Fractal and Fractional

23 pages, 1072 KiB

Open AccessArticle

Bridging Pre-Invex Mappings and Fractional Integrals: A Pathway to Iterative Schemes via Error Boundaries of Maclaurin’s Rule

by Qi Liu, Rukhsar, Muhammad Uzair Awan, Bandar Bin-Mohsin, Muhammad Zakria Javed, Loredana Ciurdariu and Badreddine Meftah

Fractal Fract. 2024, 8(12), 734; https://doi.org/10.3390/fractalfract8120734 - 13 Dec 2024

Abstract

In this paper, we aim to investigate corrected Euler–Maclaurin inequalities involving pre-invex mappings within the framework of fractional calculus. We want to find a number of important results for differentiable pre-invex mappings and Riemann–Liouville (RL) fractional integrals so that we can make more [...] Read more.

In this paper, we aim to investigate corrected Euler–Maclaurin inequalities involving pre-invex mappings within the framework of fractional calculus. We want to find a number of important results for differentiable pre-invex mappings and Riemann–Liouville (RL) fractional integrals so that we can make more accurate error estimates. Additionally, we present examples with graphical illustrations to substantiate our major findings and deduce several special cases under certain conditions. Afterwards, we introduce applications such as the linear combination of means, composite corrected Maclaurin’s rule, modified Bessel mappings, and novel iterative methods for solving nonlinear equations. Full article

(This article belongs to the Section General Mathematics, Analysis)

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21 pages, 366 KiB

Open AccessArticle

Approximate and Exact Controllability for Hilfer Fractional Stochastic Evolution Equations

by Qien Li and Danfeng Luo

Fractal Fract. 2024, 8(12), 733; https://doi.org/10.3390/fractalfract8120733 - 13 Dec 2024

Abstract

This paper investigates the controllability of Hilfer fractional stochastic evolution equations (HFSEEs). Initially, we obtain a conclusion regarding the approximate controllability of HFSEEs by employing the Tikhonov-type regularization method and Schauder′s fixed-point theorem. Additionally, the conditions for the exact controllability of HFSEEs are [...] Read more.

This paper investigates the controllability of Hilfer fractional stochastic evolution equations (HFSEEs). Initially, we obtain a conclusion regarding the approximate controllability of HFSEEs by employing the Tikhonov-type regularization method and Schauder′s fixed-point theorem. Additionally, the conditions for the exact controllability of HFSEEs are explored, utilizing the Mönch′s fixed-point theorem and measure of noncompactness. Finally, the proposed method is validated through an example, thereby demonstrating its effectiveness. Full article

20 pages, 22977 KiB

Open AccessArticle

Introducing an Enhanced Cumulative Size Distribution Model to Unearth the Origins of Macropore Heterogeneity of the Chang-7 Shale Oil Reservoirs in the Ordos Basin

by Ziyi Wang, Lin Dong, Zhijun Jin, Xubin Wang, Jinhua Fu, Xianyang Liu and Rukai Zhu

Fractal Fract. 2024, 8(12), 732; https://doi.org/10.3390/fractalfract8120732 - 13 Dec 2024

Abstract

Characterizing the fractal nature of shale pores is critical for understanding their heterogeneity, which significantly impacts hydrocarbon mobility. A critical technique in this analysis involves the digital evaluation of scanning electron microscopy (SEM) images. While the cumulative size distribution model is a viable [...] Read more.

Characterizing the fractal nature of shale pores is critical for understanding their heterogeneity, which significantly impacts hydrocarbon mobility. A critical technique in this analysis involves the digital evaluation of scanning electron microscopy (SEM) images. While the cumulative size distribution model is a viable method in SEM fractal analysis, its accurate application necessitates exhaustive microimaging of entire sample cross-sections, rendering it impractical. To address this challenge, we propose an innovative approach that develops a universal equation for this model, along with numerical solving methods. This approach expands the model’s applicability, enabling accurate fractal analysis of entire samples using only randomly selected microscopic fields. It enhances the existing framework for fractal modeling and potentially serves as a valuable tool for studying fractal structures. Using this method, we investigated the fractal characteristics of various macropore types across different lithological units in the Chang-7 shale of the Ordos Basin. Our findings indicate that macropore heterogeneity is influenced by variations in pore type composition, their response under compaction, and organic acid dissolution, all of which are driven by differences in lithology. These findings provide new insights into the mechanisms driving macropore heterogeneity from the perspective of lithological and pore-type variations, potentially yielding valuable implications for the exploration and development of Chang-7 shale oil. Full article

(This article belongs to the Special Issue Pore Structure and Fractal Characteristics in Unconventional Oil and Gas Reservoirs)

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19 pages, 46312 KiB

Open AccessArticle

Persistent Homology Analysis of AI-Generated Fractal Patterns: A Mathematical Framework for Evaluating Geometric Authenticity

by Minhyeok Lee and Soyeon Lee

Fractal Fract. 2024, 8(12), 731; https://doi.org/10.3390/fractalfract8120731 - 13 Dec 2024

Abstract

We present a mathematical framework for analyzing fractal patterns in AI-generated images using persistent homology. Given a text-to-image mapping

M : T \to I

, we demonstrate that the persistent homology groups

H_{k} (t)

of sublevel set filtrations [...] Read more.

