# Copious Closed Forms of Solutions for the Fractional Nonlinear Longitudinal Strain Wave Equation in Microstructured Solids

Authors: Haiyong Qin; Mostafa M. A. Khater; Raghda A. M. Attia
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

A computational scheme is employed to investigate various types of the solution of the fractional nonlinear longitudinal strain wave equation. The novelty and advantage of the proposed method are illustrated by applying this model. A new fractional definition is used to convert the fractional formula of these equations into integer-order ordinary differential equations. Soliton, rational functions, the trigonometric function, the hyperbolic function, and many other explicit wave solutions are obtained.


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## Body

## 1. Introduction

Fractional nonlinear evolution equation is one of the noticeable branches of science, particularly in recent years. Fractional calculus has a great profound physical background where it is able to formulate many various phenomena in distinct fields such as physics, mechanical engineering, economics, chemistry, signal processing, food supplement, applied mathematics, quasichaotic dynamical systems, hydrodynamics, system identification, statistics, finance, fluid mechanics, solid-state biology, dynamical systems with chaotic dynamical behavior, optical fibers, electric control theory, and economics and diffusion problems. The mathematical modeling of these phenomena will contain a fractional derivative which provides a great explanation of the nonlocal property of these models since it depends on both historical and current states of the problem in contrast with the classical calculus which depends on the current state only. Based on the importance of this kind of calculus, many definitions have been being derived such as conformable fractional derivative, fractional Riemann–Liouville derivatives, Caputo, and Caputo–Fabrizio definition [1–17]. These definitions have been being employed to convert the fractional nonlinear partial differential equations to nonlinear integer-order ordinary differential equation and then the computational and numerical schemes can be applied to get various types of solutions for these models and the examples of these schemes [18–30].Recently, the mK method is formulated and applied to distinct physical models such as the complex Ginzburg–Landau model, the (2 + 1)-dimensional KD equation and KdV equation, fractional (N + 1) sinh-Gordon, biological population, equal width, modified equal width, and Duffing equations [31–40].This method depends on a new auxiliary equation, which is equal to the Riccati equation [41]. The auxiliary equation of the mK method is given by(1)ℰ′φ=1lnQδQℰφ+ϱQ−ℰφ+X,where δ,ϱ,χ,andQ are the arbitrary constants. The Riccati equation is given by(2)ℛ′φ=A0+A1ℛφ+A2ℛ2φ,where A0,A1,andA2 are the arbitrary constants. So, equations (1) and (2) are equal when ℰφ=ℛφ,χ=A1,ϱ=A0,δ=A2. Using this technique, it leads to the equalling of the mK auxiliary equation with many other analytical methods, but the mK method can obtain more solutions than most of them. This equivalence shows superiority, power, and productivity of the mK method.In this context, the mK method is employed to construct new formulas of solutions for the fractional nonlinear longitudinal strain wave equation which is given in [42–48]:(3)Dtt2ϑℳ+ℳxx−γℒ1ℳ2xx−ℒ2ℳxxxx+ℒ3Dtt2ϑℳxx==0,where γ,ℒ1,ℒ2,ℒ3 are the arbitrary constants.This model is considered as one of the fundamental models in the microstructure of a material that is used to determine the elasticity, which is caused by the dissipation/energy input in the content. This model contains nonlinear and dissipation terms which are the officials of construct the kink and shock waves. This shows the permanent form in medium points to a possible presence of dispersion or dissipation. The following definition of theAℬℛ fractional operator [49–52] is applied to equation (3).Definition 1.
It is given in [17] that(4)DAℬℛa+αℱt=ℬα1−αddt∫atℱxGα−αt−αα1−αdx,where Gα is the Mittag–Leffler function defined by the following formula:(5)Gα−αt−αα1−α=∑n=0∞−α/1−αnt−xαnΓαn+1,and ℬα is a normalisation function. Thus,(6)DAℬℛa+αℱx=ℬα1−α∑n=0∞−α1−αnℐRLaαnℱx,leads to(7)ℳx,t=ℳφ,φ=x+c1−αt−αnBα∑n=0∞−α/1−αnΓ1−αn,where c is the arbitrary constant. This wave transformation converts equation (3) to ODE. Integrating the obtained ODEs twice with zero constant of the integration gives(8)ℳ−αℳ2+λℳ′′=0,where α=γℒ1/c2−1,λ=γℒ2−c2ℒ3/c2−1. Calculating the homogeneous balance value in equation (8) yields n=2. Thus, both equations have same general formula of solution and it is given according to the mK method by(9)ℳφ=∑i=1naiQiℰφ+∑i=1nbiQ−iℰφ+a0=a1Qℰφ+a2Q2ℰφ+a0+b2Q−2ℰφ+b1Q−ℰφ,where a0,a1,a2,b1,andb2 are the arbitrary constants.
The order for the rest of this article is shown as follows: Section2   applies the mK method to the nonlinear fractional strain wave equation. Section 3   discusses the obtained computational results and explains the comparison between them and that obtained in previous work. Moreover, it shows the comparison between the obtained numerical results. Section 4   gives the conclusion of the whole research.


