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    A plethora of novel solitary wave solutions related to van der Waals equation: a comparative study
    (Nature Portfolio, 2024) Butt, Asma Rashid; Jhangeer, Adil; Akgul, Ali; Hassani, Murad Khan
    In this article, we explore exact solitary wave solutions to the van der Waals equation which is crucial for numerous applications involving a variety of physical occurrences. This system is used to define the behavior of real gases taking into consideration finite size of molecules and also has some applications in industry for granular materials. The model is studied under the effect of fractional derivatives by employing two different definitions: beta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}, and M-truncated. Further, new extended direct algebraic method is employed to construct the solitary wave solutions for the model. The solutions transmit several novel solutions, such as dark-singular, dark-bright, singular-periodic and dark solutions, and this method establishes the conditions required for the formation of these structures. To show the comparative analysis between two different fractional operators, results are graphically represented in the form of 2-dimensional and 3-dimensional visualizations.
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    Öğe
    Analytical study of soliton solutions for an improved perturbed Schrödinger equation with Kerr law non-linearity in non-linear optics by an expansion algorithm
    (Elsevier B.V., 2021) Jhangeer, Adil; Faridi, Waqas Ali; Asjad, Muhammad Imran; Akgül, Ali
    This paper aims to study an improved perturbed Schrödinger equation (IPSE) with a kind of Kerr law non-linearity equation governing the propagation dynamics of soliton in optical fibers through the nano-optical fiber. The considered model predicts the influence of quantic non-linearity on the motion of ultrashort optical pulses. The integrability of the model is accompanied by the transformed rational function V-expansion method (for simplicity [Formula presented]). This proposed method is a significant mathematical tool to obtain the exact travelings wave solutions of non-linear complex partial differential equations (PDEs). A bunch of soliton solutions like dark, dark singular, plane wave solution, and periodic are retrieved along with suitable parametric values. The graphical analysis is also presented for the description of propagation of waves expressed by rational functions, hyperbolic functions, and trigonometric functions. © 2021 The Author(s)