Fractal Calculus Research Articles

Witte’s conditions for uniqueness of solutions to a class of Fractal-Fractional ordinary differential equations

In this paper, Witte's conditions for the uniqueness solution of nonlinear differential equations with integer and non-integer order derivatives are investigated. We present a detailed analysis of the uniqueness solutions of four classes of nonlinear differential equations with nonlocal operators. These classes include classical and fractional ordinary differential equations in fractal calculus. For each case, theorems and lemmas and their proofs are presented in detail.

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 [Formula: see text]-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 [Formula: see text]-order linear fractal differential equations is defined, showcasing its non-integer dimensionality. The solutions to [Formula: see text]-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.

Schwarzschild spacetime in fractal dimensions: Deflection of light, supermassive black holes and temperature effects

In this paper, we construct a Schwarzschild metric in the fractal dimension based on fractal calculus, and we study the deflection of light near the sun. It was observed that the new fractal spherically symmetric spacetime metric may describe supermassive black holes, which have been detected through various astronomical observations such as SDSS. We have discussed the deflection of light by the gravitational field of the sun. It was observed that the total deflection of light is affected by the fractal dimension. To estimate the numerical value of the fractal dimension, we have used the 2004[Formula: see text]analysis of almost 2 million VLBI observations of 541 radio sources made by 87 VLBI sites, which estimates the factor [Formula: see text] based on PNN formalism. It was observed that the fractal dimension is roughly less than unity. However, our analysis showed that large bending gravity gives fractal dimensions lower than unity to some extent. We also gave a naïve idea about the emission thermal Unruh process in fractal dimension. It was observed that both the temperature and the entropy of black holes are affected by the fractal dimension, and that for low numerical values, the entropy of the black hole is enhanced.

Efficient numerical modeling of time-fractal tangent hyperbolic fluid flow with heat and mass transfer

This paper introduces a precise method for solving time-fractal parabolic equations specifically designed to represent complex physical phenomena using fractal calculus. The suggested technique attains third-order accuracy by utilizing the fractal Taylor series and implements a concise scheme for spatial discretization. Stability and convergence analyses are offered for scalar and system fractal parabolic equations. The mathematical model analyzes fluid flow over both flat and oscillatory surfaces, considering the effects of heat and mass transfer. This involves transforming the governing equations into dimensionless partial differential equations, which are then solved using the suggested method. From the calculated results, it can be verified that the proposed scheme produces less error than the existing two schemes. This study enhances computational techniques for fractal calculus and introduces novel opportunities for comprehending non-Newtonian fluids in diverse physical contexts.

Hermite-Hadamard inequalities for generalized (m-F)-convex function in the framework of local fractional integrals

This work presents new versions of the Hermite-Hadamard Inequality, for (m−F)-convex functions, defined on fractal sets Rς ( 0 ς ≤ 1). So, we show some new results for twice differentiable functions using local fractional calculus, as well as some new definitions. We will construct these new integral inequality using the generalized H¨older-integral inequality and the power mean integral inequality. Furthermore, we present some new inequalities for the midpoint and trapezoid formulas in a novel type of fractal calculus. The conclusions in this paper are substantial advancements and generalizations of prior research reported in the literature.

Stochastic processes and mean square calculus on fractal curves

Abstract In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define the cumulative distribution function, probability density function, moments, variance, and correlation function of stochastic processes on fractal curves. A new framework, which is a generalization of mean square calculus, is formulated. The sequence of random variables on the fractal curve, fractal mean square continuity, mean square F α {F^{\alpha}} -derivative, and fractal mean square integral are discussed. The mean square solution of a fractal stochastic equation is derived and plotted to illustrate the details.

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.

Existence Theorem for a Fractal Sturm-Liouville Problem

In this article, using a new calculus defined on fractal subsets of the set of real numbers, a Sturm--Lioville type problem is discussed, namely the fractal Sturm--Liouville problem. The existence and uniqueness theorem has been proved for such equations. In this context, the historical development of the subject is discussed in the introduction. In Section 2, the basic concepts of $F^{\alpha}$-calculus defined on fractal subsets of real numbers are given, i. e., $F^{\alpha}$-continuity, $F^{\alpha}$-derivative and fractal integral definitions are given and some theorems to be used in the article are given. In Section 3, the existence and uniqueness of the solutions for the fractal Sturm--Liouville problem are obtained by using the successive approximations method. Thus, the well-known existence and uniqueness problem for Sturm--Liouville equations in ordinary calculus is handled on the fractal calculus axis, and the existing results are generalized.

On the analytical modeling of fractal memelement and inverse memelement

SummaryIn this work, an improved analytical model of fractal memelement in which the pinched point shifting has been considered and the original analytical model of fractal inverse memelement have been proposed. These fractal circuit elements are the memelement, and inverse memelement operates based on the principle of electromagnetic in fractal time/space, which must be applied whenever the current flows through fractal media. These models are important because these memory elements can be realized based on the porous material, which is a fractal media. In addition, they can employ self‐similarity, which is hard to be simulated by using the traditional models. This is because such self‐similarity can be well explained by the fractal set‐based model, yet those traditional models are based on the set of real values. Therefore, for deriving the proposed models, the fractal calculus, which is oriented to the fractal set, has been adopted as the mathematical basis. From the analytical and numerical analyses based on the derived models, it has been found that both memelement and inverse memelement can retain their unique frequency characteristics despite being operated based on the abovementioned principle. In addition, their input–output relationships are mathematically differentiable albeit the inputs and outputs themselves are not.

