# Exact Analytical Solutions of Nonlinear Fractional Liouville Equation by Extended Complex Method

Authors: Mehvish Fazal Ur Rehman; Yongyi Gu; Wenjun Yuan
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

The extended complex method is investigated for exact analytical solutions of nonlinear fractional Liouville equation. Based on the work of Yuan et al., the new rational, periodic, and elliptic function solutions have been obtained. By adjusting the arbitrary values to the constants in the constructed solutions, it can describe the physical phenomena to the traveling wave solutions, since traveling wave has significant value in applied sciences and engineering. Our results indicate that the extended complex technique is direct and easily applicable to solve the nonlinear fractional partial differential equations (NLFPDEs).


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## Body

## 1. Introduction

The idea of FPDEs has been an area of focus not only among mathematicians but also among physicists and engineers. FPDEs have played an important task in numerous fields like physics, biology, biogenetics, and fluid mechanics [1–3].Because of their several applications, numerous techniques have been evolved to attain analytical and numerical solutions for FPDEs, for example, modified extended tanh method [4], differential transform method [5], Ansatz method [6], Sine-Gordon expansion method [7], unified method [8], Q-homotopy analysis Sumudu transform technique [9], implicit Riesz wavelet-based method [10], first integral method [11], Backlund transformation [12], Chebyshev wavelet operational matrix [13], F-expansion method [14], homogeneous balance method [15], a generalization of truncated M-fractional derivative [16], fractional natural decomposition method [17], iterative method [18], and Adam’s-type predictor-corrector method [19].The nonlinear evolution equations have been solved symbolically and numerically by using various methods, for example, Durur et al. have appliedm+1/G′-expansion method to attain analytical solutions of the hyperbolic nonlinear Schro¨dinger′s equation (NLSE) [20]. Taher has applied simple equation method to the Kadomtsev-Petviashvili (KP) equation [21]. Miao and Zhang have introduced modified G′/G-expansion method to attain the exact analytical solutions of perturbed nonlinear Schrödinger’s equation [22]. Khatera et al. have used tanh method to investigate analytical solutions of nonlinear reaction-diffusion equation [23]. Zhang et al. have used first integral method to attain the exact analytical solutions of nonlinear Boussinesq wave packet mode and 2+1-dimensional Zoomeron equation [24]. Wazwaz has introduced Sine-cosine method to explore analytical solutions of Benjamin-Bona-Mahony equation and Phi-four equation [25].El-Danaf et al. have applied the new numerical treatment to generalized long-wave equation system [26]. Khalid et al. have introduced the nonpolynomial spline method to attain numerical solutions of Coupled Burgers equations [27]. Lakshmi and Ashish have attained numerical solutions of Burgers equation by using cubic B-spline method [28], and Raslan et al. have applied the B-spline collocation method to find numerical solutions of coupled-BBM system [29].Recently, Hadi et al. have introduced analytical methods to solve the nonlinear evolution equations [30–32]. Nestor et al. have applied modified Kudryashov method to solve perturbed nonlinear Schrödinger’s equation [33]. Nauman et al. have attained analytical solutions for various nonlinear evolution equations [34, 35]. The general form of nonlinear fractional Liouville equation [36] is given by
(1)∂2αv∂X2α+∂2βv∂T2β=e2v,0<α,β≤1.This equation plays an important task in different scientific applications such as solid-state physics, nonlinear optic, and chemical kinetics. Wherev is a function of X and T. The conformable derivative of order α and β in nonlinear fractional Liouville equation is defined as a conformable derivative sense, introduced by Khalil et al. [37]. The main advantages of conformable derivatives is to satisfy the rules of ODE like quotient, product, and chain rules as well as other definitions fail to satisfy these rules, and it can be solved exactly and numerically of other FPDEs. The conformable order definition can be expressed as follows.Definition 1.
Lete: [0,∞)→R be a function, then conformable order can be expressed as
(2)Dβex=limx→0ex+δx1−β−exδ,where hβ0=limx→0hβx. From the above conformable integral function k defined as
(3)Iβlex=∫lxett1−βdt,≥0,β∈0,1.The significant properties of the conformable derivatives are introduced as below:(1)
Dβau+bv=aDβu+bDβv,∀a,b∈ℝ(2)
Dβxp=pxp−β,∀p∈ℝ(3)
DβA=0,∀ux=Aconstantfunctions(4)Duv=uDβv+vDβu(4)
Dβu/v=vDβu−uDβv/v2, moreover, if the function u is differentiable, then(5)
Dβux=x1−βdu/dxIn present work, our goal is to solve the nonlinear fractional Liouville equation by using the extended complex method based on the work of Yuan et al. [38–43]. This is a beneficial technique to attain exact analytical solutions. This method’s application can be used in the discipline of mathematical physics and engineering. The whole article is organized as mention below: In Section 2, methods and materials are explained. In Section 3, analytical solutions for the given problem are computed. Section 4 graphical structures of our results are given, and we have mentioned the conclusions in Section 5.


