Dynamics of Nonlinear Time Fractional Equations in Shallow Water Waves

This study investigates the modified nonlinear time fractional Harry Dym equation, incorporating the conformable fractional derivative. Functioning as a mathematical framework for examining nonlinear phenomena in shallow water waves, particularly solitons, this model elucidates the intricate effects of dispersion and nonlinear steepening on wave dynamics. Employing a blend of analytical and numerical methodologies, the research aims to decipher the physical implications of the equation and its interconnectedness with other nonlinear evolution equations. The model delineates the evolution of a nonlinear wave in 1+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+1$$\end{document} dimensions (one spatial dimension x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x$$\end{document} and time t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document}). The proposed methodology encompasses the G ' G,1G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \frac{G'}{G},\, \frac{1}{G}\right) $$\end{document} expansion method, an analytical technique, alongside three numerical schemes utilizing B-spline methods. These methodologies facilitate the exploration of the equation's behavior and enable precise computations of its solutions. The principal findings underscore the effective application of the proposed methodologies in resolving the modified nonlinear time fractional Harry Dym equation, furnishing valuable insights into its dynamics and significantly contributing to its physical interpretation. The significance of these discoveries lies in their contribution to the broader comprehension of nonlinear evolution equations and their pertinence across various scientific and engineering domains. This study provides novel insights into the modified nonlinear time fractional Harry Dym equation through a combined analytical and numerical approach. It advances the field of nonlinear dynamics and carries implications for analyzing analogous nonlinear evolution equations. These findings deepen our understanding of the equation's physical interpretation and lay the groundwork for future explorations in related domains.