## 2. Abundant Wave Solutions of the Fractional Strain Wave Equation

Applying the mK method with its auxiliary equation and the suggested general solutions for the fractional strain wave equation leads to a system of algebraic equations. Using Mathematica 11.2 to find the values of the parameters in this system leads to the following.


### 2.1. Family I

(10)
a
0
⟶
b
2
−
χ
2
−
4
δ
ϱ
2
+
8
δ
ϱ
+
χ
2
12
ϱ
2
,
a
1
⟶
0
,
a
2
⟶
0
,
b
1
⟶
b
2
χ
ϱ
,
α
⟶
−
6
ϱ
2
b
2
4
δ
ϱ
−
χ
2
2
,
λ
⟶
−
1
χ
2
−
4
δ
ϱ
2
,
where
ϱ
≠
0
,
χ
2
≠
4
δ
ϱ
.Consequently, the closed forms of solutions for the fractional strain waves model are given as follows:Whenχ2−4δϱ<0&δ≠0,(11)ℳ1x,t=b21248δ2χ−4δϱ−χ2tan1/24δϱ−χ2x−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn2−24δχχϱ−ϱ4δϱ−χ2tan1/24δϱ−χ2x−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+−χ2−4δϱ2+8δϱ+χ2ϱ2,ℳ2x,t=b21248δ2χ−4δϱ−χ2cot1/24δϱ−χ2x−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn2−24δχχϱ−ϱ4δϱ−χ2cot1/24δϱ−χ2x−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+−χ2−4δϱ2+8δϱ+χ2ϱ2.Whenχ2−4δϱ>0&δ≠0,(12)ℳ3x,t=b21248δ2χ−4δϱ−χ2tanh1/2χ2−4δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+χ2−24δχχϱ−ϱ4δϱ−χ2tanh1/2χ2−4δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+χϱ+−χ2−4δϱ2+8δϱ+χ2ϱ2,ℳ4x,t=b21248δ2χ−4δϱ−χ2coth1/2χ2−4δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+χ2−24δχχϱ−ϱ4δϱ−χ2coth1/2χ2−4δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+χϱ+−χ2−4δϱ2+8δϱ+χ2/ϱ2.Whenδϱ>0&ϱ≠0&δ≠0&χ=0,(13)ℳ5x,t=b2δϱ3cot2δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+2−δ2ϱ23ϱ2,ℳ6x,t=b2δϱ3tan2δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+2−δ2ϱ23ϱ2.Whenδϱ<0&ϱ≠0&δ≠0&χ=0,(14)ℳ7x,t=−b23δϱcsch2−δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+δ2ϱ2+δϱ3ϱ2,ℳ8x,t=−b2δϱ3tanh2−δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−2+δ2ϱ23ϱ2.Whenχ=0&ϱ=−δ,(15)ℳ9x,t=b2312exp2ϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+12+6tanhϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn−ϱ4ϱ2−5a.Whenχ=ϱ/2=κ&δ=0,(16)ℳ10x,t=b21expκx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−22+12expκx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−4−κ448κ2+148.Whenχ=δ=0&ϱ≠0,(17)ℳ11x,t=b2ϱ2x−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn2.Whenχ=0&ϱ=δ,(18)ℳ12x,t=13b23cot2−α−1cϱt−2αBα∑n=0∞−α/1−αnΓ1−αn+C+xϱ−ϱ4ϱ2+2.Whenδ=0&χ≠0&ϱ≠0,(19)ℳ13x,t=b212χ21−12ϱ/ϱ−χexpχx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−χ4ϱ2+12expχx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−ϱ/χ2.


