Browsing by Author "Golmankhaneh, Alireza Khalili"

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Analogues To Lie Method and Noether's Theorem in Fractal Calculus
(Mdpi, 2019) Golmankhaneh, Alireza Khalili; Tunc, Cemil
In this manuscript, we study symmetries of fractal differential equations. We show that using symmetry properties, one of the solutions can map to another solution. We obtain canonical coordinate systems for differential equations on fractal sets, which makes them simpler to solve. An analogue for Noether's Theorem on fractal sets is given, and a corresponding conservative quantity is suggested. Several examples are solved to illustrate the results.
Analyzing the Stability of Fractal Delay Differential Equations
(Pergamon-elsevier Science Ltd, 2024) Golmankhaneh, Alireza Khalili; Tunc, Cemil
In this paper, we provide a comprehensive overview of fractal calculus and investigate the stability of both linear and non-linear fractal delay differential equations with fractal support. Our analysis encompasses the stability of the fractal Mackey-Glass equation as well as fractal differential equations with single and dual delays. Additionally, we introduce a predictor-corrector scheme to solve the fractal one-delay differential equation. Several examples are presented to illustrate the effects of fractal-order differentiation, which arise from the dimensionality of the fractal support, and the impact of fractal delays.
Classical Mechanics on Fractal Curves
(Springer Heidelberg, 2023) Golmankhaneh, Alireza Khalili; Welch, Kerri; Tunc, Cemil; Gasimov, Yusif S.
Fractal analogue of Newton, Lagrange, Hamilton, and Appell's mechanics are suggested. The fractal alpha-velocity and alpha-acceleration are defined in order to obtain the Langevin equation on fractal curves. Using the Legendre transformation, Hamilton's mechanics on fractal curves is derived for modeling a non-conservative system on fractal curves with fractional dimensions. Fractal differential equations have solutions that are non-differentiable in the sense of ordinary derivatives and explain space and time with fractional dimensions. The illustrated examples with graphs present the details.
Extending Dirac and Faddeev-Jackiw Formalisms To Fractal First Α-Order Lagrangian Systems
(Springer Basel Ag, 2025) Golmankhaneh, Alireza Khalili; Sevli, Hamdullah; Tavares, Dina; Jorgensen, Palle E. T.
This paper presents the foundational concepts of fractal calculus before generalizing the Dirac Constraint Formalism and the Faddeev-Jackiw Formalism for first alpha-order Lagrangian systems in fractal spaces with non-integer dimensions. We provide a detailed analysis of the generalization process, highlighting the theoretical framework and key results, including the extended structure of the constraint systems in these Lagrangian formulations. Specific examples are discussed to demonstrate the practical application of the generalized formalism and to validate the consistency of our results. Moreover, graphical visualizations are included to enhance clarity, offering a visual interpretation of the findings and illustrating the relationship between the theory and its real-world implications.
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 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 Complex Analysis
(Univ Guilan, 2025) Golmankhaneh, Alireza Khalili; Rodriguez-Lopez, Rosana; Stamova, Ivanka M.; Celik, Ercan
In this paper, we begin by providing a concise overview of fractal calculus. We then explore the concepts of fractal complex numbers and functions, define the fractal complex derivative, and derive the fractal Cauchy-Riemann equations. Additionally, we introduce fractal contour integrals, offer illustrative examples, and present their visualizations. Finally, we examine and visualize the transformations of circles under fractal complex functions.
Fractal Differential Equations of 2α-Order
(Mdpi, 2024) Golmankhaneh, Alireza Khalili; Bongiorno, Donatella
In this research paper, we provide a concise overview of fractal calculus applied to fractal sets. We introduce and solve a 2 alpha-order fractal differential equation with constant coefficients across different scenarios. We propose a uniqueness theorem for 2 alpha-order fractal linear differential equations. We define the solution space as a vector space with non-integer orders. We establish precise conditions for 2 alpha-order fractal linear differential equations and derive the corresponding fractal adjoint differential equation.
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 Sturm-Liouville Problems
(Taylor & Francis Ltd, 2025) Allahverdiev, Bilender P.; Tuna, Huseyin; Golmankhaneh, Alireza Khalili
In this study, fractal Sturm-Liouville problems are considered. First, minimal and maximal operators corresponding to such problems are defined and a symmetric operator is obtained. Then Green's function is constructed and an eigenfunction expansion theorem is obtained. Finally, some examples are given.