Fractal calculus in tumor growth simulations: The proof is in the pudding

Mathematical modeling in oncology has a long history. Recently, mathematical models and their predictions have made inroads into prospective clinical trials with encouraging results. The goal of many such modeling efforts is to make predictions, either to clinician's choice therapy or into “optimal” therapy – often for individual patients. The mathematical oncology community rightfully puts great hope into predictive modeling and mechanistic digital twins – but with this great opportunity comes great responsibility. Mathematical models need to be rigorously calibrated and validated, and their predictive performance ascertained, before conclusions about predictions into the unknown can be drawn. The recent article “Modeling tumor growth using fractal calculus: Insights into tumor dynamics” (Golmankhaneh et al., 2023), applied fractal calculus to tumor growth data. In this short commentary, I raise concerns about the study design and interpretation. In its current form, this study is poised to put cancer patients at risk if interpreted as concluded by the authors.

Local Fuzzy Fractional Partial Differential Equations in the Realm of Fractal Calculus with Local Fractional Derivatives

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration method (LFFSVIM); these are applied when solving LFFPDEs. The working procedure shows how effective solutions for specific LFFPDEs can be obtained using the applied approaches. Moreover, we present a comparison of the local fuzzy fractional Laplace variational iteration method (LFFLIM), the local fuzzy fractional series expansion method (LFFSEM), the local fuzzy fractional variation iteration method (LFFVIM), and the local fuzzy fractional Adomian decomposition method (LFFADM), which are applied to obtain fuzzy fractional diffusion and wave equations on Cantor sets. To demonstrate the effectiveness of the used techniques, some examples are given. The results demonstrate the major advantages of the approaches, which are equally efficient and simple to use in order to solve fuzzy differential equations with local fractional derivatives.

Fractal Mellin transform and non-local derivatives

Abstract This paper provides a comparison between the fractal calculus of fractal sets and fractal curves. There are introduced the analogues of the Riemann–Liouville and Caputo integrals and derivatives for fractal curves, which are non-local derivatives. Moreover, the concepts analogous to the fractional Laplace operator to address fractal non-local differential equations on fractal curves are defined. Additionally, in the paper it is introduced the fractal local Mellin transform and fractal non-local transform as tools for solving fractal differential equations. The results are supported with tables and examples to demonstrate the findings.

Modeling tumor growth using fractal calculus: Insights into tumor dynamics

Important concepts like fractal calculus and fractal analysis, the sum of squared residuals, and Aikaike’s information criterion must be thoroughly understood in order to correctly fit cancer-related data using the proposed models. The fractal growth models employed in this work are classified in three main categories: Sigmoidal growth models (Logistic, Gompertz, and Richards models), Power Law growth model, and Exponential growth models (Exponential and Exponential-Lineal models)”. We fitted the data, computed the sum of squared residuals, and determined Aikaike’s information criteria using Matlab and the web tool WebPlotDigitizer. In addition, the research investigates “double-size cancer” in the fractal temporal dimension with respect to various mathematical models.

A numerical study of a new non-linear fractal fractional mathematical model of malicious codes propagation in wireless sensor networks

This paper presents a novel non-linear fractal fractional mathematical model for analyzing the spread of malicious codes in wireless sensor networks (WSNs). The model incorporates fractional calculus and non-linear interactions among nodes in a WSN. It is based on the Fractal-Fractional (FF) operator, a powerful mathematical tool that combines fractional and fractal calculus. This study explores the potential of fractional derivatives in addressing computer virus-related issues and proposes a model that elucidates the spread of viruses in a vulnerable system and potential countermeasures. The Fractal-Fractional operator is utilized to fractionalize the model, and the fixed point theory of Schauder and Banach is applied to establish the existence and uniqueness of a solution. Numerical simulations, employing MATLAB and the Adams-Bashforth method, validate the effectiveness of the proposed model in capturing the propagation dynamics of malicious codes. Ulam-Hyers stability techniques are also employed for stability analysis. The model's insights contribute to enhancing the security and robustness of WSNs against malicious code propagation, paving the way for more secure network designs. This research provides a unique perspective on analyzing the dynamics of WSNs and underscores the significance of fractional derivatives in addressing security threats in network systems.

Fractal calculus approach to diffusion on fractal combs

In this paper, we present a generalization of diffusion on fractal combs using fractal calculus. We introduce the concept of a fractal comb and its associated staircase function. To handle functions supported on these combs, we define derivatives and integrals using the staircase function. We then derive the Fokker–Planck equation for a fractal comb with dimension α, incorporating fractal time, and provide its solution. Additionally, we explore α-dimensional and (2α)-dimensional Brownian motion on fractal combs with drift and fractal time. We calculate the corresponding fractal mean square displacement for these processes. Furthermore, we propose and solve the heat equation on an α-dimensional fractal comb space. To illustrate our findings, we include graphs that showcase the specific details and outcomes of our results.