### 2.2. Family II

(20)
a
0
⟶
χ
2
−
4
δ
ϱ
2
−
8
δ
ϱ
−
χ
2
2
α
χ
2
−
4
δ
ϱ
2
,
a
1
⟶
−
6
δ
χ
α
χ
2
−
4
δ
ϱ
2
,
a
2
⟶
−
6
δ
2
α
4
δ
ϱ
−
χ
2
2
,
b
1
⟶
0
,
b
2
⟶
0
,
λ
⟶
−
1
χ
2
−
4
δ
ϱ
2
,
where
δ
≠
0
,
α
≠
0
,
4
δ
ϱ
≠
χ
2
.Consequently, the closed forms of solutions for the fractional strain waves model are given as follows:Whenχ2−4δϱ<0&δ≠0,(21)ℳ14x,t=3χ2−4δϱtan2124δϱ−χ2x−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+χ2−4δϱ2−8δϱ+2χ2×12αχ2−4δϱ2,ℳ15x,t=3χ2−4δϱcot2124δϱ−χ2x−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+χ2−4δϱ2−8δϱ+2χ2×12αχ2−4δϱ2.Whenχ2−4δϱ>0&δ≠0,(22)ℳ16x,t=3χ2−4δϱsech212χ2−4δϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+χ2−4δϱ2+4δϱ−χ2×12αχ2−4δϱ2,ℳ17x,t=34δϱ−χ2coth212χ2−4δϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+χ2−4δϱ2−8δϱ+2χ2×12αχ2−4δϱ2.Whenδϱ>0&ϱ≠0&δ≠0&χ=0,(23)ℳ18x,t=12αδ2ϱ2−3δϱsec2δϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+δ2ϱ2+δϱ,ℳ19x,t=12αδ2ϱ2−3δϱcsc2δϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+δ2ϱ2+δϱ.Whenδϱ<0&ϱ≠0&δ≠0&χ=0,(24)ℳ20x,t=12αδϱ3tanh2−δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−2δ2ϱ2+1,ℳ21x,t=12αδ2ϱ23δϱcsch2−δϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+δ2ϱ2+δϱ.Whenχ=0&ϱ=−δ,(25)ℳ22x,t=12αϱ4−3ϱ2coth2ϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+ϱ4+2ϱ2.Whenχ=δ=κ&ϱ=0,(26)ℳ23x,t=12α1−κ43csch21/2κx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+1κ2.Whenϱ=0&χ≠0&δ≠0,(27)ℳ24x,t=−12αχ4χ248δexpχx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−22+24δexpχx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−2+1−1.Whenχ=0&ϱ=δ,(28)ℳ25x,t=12αϱ4−3ϱ2sec2−α−1cϱt−2αBα∑n=0∞−α/1−αnΓ1−αn+C+xϱ+ϱ4+ϱ2.


## 2.1. Family I

(10)
a
0
⟶
b
2
−
χ
2
−
4
δ
ϱ
2
+
8
δ
ϱ
+
χ
2
12
ϱ
2
,
a
1
⟶
0
,
a
2
⟶
0
,
b
1
⟶
b
2
χ
ϱ
,
α
⟶
−
6
ϱ
2
b
2
4
δ
ϱ
−
χ
2
2
,
λ
⟶
−
1
χ
2
−
4
δ
ϱ
2
,
where
ϱ
≠
0
,
χ
2
≠
4
δ
ϱ
.Consequently, the closed forms of solutions for the fractional strain waves model are given as follows:Whenχ2−4δϱ<0&δ≠0,(11)ℳ1x,t=b21248δ2χ−4δϱ−χ2tan1/24δϱ−χ2x−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn2−24δχχϱ−ϱ4δϱ−χ2tan1/24δϱ−χ2x−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+−χ2−4δϱ2+8δϱ+χ2ϱ2,ℳ2x,t=b21248δ2χ−4δϱ−χ2cot1/24δϱ−χ2x−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn2−24δχχϱ−ϱ4δϱ−χ2cot1/24δϱ−χ2x−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+−χ2−4δϱ2+8δϱ+χ2ϱ2.Whenχ2−4δϱ>0&δ≠0,(12)ℳ3x,t=b21248δ2χ−4δϱ−χ2tanh1/2χ2−4δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+χ2−24δχχϱ−ϱ4δϱ−χ2tanh1/2χ2−4δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+χϱ+−χ2−4δϱ2+8δϱ+χ2ϱ2,ℳ4x,t=b21248δ2χ−4δϱ−χ2coth1/2χ2−4δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+χ2−24δχχϱ−ϱ4δϱ−χ2coth1/2χ2−4δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+χϱ+−χ2−4δϱ2+8δϱ+χ2/ϱ2.Whenδϱ>0&ϱ≠0&δ≠0&χ=0,(13)ℳ5x,t=b2δϱ3cot2δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+2−δ2ϱ23ϱ2,ℳ6x,t=b2δϱ3tan2δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+2−δ2ϱ23ϱ2.Whenδϱ<0&ϱ≠0&δ≠0&χ=0,(14)ℳ7x,t=−b23δϱcsch2−δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+δ2ϱ2+δϱ3ϱ2,ℳ8x,t=−b2δϱ3tanh2−δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−2+δ2ϱ23ϱ2.Whenχ=0&ϱ=−δ,(15)ℳ9x,t=b2312exp2ϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+12+6tanhϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn−ϱ4ϱ2−5a.Whenχ=ϱ/2=κ&δ=0,(16)ℳ10x,t=b21expκx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−22+12expκx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−4−κ448κ2+148.Whenχ=δ=0&ϱ≠0,(17)ℳ11x,t=b2ϱ2x−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn2.Whenχ=0&ϱ=δ,(18)ℳ12x,t=13b23cot2−α−1cϱt−2αBα∑n=0∞−α/1−αnΓ1−αn+C+xϱ−ϱ4ϱ2+2.Whenδ=0&χ≠0&ϱ≠0,(19)ℳ13x,t=b212χ21−12ϱ/ϱ−χexpχx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−χ4ϱ2+12expχx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−ϱ/χ2.