Fractal Sturm-Liouville Theory
(Mdpi, 2025) Golmankhaneh, Alireza Khalili; Vidovic, Zoran; Tuna, Hueseyin; Allahverdiev, Bilender P.
This paper provides a short summary of fractal calculus and its application to generalized Sturm-Liouville theory. It presents both the fractal homogeneous and non-homogeneous Sturm-Liouville problems and explores the theory's applications in optics. We include examples and graphs to illustrate the effect of fractal support on the solutions and propose new models for fractal structures.
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.
Homotopy Perturbation Method for a System of Fractal Schrödinger-Korteweg Vries Equations
(Springer Heidelberg, 2025) Golmankhaneh, Alireza Khalili; Pham, Diana; Stamova, Ivanka; Ramazanova, Aysel; Rodriguez-Lopez, Rosana
This paper presents a novel application of the Homotopy Perturbation Method (HPM) to a system of coupled fractal Schr & ouml;dinger-Korteweg-de Vries (S-KdV) equations, formulated within the framework of fractal calculus. By extending classical S-KdV equations and diffusion-reaction systems to fractal space, we introduce a new mathematical model that captures the complex behavior of nonlinear wave interactions and reaction-diffusion processes in media with fractal geometries. The main contribution of this work lies in deriving approximate analytical solutions for these fractal systems using HPM, demonstrating both its effectiveness and accuracy in handling fractal differential equations. The influence of fractal time and space on the system dynamics is examined and visualized through detailed graphical analysis. This study provides a foundation for further exploration of fractal models in physical and engineering contexts, offering insights into how fractality alters classical system behavior.
Hyers-Ulam Stability on Local Fractal Calculus and Radioactive Decay
(Springer Heidelberg, 2021) Golmankhaneh, Alireza Khalili; Tunc, Cemil; Sevli, Hamdullah
In this paper, we summarize the local fractal calculus, called F-alpha-calculus, which defines derivatives and integrals of functions with fractal domains of non-integer dimensions, functions for which ordinary calculus fails. Hyers-Ulam stability provides a method to find approximate solutions for equations where the exact solution cannot be found. Here, we generalize Hyers-Ulam stability to be applied to oi-order linear fractal differential equations. The nuclear decay law involving fractal time is suggested, and it is proved to be fractally Hyers-Ulam stable.
On Initial Value Problems of Fractal Delay Equations
(Elsevier Science inc, 2023) Golmankhaneh, Alireza Khalili; Tejado, Ines; Sevli, Hamdullah; Valdes, Juan E. Napoles
In this paper, we give a brief summary of fractal calculus. Fractal functional differential equations are formulated as a framework that provides a mathematical model for the phe-nomena with fractal time and fractal structure. Fractal retarded, neutral, and renewal delay differential equations with constant coefficients are solved by the method of steps and us-ing Laplace transform. The graphs of solutions are given to show the details.(c) 2023 Elsevier Inc. All rights reserved.
On Stability of a Class of Second Alpha-Order Fractal Differential Equations
(Amer inst Mathematical Sciences-aims, 2020) Tunc, Cemil; Golmankhaneh, Alireza Khalili
In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions that are not differentiable or integrable on totally disconnected fractal sets such as middle-mu Cantor sets. Analogues of the Lyapunov functions and their features are given for asymptotic behaviors of fractal differential equations. The stability of fractal differentials in the sense of Lyapunov is defined. For the suggested fractal differential equations, sufficient conditions for the stability and uniform boundedness and convergence of the solutions are presented and proved. We present examples and graphs for more details of the results.
Regular Fractal Dirac Systems
(World Scientific Publ Co Pte Ltd, 2025) Allahverdiev, Bilender P.; Tuna, Huseyin; Golmankhaneh, Alireza Khalili
In this paper, the classical one-dimensional Dirac equation is considered under the framework of fractal calculus. First, the maximal and minimal operators corresponding to the problem are defined. Then the symmetric operator is obtained, the Green's function corresponding to the problem is constructed, and the eigenfunction expansion is given. Finally, some examples are given.
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.