New Families of Solutions for the Space Time Fractional Burgers Equation
Abaker A. Hassaballa, Ahmed M. A. Adam, Eltayeb A. Yousif, Mohamed I. Nouh
11 New Families of Solutions for the Space Time Fractional Burgers’ Equation
Abaker A. Hassaballa , Ahmed M. A. Adam , Eltayeb A. Yousif , and Mohamed I. Nouh Department of Mathematics, Faculty of Science, Northern Border University, Arar 91431, Saudi Arabia Department of Mathematics, College of Applied & Industrial Sciences, Bahri University, Khartoum, Sudan Faculty of Engineering, Alzaiem Alazhari University, Khartoum North 13311, Sudan Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Khartoum, Khartoum 11111, Sudan Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), 11421 Helwan, Cairo, Egypt
Abstract
In this paper, the hyperbolic tangent function method is applied for constructing exact solutions for space-time conformal fractional Burger’s equation. Furthermore, the space-time conformal fractional Burgers’ equation is tested for the Painlevé property and consequently new numerous exact solutions are generated via Bäcklund transform.
Keywords:
Conformal fractional Burgers’ equation, hyperbolic tangent function method, Painlevé property, Bäcklund transform, fractional differential equations.
1. Introduction
Recently, fractional Calculus has made a very important impact to the most fields of science, such as mathematics, engineering, physics, economics, etc. [1-21]. The applications of fractional calculus are contemporary [4, 19]. Historically, the fractional derivatives introduced in different ways, for example, Riemann–Liouville, Riesz, Caputo, Modified Riemann–Liouville [1, 19]. In the year (2014), a new fractional derivative is defined by the authors Khalil et al [21], named conformal fractional derivative (CFD). The definition and basic concepts of CFD is developed [22]. Furthermore, the interpretations of CFD in engineering and physics applications are introduced and discussed [23]. The definition of CFD of a function 𝑓: (0, ∞) → ℝ of order 𝛼 , where is mainly given by the following limit 𝐷 𝛼 𝑓(𝑡) = lim 𝜀→0 𝑓(𝑡 + 𝜀𝑡 ) − 𝑓(𝑡)𝜀 (1.1) The function 𝑓 is said to be 𝛼 - differentiable. The fractional derivative at 𝑡 = 0 is given as 𝑓 𝛼 (0) = lim 𝑡→0 + 𝐷 𝛼 𝑓(𝑡) (1.2) The advantages of CFD is that satisfying the properties of the classical integer derivatives [21-23]. Assume that 𝑓 and 𝑔 are 𝛼 - differentiable functions and 𝜆, 𝑎, 𝑏 are constants, then the CFD satisfies the following properties: i. 𝐷 𝛼 𝑡 𝑝 = 𝑝𝑡 𝑝−𝛼 , 𝑝 ∈ ℝ . ii. 𝐷 𝛼 𝜆 = 0. iii. 𝐷 𝛼 𝑓(𝑡) = 𝑡 𝑑𝑓𝑑𝑡 . iv. 𝐷 𝛼 (𝑎𝑓 + 𝑏𝑔) = 𝑎𝐷 𝛼 𝑓 + 𝑏𝐷 𝛼 𝑔. v. 𝐷 𝛼 (𝑓𝑔) = 𝑓𝐷 𝛼 𝑔 + 𝑔𝐷 𝛼 𝑓. vi. 𝐷 𝛼 (𝑓𝑔) = 𝑔𝐷 𝛼 𝑓 − 𝑓𝐷 𝛼 𝑔𝑔 . vii. 𝐷 𝛼 (𝑓(𝑔(𝑡))) = 𝑑𝑓𝑑𝑔 𝐷 𝛼 𝑔(𝑡) = 𝑡 𝑑𝑓𝑑𝑔 𝑑𝑔𝑑𝑡 . In mathematical physics and many other phenomena in various fields of applied science are described by nonlinear models, particularly by nonlinear partial differential equations (NLPDEs), such as astrophysics, optics, fluid dynamics, mathematical biology, plasma physics and so on. It is important to search for the solutions of the concerned models to understand and interpret their physical mechanism and behavior. The search for exact solutions to NLPDE has become of great interest by mathematicians and physicists and they have exerted great efforts for that . There are many powerful and efficient methods for finding exact solutions have been introduced, such as the inverse scattering method [24, 25], Hirota method [26], Bäcklund transformation [27-29], Darboux transformation [30], hyperbolic tangent function method [31, 32], Jacobi elliptic function method [33, 34], truncated Painlevé expansion method [35-38], homogenous balance method [39, 40], and there are other various methods in the literature. The Bäcklund transformations (BT) are considered as powerful tools for integrable systems to relate NLPDEs and their solutions [27, 28, 29, 41, 42]. Up to now, the research is still devoted and ongoing for finding the BT, e.g from the Painlevé property [35-38], the Ablowitz-Kaup-Newell-Segur (AKNS) system [41], the nonclassical symmetries [42], etc. One of the simple NLPDEs in mathematical physics is the Burger’s equation that arising in many areas of science such as Navier-Stokes equations, traffic flow, and acoustics [43, 44]. Consider the space-time conformable fractional Burger’s equation (CFBE) 𝐷 𝑡𝛼 𝑢 + 𝑢𝐷 𝑥𝛼 𝑢 = 𝜎𝐷 𝑥𝛼𝛼 𝑢, (1.3) where 𝜎 is arbitrary constants, 𝛼 ∈ (0,1], and 𝐷 𝑥𝛼𝛼 𝑢 = 𝐷 𝑥𝛼 (𝐷 𝑥𝛼 𝑢) . In a few years ago, many researchers are introduced the solutions of CFBE by different methods. The Hopf-Cole transform is applied to a time CFBE, subsequently the approximate analytical solution is founded by applying a Homotopy Analysis Method [45]. Also, the exact solution is obtained by using Fourier transform [46]. The residual power series method is introduced for finding approximate solutions of a time CFBE [47]. Also, the residual power series method is used in finding the solution of the space-time conformable fractional KdV-Burgers equation [48]. The solution of regular and singular space-time coupled CFBEs is formulated by applying double Laplace transform [49]. The rest of the paper is organized as follows: In Sec. 2, we show that the space-time CFBE possesses the Painlevé property. In Sec. 3 the exact solutions for the space-time CFBE is constructed based on the hyperbolic tangent function method. In Sec. 4, the BT is used for generating abundant new exact solutions for the space-time CFBE. The paper is concluded in Sec. 5.