We present a mathematical framework for analyzing fractal patterns in AI-generated images using persistent homology. Given a text-to-image mapping

M : T \to I

, we demonstrate that the persistent homology groups

H_{k} (t)

of sublevel set filtrations

{f^{- 1} ((- \infty, t])}_{t \in R}

characterize multi-scale geometric structures, where

f : M (p) \to R

is the grayscale intensity function of a generated image. The primary challenge lies in quantifying self-similarity in scales, which we address by analyzing birth–death pairs

(b_{i}, d_{i})

in the persistence diagram

P D (M (p))

. Our contribution extends beyond applying the stability theorem to AI-generated fractals; we establish how the self-similarity inherent in fractal patterns manifests in the persistence diagrams of generated images. We validate our approach using the Stable Diffusion 3.5 model for four fractal categories: ferns, trees, spirals, and crystals. An analysis of guidance scale effects

γ \in [4.0, 8.0]

reveals monotonic relationships between model parameters and topological features. Stability testing confirms robustness under noise perturbations

η \leq 0.2

, with feature count variations

Δ μ_{f} < 0.5

. Our framework provides a foundation for enhancing generative models and evaluating their geometric fidelity in fractal pattern synthesis. Full article

(This article belongs to the Special Issue Advances in Pattern Recognition—Image and Time Series Analyses—through Fractal Geometry and Complexity Theory)

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15 pages, 1845 KiB

Open AccessArticle

Box-Counting Dimension Sequences of Level Sets in AI-Generated Fractals

by Minhyeok Lee and Soyeon Lee

Fractal Fract. 2024, 8(12), 730; https://doi.org/10.3390/fractalfract8120730 - 12 Dec 2024

Abstract

We introduce a mathematical framework to characterize the hierarchical complexity of AI-generated fractals within the finite resolution constraints of digital images. Our method analyzes images produced by text-to-image models at multiple intensity thresholds, employing a discrete level set approach and box-counting dimension estimates. [...] Read more.

We introduce a mathematical framework to characterize the hierarchical complexity of AI-generated fractals within the finite resolution constraints of digital images. Our method analyzes images produced by text-to-image models at multiple intensity thresholds, employing a discrete level set approach and box-counting dimension estimates. By conducting experiments on fractals created with the FLUX model at a resolution of

128 \times 128

, we identify a fully monotonic behavior in the dimension sequences for various box sizes, with inter-scale correlations surpassing 0.95. Pattern-specific dimensional gradients quantify how fractal complexity changes with threshold levels, offering insights into how text-to-image models encode fractal-like geometry through dimensional sequences. Full article

(This article belongs to the Topic AI and Computational Methods for Modelling, Simulations and Optimizing of Advanced Systems: Innovations in Complexity)

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25 pages, 3204 KiB

Open AccessArticle

Fractional Partial Differential Equation Modeling for Solar Cell Charge Dynamics

by Waleed Mohammed Abdelfattah, Ola Ragb, Mohamed Salah, Mohamed S. Matbuly and Mokhtar Mohamed

Fractal Fract. 2024, 8(12), 729; https://doi.org/10.3390/fractalfract8120729 - 12 Dec 2024

Abstract

This paper presents a groundbreaking numerical approach, the fractional differential quadrature method (FDQM), to simulate the complex dynamics of organic polymer solar cells. The method, which leverages polynomial-based differential quadrature and Cardinal sine functions coupled with the Caputo-type fractional derivative, offers a significant [...] Read more.

This paper presents a groundbreaking numerical approach, the fractional differential quadrature method (FDQM), to simulate the complex dynamics of organic polymer solar cells. The method, which leverages polynomial-based differential quadrature and Cardinal sine functions coupled with the Caputo-type fractional derivative, offers a significant improvement in accuracy and efficiency over traditional methods. By employing a block-marching technique, we effectively address the time-dependent nature of the governing equations. The efficacy of the proposed method is validated through rigorous numerical simulations and comparisons with existing analytical and numerical solutions. Each scheme’s computational characteristics are tailored to achieve high accuracy, ensuring an error margin on the order of

10^{- 8}

or less. Additionally, a comprehensive parametric study is conducted to investigate the impact of key parameters on device performance. These parameters include supporting conditions, time evolution, carrier mobilities, charge carrier densities, geminate pair distances, recombination rate constants, and generation efficiency. The findings of this research offer valuable insights for optimizing and enhancing the performance of organic polymer solar cell devices. Full article

