeron equation, have been obtained by using the auxiliary equation method. Conformable fractional derivative definitions were used to cope with the fractional terms in fractional partial differential equation. We have used the new definition of wave transformation for converting the nonlinear CFPDEs into the ordinary differential equation. We have obtained a variety of new solutions of the mentioned equations. Since the technique is efficient and powerful, it can be used to handle a variety of equations which appears in applications in several branches of the nonlinear sciences.
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