2. Painlevé property for the space-time conformal fractional Burger’s equation
In this section, we intent to test the Painlevé property for the space-time CFBE given by Eq. (1.3) following the approach introduced in the Refs. [35- 37, 50]. The space-time CFBE has the Painlevé property when all the movable singularities are simple poles. For Eq. (1.3) we let 𝑢 = 𝑢 (𝑥 𝛼 𝛼 , 𝑡 𝛼 𝛼 ) = 𝜙 𝑛 ∑ 𝜙 𝑗 𝑢 𝑗∞𝑗=0 , (2.1) where 𝑢 ≠ 0 , 𝑛 is an integer, 𝜙 = 𝜙 ( 𝑥 𝛼 𝛼 , 𝑡 𝛼 𝛼 ) and 𝑢 𝑗 = 𝑢 𝑗 ( 𝑥 𝛼 𝛼 , 𝑡 𝛼 𝛼 ) are analytic functions of ( 𝑥 𝛼 𝛼 , 𝑡 𝛼 𝛼 ) in a neighborhood of 𝑀 = {( 𝑥 𝛼 𝛼 , 𝑡 𝛼 𝛼 ) ∶ 𝜙 ( 𝑥 𝛼 𝛼 , 𝑡 𝛼 𝛼 ) = 0} . To determine the values of 𝑛 we consider the ansätz 𝑢 = 𝜙 𝑛 𝑢 (2.2) By using chain rule on Eq. (2.2) we obtain 𝐷 𝑡𝛼 𝑢 = 𝑛𝑢 𝜙 𝑛−1 𝐷 𝑡𝛼 𝜙 + 𝜙 𝑛 𝐷 𝑡𝛼 𝑢 𝐷 𝑥𝛼 𝑢 = 𝑛𝑢 𝜙 𝑛−1 𝐷 𝑥𝛼 𝜙 + 𝜙 𝑛 𝐷 𝑥𝛼 𝑢 𝐷 𝑥𝛼𝛼 𝑢 = 𝑛𝑢 𝜙 𝑛−1 𝐷 𝑥𝛼𝛼 𝜙 + 𝑛(𝑛 − 1)𝑢 𝜙 𝑛−2 (𝐷 𝑥𝛼 𝜙) + 2𝑛𝜙 𝑛−1 𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼 𝑢 + 𝜙 𝑛 𝐷 𝑥𝛼𝛼 𝑢 . } (2.3) By substituting Eqs. (2.2) and (2.3) into Eq. (1.3), we get 𝑛𝑢 𝜙 𝑛−1 𝐷 𝑡𝛼 𝜙 + 𝜙 𝑛 𝐷 𝑡𝛼 𝑢 + 𝑛𝑢 𝜙 𝐷 𝑥𝛼 𝜙 + 𝑢 𝜙 𝐷 𝑥𝛼 𝑢 = 𝑛𝜎𝑢 𝜙 𝑛−1 𝐷 𝑥𝛼𝛼 𝜙 + 𝑛(𝑛 − 1)𝜎𝑢 𝜙 𝑛−2 (𝐷 𝑥𝛼 𝜙) + 2𝑛𝜎𝜙 𝑛−1 𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼 𝑢 + 𝜎𝜙 𝑛 𝐷 𝑥𝛼𝛼 𝑢 . (2.4) The dominant terms of Eq. (2.4) are 𝜙 & 𝜙 𝑛−2 . Balancing of these gives 𝑛 = −1. Thus, 𝑢 = −2𝜎𝐷 𝑥𝛼 𝜙 (2.5) Back to Eq. (2.1), we can consider the following Ansӓtz with resonance 𝑟 as 𝑢 = 𝑢 𝜙 −1 + 𝑏𝜙 𝑟−1 = −2𝜎𝜙 −1 𝐷 𝑥𝛼 𝜙 + 𝑏𝜙 𝑟−1 (2.6) From Eq. (2.6) we get 𝐷 𝑥𝛼 𝑢 = 2𝜎𝜙 −2 (𝐷 𝑥𝛼 𝜙) − 2𝜎𝜙 −1 𝐷 𝑥𝛼𝛼 𝜙 + 𝑏(𝑟 − 1)𝜙 𝑟−2 𝐷 𝑥𝛼 𝜙 = 2𝜎𝜙 −2 (𝐷 𝑥𝛼 𝜙) + 𝑏(𝑟 − 1)𝜙 𝑟−2 𝐷 𝑥𝛼 𝜙 + ⋯ 𝐷 𝑥𝛼𝛼 𝑢 = −4𝜎𝜙 −3 (𝐷 𝑥𝛼 𝜙) + 4𝜎𝜙 −2 𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼𝛼 𝜙 + 2𝜎𝜙 −2 𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼𝛼 𝜙 − 2𝜎𝜙 −1 𝐷 𝑥𝛼𝛼𝛼 𝜙 +𝑏(𝑟 − 1)(𝑟 − 2)𝜙 𝑟−3 (𝐷 𝑥𝛼 𝜙) + 𝑏(𝑟 − 1)𝜙 𝑟−2 𝐷 𝑥𝛼𝛼 𝜙 = −4𝜎𝜙 −3 (𝐷 𝑥𝛼 𝜙) + 𝑏(𝑟 − 1)(𝑟 − 2)𝜙 𝑟−3 (𝐷 𝑥𝛼 𝜙) + ⋯ } (2.7) The fractional derivatives are rearranged in terms of powers of 𝐷 𝑥𝛼 𝜙 . Now. By substituting Eqs. (2.6) and (2.7) into 𝑢𝐷 𝑥𝛼 𝑢 = 𝜎𝐷 𝑥𝛼𝛼 𝑢 , we have (−2𝜎𝜙 −1 𝐷 𝑥𝛼 𝜙 + 𝑏𝜙 𝑟−1 )(2𝜎𝜙 −2 (𝐷 𝑥𝛼 𝜙) + 𝑏(𝑟 − 1)𝜙 𝑟−2 𝐷 𝑥𝛼 𝜙 + ⋯ ) = 𝜎(−4𝜎𝜙 −3 (𝐷 𝑥𝛼 𝜙) + 𝑏(𝑟 − 1)(𝑟 − 2)𝜙 𝑟−3 (𝐷 𝑥𝛼 𝜙) + ⋯ ) ⟹ −2𝑏(𝑟 − 1)𝜎𝜙 𝑟−3 (𝐷 𝑥𝛼 𝜙) + 2𝑏𝜎𝜙 𝑟−3 (𝐷 𝑥𝛼 𝜙) = 𝜎𝑏(𝑟 − 1)(𝑟 − 2)𝜙 𝑟−3 (𝐷 𝑥𝛼 𝜙) + ⋯ ⟹ (−2𝑟 + 2 + 2 − 𝑟 + 3𝑟 − 2)𝜎𝑏𝜙 𝑟−3 (𝐷 𝑥𝛼 𝜙) = ⋯ ⟹ −(𝑟 + 1)(𝑟 − 2)𝜎𝑏𝜙 𝑟−3 (𝐷 𝑥𝛼 𝜙) = ⋯ ⟹ 𝑟 = −1 and 𝑟 = 2 . If 𝑀 is a singularity manifold, it is obtained that 𝑛 = −1 . By leading order analysis, 𝑢 = 𝜙 −1 ∑ 𝜙 𝑗 𝑢 𝑗∞𝑗=0 = ∑ 𝜙 𝑗−1 𝑢 𝑗∞𝑗=0 , (2.8) From (2.8) we get 𝐷 𝑡𝛼 𝑢 = [∑[(𝑗 − 1)𝜙 𝑗−2 𝑢 𝑗 𝐷 𝑡𝛼 𝜙 + 𝜙 𝑗−1 𝐷 𝑡𝛼 𝑢 𝑗 ] ∞𝑗=0 ] = [∑[(𝑗 − 2)𝜙 𝑗−3 𝑢 𝑗−1 𝐷 𝑡𝛼 𝜙 + 𝜙 𝑗−3 𝐷 𝑡𝛼 𝑢 𝑗−2 ] ∞𝑗=0 ], 𝐷 𝑥𝛼 𝑢 = [∑[(𝑗 − 1)𝜙 𝑗−2 𝑢 𝑗 𝐷 𝑥𝛼 𝜙 + 𝜙 𝑗−1 𝐷 𝑥𝛼 𝑢 𝑗 ] ∞𝑗=0 ] = [∑[(𝑗 − 1)𝜙 𝑗−2 𝑢 𝑗 𝐷 𝑥𝛼 𝜙 + 𝜙 𝑗−2 𝐷 𝑥𝛼 𝑢 𝑗−1 ] ∞𝑗=0 ], 𝐷 𝑥𝛼𝛼 𝑢 = [∑[(𝑗 − 1)(𝑗 − 2)𝑢 𝑗 𝜙 𝑗−3 (𝐷 𝑥𝛼 𝜙) + (𝑗 − 1)𝜙 𝑗−2 𝑢 𝑗 𝐷 𝑥𝛼𝛼 𝜙 + 2(𝑗 − 1)𝜙 𝑗−2 𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼 𝑢 𝑗∞𝑗=0 + 𝜙 𝑗−1 𝐷 𝑥𝛼𝛼 𝑢 𝑗 ]], = [∑[(𝑗 − 1)(𝑗 − 2)𝑢 𝑗 𝜙 𝑗−3 (𝐷 𝑥𝛼 𝜙) + (𝑗 − 2)𝜙 𝑗−3 𝑢 𝑗−1 𝐷 𝑥𝛼𝛼 𝜙 ∞𝑗=0 + 2(𝑗 − 2)𝜙 𝑗−3 𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼 𝑢 𝑗−1 + 𝜙 𝑗−3 𝐷 𝑥𝛼𝛼 𝑢 𝑗−2 ]]. The recursion relations for 𝑢 𝑗 are found to be 𝐷 𝑡𝛼 𝑢 𝑗−2 + (𝑗 − 2)𝑢 𝑗−1 𝐷 𝑡𝛼 𝜙 + ∑ 𝑢 𝑗−𝑚 [𝐷 𝑥𝛼 𝑢 𝑚−1 + (𝑚 − 1)𝑢 𝑚 𝐷 𝑥𝛼 𝜙] 𝑗𝑚=0 = 𝜎[(𝑗 − 1)(𝑗 − 2)𝑢 𝑗 (𝐷 𝑥𝛼 𝜙) + (𝑗 − 2)𝑢 𝑗−1 𝐷 𝑥𝛼𝛼 𝜙 + 2(𝑗 − 2)𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼 𝑢 𝑗−1 + 𝐷 𝑥𝛼𝛼 𝑢 𝑗−2 ], (2.9) where 𝑢 𝑘 = 0 for 𝑘 = −1, −2, −3, … From the recurrence formula (2.9) we obtain for: 𝑗 = 0 ⟹ −𝑢 𝐷 𝑥𝛼 𝜙 = 2𝜎𝑢 (𝐷 𝑥𝛼 𝜙) ⟹ 𝑢 = −2𝜎𝐷 𝑥𝛼 𝜙. (2.10) Eq. (2.10) is identical to Eq. (2.5). 𝑗 = 1 ⟹ −𝑢 𝐷 𝑡𝛼 𝜙 + ∑ 𝑢 [𝐷 𝑥𝛼 𝑢 𝑚−1 + (𝑚 − 1)𝑢 𝑚 𝐷 𝑥𝛼 𝜙] = −𝜎[𝑢 𝐷 𝑥𝛼𝛼 𝜙 + 2𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼 𝑢 ] ⟹ −𝑢 𝐷 𝑡𝛼 𝜙 − 𝑢 𝑢 𝐷 𝑥𝛼 𝜙 + 𝑢 𝐷 𝑥𝛼 𝑢 = −𝜎[𝑢 𝐷 𝑥𝛼𝛼 𝜙 + 2𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼 𝑢 ] = 𝜎[2𝜎𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼𝛼 𝜙 + 4𝜎𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼𝛼 𝜙] = 6𝜎 𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼𝛼 𝜙 ⟹ −𝑢 𝐷 𝑡𝛼 𝜙 − 𝑢 𝑢 𝐷 𝑥𝛼 𝜙 + 4𝜎 𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼𝛼 𝜙 = 6𝜎 𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼𝛼 𝜙 ⟹ −𝑢 𝐷 𝑡𝛼 𝜙 − 𝑢 𝑢 𝐷 𝑥𝛼 𝜙 = 2𝜎 𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼𝛼 𝜙 = −𝑢 𝜎𝐷 𝑥𝛼𝛼 𝜙 ⟹ ⟹ 𝐷 𝑡𝛼 𝜙 + 𝑢 𝐷 𝑥𝛼 𝜙 − 𝜎𝐷 𝑥𝛼𝛼 𝜙 = 0 (2.11) 𝑗 = 2 and considering Eqs. (2.10, 2.11), then from Eq. (2.9) we get 𝐷 𝑡𝛼 𝑢 + ∑ 𝑢 [𝐷 𝑥𝛼 𝑢 𝑚−1 + (𝑚 − 1)𝑢 𝑚 𝐷 𝑥𝛼 𝜙] = 𝜎𝐷 𝑥𝛼𝛼 𝑢 ⟹ 𝐷 𝑡𝛼 𝑢 + 𝑢 𝐷 𝑥𝛼 𝑢 + 𝑢 𝐷 𝑥𝛼 𝑢 = 𝜎𝐷 𝑥𝛼𝛼 𝑢 ⟹ 𝐷 𝑥𝛼 (𝐷 𝑡𝛼 𝜙 + 𝑢 𝐷 𝑥𝛼 𝜙 − 𝜎𝐷 𝑥𝛼𝛼 𝜙) = 0 (2.12) By Eq. (2.11) the compatability condition Eq. (2.12) at 𝑗 = 2 is satisfied identically. For 𝑗 = 3 , then from recurrence formula given by Eq. (2.9) we have 𝐷 𝑡𝛼 𝑢 + 𝑢 𝐷 𝑡𝛼 𝜙 + ∑ 𝑢 [𝐷 𝑥𝛼 𝑢 𝑚−1 + (𝑚 − 1)𝑢 𝑚 𝐷 𝑥𝛼 𝜙] = 𝜎[2𝑢 (𝐷 𝑥𝛼 𝜙) + 𝑢 𝐷 𝑥𝛼𝛼 𝜙 + 2𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼 𝑢 + 𝐷 𝑥𝛼𝛼 𝑢 ] ⟹ [𝐷 𝑡𝛼 𝑢 + 𝑢 𝐷 𝑥𝛼 𝑢 − 𝜎𝐷 𝑥𝛼𝛼 𝑢 ] + 𝑢 [𝐷 𝑡𝛼 𝜙 + 𝑢 𝐷 𝑥𝛼 𝜙 − 𝜎𝐷 𝑥𝛼𝛼 𝜙]+ [𝑢 𝐷 𝑥𝛼 𝑢 + 𝑢 𝐷 𝑥𝛼 𝑢 − 2𝜎𝐷 𝑥𝛼 𝜙𝐷 𝑥𝛼 𝑢 ] + 𝑢 𝐷 𝑥𝛼 𝜙[𝑢 − 2𝜎𝐷 𝑥𝛼 𝜙] = 0 ⟹ [𝐷 𝑡𝛼 𝑢 + 𝑢 𝐷 𝑥𝛼 𝑢 − 𝜎𝐷 𝑥𝛼𝛼 𝑢 ] + [𝑢 𝐷 𝑥𝛼 𝑢 + 2𝑢 𝐷 𝑥𝛼 𝑢 ] + 2𝑢 𝑢 𝐷 𝑥𝛼 𝜙 = 0 (2.13) Since the resonances occur at 𝑟 = −1, 2, and ( 𝜙 , 𝑢 ) are arbitrary functions of ( 𝑥 𝛼 𝛼 , 𝑡 𝛼 𝛼 ) in the expansion (2.13). If we let the arbitrary functions 𝑢 = 𝑢 = 0 , then we get 𝐷 𝑡𝛼 𝑢 + 𝑢 𝐷 𝑥𝛼 𝑢 = 𝜎𝐷 𝑥𝛼𝛼 𝑢 (2.14) For 𝑗 = 3 then Eq. (2.14) is automatically satisfied for 𝑢 . for 𝑗 = 4 , 𝑢 = −2𝜎𝐷 𝑥𝛼 𝜙, and 𝑢 = 𝑢 = 0 then from Eq. (2.9) we obtain ∑ 𝑢 [𝐷 𝑥𝛼 𝑢 𝑚−1 + (𝑚 − 1)𝑢 𝑚 𝐷 𝑥𝛼 𝜙] = 6𝜎𝑢 (𝐷 𝑥𝛼 𝜙) ⟹ 𝑢 𝐷 𝑥𝛼 𝜙(2𝑢 − 6𝜎𝐷 𝑥𝛼 𝜙) = 0 ⟹ 𝑢 = 0. Thus, we conclude that all 𝑢 𝑗 = 0, 𝑗 ≥ 2, (2.15) providing 𝑢 satisfies space-time CFBE (Eq. (2.14)). Hence, the space-time CFBE possesses the Painlevé property and the truncation of the Painlevé expansion Eq. (2.8) then takes the form 𝑢 = − 2𝜎𝜙 𝐷 𝑥𝛼 𝜙 + 𝑢 , (2.16) which is the Bäcklund transform for the space-time CFBE. When 𝑢 = 0 then from Eq. (2.16) we obtain 𝑢 = − 2𝜎𝜙 𝐷 𝑥𝛼 𝜙, (2.17) Eq. (2.17) yields the fractional Cole-Hopf transform. When 𝑢 = 𝜙 then we have 𝑢 = − 2𝜎𝜙 𝐷 𝑥𝛼 𝜙 + 𝜙, (2.18) where 𝐷 𝑡𝛼 𝜙 + 𝜙𝐷 𝑥𝛼 𝜙 = 𝜎𝐷 𝑥𝛼𝛼 𝜙 (2.19) Eqs. (2.18) and (2.19) represent the Bäcklund transform for the space-time CFBE.