## 2. Methods and Materials

In this section, we introduce the extended complex method for solving FPDE. Let us consider that the general FPDE form is expressed as(5)fv,DTαv,DX1αv,DX2βv,DX3γv,DX4δv⋯⋯=0,0<α,γ,δ,β≤1.Wherev is an unknown function of X1,X2,X3⋯⋯Xn and T, f is a polynomial of v and its fractional partial derivatives. The FPDE has been solved by the following steps.into Eq. (8), respectively, then the systems of algebraic equations are calculated by equating the coefficient to zero. These algebraic equations are solved with the help of maple packages. Finally, elliptic function solutions, rational function solutions, and simply periodic solutions with the pole at z=0 are determined. As γ−ij are obtained by (9), Ei2=4Fi3−g2Fi−g3 and ∑i=1γ−i1=0 and Vz,Veαzα∈ℂ have r≤p distinct poles of multiplicity q.Step 1.
The transformationz=Tα/α+X1β/β+⋯⋯z0 is applied to Eq. (5), it becomes NPDE and this equation can be expressed as
(6)sv,vt,vx,vy,vz,vtt,vxx⋯⋯⋯.Step 2.
A transformationT:vx,t→Vz is applied, x,t can be explained in different criteria. By this way, we have defined the following transformation
(7)vx,t=Vz,z=x+νt.Step 3.
This transformation changes Eq. (6) into nonlinear ODE:
(8)SV,V′,V′′,V′′′,⋯=0,where V prime are the derivatives in Eq. (8) with respect to z. Equation (8) can be reduced by integrating with respect to z.Step 4.
The weakp,q condition is determined. Let p,q∈ℤ, and let us consider the meromorphic solutions V of Eq. (8) have at least one pole. Inserting the Laurent series
(9)Vz=∑k=−q∞Bkzk,q>0,B−q≠0,into Eq. (8), if it has been found out p distinct Laurent singular parts
(10)∑k=−q−1Bkzk,then the weak p,q condition of Eq. (8) holds. Weierstrass elliptic function ℘z≔℘z,k2,k3 with double periods of the following equation as mentioned below:
(11)℘′z2=4℘z3−k2℘z−k3.Step 5.
Substitute the indeterminate formulas(12)Vz=∑i=1r−1∑j=2q−1jγ−ijj−1!dj−2dzj−214℘′z+Ei℘z−Fi2−℘z+∑i=1r−1γ−i12℘′z+Ei℘z−Fi+∑j=2q−1jγ−rjj−1!dj−2dzj−2℘z+γ0,(13)Vz=∑i=1∑j=1γijz−zij+γ0,(14)Veμz=∑i=1s∑j=1qγijeμz−eμzij+γ0,Step 6.
Meromorphic solutions are obtained with an arbitrary pole, then the inverse transformationT−1 is inserted into meromorphic solutions, and exact analytical solutions of FPDE are explored.


