Traveling wave solutions of conformable time-fractional Klien-Fock-Gordon equation by the improved $\tan (\mathrm{\Psi }(\zeta )/2)$ -expansion method

1 Introduction

The Klien-Fock-Gordon equation (KFG) models various physical phenomena in quantum field theory, condensed matter physics, nonlinear equations, correlation of solitons, collisionless plasma and theory of relativity etc. It was established by Osker Klien, Viladimir Fock and Walter Gordon for determining relativistic electrons. This equation is calibrated version of the relativistic energy–momentum correspondence. It is associated with Schr $\ddot{o}$ dinger equation and also referred to as relativistic wave equation.

In recent times, various research scholars of science and telecommunication have taken fractional order mathematical models under consideration. Fractional calculus has become a preeminent tool to analyze and characterize the nonlinear complex phenomena. Many partial differential equations (PDEs) that exhibit different physical phenomena are remodeled in terms of fractional derivative and their behavior is observed along with their solutions by developed methods (Akgül, 2018; Mamun et al., 2021). Fractional differential equations are widely studied in engineering mathematics and their fractional effects are observed for particular choices of fractional order parameters (Shahen et al., 2020; Shahen et al., 2021; Mamun et al., 2021; Hosseini et al., 2020).

This study considers the conformable time-fractional form of the Klein-Fock-Gordon equation (Alam et al., 2021)

Keeping in view the importance of fractional effects, the conformable time-fractional Klien-Fock-Gordon equation is solved to construct solitary wave solutions by $\tan (\mathrm{\Psi }(\zeta )/2)$ method for quadratic state. The conformable fractional derivative is an extension of Caputo derivative and provides a new designated memory length. It coincides with classic definition of first derivative for $\alpha =1$ . Furthermore, the conformable derivative satisfies the property of product rule, chain rule and quotient rule of two functions which is not usually satisfied by all the fractional derivative (Zheng et al., 2019). The real applications of the conformable time-fractional form of the Klein-Fock-Gordon equation include the study of different varieties of matter, spread of deviation in crystals, properties of fundamental particles and the effects and preservation of inherited memory.

Various mathematical techniques have been developed to construct the traveling wave solutions of nonlinear evolution equations (NLEEs). Some recently proposed methods include $\exp$ - $(\varphi (\xi ))$ -expansion method (Elboree, 2020), generalized projective Riccati equations method (Akram et al., 2021), modified auxiliary equation mapping method (Sajid and Akram, 2021), Legendre-homotopy analysis method (Sadaf and Akram, 2020), first integral method (Lu, 2012), extended rational sine–cosine method (Mahak and Akram, 2019), ${\varphi }^6$ -expansion method (Sajid and Akram, 2020), modified extended $\tanh$ -function method (Mamun et al., 2020; Alam et al., 2021) and several others. Various nonlinear partial differential equations have been solved to extract traveling wave solutions for studying real world phenomena (Hosseini et al., 2021; Hosseini et al., 2021; Mamun et al., 2018). From the past few decades, a variety of NLDEs have been considered for modeling complex nonlinear physical problems and their solutions have diverse applications in different fields of nonlinear sciences (Mamun et al., 2021; Ananna and Mamun, 2020; Mamun et al., 2019; Mamun and Asaduzzaman, 2019).

This article appraises the $\tan (\mathrm{\Psi }(\zeta )/2)$ -expansion method that was first introduced by Manafian and Lakestani (2015) in 2015. This is a powerful method and has been used to obtain closed form solution of many nonlinear PDEs with accuracy. The method possesses more novel solutions with wider range of implementation to deal with NLEEs. The method has been applied to a variety of NLEEs for finding their closed form solutions. A few of which are the time-fractional Kuramoto–Sivashinsky equation (Manafian and Foroutan, 2017), $(2+1)$ -dimensional KP-BBM wave equation (Khan et al., 2018), Biswas-Milovic equation (Manafian and Lakestani, 2016), Gerdjikov-Ivanov equation (Manafian and Lakestani, 2016) and generalised Hirota–Satsuma coupled KdV (HScKdV) equation (Özkan and Yaşar, 2020).

This paper is coordinated as follows: In Section 2, the proposed model is introduced. In Section 3, the improved $\tan (\mathrm{\Psi }(\zeta )/2)$ -expansion method is described. In Section 4, mathematical analysis is applied on quadratic KFG equation with conformable derivative. In Section 5, graphical illustration of some solutions is given. The conclusion is presented in Section 6.

2 Governing model

The quadratic conformable time-fractional KFG equation is considered, as

Conformable Derivative A new definition of conformable fractional derivative was proposed by Khalil et al. (2014) in 2014.