## 2.2. Family II

(20)
a
0
⟶
χ
2
−
4
δ
ϱ
2
−
8
δ
ϱ
−
χ
2
2
α
χ
2
−
4
δ
ϱ
2
,
a
1
⟶
−
6
δ
χ
α
χ
2
−
4
δ
ϱ
2
,
a
2
⟶
−
6
δ
2
α
4
δ
ϱ
−
χ
2
2
,
b
1
⟶
0
,
b
2
⟶
0
,
λ
⟶
−
1
χ
2
−
4
δ
ϱ
2
,
where
δ
≠
0
,
α
≠
0
,
4
δ
ϱ
≠
χ
2
.Consequently, the closed forms of solutions for the fractional strain waves model are given as follows:Whenχ2−4δϱ<0&δ≠0,(21)ℳ14x,t=3χ2−4δϱtan2124δϱ−χ2x−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+χ2−4δϱ2−8δϱ+2χ2×12αχ2−4δϱ2,ℳ15x,t=3χ2−4δϱcot2124δϱ−χ2x−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+χ2−4δϱ2−8δϱ+2χ2×12αχ2−4δϱ2.Whenχ2−4δϱ>0&δ≠0,(22)ℳ16x,t=3χ2−4δϱsech212χ2−4δϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+χ2−4δϱ2+4δϱ−χ2×12αχ2−4δϱ2,ℳ17x,t=34δϱ−χ2coth212χ2−4δϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+χ2−4δϱ2−8δϱ+2χ2×12αχ2−4δϱ2.Whenδϱ>0&ϱ≠0&δ≠0&χ=0,(23)ℳ18x,t=12αδ2ϱ2−3δϱsec2δϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+δ2ϱ2+δϱ,ℳ19x,t=12αδ2ϱ2−3δϱcsc2δϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+δ2ϱ2+δϱ.Whenδϱ<0&ϱ≠0&δ≠0&χ=0,(24)ℳ20x,t=12αδϱ3tanh2−δϱx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−2δ2ϱ2+1,ℳ21x,t=12αδ2ϱ23δϱcsch2−δϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+δ2ϱ2+δϱ.Whenχ=0&ϱ=−δ,(25)ℳ22x,t=12αϱ4−3ϱ2coth2ϱx−α−1ct−2αBα∑n=0∞−α/1−αnΓ1−αn+ϱ4+2ϱ2.Whenχ=δ=κ&ϱ=0,(26)ℳ23x,t=12α1−κ43csch21/2κx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn+1κ2.Whenϱ=0&χ≠0&δ≠0,(27)ℳ24x,t=−12αχ4χ248δexpχx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−22+24δexpχx−α−1ct−2α/Bα∑n=0∞−α/1−αnΓ1−αn−2+1−1.Whenχ=0&ϱ=δ,(28)ℳ25x,t=12αϱ4−3ϱ2sec2−α−1cϱt−2αBα∑n=0∞−α/1−αnΓ1−αn+C+xϱ+ϱ4+ϱ2.


## 3. Results and Discussion

This section is divided into two main parts. The first part shows studying the obtained computational solutions for the fractional suggested model, while the second part presents a comparison between them and other obtained results in previous work.(1)
The solutions obtained in this paper:(i)
In this research paper, the fractional nonlinear longitudinal strain wave equation is investigated by the employment of the mK method and a new fractional definitionAℬℛ. Abundant explicit closed forms of solutions are obtained (twenty five solutions).(2)
The previously obtained solutions in previous work:(i)
In [53], two analytical methods were applied to three different models involving our two investigated models. However, they used two schemes but a very few special solutions were obtained.(ii)
The two analytical schemes in [53] are just a special case of the mK method when Qℳφ=G′/G,ϱ=−μ,χ=−λ,δ=1.(iii)
Equation (22) is equal to equation (3.9) in [53] when e0=−12δμ+dd−λ,−3λ2−4μ=δλϱ.(iv)
All other solutions obtained in this paper are considered new solutions when compared with those obtained in [53].


## 4. Conclusion

In our research paper, we solved the flaws and disadvantages of theG′/G-expansion methods that are used in [53] by M. Ali Akbar et al., and as shown in the previous section, it is just a particular case of our applied method in this research paper. Moreover, a new definition of fractional derivative is used, successfully converting the fractional from our abovementioned models to integer-order ordinary differential equations. Abundance new solutions for both the models were obtained.



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*Source: 3498796-2020-04-30.xml*