Browsing by Author "Li, Shuming"

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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.
Formulation and Quantization of Field Equations on Fractal Space-Time
(Springer/plenum Publishers, 2025) Golmankhaneh, Alireza Khalili; Pasechnik, Roman; Jorgensen, Palle E. T.; Li, Shuming
This paper explores the framework of fractal calculus and its application to classical and quantum field theories. We begin with a brief overview of the fundamental concepts of fractal calculus. Building on this foundation, we introduce the formulation of the classical scalar field within a fractal space. The study is then extended to the quantization of the fractal field, where we examine how fractal geometry influences the quantization process. As a key example, we consider the fractal version of the Klein-Gordon equation and analyze how the fractal dimension affects the behavior of the field. Graphical representations are provided to illustrate the impact of fractal dimensions on the solutions. The paper concludes with a summary of the results and their potential implications for future research in fractal field theory.
Fractal Quantum Nambu Mechanics
(Springer, 2025) Khalili Golmankhaneh, Alireza; Pasechnik, Roman; Jorgensen, Palle E. T.; Li, Shuming
This paper develops a comprehensive framework for the extension of classical and quantum mechanics to fractal settings. We begin by summarizing the classical formulation of Fractal Nambu Mechanics and then introduce its quantization. The Fractal Hamilton-Jacobi Theory is established to describe dynamical systems evolving over fractal time and space, followed by a fractal generalization of the quantum Hamilton-Jacobi framework. We further formulate the Fractal Nambu-Hamilton-Jacobi Theory and propose its quantum counterpart-the Quantum Fractal Nambu-Hamilton-Jacobi Theory. These constructions demonstrate how the structure of Nambu mechanics, when combined with local fractal calculus, can provide new insights into systems with multiple invariants and non-smooth geometric evolution.