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

School of Mathematics, Qilu Normal University, Jinan 250200, Shandong Province, China School of Control Science and Engineering, Shandong University, Jinan 250061, Shandong, China Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang, China Department of Mathematics, Obour Institutes, Cairo, Egypt Department of Basic Science, Higher Technological Institute of 10 Ramadan City, Ramadan City, Egypt


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. e 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][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].
is method depends on a new auxiliary equation, which is equal to the Riccati equation [41]. e auxiliary equation of the mK method is given by where δ, ϱ, χ, and Q are the arbitrary constants. e Riccati equation is given by where A 0 , A 1 , and A 2 are the arbitrary constants. So, equations (1) and (2) 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. is 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][43][44][45][46][47][48]: where [c, L 1 , L 2 , L 3 ] are the arbitrary constants. is 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. is model contains nonlinear and dissipation terms which are the officials of construct the kink and shock waves. is shows the permanent form in medium points to a possible presence of dispersion or dissipation. e following definition of the ABR fractional operator [49][50][51][52] is applied to equation (3).

Definition 1. It is given in [17] that
where G α is the Mittag-Leffler function defined by the following formula: and B(α) is a normalisation function. us, where c is the arbitrary constant. is wave transformation converts equation (3) to ODE. Integrating the obtained ODEs twice with zero constant of the integration gives where Calculating the homogeneous balance value in equation (8) yields n � 2. us, both equations have same general formula of solution and it is given according to the mK method by where a 0 , a 1 , a 2 , b 1 , and b 2 are the arbitrary constants. e order for the rest of this article is shown as follows: Section 2 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.

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.

Family I.
2

Mathematical Problems in Engineering
Consequently, the closed forms of solutions for the fractional strain waves model are given as follows:

Mathematical Problems in Engineering 3
When When

Mathematical Problems in Engineering
Consequently, the closed forms of solutions for the fractional strain waves model are given as follows: When Mathematical Problems in Engineering 5 When

Results and Discussion
is section is divided into two main parts. e 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.

Conclusion
In our research paper, we solved the flaws and disadvantages of the (G ′ /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.
Data Availability e data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.