## 3. Application of the Method

In this section, we will seek exact analytical solutions of nonlinear fractional Liouville equation by the extended complex technique. Suppose nonlinear fractional Liouville equation is given(15)∂2αv∂X2α+∂2βv∂T2β=e2v,0<α,β≤1.By applying the transformation (16) into Eq. (15)
(16)x=Xαα,t=Tββ,then, we have
(17)∂2v∂x2+∂2v∂t2=e2v,if we consider u=ev, it represents Eq. (17) into
(18)ux2+ut2−uuxx−vvtt+u4=0,using wave transformation vx,t=Vz,z=x+νt, Eq. (18) is reduced to
(19)1+ν2VV′′−1+ν2V′2−V4=0.Substituting (9) into (19), then we have p=2 and q=1 and then the weak 2,1 condition of (19) holds. By the weak 2,1 and (13), we have defined the forms of the rational solutions
(20)Vrz=γ12z+γ11z−z1+γ10,with at z=0. Putting the Vrz into the Eq. (19), then we have
(21)∑i=19c1iz−i−3z−z1−4=0,where
(22)c11=γ124z14−γ122v2z14−γ122z14,c12=4γ123γ10z14−2γ12γ10v2z14−4γ124z13−4γ123γ11z13+4γ122v2z13+2γ12γ11v2z13−2γ12γ10z14+4γ122z13+2γ12γ11z13,c13=6γ122γ102z14−16γ123γ11z13−12γ122γ11γ10z13+8γ12γ10v2z13+6γ124z12+12γ123γ11z12+6γ102γ112z12−6γ122v2z12−4γ12γ11v2z12+8γ12γ10z13−6γ122z12−4γ12γ11z12,c14=4γ12γ113z14−24γ122γ102z13−12γ12γ11γ102z13+24γ123γ10z12+36γ122γ11γ10z12+12γ12γ112γ10z12−12γ12γ10v2z12−4γ124z1−12γ123γ11z1−12γ122γ112z1+4γ122v2z1−4γ12γ113z1+4γ12γ11v2z1−12γ12γ10z12+4γ122z1+4γ12γ11z1,c15=γ104z14−16γ12γ103z13−4γ11γ103z13+36γ122γ10z12+36γ12γ11γ102z12+6γ112γ102z12−16γ123γ10z1−36γ122γ11γ10z1−24γ12γ112γ10z1+8γ12γ10v2z1−4γ113γ10z1+2γ11γ10v2z1+γ124+4γ123γ11+6γ122γ112−γ122v2+4γ12γ113−2γ12γ11v2+γ114−γ112v2+8γ12γ10z1+2γ11γ10z1−γ122−2γ12γ11−γ112,c16=−4γ104z13+24γ12γ103z12+12γ11γ103z12−24γ122γ102z1−36γ12γ11γ102z1−12γ112γ102z1+4γ123γ10+12γ122γ11γ10+12γ12γ112γ10−2γ12γ10v2+4γ113γ10−2γ12γ10v2−2γ12γ10−2γ11γ10,c17=6γ104z12−16γ12γ103z1−12γ11γ103z1+6γ122γ102+12γ12γ11γ102+6γ112γ102,c18=−4γ104z1+4γ12γ103+4γ11γ103,c19=γ104.Setting the coefficients of the identical powers aboutz in Eq. (21) to zero, then attain the number of following equations:
(23)c1i=0,i=1,2,⋯,9.By solving the following equations, we attain(24)γ12=v2+1,γ10=0,γ11=v2+1,γ12=v+1−v2+1,γ11=v+1−v2+1,then
(25)Vr10z=v2+1z+v2+1z−z1,Vr20z=v+1−v2+1z+v+1−v2+1z−z1,where γ12/a−1=v is the first case and γ11/a−1=v is the second case. Applying Vz=Rη into Eq. (19), then
