arXiv subjects

Alireza Khalili Golmankhaneh

Publications and source records attributed to Alireza Khalili Golmankhaneh.

At least 19 recordsLinked to original sources

arXiv2026

Non-homogeneous Partial Differential Wave Equation for the Description of Correlated Pair States in the Microworld

In this paper, a new non-homogeneous linear second-order partial differential equation (PDE) is proposed and investigated, designed to describe the dynamics of correlated pair objects of the microworld (analogues of Cooper pairs or biphotons). The fundamental complex solution of the proposed equation initially contains a spatial background and a doubled phase, which allows for a natural localization of the pair condensate in space. In this work, the continuity and energy balance equations are derived, the discrete spectrum in potential fields is calculated, and specific effects of tunneling and nonlinear interaction of the pair conglomerate with electromagnetic radiation are described.

math.GM↗

arXiv2025

The Fractal Lie Derivative: Theory and Applications

This paper presents a new Lie theoretic approach to fractal calculus, which in turn yields such new results as a Fractal Noether's Theorem, a setting for fractal differential forms, for vector fields, and Lie derivatives, as well as k-fractal jet space, and algorithms for k-th fractal prolongation. The symmetries of the fractal nonlinear $n$-th $α$-order differential equation are examined, followed by a discussion of the symmetries of the fractal linear $n$-th $α$-order differential equation. Additionally, the symmetries of the fractal linear first $α$-order differential equation are derived. Several examples are provided to illustrate and highlight the details of these concepts.

math.GM↗

arXiv2024

Second $α$-Order Fractal Differential Equations

In this research paper, we provide a concise overview of fractal calculus applied to fractal sets. We introduce and solve a second $α$-order fractal differential equation with constant coefficients across different scenarios. We propose a uniqueness theorem for second $α$-order fractal linear differential equations. We define the solution space as a vector space with non-integer orders. We establish precise conditions for second $α$-order fractal linear differential equations and derive the corresponding fractal adjoint differential equation.

math.GM↗

arXiv2024

On Homogeneous System of Fractal Differential Equations

In this paper, a homogeneous system of n $α$-order linear fractal differential equation is defined and the set of its fundamental solutions through the corresponding Wronskian matrix is described. Finally, the solutions of some assigned autonomous homogeneous system are plotted to show the details previously proved.

math.GM↗

arXiv2024

Higher Order Fractal Differential Equations

This paper provides a summary of the fractal calculus framework. It presents higher-order homogeneous and nonhomogeneous linear fractal differential equations with $α$-order. Solutions for these equations with constant coefficients are obtained through the method of variation of parameters and the method of undetermined coefficients. The solution space for higher $α$-order linear fractal differential equations is defined, showcasing its non-integer dimensionality. The solutions to $α$-order linear fractal differential equations are graphically depicted to illustrate their non-differentiability. Additionally, equations of motion governing the behavior of two masses in fractal time are proposed and solved.

math.GM↗

arXiv2024

Fractal Calculus to Derive Fractal Frenet Equations for Fractal Curves

This paper introduces the concept of Fractal Frenet equations, a set of differential equations used to describe the behavior of vectors along fractal curves. The study explores the analogue of arc length for fractal curves, providing a measure to quantify their length. It also discusses fundamental mathematical constructs, such as the analogue of the unit tangent vector, which indicates the curve's direction at different points, and the analogue of curvature vector or fractal curvature vector, which characterizes its curvature at various locations. The concept of torsion, describing the twisting and turning of fractal curves in three-dimensional space, is also explored. Specific examples, like the fractal helix and the fractal snowflake, illustrate the application and significance of the Fractal Frenet equations.

math.GM↗

arXiv2024

Fractal Nambu Mechanics: Extending Dynamics with Fractal Calculus

In this paper, we extend the principles of Nambu mechanics by incorporating fractal calculus. This extension introduces Hamiltonian and Lagrangian mechanics that incorporate fractal derivatives. By doing so, we broaden the scope of our analysis to encompass the dynamics of fractal systems, enabling us to capture their intricate and self-similar properties. This novel approach opens up new avenues for understanding and modeling complex fractal structures, thereby advancing our comprehension of these intricate phenomena.

math.GM↗

arXiv2023

Solving and Applying Fractal Differential Equations: Exploring Fractal Calculus in Theory and Practice

In this paper, we delve into the fascinating realm of fractal calculus applied to fractal sets and fractal curves. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for solving $α$-order differential equations. Notably, we extend our analysis to solve Fractal Bernoulli differential equations. The applications of our findings are then showcased through the solutions of problems such as fractal compound interest, the escape velocity of the earth in fractal space and time, and estimation of time of death incorporating fractal time. Visual representations of our results are also provided to enhance understanding.

