Browsing by Author "Khalili Golmankhaneh, Alireza"

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On the Generalized Fractal Calculus of Variations
(Springer Int Publ Ag, 2025) Khalili Golmankhaneh, Alireza; Cattani, Carlo; Park, Choonkil; Furuichi, Shigeru
In this paper, we provide a brief overview of fractal calculus and present a comprehensive study of the calculus of variations for functionals on fractal sets. We begin by introducing the calculus of variations for functionals with several dependent variables on fractal sets. We then explore the calculus of variations for functionals with several independent variables on fractal sets. Subsequently, we investigate the calculus of variations for functionals with both several independent and dependent variables on fractal sets. Finally, we suggest applications of fractal calculus of variations in physics, providing examples and plots to illustrate the details.
Fractal Calculus: Nonhomogeneous Linear Systems
(Walter de Gruyter GmbH, 2025) Khalili Golmankhaneh, Alireza; Bongiorno, Donatella; Ramazanova, Aysel T.
In this paper, we present a concise overview of fractal calculus and explore the solution of non-homogeneous fractal differential equations. We analyze fractal homogeneous linear systems with initial conditions, introducing the fundamental matrix and special fundamental matrix, and demonstrate their applications in solving systems and analyzing the Jordan form of matrices. We propose the method of undetermined coefficients for solving non-homogeneous fractal linear differential equations and introduce the method of variation of parameters as a supplementary technique. To illustrate these methods, we apply them to the differential equations of resistor-inductor-capacitor (RLC) circuits, successfully solving the corresponding fractal differential equations. Additionally, we provide examples, solve systems with initial conditions, and present the results through plotted graphs.
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.