Browsing by Author "Myrzakulov, Ratbay"

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    About the Fractal Navier-Stokes Equations
    (Springer Int Publ Ag, 2025) Golmankhaneh, Alireza Khalili; Myrzakulov, Ratbay; Li, Shuming
    This paper presents a novel formulation of the Navier-Stokes equations within a fractal space-time framework by incorporating the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F<^>{\alpha }$$\end{document}-derivative to model fluid behavior in media with non-integer spatial and temporal dimensions. We derive the generalized fractal Navier-Stokes momentum equation and introduce a corresponding fractal Reynolds number that captures the effects of both spatial fractal dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and temporal fractal dimension \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}. Analytical solutions are obtained for several classical flow problems adapted to fractal geometries, including fractal Poiseuille flow, planar and generalized Couette flow, and their multi-dimensional extensions. The results reveal that increasing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} leads to nonlinear distortions in velocity profiles, while increasing \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} alters the relaxation time and can induce temporal instabilities. Graphical illustrations are provided to demonstrate the influence of fractal dimensions on flow characteristics, offering new insight into the behavior of fluids in complex fractal environments.
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    Article
    Solving Fractal Differential Equation Via Numerical Methods
    (Springer Heidelberg, 2025) Golmankhaneh, Alireza Khalili; Zayed, Ahmed I.; Myrzakulov, Ratbay
    In this paper, the fractal numerical methods for solving fractal differential equations are given such as the Fractal Euler's Method, the Fractal Heun Method, and the Fractal Runge-Kutta Method. The errors of the fractal numerical approximate solution of the fractal differential equation are presented and compared.