(26)1+ν2α2RηR′+η2R′′−1+ν2α2R′η2−R4=0,inserting
(27)Vsη=γ12η−1+γ11η−η1+γ10,into the Eq. (26), we attain that
(28)∑i=19c2iα2ηiη−14η−η14=0,where
(29)c21=γ124η14−4γ123γ10η14+6γ122γ102η14−4γ12γ103η14+γ104η14+4γ123γ11η13−12γ122γ11γ10η13+12γ12γ11γ102η13−4γ11γ103η13+6γ102γ112η12−12γ12γ112γ10η12+6γ112γ102z12+4γ12γ113η1−4γ113γ10η1+γ114,c22=−γ122α2η14v2+γ12α2γ10η14v2−γ12α2γ11η13v2+4γ123γ10η14−γ122α2η14−12γ122γ102η14−γ12α2γ11η12v2+γ12α2γ10η14+12γ12γ103η14+α2γ11γ10η12v2−4γ104η14−4γ124η13−4γ123γ11η13+16γ123γ10η13+24γ122γ11γ10η13−24γ122γ102η13−γ12α12γ11η13−36γ12γ11γ102η13+16γ12γ103η13−α2γ112η1v2+16γ11γ103η13−4γ104η13−12γ103γ11η12−12γ122γ112η12+36γ122γ11γ10η12−γ12α2γ11η12+36γ12γ112γ10η12−36γ12γ11γ102η12+α2γ11γ10η12−24γ112γ102η12+12γ11γ103η12−12γ122γ112η1−12γ12γ113η1+24γ12γ112γ10η1−α2γ112η1+16γ113γ10η1−12γ112γ102η1−4γ12γ113−4γ114+4γ113γ10,c23=4γ122α2η13v2−4γ12α2γ10η13v2+6γ122γ102ηl4+8γ12α2γ11η12v2−12γ12γ103η14−4α2γ11γ10η12v2+6γ104η14−16γ123γ10η13+4γ102α2η13−12γ122γ11γ10η13+48γ122γ102η13−4γ12α2γ10η13+36γ12γ11γ102η13−48γ12γ103η13+4α2γ112η1v2−24γ11γ103η13+16γ104η13+6γ124η12+12γ123γ11η12−24γ123γ10η12+6γ122γ112η12−72γ122γ11γ10η12+36γ122γ102η12+8γ12α2γ11η12−36γ12γ112γ10η12+108γ12γ11γ102η12−24γ12γ103η12−4α2γ11γ10η12+36γ112γ102η12−48γ11γ103η12+6γ104η12+12γ123γ11η1+24γ122γ112η1−36γ122γ11γ10η1+12γ12γ113η1−72γ12γ112γ10η1+36γ12γ11γ102η1+4α2γ112η1−24γ113γ10η1+48γ112γ102η1−12γ11γ103η1+6γ122γ112+12γ12γ113−12γ12γ112γ10+6γ114−16γ113γ10+6γ112γ102,c24=−γ12α2γ10η14v2+γ12α2γ11η13v2−6γ122α2η12v2−7γ12α2γ11η12v2−γ12α2γ10η14+6γ12α2γ10η12v2+4γ12γ103η14+6α2γ11γ10z12v2−4γ104η14−24γ122γ102η13+γ12α2γ11η13−7γ12α2γ11η1v2−12γ12γ11γ102η13+48γ12γ103η13−6α2γ112η1v2+16γ11γ103η13−24γ104η13+24γ123γ10η12−6γ122α2η12+36γ122γ11γ10η12−72γ122γ102η12−7γ12α2γ11η12+γ12α2γ11v2+6γ12α2γ10η12+12γ12γ112γ10η12−108γ12γ11γ102η12+72γ12γ103η12+6α2γ11γ10η12−α2γ11γ10v2−24γ112γ102η12+72γ11γ103η12−24γ104η12−4γ124η1−12γ123γ11η1+16γ123γ10η1−12γ122γ112η1+72γ122γ11γ10η1−24γ122γ102η1−7γ12α2γ11η1−4γ12γ113η1+72γ12γ112γ10η1−108γ12γ11γ102η1+16γ12γ103η1−6α2γ112η1+16γ113γ10η1−72γ112α2η1+48γ11γ103η1−4γ104η1−4γ123γ11−12γ122γ112+12γ122γ11γ10+γ12α2γ11−12γ12γ113+36γ12γ112γ10−12γ12γ11γ102−α2γ11γ10−4γ114+24γ113γ10−24γ112γ102+4γ11γ103,c25=4γ12α2γ10η13v2−4α2γ11γ10η12v2+γ104η14+4γ122α2η1v2+8γ12α2γ11η1v2+4γ12α2γ11η13−4γ12α2γ11η1v2−16γ12