Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class

1 Introduction

Even though some nonlinear partial differential equations are integrable, it may be not easy to integrate them. Instead, a predicted solution with parameters are assumed to be a solution of governing equations and the relations among the parameters are investigated. The logic is simply based on the similarity with exponential-type solutions to the ordinary differential equations with constant coefficients. These predicted solutions are of various forms covering exponential, hyperbolic, trigonometric or rational functions, and more. Moreover the interaction of various types of waves were also determined by using analytical techniques (Ma and Zhou, 2018; Zhang and Ma, 2017; Zhao and Ma, 2017; Yang et al., 2017; Ma et al., 2018). Parallel to the recent developments in computer algebra in the last four decades, a tendency has been observed to determine exact solutions to nonlinear PDEs by following the procedure that starts with a predicted solution. Recently, this tendency has focused on exact solutions to fractional nonlinear partial differential equations. Many of techniques implemented to nonlinear PDEs to find exact solutions have been adapted for fractional nonlinear PDES (Korkmaz and Hosseini, 2017; Hosseini and Ansari, 2017; Kumar et al., 2017; Korkmaz, 2017a,b; Hosseini et al., 2017a,b; Kumar et al., 2018a,b; Rezazadeh et al., 2018; Singh et al., 2017a,b; Guner et al., 2017a,b). We also derive exact solutions to some conformable fractional equations in RLW-class modeling various wave phenomena both in nature or technology implementations. Different from previous studies, we adapt Sine-Gordon expansion approach to determine exact solutions to governing equations in fractional RLW-class.

The first equation considered in this study is the RLW equation

The second equation to discuss exact solutions here is the modified RLW (mRLW) equation of the form

The last equation to derive the exact solutions is the symmetric RLW (sRLW) equation

Before starting the solution procedure, we should give some significant properties of conformable fractional derivative. Thus, the next section focuses on conformable fractional derivative definition and some important properties. We explain the solution procedure in the third section. The following sections cover implementations of the proposed procedure to some conformable fractional PDEs in RLW-class.

2 Conformable fractional derivative

The conformable derivative of order $\alpha$ with respect to the independent variable t is defined as

Conformable fractional differential operator satisfies some critical fundamental properties like the chain rule, Taylor series expansion and Laplace transform (Abdeljawad, 2015).

3 Sine-Gordon expansion method

The compatible fractional form $u(x,t)=U(\xi )$ with $\xi =a(x-\nu t^{\alpha }/\alpha )$ of the classical wave transform (Ma and Fuchssteiner, 1996; Ma and Lee, 2009; Ma et al., 2010; Korkmaz, 2018) reduces the fractional Sine–Gordon equation in one dimension of the form

4 Solutions to the conformable time fractional RLW equation

The traveling wave transform $u(x,t)\to U(\xi )$ , $\xi =a(x-\nu t^{\alpha }/\alpha )$ reduces the time fractional RLW equation to

Set 1:

Set 2:

Set 3:

Set 4:

Set 5:

Set 6:

Using these solution sets of algebraic equations, we construct the solutions to (1) as $$u_1(x,t)=-\frac{6a^{2}r\nu }{q}+\frac{6a^{2}r\nu }{q}i\tanh \left(a\left(x-\nu \frac{t^{\alpha }}{\alpha }\right)\right)\text{sech}\left(a\left(x-\nu \frac{t^{\alpha }}{\alpha }\right)\right)+\frac{6a^{2}r\nu }{q}{\tanh }^2\left(a\left(x-\nu \frac{t^{\alpha }}{\alpha }\right)\right)$$ $$u_2(x,t)=-\frac{6a^{2}r\nu }{q}+\frac{6a^{2}r\nu }{q}i\tanh \left(a\left(x-\nu \frac{t^{\alpha }}{\alpha }\right)\right)\text{sech}\left(a\left(x-\nu \frac{t^{\alpha }}{\alpha }\right)\right)-\frac{6a^{2}r\nu }{q}{\tanh }^2\left(a(x-\nu \frac{t^{\alpha }}{\alpha })\right)$$ for $\nu =p/({\mathit{ra}}^2+1)$ , $$u_3(x,t)=-\frac{4a^{2}r\nu }{q}+\frac{6a^{2}r\nu }{q}i\tanh \left(a\left(x-\nu \frac{t^{\alpha }}{\alpha }\right)\right)\text{sech}\left(a\left(x-\nu \frac{t^{\alpha }}{\alpha }\right)\right)-\frac{6a^{2}r\nu }{q}{\tanh }^2\left(a\left(x-\nu \frac{t^{\alpha }}{\alpha }\right)\right)$$ $$u_4(x,t)=-\frac{4a^{2}r\nu }{q}+\frac{6a^{2}r\nu }{q}i\tanh \left(a\left(x-\nu \frac{t^{\alpha }}{\alpha }\right)\right)\text{sech}\left(a\left(x-\nu \frac{t^{\alpha }}{\alpha }\right)\right)+\frac{6a^{2}r\nu }{q}{\tanh }^2\left(a\left(x-\nu \frac{t^{\alpha }}{\alpha }\right)\right)$$ for $\nu =-p/({\mathit{ra}}^2-1)$ , $$u_5(x,t)=\frac{(8{\mathit{ra}}^2-1)\nu +p}{q}+\frac{12a^{2}r\nu }{q}{\tanh }^2\left(a\left(x-\nu \frac{t^{\alpha }}{\alpha }\right)\right)$$ for $\nu =p/(4{\mathit{ra}}^2+1)$ . $$u_6(x,t)=-\frac{(8{\mathit{ra}}^2-1)\nu +p}{q}+\frac{12a^{2}r\nu }{q}{\tanh }^2\left(a(x-\nu \frac{t^{\alpha }}{\alpha })\right)$$ for $\nu =-p/(4{\mathit{ra}}^2+1)$ .

A particular form of $u_5(x,t)$ is depicted for various values of $\alpha \in \{0.25,0.5,0.75,1\}$ in a finite domain in Fig. 1(a)–(d). It is observed that the propagation of the initial pulse propagates along the $x$ -axis as time proceeds by preserving its shape and amplitude in all cases. $\alpha$ affects only propagation velocity. The propagation is faster in small times but later it gets slower when $\alpha$ is less than 1. $\alpha =1$ choice gives a constant propagation velocity to the pulse, Fig. 1(d).

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Fig. 1

The solution $u_5(x,t)$ for $\{a=1/4,q=-1,r=1,p=10\}$ .

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5 Solutions to the conformable time fractional mRLW equation

The traveling wave transform $u(x,t)\to U(\xi ),\xi =a(x-\nu t^{\alpha }/\alpha )$ reduces the mRLW Eq. (2) to

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Fig. 2

The solution $u_{11}(x,t)$ for $\{a=1,q=-1,r=1/3,p=10\}$ .

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6 Solutions to the conformable time fractional sRLW equation

The traveling wave transform $u(x,t)\to U(\xi )$ , $\xi =a(x-\nu t^{\alpha }/\alpha )$ reduces the sRLW Eq. (2) to