3. Exact solutions for the space-time conformal fractional Burger’s equation
The hyperbolic tangent function method (tanh-method) is an efficient method that used for constructing the exact traveling wave solutions for NLPDEs. The Ansӓtz is considered as a power series in tanh, where tanh is introduced as a new variable. Moreover, the derivatives of tanh are also given in terms of tanh itself [31, 32]. In the following steps we seek to describe the general tanh-method for constructing the solutions of the space-time conformal fractional partial deferential equations (CFPDEs). Consider a general space-time conformal fractional partial differential equation (space-time CFPDE):
𝐻(𝑢, 𝐷 𝑡𝛼 𝑢, 𝐷 𝑥𝛼 𝑢, 𝐷 𝑡𝛼𝛼 𝑢, 𝐷 𝑡𝛼 (𝐷 𝑥𝛼 𝑢), 𝐷 𝑥𝛼𝛼 𝑢, . . . ) = 0. (3.1) In the following steps, we summarize the tanh-method for solving Eq. (1.3): Step 1: consider the traveling wave solution of Eq. (1.3) as 𝑢(𝑥, 𝑡) = 𝑢(𝜉) 𝜉 = 𝑘𝛼 (𝑥 𝛼 − 𝜔𝑡 𝛼 ), (1.3) where 𝑘 and 𝜔 are the wave number and wave velocity, respectively. Substitution of Eq. (1.3) into Eq. (1.3) produces the following ordinary differential equation for 𝑢(𝜉) 𝐻̃(𝑢, 𝑢 ′ , 𝑢 ′′ , … ) = 0 𝑢 ′ = 𝑑𝑢𝑑𝜉 , … 𝑒𝑡𝑐. (1.1) Step 2: Assume that the solution of Eq. (3.3) can be expressed as a finite power series of 𝐹(𝜉) 𝑢(𝜉) = 𝑎 + ∑ 𝑎 𝑗𝑠𝑗=1 𝐹 𝑗 (𝜉), 𝑎 𝑠 ≠ 0, (3.4) where 𝑠 ∈ ℕ , which is determined by balancing the highest power of the linear term with the highest power of the nonlinear term in Eq. (1.1), and 𝑎 𝑗 are constants to be determined. The new exact solutions of the space-time CFPDE can be obtained via the solutions of the Riccati equation that satisfied by tanh function. Consider the required Riccati equation to be 𝐹 ′ = 𝐴 + 𝐵𝐹 + 𝐶𝐹 , ′ ≡ 𝑑𝑑𝜉 , (3.5) where 𝐴, 𝐵 and 𝐶 are constants. Step 3: Substitution of Eq. (3.4) into Eq. (3.3), generates system of algebraic equations for 𝑎 , 𝑎 , … , 𝑎 𝑠 , 𝜔, and 𝑘 . Step 4: Solution of the system obtained in step 3, produces the values of 𝑎 , 𝑎 , … , 𝑎 𝑠 , 𝜔 and 𝑘 in terms of 𝐴, 𝐵 and 𝐶 . By substituting these results into Eq. (3.4), we obtain the general form of traveling wave solution of Eq. (3.1). Choosing of each proper value for 𝐴, 𝐵 and 𝐶 in Eq. (3.5) corresponds a solution 𝐹(𝜉) of Eq. (3.5) that is could be one of the hyperbolic function or triangular function as follows. Case 1: If
𝐴 = 𝐶 = 1, and
𝐵 = 0 , then Eq. (3.5) has the solutions, 𝑡𝑎𝑛𝜉.
Case 2: If
𝐴 = 𝐶 = −1, and
𝐵 = 0 , then Eq. (3.5) has the solutions, 𝑐𝑜𝑡𝜉 . Case 3: If
𝐴 = 1, 𝐵 = 0, and
𝐶 = −1 , then Eq. (3.5) has the solutions, 𝑡𝑎𝑛ℎ𝜉, 𝑐𝑜𝑡ℎ𝜉 . Case 4: If
𝐴 = , 𝐵 = 0 and, 𝐶 = − , then Eq. (3.5) has the solutions, 𝑡𝑎𝑛ℎ𝜉 ± 𝑖 𝑠𝑒𝑐ℎ𝜉, 𝑐𝑜𝑡ℎ𝜉 ± 𝑐𝑠𝑐ℎ𝜉, 𝑡𝑎𝑛ℎ𝜉1 ± 𝑠𝑒𝑐ℎ𝜉 , 𝑐𝑜𝑡ℎ𝜉1 ± 𝑖 𝑐𝑠𝑐ℎ𝜉 , 𝑖 = −1. Case 5: If
𝐴 = 𝐶 = , and 𝐵 = 0, then Eq. (3.5) has the solutions, 𝑡𝑎𝑛𝜉 ± 𝑠𝑒𝑐𝜉, 𝑐𝑠𝑐𝜉 − 𝑐𝑜𝑡𝜉, 𝑡𝑎𝑛𝜉1± 𝑠𝑒𝑐𝜉 . Case 6: If
𝐴 = 𝐶 = − , and 𝐵 = 0, then Eq. (3.5) has the solutions, 𝑐𝑜𝑡𝜉 ± 𝑐𝑠𝑐𝜉, 𝑠𝑒𝑐𝜉 − 𝑡𝑎𝑛𝜉, 𝑐𝑜𝑡𝜉1± 𝑐𝑠𝑐𝜉 . Case 7: If
𝐴 = 1, 𝐶 = −4, and
𝐵 = 0 , then Eq. (3.5) has the solutions, 𝑡𝑎𝑛ℎ𝜉1+𝑡𝑎𝑛ℎ 𝜉 . Case 8: If
𝐴 = 1, 𝐶 = 4, and
𝐵 = 0 , then Eq. (3.5) has the solutions, 𝑡𝑎𝑛𝜉1−𝑡𝑎𝑛 𝜉 . Case 9: If
𝐴 = −1, 𝐶 = −4, and
𝐵 = 0 , then Eq. (3.5) has the solutions, 𝑐𝑜𝑡𝜉 1−𝑐𝑜𝑡 𝜉 . Case 10: If
𝐴 = 1, 𝐵 = −2, and
𝐶 = 2 , then Eq. (3.5) has the solutions, 𝑡𝑎𝑛𝜉1+𝑡𝑎𝑛𝜉 . Case 11: If
𝐴 = 1 and
𝐵 = 𝐶 = 2 , then Eq. (3.5) has the solutions, 𝑡𝑎𝑛𝜉1−𝑡𝑎𝑛𝜉 . Case 12: If
𝐴 = −1, 𝐵 = 2, and
𝐶 = −2 , then Eq. (3.5) has the solutions, 𝑐𝑜𝑡𝜉 1+𝑐𝑜𝑡𝜉 . Case 13: If
𝐴 = −1, 𝐵 = 𝐶 = −2 , then Eq. (3.5) has the solutions, 𝑐𝑜𝑡𝜉 1−𝑐𝑜𝑡𝜉 . Case 14: If
𝐴 = 𝐵 = 0 and
𝐶 ≠ 0 , then Eq. (3.5) has the solutions, −1 𝐶𝜉+𝐶 . Case 15: If
𝐴 ≠ 0, 𝐶 = 0, and