γ103η13+4α2γ112η1v2−4γ11γ103η13+16γ104η13+36γ122γ102η12+36γ12γ11γ102η12−72γ12γ103η12−4α2γ11γ10η12+4α2γ11γ10v2+6γ112γ102η12−48γ11γ103η12+36γ104z12−16γ123γ10η1+4γ122α2η1−36γ122γ11γ10η1+48γ122γ102η1+8γ12α2γ11η1−4γ12α2γ10η1−24γ12γ112γ10η1+108γ12γ11γ102η1−48γ12γ103η1+4α2γ112η1−4γ113γ10η1+48γ122γ102η1−72γ11γ103η1+16γ104η1+γ124+4γ123γ11−4γ123γ10+6γ122γ112−24γ122γ11γ10+6γ122γ102+4γ12γ113−36γ12γ112γ10+36γ12γ11γ102−4γ12γ103+4α2γ11γ10+γ114−16γ113γ10+36γ112γ102−16γ11γ103+γ104,c26=−6γ12α2γ10η12v2+α2γ11γ10η12v2−γ12α2γ11η1v2−α2γ112η1v2−4γ104η13−γ122α2v2−γ12α2γ11v2−6γ12α2γ10η12+γ12α2γ10v2+24γ12γ103η12+α2γ11γ10η12−6α2γ11γ10v2+12γ11γ103η12−24γ104η12−24γ122γ102η1−γ12α2γ11η1−36γ12γ11γ102η1+48γ12γ103η1−α2γ112η1−12γ112γ102η1+48γ11γ103η1−24γ104η1+4γ123γ10−γ122α2+12γ122γ11γ10−12γ122γ102−γ12α2γ11+γ12α2γ10+12γ12γ112γ10−36γ12γ11γ102+12γ12γ103−6α2γ11γ10+4γ113γ10−24γ112γ102+24γ12γ103−4γ104,c27=4γ12α2γ10η1v2+4α2γ11γ10v2+6γ104η12+4γ12α2γ10η1−16γ12γ103η1−12γ11γ103η1+16γ104η1+6γ122γ102+12γ12γ11γ102−12γ12γ103+4α2γ11γ10+6γ112γ102−16γ11γ103+6γ104,c28=−γ12α2γ10v2−α2γ11γ10v2−4γ104η1−γ12α2γ10+4γ12γ103−α2γ11γ10+4γ11γ103−4γ104,Setting the coefficients of the identical powers aboutη in Eq. (28) to zero, then obtain the number of following equations:
(30)c2i=0,i=1,2,⋯,9.Solve the number of following equations, then attain(31)γ12=v2+1α,γ11=v2+1α,where η=eαzα∈ℂ.
(32)Vseαz=v2+1αeαz−1+v2+1αeαz−eαz1,hence, we get the simply periodic solutions of Eq. (19) with pole at z=0(33)Vs10z=v2+1α2cothα2z−cothα2z−z1,where γ12/a−1=v is the first case and γ11/a−1=v is the second case.According to the weak ⟨2.1⟩ condition, we introduce here elliptic solutions of (12) with z=0 pole.
(34)Vd,1z=±v2+1α2℘′z+E1℘z−F1+γ10,where E12=4F13−k2F1−k3. We can rewrite the Vd,1z and considering the above mentioned results, we have γ10=0, k3=0,andE1=F1=0. So, we have
(35)Vd,1z=±v2+1α2℘′z℘z,where k3=0, hence, the elliptic function solutions of Eq. (19) are
(36)Vd,1z=±v2+1α2℘′z−z0:k2,0℘z−z0:k2,0,where z0 and k2 are arbitrary constants. By further process, we attain meromorphic solutions of Eq. (19) with an arbitrary pole as follows:
(37)Vr,1z=v2+1z−z0+v2+1z−z1−z0,Vr,2z=v+1−v2+1z−z0+v+1−v2+1z−z1−z0,Vs,1z=v2+1α2cothα2z−z0−cothα2z−z1−z0,where γ12/a−1=v is the first case and γ1a/a−1=v is the second case.


