New applications of the fractional derivative to extract abundant soliton solutions of the fractional order PDEs in mathematics physics

• Published in: Partial differential equations in applied mathematics : a spin-off of Applied Mathematics Letters 2024-03, Vol.9, p.100597, Article 100597
• Main Authors: Iqbal, M. Ashik, Miah, M. Mamun, Ali, H. M. Shahadat, Shahen, Nur Hasan Mahmud, Deifalla, Ahmed
• Format: Article
• Language: English
• Subjects: Nonlinear PDEs; The (G′/G′+G+A)-expansion method; The fractional order (2+1)-dimensional breaking soliton equation; The fractional order (4+1)-dimensional Fokas equation; The soliton solutions; Wave propagation
• ISSN: 2666-8181; 2666-8181
• DOI: 10.1016/j.padiff.2023.100597

The motive of this research work is to unravel the mysteries of nature through fractional-order partial differential equations (PDEs). Here, we focus on two important fractional order nonlinear PDEs, namely the fractional order (4+1)-dimensional Fokas equation, which is used to give the model of many physical phenomena and dynamical processes, and the other one is the fractional order (2+1)-dimensional breaking soliton equation which is used to analyze the nonlinear problems like optical fiber communications, ocean engineering, etc. Recently, it has been an essential topic to extract the new soliton solutions which are used to investigate the hidden physical conditions of the nonlinear fractional PDEs. So, it is essential to solve those nonlinear fractional PDEs which have a physical impact in the fields of science and modern engineering. In our investigation, we attempt to provide nonlinear wave propagation patterns and investigate the equations, as mentioned earlier, through a computational method. A computing operating software called Mathematica has been applied to get a clear visualization of our gained outcomes, and we ascertain such types of shapes as the bell shape soliton, the anti-bell shape soliton, the singular bell shape soliton, the periodic solution, and the singular periodic solution. Our obtained results can keep an indispensable role in explaining various physical phenomena of nature shortly, and the applied method is the very cogent, efficient, and interesting to extract such types of solutions. Since we extract abundant solutions of these models, so we hope that this article is the best applications of mention method.