𝐵 ≠ 0 , then Eq. (3.5) has the solutions, (exp(𝐵𝜉) − 𝐴). Case 16: If
𝐴 = 0 and
𝐵 = 𝐶 = 1, then Eq. (3.5) has the solutions, exp(𝜉)1−exp(𝜉) . Case 17: If 𝐴 = 0 , and 𝐵 = 𝐶 = , then Eq. (3.5) has the solutions, exp( 𝜉2 )1−exp( 𝜉2 ) . Case 18: If
𝐴 = − , 𝐵 = 0 , and
𝐶 = , then Eq. (3.5) has the solutions, −𝑡𝑎𝑛ℎ ( 𝜉2 ) , −𝑐𝑜𝑡ℎ ( 𝜉2 ) . Case 19: If 𝐴 = −1 , and
𝐵 = 𝐶 = 2 , then Eq. (3.5) has the solutions, − − √3𝑡𝑎𝑛ℎ(√3𝜉)2 , − − √3𝑐𝑜𝑡ℎ(√3𝜉)2 . Case 20: If 𝐴 = 1 , 𝐵 = 1 , and
𝐶 = 1 , then Eq. (3.5) has the solutions, − + √3𝑡𝑎𝑛( √32 𝜉)2 , − − √3𝑐𝑜𝑡( √32 𝜉)2 . Case 21: If 𝐴 = −4 , 𝐵 = 0 , and
𝐶 = 4 , then Eq. (3.5) has the solutions, −𝑡𝑎𝑛ℎ(4𝜉), −𝑐𝑜𝑡ℎ(4𝜉) . Case 22: If
𝐴 = , 𝐵 = −1 , and
𝐶 = 1 , then Eq. (3.5) has the solutions, + 𝑡𝑎𝑛 ( 𝜉2 ) , − 𝑐𝑜𝑡 ( 𝜉2 ) . Now, we need to implement the tanh-method technique into the space-time CFBE given by Eq. (1.3) to generate new exact solutions. Firstly, substituting the traveling wave solution given by Eq. (3.2) into Eq. (1.3) we obtain −𝜔𝑢 ′ + 𝑢𝑢 ′ − 𝑘𝜎𝑢 ′′ = 0, (3.6) where 𝐷 𝑡𝛼 𝑢 = −𝜔𝑘𝑢 ′ , 𝐷 𝑥𝛼 𝑢 = 𝑘𝑢 ′ , 𝐷 𝑥𝛼𝛼 𝑢 = 𝑘 𝑢 ′′ , , by balancing 𝑢 ′′ , with 𝑢𝑢 ′ gives 𝑠 = 1 . Use 𝑠 = 1 in Eq. (3.4), then the solution of Eq. (1.3) can be expressed as 𝑢 = 𝑎 + 𝑎 𝐹, (3.7) substituting Eq. (3.7) into Eq. (3.6) and using Eq. (3.5), then we obtain a set of algebraic equations with respect to 𝐹 𝑖 (𝑖 = 0,1,2,3). Equating the coefficients of 𝐹 𝑖 (𝑖 = 0,1,2,3) to zero. The solution of the resulting system is given by 𝑎 = 𝜔 + 𝜎𝐵𝑘, 𝑎 = 2𝜎𝐶𝑘, (3.8) 0 with 𝜔 and 𝑘 are arbitrary constants. Inserting Eq. (3.8) into Eq. (3.7) and using the special solutions of Eq. (3.5), we obtain the following soliton like-solution and triangular periodic solutions of the space-time CFBE: 𝑢 = 𝜔 + 2𝜎𝑘 𝑡𝑎𝑛𝜉, (3.9) 𝑢 = 𝜔 − 2𝜎𝑘 𝑐𝑜𝑡𝜉, (3.10) 𝑢 = 𝜔 − 2𝜎𝑘 𝑡𝑎𝑛ℎ𝜉, (3.11) 𝑢 = 𝜔 − 2𝜎𝑘 𝑐𝑜𝑡ℎ𝜉, (3.12) 𝑢 = 𝜔 − 𝜎𝑘(𝑡𝑎𝑛ℎ𝜉 ± 𝑖 𝑠𝑒𝑐ℎ𝜉), (3.13) 𝑢 = 𝜔 − 𝜎𝑘(𝑐𝑜𝑡ℎ𝜉 ± 𝑐𝑠𝑐ℎ𝜉), (3.14) 𝑢 = 𝜔 + 𝜎𝑘(𝑡𝑎𝑛𝜉 ± 𝑠𝑒𝑐𝜉), (3.15) 𝑢 = 𝜔 − 𝜎𝑘(𝑐𝑜𝑡𝜉 ± 𝑐𝑠𝑐𝜉), (3.16) 𝑢 = 𝜔 − 𝜉 , (3.17) 𝑢 = 𝜔 + 𝜉 , (3.18) 𝑢 = 𝜔 − 𝜉 , (3.19) 𝑢 = 𝜔 − 𝜎𝑘 𝑡𝑎𝑛ℎ𝜉1+𝑠𝑒𝑐ℎ𝜉 , (3.20) 𝑢 = 𝜔 − 𝜎𝑘 𝑐𝑜𝑡ℎ𝜉1+𝑖 𝑐𝑠𝑐ℎ𝜉 , (3.21) 𝑢 = 𝜔 + 𝜎𝑘 𝑡𝑎𝑛𝜉1+𝑠𝑒𝑐𝜉 , (3.22) 𝑢 = 𝜔 − 𝜎𝑘 𝑐𝑜𝑡𝜉1+𝑐𝑠𝑐𝜉 , (3.23) 𝑢 = 𝜔 − 2𝜎𝑘 + , (3.24) 𝑢 = 𝜔 + 2𝜎𝑘 − , (3.25) 𝑢 = (𝜎𝑘+𝜔)+(𝜎𝑘−𝜔)𝑒 𝜉 𝜉 , (3.26) 𝑢 = (𝜎𝑘/2+𝜔)+(𝜎𝑘/2−𝜔)𝑒 𝜉/2 𝜉/2 , (3.27) 𝑢 = 𝜔 − 𝜎𝑘 𝑡𝑎𝑛ℎ 𝜉2 , (3.28) 𝑢 = 𝜔 − 𝜎𝑘 𝑐𝑜𝑡ℎ 𝜉2 , (3.29) 𝑢 = 𝜔 − 2√3𝜎𝑘 𝑡𝑎𝑛ℎ√3𝜉, (3.30) 1 𝑢 = 𝜔 − 2√3𝜎𝑘 𝑐𝑜𝑡ℎℎ√3𝜉, (3.31) 𝑢 = 𝜔 + √3𝜎𝑘 𝑡𝑎𝑛 ( √32 𝜉), (3.32) 𝑢 = 𝜔 − √3𝜎𝑘 𝑐𝑜𝑡 ( √32 𝜉), (3.33) 𝑢 = 𝜔 − 8𝜎𝑘 𝑡𝑎𝑛ℎ(4𝜉), (3.34) 𝑢 = 𝜔 − 8𝜎𝑘 𝑐𝑜𝑡ℎ(4𝜉), (3.35) 𝑢 = 𝜔 + 𝜎𝑘 𝑡𝑎𝑛 ( 𝜉2 ), (3.36) 𝑢 = 𝜔 − 𝜎𝑘 𝑐𝑜𝑡 ( 𝜉2 ), (3.37) Remarks: 1- Making the transformation 𝑘 → 2𝑘 then Eq. (3.28) and Eq. (3.29) transformed to Eq. (3.11) and Eq. (3.12) respectively, and if 𝑘 → 2𝑘𝑖 where 𝑖 = √−1 , they transformed to Eq. (3.9) and Eq. (3.10) respectively. 2- Making the transformation 𝑘 → 𝑘√3 then Eq. (3.30) and Eq. (3.31) transformed to Eq. (3.11) and Eq. (3.12) respectively. 3- Making the transformation 𝑘 → then Eq. (3.32) and Eq. (3.33) transformed to Eq. (3.9) and Eq. (3.10) respectively. 4- Making the transformation 𝑘 → 2𝑘 then Eq. (3.27) transformed to Eq. (3.26). 5- Making the transformation 𝑘 → 𝑘4 then Eq. (3.34) and Eq. (3.35) transformed to Eq. (3.11) and Eq. (3.12) respectively. 6- Making the transformation 𝑘 → 2𝑘 then Eq. (3.36) and Eq. (3.37) transformed to Eq. (3.9) and Eq. (3.10) respectively. Also, we can get two new exact solutions 𝑢 = 𝜔 − 2𝜎𝑘 (𝑡𝑎𝑛ℎ𝜉 + 𝑐𝑜𝑡ℎ𝜉), (3.38) 𝑢 = 𝜔 + 2𝜎𝑘 (𝑡𝑎𝑛𝜉 − 𝑐𝑜𝑡𝜉), (3.39) with 𝜉 = 𝑘𝛼 (𝑥 𝛼 − 𝜔𝑡 𝛼 ). In the following section we use the obtained solutions Eqs. (3.9) - (3.39) to generate new abundant exact solutions via BT.
