Browsing by Author "Cattani, Carlo"

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Fractal Calculus of Variations for Problems With Constraints
(World Scientific Publ Co Pte Ltd, 2025) Golmankhaneh, Alireza Khalili; Cattani, Carlo; Pasechnik, Roman; Furuichi, Shigeru; Jorgensen, Palle E. T.
In this paper, we present a summary of fractal calculus and propose the use of Lagrange multipliers for both fractal calculus and fractal variational calculus with constraints. We examine the application of these methods across various branches of physics. By employing fractal variational calculus with constraints, we derive fundamental equations such as the fractal mechanical wave equation, the fractal Schr & ouml;dinger equation in quantum mechanics, Maxwell's equations in fractal electromagnetism, and the Lagrange equation for constraints in fractal classical mechanics. Several examples are provided to illustrate these concepts in detail.
Fractal Green Function Theory
(Elsevier, 2026) Golmankhaneh, Alireza Khalili; Cattani, Carlo; Kalita, Hemanta; Furuichi, Shigeru; Jorgensen, Palle E. T.
This paper provides a comprehensive study of fractal calculus and its application to differential equations within fractal spaces. It begins with a review of fractal calculus, covering fundamental definitions and measures related to fractal sets. The necessary preliminaries for understanding fractal Green's functions are introduced, laying the groundwork for further exploration. We develop the fractal Green's function for inhomogeneous fractal differential equations and extend this to the fractal Helmholtz equation. The application of the fractal Green's function to the Schr & ouml;dinger equation is also investigated, focusing on the fractal Schr & ouml;dinger-type differential equation with a fractal mesonic potential. Additionally, the scattering amplitude is derived within the fractal Born approximation, offering insights into scattering phenomena in fractal spaces. The findings highlight the significant impact of fractal geometry on classical and quantum mechanics and present new methods for addressing problems in fractal environments.
Fractal Hankel Transform
(Mdpi, 2025) Golmankhaneh, Alireza Khalili; Sevli, Hamdullah; Cattani, Carlo; Vidovic, Zoran
This paper explores the extension of classical transforms to fractal spaces, focusing on the development and application of the Fractal Hankel Transform. We begin with a concise review of fractal calculus to set the theoretical groundwork. The Fractal Hankel Transform is then introduced, along with its formulation and properties. Applications of this transform are presented to demonstrate its utility and effectiveness in solving problems within fractal spaces. Finally, we conclude by summarizing the key findings and discussing potential future research directions in the field of fractal analysis and transformations.
Fractal Telegraph Equation
(Springer int Publ Ag, 2024) Golmankhaneh, Alireza Khalili; Cattani, Carlo; O'Regan, Donal; Tejado, Ines; Vidovic, Zoran
In this paper, we provide a brief review of fractal calculus. We introduce the fractal telegraph equation, which generalizes both the fractal heat and wave equations, and derive its solution. The solutions are plotted to highlight the differences between fractal differential equations and standard differential equations, demonstrating the effects of fractal time and space on the solutions.
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.