Well-posedness of a nonlinear Hilfer fractional derivative model for the Antarctic circumpolar current

This article explores the Hilfer fractional derivative within the context of fractional differential equations and investigates a mathematical model formulated as a three-point boundary value problem (BVP). The primary focus is on the application of these models to analyze the jet flow of the Antarctic Circumpolar Current. The study establishes the existence of stream functions using Schaefer's fixed point theorem under the assumption of the continuity of the vorticity function phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document}. Furthermore, the article delves into the existence and uniqueness results of the stream functions by employing the Banach fixed point theorem. This analysis is conducted under the condition that the vorticity function phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} is Lipschitz continuous with respect to the stream function. Additionally, the stability of the stream functions of the BVP is explored through Ulam-Hyers and generalized Ulam-Hyers stability analyses. In contrast to the foundational results presented for the three-point BVP, the article includes illustrative examples aimed at validating the findings.