Butt, Asma RashidJhangeer, AdilAkgul, AliHassani, Murad Khan2024-12-242024-12-2420242045-2322https://doi.org/10.1038/s41598-024-65218-7https://hdl.handle.net/20.500.12604/6858In 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.eninfo:eu-repo/semantics/openAccessVan der Waals equationSoliton solutionsM-truncated derivativeBeta-derivativeFractional wave transformNew extended direct algebraic methodArticle141N/AWOS:001317187900004Q12-s2.0-852043112683928941310.1038/s41598-024-65218-7