A Mechanical Picture of Fractal Darcy’s Law

The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this purpose, the inherent features of fractal sets are scrutinized. A set of fractal dimensions is incorporated to describe the geometry, morphology, and fractal topology of the domain under study. These characteristics are known through their Hausdorff, chemical, shortest path, and elastic backbone dimensions. Afterward, fractal continuum Darcy’s law is suggested based on the mapping of the fractal reservoir domain given in Cartesian coordinates xi into the corresponding fractal continuum domain expressed in fractal coordinates ξi by applying the relationship ξi=ϵ0(xi/ϵ0)αi−1, which possesses local fractional differential operators used in the fractal continuum calculus framework. This generalized version of Darcy’s law describes the relationship between the hydraulic gradient and flow velocity in fractal porous media at any scale including their geometry and fractal topology using the αi-parameter as the Hausdorff dimension in the fractal directions ξi, so the model captures the fractal heterogeneity and anisotropy. The equation can easily collapse to the classical Darcy’s law once we select the value of 1 for the alpha parameter. Several flow velocities are plotted to show the nonlinearity of the flow when the generalized Darcy’s law is used. These results are compared with the experimental data documented in the literature that show a good agreement in both high-velocity and low-velocity fractal Darcian flow with values of alpha equal to 0<α1<1 and 1<α1<2, respectively, whereas α1=1 represents the standard Darcy’s law. In that way, the alpha parameter describes the expected flow behavior which depends on two fractal dimensions: the Hausdorff dimension of a porous matrix and the fractal dimension of a cross-section area given by the intersection between the fractal matrix and a two-dimensional Cartesian plane. Also, some physical implications are discussed.

The generalized nonlocal fractal calculus: an efficient tool for fractal circuit analysis

PurposeThe purpose of this paper is to propose a novel nonlocal fractal calculus scheme dedicated to the analysis of fractal electrical circuit, namely, the generalized nonlocal fractal calculus.Design/methodology/approachFor being generalized, an arbitrary kernel function has been adopted. The condition on order has been derived so that it is not related to the γ-dimension of the fractal set. The fractal Laplace transforms of our operators have been derived.FindingsUnlike the traditional power law kernel-based nonlocal fractal calculus operators, ours are generalized, consistent with the local fractal derivative and use higher degree of freedom. As intended, the proposed nonlocal fractal calculus is applicable to any kind of fractal electrical circuit. Thus, it has been found to be a more efficient tool for the fractal electrical circuit analysis than any previous fractal set dedicated calculus scheme.Originality/valueA fractal calculus scheme that is more efficient for the fractal electrical circuit analysis than any previous ones has been proposed in this work.

Monotone Data Modelling Using Rational Cubic Fractal Interpolation Function

Geometric modelling of several intricate and complex structures such as trees, mountains, clouds, ferns, geographic topography, and coastlines is challenging in computer graphics. Traditional splines such astrigonometric, polynomial, exponential, and rational fail to simulate this significant class of complex structures, which are highly irregular in nature. For this purpose, this research develops a novel cutting-edgemethod for synthesizing and modelling structures. The proposed technique; C 1 fractal interpolation function (FIF) builds an iterated function system (IFS) by integrating fractal calculus and rational cubic polynomial functions. Appropriate conditions on scaling and shape parameters are derived to help maintain the inherited shape qualities of the data. Experiments in numerous scientific domains, such as the pharmaceutical and chemical industries have been presented as an example, to confirm the usefulness of the suggested model. Moreover, the graphic results demonstrated that the developed monotone hybrid model (MHM) offers a heterogeneous method for gathering data with a monotone structure.

Application of Fixed Points in Bipolar Controlled Metric Space to Solve Fractional Differential Equation

Fixed point results and metric fixed point theory play a vital role to find the unique solution to differential and integral equations. Likewise, fractal calculus has vast physical applications. In this article, we introduce the concept of bipolar-controlled metric space and prove fixed point theorems. The derived results expand and extend certain well-known results from the research literature and are supported with a non-trivial example. We have applied the fixed point result to find the analytical solution to the integral equation and fractional differential equation. The analytical solution has been supplemented with numerical simulation.

Propagation of waves in fractal spaces

In this study, we study waves propagation in fractal spaces based on two independent variational approaches: the first one is based on the ‘product-like fractal measure’ introduced by Li and Ostoja-Starzewski in their analysis of nonlinear fractal dynamics in anisotropic porous media whereas the second and another is based on fractal calculus, which includes analytical functions with fractal support. The first approach is motivating since it describes physics in anisotropic media characterized by fractal dimensions. The second one is also of particular importance since it offers a new framework to reformulate the laws of physics in spaces with fractional dimensions. In both approaches, the fractional Laplace equation is derived and solved using plausible boundary conditions. This study proves the importance of fractals in wave motion We show that these models are able to describe the propagation of waves in a fractal Hausdorff dimensional space.