## 4. Description about Figures

In this section, we represent traveling wave solutions forVr,1z, Vr,2z and Vs,1z by 3D graphs as in Figures 1–6. These figures are displayed by maple software to justify our main results physically.Figure 1
3D graph ofVr,1z for the fixed values v=1, z0=0.5, and z1=1 represent the exact traveling wave solutions.Figure 2
3D graph ofVr,1z for the fixed values v=1, z0=−0.5, and z1=1 represent the exact traveling wave solution.Figure 3
3D graph ofVr,2z for the fixed values v=1, z0=0.2, and z1=1 represent the exact traveling wave solutions.Figure 4
3D graph ofVr,2z for the fixed values v=1, z0=−0.2, and z1=1 represent the exact traveling wave solutions.Figure 5
3D graph ofVs,1z for the fixed values v=1, z0=1/6, z1=1, and α=1 represent traveling wave solutions.Figure 6
3D graph ofVs,1z for the fixed values v=1, z0=−1/6, z1=1, and α=1 represent traveling wave solutions.Figures1 and 2 show exact traveling wave solutions for Vr,1z, set as fixed values v=1, z0=0.5, and z1=1 and v=1, z0=−0.5, and z1=1.Figures3 and 4 show exact traveling wave solutions for Vr,2z, set as fixed values v=1, z0=0.2, and z1=1 and v=1, z0=−0.2, and z1=1.Figures5 and 6 show exact traveling wave solutions for Vs,1z, set as fixed values v=1, z0=1/6, z1=1, and α=1 and v=1, z0=−1/6, z1=1, and α=1.


## 5. Comparison and Conclusion

Biswas et al. [36] introduced G′/G-expansion technique to explore the analytical solutions of nonlinear fractional Liouville equation. The exact analytical solutions attained by this proposed method show the inner structure of physical phenomena in applied science. The results indicate the different types of traveling wave solutions of this proposed method. For this reason, we compare G′/G-expansion method with extended complex technique.We have applied the extended complex technique to attain the exact analytical solutions of the nonlinear fractional Liouville equation. The 3D graphs are represented by adjusting the values of arbitrary parameters, and these graphs have demonstrated the different physical structures such as Figures1 and 2 represent multisolitary traveling wave solutions, Figures 3 and 4 represent soliton type traveling wave solutions, and Figures 5 and 6 show solitary traveling wave solutions. Figures 1–6 depict that traveling wave solutions are attained for smaller values of z0, whereas other parameters are fixed. The extended complex method has been solved by means of maple packages. The proposed technique is an effective analytical approach in comparison with other analytical techniques via tanh method and first integral method [23, 24] because it gives different new traveling wave solutions which have been shown in terms of rational, periodic, and elliptic function solutions. The results that are attained by the proposed method play a significant task to show the deep mechanism of physical phenomena and give tedious solutions of higher degree NFPDEs.By extended complex technique, we could find meromorphic solutions of numerous DEs which do not satisfy Briot-Bouquet equation. Therefore, in the future, this proposed approach plays an important task in mathematical physics. The intermediate forms of the solutions are applied to explore meromorphic solutionsVz for the DEs with a pole at z=0. After that solve meromorphic solutions for DEs with an arbitrary pole.



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