4. BT and abundant exact solutions
From section 2, we show that the space-time CFBE possesses the Painlevé property and it has a BT in the form 𝑢 = − 𝐷 𝑥𝛼 𝜙 + 𝑤, (4.1) where 𝜙 = 𝜙 ( 𝑥 𝛼 𝛼 , 𝑡 𝛼 𝛼 ) is the singular manifold variable, w is a function of 𝑥 𝛼 𝛼 and 𝑡 𝛼 𝛼 . Also, the function w solves the space-time CFBE given by Eq. (1.3) and the function 𝜙 satisfies the FDE 𝐷 𝑡𝛼 𝜙 + 𝑤𝐷 𝑥𝛼 𝜙 = 𝜎𝐷 𝑥𝛼𝛼 𝜙. (4.2) Now, if we take 𝑤 = 𝜙 then the function 𝜙 satisfies also the space-time CFBE 𝐷 𝑡𝛼 𝜙 + 𝜙𝐷 𝑥𝛼 𝜙 = 𝜎𝐷 𝑥𝛼𝛼 𝜙, (4.3) thus the BT for the space-time CFBE takes the following recurrence form 𝑢 𝑛+1 = − 𝑛 𝜕 𝛼 𝑢 𝑛 𝜕𝑥 𝛼 + 𝑢 𝑛 . (4.4) We turn to the application of the BT for the FDEs. Their power lies in that they may be used to generate additional solutions of the FDEs. Here 𝑢 𝑛+1 quantities refer to new solution and 𝑢 𝑛 quantities refer to old solution. This means that, on the basis of a known solution to the space-time CFBE, we are able to find new solution of space-time CFBE. To construct the new solution of the space-time CFBE one can start with the solution 𝑢 obtained in Eq. (3.9) and using BT given in Eq. (4.4) we get the following set of new solutions: 𝑢 = −4𝜎 𝑘 +𝜔 +4𝜎𝑘𝜔 𝑡𝑎𝑛𝜉𝜔+2𝜎𝑘 𝑡𝑎𝑛𝜉 , (4.5) Inserting 𝑢 into the BT given in Eq. (4.4) we have 𝑢 = 𝑘 𝜔−𝜔 +2𝜎𝑘(4𝜎 𝑘 −3𝜔 ) 𝑡𝑎𝑛𝜉4𝜎 𝑘 −𝜔 −4𝜎𝑘𝜔 𝑡𝑎𝑛𝜉 , (4.6) furthermore, using 𝑢 and BT given in Eq. (4.4) we get 𝑢 = −16𝜎 𝑘 +24𝜎 𝑘 𝜔 −𝜔 +(32𝜎 𝑘 𝜔−8𝜔 𝜎𝑘) 𝑡𝑎𝑛𝜉12𝜎 𝑘 𝜔−𝜔 +2𝜎𝑘(4𝜎 𝑘 −3𝜔 ) 𝑡𝑎𝑛𝜉 , (4.7) and so on, we can get a sequences of exact solutions generated by the known tan-function solution Eq. (3.9) of the space-time CFBE. Starting from 𝑢 obtained in Eq. (3.10) and using BT given in Eq. (4.4) we get 𝑢 = −4𝜎 𝑘 +𝜔 −4𝜎𝑘𝜔 𝑐𝑜𝑡𝜉𝜔−2𝜎𝑘 𝑐𝑜𝑡𝜉 , (4.8) Inserting 𝑢 into the BT given in Eq. (4.4) we have 3 𝑢 = −12𝜎 𝑘 𝜔+𝜔 +2𝜎𝑘(4𝜎 𝑘 −3𝜔 ) 𝑐𝑜𝑡𝜉−4𝜎 𝑘 +𝜔 −4𝜎𝑘𝜔 𝑐𝑜𝑡𝜉 , (4.9) furthermore, using 𝑢 and BT given in Eq. (4.4) we get 𝑢 = 𝑘 −24𝜎 𝑘 𝜔 +𝜔 +(32𝜎 𝑘 𝜔−8𝜔 𝜎𝑘) 𝑐𝑜𝑡𝜉−12𝜎 𝑘 𝜔+𝜔 +2𝜎𝑘(4𝜎 𝑘 −3𝜔 ) 𝑐𝑜𝑡𝜉 , (4.10) and so on, we can get a sequences of exact solutions generated by the cot-function solution Eq. (3.10) of the space time fractional CFBE. Starting from 𝑢 obtained in Eq. (3.11) and using BT given in Eq. (4.4) we get 𝑢 = −4𝜎 𝑘 −𝜔 +4𝜎𝑘𝜔 tanh 𝜉−𝜔+2𝜎𝑘 tanh 𝜉 , (4.11) Inserting 𝑢 into the BT given in Eq. (4.4) we have 𝑢 = 𝑘 𝜔+𝜔 −2𝜎𝑘(4𝜎 𝑘 +3𝜔 ) tanhξ4𝜎 𝑘 +𝜔 −4𝜎𝑘𝜔 tanh 𝜉 , (4.12) furthermore, using 𝑢 and BT given in Eq. (4.4) we get 𝑢 = 𝑘 +24𝜎 𝑘 𝜔 +𝜔 −(32𝜎 𝑘 𝜔+8𝜔 𝜎𝑘) tanh 𝜉12𝜎 𝑘 𝜔+𝜔 −2𝜎𝑘(4𝜎 𝑘 +3𝜔 ) tanh 𝜉 , (4.13) and so on, we can get a sequences of exact solutions generated by the tanh-function solution Eq. (3.11) of the space-time CFBE. Starting from 𝑢 obtained in Eq. (3.12) and using BT given in Eq. (4.4) we get 𝑢 = −4𝜎 𝑘 −𝜔 +4𝜎𝑘𝜔 coth 𝜉−𝜔+2𝜎𝑘 coth 𝜉 , (4.14) Inserting 𝑢 into the BT given in Eq. (4.4) we have 𝑢 = 𝑘 𝜔+𝜔 −2𝜎𝑘(4𝜎 𝑘 +3𝜔 ) coth 𝜉4𝜎 𝑘 +𝜔 −4𝜎𝑘𝜔 coth 𝜉 , (4.15) furthermore, using 𝑢 and BT given in Eq. (4.4) we get 𝑢 = 𝑘 +24𝜎 𝑘 𝜔 +𝜔 −(32𝜎 𝑘 𝜔+8𝜔 𝜎𝑘) coth 𝜉12𝜎 𝑘 𝜔+𝜔 −2𝜎𝑘(4𝜎 𝑘 +3𝜔 ) coth 𝜉 , (4.16) and so on, we can get a sequences of exact solutions generated by the coth-function solution Eq. (3.12) of the space-time CFBE. Starting from 𝑢 obtained in Eq. (3.13) and using BT given in Eq. (4.4) we get 𝑢 = 𝜎 𝑘 +𝜔 −2𝜎𝑘𝜔 (tanh 𝜉±𝑖 sech 𝜉)𝜔−𝜎𝑘 (tanh 𝜉±𝑖 sech 𝜉) , (4.17) and so on. Starting from 𝑢 obtained in Eq. (3.14) and using BT given in Eq. (4.4) we get 𝑢 = 𝜎 𝑘 +𝜔 −2𝜎𝑘𝜔 (coth 𝜉±csch 𝜉)𝜔−𝜎𝑘 (coth 𝜉±csch 𝜉) , (4.18) and so on. Starting from 𝑢 obtained in Eq. (3.15) and using BT given in Eq. (4.4) we get 4 𝑢 = 𝜎 𝑘 −𝜔 −2𝜎𝑘𝜔 (tan 𝜉±sec 𝜉)−𝜔−𝜎𝑘 (tan 𝜉±sec 𝜉) , (4.19) and so on. Starting from 𝑢 obtained in Eq. (3.16) and using BT given in Eq. (4.4) we get 𝑢 = −𝜎 𝑘 +𝜔 −2𝜎𝑘𝜔 (cot 𝜉±csc 𝜉)𝜔−𝜎𝑘 (cot 𝜉±csc 𝜉) , (4.20) and so on. Starting from 𝑢 obtained in Eq. (3.17) and using BT given in Eq. (4.4) we get 𝑢 = (16𝜎 𝑘 +𝜔 ) tanh 𝜉−16𝜎𝑘𝜔 tanh 𝜉+16𝜎 𝑘 +𝜔 𝜔 tanh 𝜉−8𝜎𝑘 tanh 𝜉+𝜔 , (4.21) Inserting 𝑢 into the BT given in Eq. (4.4) we have 𝑢 = (48𝜎 𝑘 𝜔+𝜔 ) tanh 𝜉−(24𝜎𝑘𝜔 +128𝜎 𝑘 ) tanh 𝜉+48𝜎 𝑘 𝜔+𝜔 (16𝜎 𝑘 +𝜔 ) tanh 𝜉−16𝜎𝑘𝜔 tanh 𝜉+16𝜎 𝑘 +𝜔 , (4.22) and so on. Starting from 𝑢 obtained in Eq. (3.18) and using BT given in Eq. (4.4) we get 𝑢 = (16𝜎 𝑘 −𝜔 ) tan 𝜉+16𝜎𝑘𝜔 tan 𝜉−16𝜎 𝑘 +𝜔 𝜔−𝜔 tan 𝜉+8𝜎𝑘 tan 𝜉 , (4.23) Inserting 𝑢 into the BT given in Eq. (4.4) we have 𝑢 = (48𝜎 𝑘 𝜔−𝜔 ) tan 𝜉+(24𝜎𝑘𝜔 −128𝜎 𝑘 ) tan 𝜉−48𝜎 𝑘 𝜔+𝜔 (16𝜎 𝑘 −𝜔 ) tan 𝜉+16𝜎𝑘𝜔 tan 𝜉−16𝜎 𝑘 +𝜔 , (4.24) and so on. Starting from 𝑢 obtained in Eq. (3.19) and using BT given in Eq. (4.4) we get 𝑢 = (16𝜎 𝑘 −𝜔 ) cot 𝜉−16𝜎𝑘𝜔 cot 𝜉−16𝜎 𝑘 +𝜔 𝜔−𝜔 cot 𝜉−8𝜎𝑘 cot 𝜉 , (4.25) Inserting 𝑢 into the BT given in Eq. (4.4) we have 𝑢 = (𝜔 −48𝜎 𝑘 𝜔) cot 𝜉+(24𝜎𝑘𝜔 −128𝜎 𝑘 ) cot 𝜉+48𝜎 𝑘 𝜔−𝜔 (𝜔 −16𝜎 𝑘 ) cot 𝜉+16𝜎𝑘𝜔 cot 𝜉+16𝜎 𝑘 −𝜔 , (4.26) and so on. Starting from 𝑢 obtained in Eq. (3.20) and using BT given in Eq. (4.4) we get 𝑢 = 𝜎 𝑘 +𝜔 +(𝜎 𝑘 +𝜔 ) 𝑠𝑒𝑐ℎ 𝜉−2𝜎𝑘𝜔 tanh 𝜉𝜔+𝜔 𝑠𝑒𝑐ℎ 𝜉−𝜎𝑘 tanh 𝜉 , (4.27) and so on. Starting from 𝑢 obtained in Eq. (3.21) and using BT given in Eq. (4.4) we get 𝑢 = 𝜎 𝑘 +𝜔 +(𝜎 𝑘 +𝜔 ) 𝑖 𝑐𝑠𝑐ℎ 𝜉−2𝜎𝑘𝜔 coth 𝜉𝜔+𝜔𝑖 𝑐𝑠𝑐ℎ 𝜉−𝜎𝑘 coth 𝜉 , (4.28) and so on. Starting from 𝑢 obtained in Eq. (3.22) and using BT given in Eq. (4.4) we get 𝑢 = − (𝜎 𝑘 −𝜔 ) 𝑐𝑜𝑠 𝜉+𝜎 𝑘 −𝜔 −2𝜎𝑘𝜔 sin 𝜉𝜔+𝜔 𝑐𝑜𝑠 𝜉+𝜎𝑘 sin 𝜉 , (4.29) and so on. Starting from 𝑢 obtained in Eq. (3.23) and using BT given in Eq. (4.4) we get 𝑢 = (𝜎 𝑘 −𝜔 ) 𝑠𝑖𝑛 𝜉+𝜎 𝑘 −𝜔 +2𝜎𝑘𝜔 cos 𝜉−𝜔−𝜔 𝑠𝑖𝑛 𝜉+𝜎𝑘 cos 𝜉 , (4.30) and so on. Starting from 𝑢 obtained in Eq. (3.24) and using BT given in Eq. (4.4) we get 5 𝑢 = 𝜔 −4𝜎 𝑘 −4𝜎𝑘𝜔+(𝜔 −4𝜎 𝑘 +4𝜎𝑘𝜔) tan 𝜉(𝜔+2𝜎𝑘) tan 𝜉+𝜔−2𝜎𝑘 , (4.31) Inserting 𝑢 into the BT given in Eq. (4.4) we have 𝑢 = 𝜔 +8𝜎 𝑘 −6𝜎𝑘𝜔 −12𝜎 𝑘 𝜔+(𝜔 −8𝜎 𝑘 +6𝜎𝑘𝜔 −12𝜎 𝑘 𝜔) tan 𝜉𝜔 −4𝜎 𝑘 −4𝜎𝑘𝜔+(𝜔 −4𝜎 𝑘 +4𝜎𝑘𝜔) tan 𝜉 , (4.32) and so on. Starting from 𝑢 obtained in Eq. (3.25) and using BT given in Eq. (4.4) we get 𝑢 = 𝜔 −4𝜎 𝑘 +4𝜎𝑘𝜔+(𝜔 −4𝜎 𝑘 −4𝜎𝑘𝜔) cot 𝜉(𝜔−2𝜎𝑘) cot 𝜉+𝜔+2𝜎𝑘 , (4.33) Inserting 𝑢 into the BT given in Eq. (4.4) we have 𝑢 = 𝜔 −8𝜎 𝑘 +6𝜎𝑘𝜔 −12𝜎 𝑘 𝜔+(𝜔 +8𝜎 𝑘 −6𝜎𝑘𝜔 −12𝜎 𝑘 𝜔) cot 𝜉𝜔 −4𝜎 𝑘 +4𝜎𝑘𝜔+(𝜔 −4𝜎 𝑘 −4𝜎𝑘𝜔) cot 𝜉 , (4.34) and so on. Starting from 𝑢 obtained in Eq. (3.26) and using BT given in Eq. (4.4) we get 𝑢 = (𝜎𝑘+𝜔) −(𝜎𝑘−𝜔) 𝑒 𝜉 𝜎𝑘+𝜔+(𝜎𝑘−𝜔)𝑒 𝜉 , (4.35) Inserting 𝑢 into the BT given in Eq. (4.4) we have 𝑢 = (𝜎𝑘+𝜔) +(𝜎𝑘−𝜔) 𝑒 𝜉 (𝜎𝑘+𝜔) −(𝜎𝑘−𝜔) 𝑒 𝜉 , (4.36) Furthermore, using 𝑢 and BT given in Eq. (4.4) we get 𝑢 = (𝜎𝑘+𝜔) −(𝜎𝑘−𝜔) 𝑒 𝜉 (𝜎𝑘+𝜔) +(𝜎𝑘−𝜔) 𝑒 𝜉 , (4.37) and so on, we can get a new sequences of exact solution of the space-time CFBE. Starting from 𝑢 obtained in Eq. (3.38) and using BT given in Eq. (4.4) we get 𝑢 = −(𝜔 +16𝜎 𝑘 ) cosh 𝜉 sinh 𝜉+8𝜎𝑘𝜔 cosh 𝜉−4𝜎𝑘𝜔−𝜔 cosh 𝜉 sinh 𝜉+4𝜎𝑘 cosh 𝜉−2𝜎𝑘 , (4.38) and so on, we can get a sequences of solutions generated by the addition of two functions tanh and coth-function solution of the space-time CFBE. Starting from 𝑢 obtained in Eq. (3.39) and using BT given in Eq. (4.4) we get 𝑢 = − (𝜔 −16𝜎 𝑘 ) cos 𝜉 sin 𝜉−8𝜎𝑘𝜔 cos 𝜉+4𝜎𝑘𝜔−𝜔 cos 𝜉 sin 𝜉+4𝜎𝑘 cos 𝜉−2𝜎𝑘 , (4.39) and so on, we can get a sequences of solutions generated by the addition of two functions tan and cot-function solution of the space-time CFBE. By means of the variational iteration method [35] and the Adomian decomposition method [36] the solution of the space-time CFBE in closed form is 𝑢 = (𝜔+𝑘)+(𝜔−𝑘) exp (𝜉/𝜎)1+exp (𝜉/𝜎) , (4.40) 6 From 𝑢 obtained in Eq. (4.40) we can get a new sequences of exact solution for the space-time CFBE by using BT given in Eq. (4.4) in the form 𝑢 = (𝜔+𝑘) +(𝜔−𝑘) exp (𝜉/𝜎) (𝜔+𝑘)+(𝜔−𝑘) exp (𝜉/𝜎) , (4.41) 𝑢 = (𝜔+𝑘) +(𝜔−𝑘) exp (𝜉/𝜎)(𝜔+𝑘) +(𝜔−𝑘) exp (𝜉/𝜎) , (4.42) 𝑢 = (𝜔+𝑘) +(𝜔−𝑘) exp (𝜉/𝜎)(𝜔+𝑘) +(𝜔−𝑘) exp (𝜉/𝜎) , (4.43) 𝑢 = (𝜔+𝑘) +(𝜔−𝑘) exp (𝜉/𝜎)(𝜔+𝑘) +(𝜔−𝑘) exp (𝜉/𝜎) , (4.44) and so on, we can get a new sequences of exact solution of the space-time CFBE.