math.GM↗

arXiv2023

Fractal Schrödinger Equation: Implications for Fractal Sets

This paper delves into the world of fractal calculus, investigating its implications for fractal sets. It introduces the Fractal Schrödinger Equation and provides insights into its consequences. The study presents a General Solution for the Time-Dependent Schrödinger Equation, unveiling its core aspects. Exploring quantum mechanics in the context of fractals, the paper analyzes the Probability Density of the Radial Hydrogen Atom, unveiling its behavior within fractal dimensions. The investigation extends to deciphering the intricate Energy Levels of the Hydrogen Atom, uncovering the interplay of quantum mechanics and fractal geometry. Innovatively, the research applies the Fractal Schrödinger Equation to Simple Harmonic Motion, leading to the introduction of the Fractal Probability Density Function for the Harmonic Oscillator. The paper employs a series of illustrative figures that enhance the comprehension of the findings. By intertwining quantum mechanics and fractal mathematics, this research paves the way for deeper insights into their relationship.

quant-ph↗

arXiv2023

Stochastic Processes and Mean Square Calculus on Fractal Curves

In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define cumulative distribution function, probability density function, moments, variance and correlation function of stochastic process on fractal curve. A new framework which is a generalization of mean square calculus is formulated. Sequence of random variable on fractal curve, fractal mean square continuity, mean square $F^α$-derivative, and fractal mean square integral. The mean square solution of a fractal stochastic equation is derived and plotted in order to show the details.

math.GM↗

arXiv2023

The Mellin Transform and Non-local Derivatives of Fractal Calculus

In this paper, the fractal calculus of fractal sets and fractal curves are compared. The analogues of the Riemann-Liouville and the Caputo integrals and derivatives are defined for the fractal curves which are non-local derivatives. The analogous for the fractional Laplace concepts are defined to solve fractal non-local differential equations on fractal curves. The fractal local Mellin and fractal non-local transforms are defined to solve fractal differential equations. We present tables and examples to illustrate the results.

math.GM↗

arXiv2023

Fuzzification of Fractal Calculus

In this manuscript, fractal and fuzzy calculus are summarized. Fuzzy calculus in terms of fractal limit, continuity, its derivative, and integral are formulated. The fractal fuzzy calculus is a new framework that includes fractal fuzzy derivatives and fractal fuzzy integral. In this framework, fuzzy number-valued functions with fractal support are the solutions of fractal fuzzy differential equations. Different kinds of fractal fuzzy differential equations are given and solved.

math.GM↗

arXiv2021

Electronic and thermoelectric properties of quasi-fractal carbon nitride nanoribbons

Recent works devoted to the synthesis of artificial molecular systems with quasi-fractal geometry provide new opportunities for the experimental study of electronic properties in atomic systems of fractional dimension. There has been a renewed interest in theoretical studies of transport phenomena in fractal nanosystems. In this work, using calculations based on the density functional theory, molecular dynamics, and the method of non-equilibrium Green functions, we study electronic and thermoelectric properties of a device based on quasi-fractal carbon nitride nanoribbon with Sierpinski triangle blocks. We estimate changes in transport properties with generation g of quasi-fractal blocks in nanoribbon.

cond-mat.mes-hall↗

arXiv2019

On the Stability of Fractal Differential Equations

In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions which are not differentiable or integrable on totally disconnected fractal sets such as middle-$μ$ Cantor sets. Analogues of the Lyapunov functions and features are given for asymptotic behaviors of fractal differential equations. 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.

math.DS↗

arXiv2018

Diffusion on middle-$ξ$ Cantor sets

In this paper, we study $C^ζ$-calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the $C^ζ$-calculus on the generalized Cantor sets known as middle-$ξ$ Cantor sets. We have suggested a calculus on the middle-$ξ$ Cantor sets for different values of $ξ$ with $0<ξ<1$. Differential equations on the middle-$ξ$ Cantor sets have been solved, and we have presented the results using illustrative examples. The conditions for super-, normal, and sub-diffusion on fractal sets are given.

math.CA↗

arXiv2018

New Derivatives for the Functions with the Fractal Tartan Support

In this manuscript, we generalize F-calculus to apply it on fractal Tartan spaces. The generalized standard F-calculus is used to obtain the integral and derivative of the functions on the fractal Tartan with different dimensions. The generalized fractional derivatives have local properties that make it more useful in modelling physical problems. The illustrative examples are used to present the details.

math.CA↗

arXiv2015

New Derivatives on Fractal Subset of Real-line

In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local Laplace transformation is given and applied for solving linear and non-linear fractal equations. The advantage of using these new nonlocal derivatives on fractals subset of real-line lies in the fact that they are used for better modelling of processes with memory effect.

math.CA↗

arXiv2013

Schrödinger Equation on Fractals Curves Imbedding in $R^3$

In this paper we have generalized the quantum mechanics on fractal time-space. The time is changing on Cantor-set like but space is considered as fractal curve like Von-Koch curve. The Feynman path method in quantum mechanics has been suggested on fractal curve. Using $F^α$-calculus and Feynman path method we found the Schrëdinger on fractal time-space. The Hamiltonian operator and momentum operator has been derived. More, the continuity equation and the probability density is given in generalized formulation.

math-ph↗