The conformable derivative of function $g:[0,\infty )$ of fractional order $\alpha$ , where $g=g(t)$ is defined on the positive half plane is expressed, as

Properties Following properties are satisfied:

Theorem The following property is satisfied by conformable differentiable function $g:(0,\infty )$ and a differentiable function h determined in the range of g

3 Description of Improved $\tan (\mathrm{\Psi }(\zeta )/2)$ -Expansion Method

In this section, the improved $\tan (\mathrm{\Psi }(\zeta )/2)$ -expansion method (Manafian and Lakestani, 2016) is briefly described. The nonlinear partial differential equation is considered in the form

Step $\mathbf{1}$ : The traveling wave transformation is considered, as

Step $\mathbf{2}$ : Eq. (3.3) has solution of the form

Eq. (3.5) provides the following families of solutions (Akram et al., 2021).

Family 1: If $l^2+m^2-n^2<0$ and $m-n\ne 0$ , then

Family 2: If $l^2+m^2-n^2>0$ and $m-n\ne 0$ , then

Family 3: If $l^2+m^2-n^2>0,m\ne 0$ and $n=0$ , then

Family 4: If $l^2+m^2-n^2<0,n\ne 0$ and $m=0$ , then

Family 5: If $l^2+m^2-n^2>0,m-n\ne 0$ and $l=0$ , then

Family 6: If $l=0$ and $n=0$ , then

Family 7: If $m=0$ and $n=0$ , then

Family 8: If $l^2+m^2=n^2$ , then

Family 9: If $l=m=n=\mathit{il}$ , then

Family 10: If $l=n=\mathit{il}$ and $m=-\mathit{il}$ , then

Family 11: If $n=l$ , then

Family 12: If $l=n$ , then

Family 13: If $n=-l$ , then

Family 14: If $m=-n$ , then

Family 15: If $m=0$ and $l=n$ , then

Family 16: If $l=0$ and $m=n$ , then

Family 17: If $l=0$ and $m=-n$ , then

Family 18: If $l=0$ and $m=0$ , then

Step $\mathbf{3}$ : A positive integer k is determined using homogeneous balance rule. If the constant k is not a positive integer, the following transformation is applied.

$(a)$ If k is a fraction of the form $k=p/q$ , then assume

Substitute Eq. (3.24) in Eq. (3.3), to determine the value of k using homogeneous balance rule.

$(b)$ On the other hand if k is a negative number, then assume

Inserting Eq. (3.25) into Eq. (3.3), the value of k is determined by homogeneous balance rule.

Step $\mathbf{4}$ : Eq. (3.4) is substituted into Eq. (3.3) using the value of k obtained in Step 3 and the coefficients of ${\left(\tan (\mathrm{\Psi }(\zeta )/2)\right)}^i$ and ${\left(\cot (\mathrm{\Psi }(\zeta )/2)\right)}^i$ are collected. Setting these coefficients to zero, a set of algebraic equations is attained in ${\alpha }_0,{\alpha }_i,{\beta }_i(i=1,2,\dots ,k),l,m,n,\lambda$ and c.

Step $\mathbf{5}$ : Putting the values of ${\alpha }_0,{\alpha }_1,{\beta }_1,\dots ,{\alpha }_i,{\beta }_i,\lambda$ and c into Eq. (3.4) the closed form solution of Eq. (3.1) is obtained.

4 Exact solutions of quadratic time-fractional KFG equation

The reduced differential equation form of Eq. (2.1) is acquired by the complex-wave transform

The application of Eq. (4.1) on Eq. (2.1) yields

Using homogeneous balance rule, the integer $m=2$ is obtained. Then, the formal solution by improved $\tan (\mathrm{\Psi }(\zeta )/2)$ -expansion method for $m=2$ becomes

Eq. (4.3) is inserted into Eq. (4.1) and the coefficients of ${\left(\tan (\frac{\mathrm{\Psi }(\zeta )}{2})\right)}^i,{\left(\cot (\frac{\mathrm{\Psi }(\zeta )}{2})\right)}^i$ are collected. Then, setting each coefficient to zero a system of algebraic equations is formed. As a result, the following sets of traveling wave solutions are obtained:

Set $\mathbf{1}$ :

By Eq. (4.5), the Families 1, 2, 3, 4, 5, 6 and 7 are written respectively, as

The above solutions are valid for $\left(-{\omega }_1+\frac{{\omega }_2}{l^2+m^2-n^2}\right)>0$ .

Using Eq. (4.5), the Families 9, 10, 11, 12, 13 and 14 can be presented respectively, as

All the above solutions hold true for $\left(-{\omega }_1+\frac{{\omega }_2}{l^2+m^2-n^2}\right)>0$ .

Using Eq. (4.5), Family 18 is reported, as

The constraint condition to be satisfied by the above solution is $\left(-{\omega }_1+\frac{{\omega }_2}{l^2+m^2-n^2}\right)>0$ .

Set $\mathbf{2}$ :

By Eq. (4.21), the Families 1, 2, 3, 4, 5, 6 and 7 are respectively given, as