5. Conclusion
In this work, we discuss the Painlevé property for non-linear conformal fractional differential equations for the first time . We apply the desired method to the space time conformal fractional Burger’s equation. Also, we derive the Bäcklund transform. The general solutions of the space-time conformal CFDEs is described based on the tanh-method, accordingly the method is successively implement to space-time CFBE. Moreover, the space-time CFBE is found to possess the Painlevé property and then Bäcklund transform. Also, we introduced a new recurrence formulae based on Bäcklund transform, that is enable us to derive analytical solution from known solution or old solution to give new solution. New numerous exact solutions are generated based on the Bäcklund transform.
Acknowledgments
The authors gratefully acknowledge the approval and the support of this research study by the grant no. SCI-2019-1-10-F-8306 from the Deanship of Scientific Research at Northern Border University, Arar, Saudi Arabia.
References [1] Kilbas, A. A., Srivastava, H. M., and Trujillo J. J., Theory and applications of fractional differential equations, Amsterdam: Elsevier B.V., 2006. [2] Hilfer, R., Applications of Fractional Calculus in Physics, World Science, Singapore, 2000. 7 [3] Miller, K. S., and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, NY, USA, 1993. [4]Abdel-Salam, E. A.-B., Yousif, E. A., and El-Aasser, M. A., Analytical Solution of The space-time fractional nonlinear Schrödinger equation. Reports on Mathematical Physics 2016; 19:77. [5]Abdel-Salam, E. A-B, Al-Muhiameed Z. I. A., Analytic solutions of the space-time fractional combined KdV-mKdV equation. Mathematical Problems in Engineering 2015; Article ID 871635. [6]Abdel-Salam, E. A-B, Hassan G.F., Solutions to class of linear and nonlinear fractional differential equations. Communications in Theoretical Physics 2016; 65: 127. [7]Abdel-Salam, E. A-B, Yousif E. A., Solution of nonlinear space-time fractional differential equations using the fractional Riccati expansion method, Mathematical Problems in Engineering 2013; Article ID 846283. [8] Nouh, M.I., Abdel-Salam, E .A-B., Approximate Solution to the Fractional Lane–Emden Type Equations. Iranian Journal of Science Technology, Transactions Science 2018;42: 2199-2206. [9]Ahmed, S. E., Mansour, M. A., Abdel-Salam, E. A-B, Mohamed, E.F, Studying the fractional derivative for natural convection in slanted cavity containing porous media. SN Applied Sciences 2019;1: 1117. [10] Abdel-Salam, E.A-B, Nouh, M.I., Conformable fractional polytropic gas spheres. New Astronomy 2020; 76: 101322. [11]Nouh, M.I., Abdel-Salam, E. A-B., Analytical solution to the fractional polytropic gas spheres. European Physics Journal Plus 2018;133: 149. [12] Abdel-Salam, E. A-B, Nouh, M.I., Approximate solution to the fractional second-type lane-emden equation. Astrophysics 2016; 59: 398. [13] Yousif, E. A., Abdel-Salam, E. A-B, El-Aasser, M. A., On the solution of the space-time fractional cubic nonlinear Schrödinger equation. Results In Physics 2018; 8: 702. [14] Mainardi F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London, 2010. [15] Abdel-Salam, E. A-B, Mourad, M. F., Fractional quasi AKNS ‐ technique for nonlinear space–time fractional evolution equations. Mathematical Methods in the Applied Sciences 2018; 42: 5953-5968. 8 [16] Nouh, M.I., Azzam, Y.A. & Abdel-Salam, E. A-B. Modeling fractional polytropic gas spheres using artificial neural network. Neural Comput & Applic (2020). https://doi.org/10.1007/s00521-020-05277-9 [17] Abo-Dahab, S. M., Kilany A. A., Abdel-Salam E A-B. and Hatem A, Fractional derivative order analysis and temperature-dependent properties on p- and SV-waves reflection under initial stress and three-phase-lag model, Results In Physics, 18 (2020) 103270. https://doi.org/10.1016/j.rinp.2020.103270. [18]
J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007. [19] Hong Guang Sun, Yong Zhang, Dumitru Baleanu, Wen Chen, Yang Quan Chen, A new collection of real world applications of fractional calculus in science and engineering, Communications in Nonlinear Science and Numerical Simulation , 213-231 (2018). [20] D. Valério, J. Machado, V. Kiryakova, Some pioneers of the applications of fractional calculus. Fract. Calc. Appl. Anal. (2), 552–578 (2014). [21] R. Khalil, M. A. Horani, A. Yousef, M. Sababheh. A new definition of fractional derivative, J. Comput. Appl. Math. , 57-66 (2015). [23] D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation. Calcolo , Chin. Phys. B Vol. 23, No. 11 (2014) 110203. [29] Mei-na Sun, Shu-fang Deng, and Deng-yuan Chen, The Bäcklund transformation and novel solutions for the Toda lattice, Chaos, Solitons and Fractals 23 (2005) 1169–1175. [
30] Matveev, V.B. and Salle, M.A, Darboux Transformations and Solitons, Springer Verlag (1991). [31] Malfliet, W , Solitary wave solutions of nonlinear wave equations , Amer. J. Phys . Phys. Scr.
563 – 568 (1996). [33] Liu, S.K. Fu, Z.T. Liu, S.D. and Zhao, Q, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,
Phys. Lett. A. , Jacobi elliptic function solutions of nonlinear wave equations via the new sinh-Gordon equation expansion method , J. Phys. A: Math. Gen . (2003). [35] Wies, J. Tabor, M. and Garnevale, G , The Painlevé property for partial differential equations,
J. Math. Phys . Peter A. Clarkson, the Painlevé conjecture, the Painlevé property for partial differential equations and complete integrability, Physica 18D (1986) 209-210. [37]
M. Tabor and J.D. Gibbon, aspects of the Painlevé property for partial differential equations, Physica 18D (1986) 180-189. [38]
Xing Lu, Tao Geng , Cheng Zhang, Hong-Wu Zhu , Xiang-Hua Meng, and Bo Tian, Multi-soliton solutions and their interactions for the (2+1)-dimensional Sawada-Kotera model with truncated Painlevé expansion, Hirota Bilinear method and symbolic computation, International Journal of Modern Physics B Vol. 23, No. 25 (2009) 5003–5015. [39] Wang, M.L. Zhou, Y.B. and Li, Z.B., Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,
Phys. Lett. A (1996) 67-75. [40]
Yang, L. Zhu, Z. and Wang, Y., Exact solutions of nonlinear equations,
Phys. Lett. A (1999) 55-59. [41] Konno, K. and Wadati, M., Simple Derivation of Bäcklund Transformation from Riccati Form of Inverse Method,
Prog. Theor. Phys . (1975) 1652-1656. 0 [42] Nucci, M.C., Pseudopotentials, Lax equations and Bäcklund transformations for nonlinear evolution equations, J. Phys. A: Math. Gen . 21 (1988) 73-79. [43] Orlandi P., The Burgers equation. Fluid Flow Phenomena. Fluid Mechanics and Its Applications, vol 55. 40-50 (2000). [44] Bonkile, M.P., Awasthi, A., Lakshmi, C. et al.
A systematic literature review of Burgers’ equation with recent advances.
Pramana - J Phys90,