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)

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21 pages, 483 KiB

Open AccessArticle

New Inequalities for GA–h Convex Functions via Generalized Fractional Integral Operators with Applications to Entropy and Mean Inequalities

by Asfand Fahad, Zammad Ali, Shigeru Furuichi, Saad Ihsan Butt, Ayesha and Yuanheng Wang

Fractal Fract. 2024, 8(12), 728; https://doi.org/10.3390/fractalfract8120728 - 12 Dec 2024

Abstract

We prove the inequalities of the weighted Hermite–Hadamard type the and Hermite–Hadamard–Mercer type for an extremely rich class of geometrically arithmetically-h-convex functions (GA-h-CFs) via generalized Hadamard–Fractional integral operators (HFIOs). The two generalized fractional integral operators (FIOs) are Hadamard proportional [...] Read more.

We prove the inequalities of the weighted Hermite–Hadamard type the and Hermite–Hadamard–Mercer type for an extremely rich class of geometrically arithmetically-h-convex functions (GA-h-CFs) via generalized Hadamard–Fractional integral operators (HFIOs). The two generalized fractional integral operators (FIOs) are Hadamard proportional fractional integral operators (HPFIOs) and Hadamard k-fractional integral operators (HKFIOs). Moreover, we also present the results for subclasses of GA-h-CFs and show that the inequalities proved in this paper unify the results from the recent related literature. Furthermore, we compare the two generalizations in view of the fractional operator parameters that contribute to the generalizations of the results and assess the better approximation via graphical tools. Finally, we present applications of the new inequalities via HPFIOs and HKFIOs by establishing interpolation relations between arithmetic mean and geometric mean and by proving the new upper bounds for the Tsallis relative operator entropy. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)

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21 pages, 506 KiB

Open AccessArticle

Study on Controllability for Ψ-Hilfer Fractional Stochastic Differential Equations

by Abdur Raheem, Fahad M. Alamrani, Javed Akhtar, Adel Alatawi, Esmail Alshaban, Areefa Khatoon and Faizan Ahmad Khan

Fractal Fract. 2024, 8(12), 727; https://doi.org/10.3390/fractalfract8120727 - 11 Dec 2024

Abstract

The goal of this paper is to study the existence of a mild solution and controllability for a class of neutral stochastic differential equations (SDEs) involving the

Ψ

-Hilfer fractional derivatives, a generalization of the well-known Riemann–Liouville fractional derivative using almost sectorial operators. [...] Read more.

The goal of this paper is to study the existence of a mild solution and controllability for a class of neutral stochastic differential equations (SDEs) involving the

Ψ

-Hilfer fractional derivatives, a generalization of the well-known Riemann–Liouville fractional derivative using almost sectorial operators. Sufficient conditions for controllability are established using the notion of measure of noncompactness (MNC) and the Mönch fixed-point theorem. An example is given to illustrate the abstract findings. Full article

(This article belongs to the Special Issue Advances in Fractional Dynamical and Control Problems with Nonlocal Operators)

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20 pages, 620 KiB

Open AccessArticle

Self-Convolution and Its Invariant Properties for the Kernel Function of the Aortic Fractal Operator

by Chaoqian Luo, Yajun Yin, Gang Peng, Tianyi Zhou, Xiaobin Yu and Dongan Li

Fractal Fract. 2024, 8(12), 726; https://doi.org/10.3390/fractalfract8120726 - 11 Dec 2024

Abstract

In this paper, we explore the self-convolution of the kernel function of the aortic fractal operator. Previous research has established a model named “physical fractal”, and confirmed that the hemodynamics of the aorta can be inscribed by a fractal operator and that the [...] Read more.

In this paper, we explore the self-convolution of the kernel function of the aortic fractal operator. Previous research has established a model named “physical fractal”, and confirmed that the hemodynamics of the aorta can be inscribed by a fractal operator and that the dominant component of the kernel function of the fractal operator is a weighted first-order Bessel function. These studies primarily focus on solving the fractal operator kernel function and examining the overall properties of the physical fractal. As we began to investigate the internal structure of physical fractals, we discovered that studying the powers of fractal operators is a necessary step. In this paper, we introduce the concept of kernel function self-convolution, establish its connection with the power of the fractal operator, and derive a series of invariant properties for the self-convolution of the aortic operator kernel function. These invariant properties, in turn, are deeply and intrinsically related to the invariant properties of the Bessel functions. The research findings of this paper enrich hemodynamics and biomechanics in physical fractal space and extend the scope of using fractal operators to characterize the dynamics of living organisms. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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13 pages, 1858 KiB

Open AccessArticle

Collinear Fractals and Bandt’s Conjecture

by Bernat Espigule, David Juher and Joan Saldaña

Fractal Fract. 2024, 8(12), 725; https://doi.org/10.3390/fractalfract8120725 - 11 Dec 2024

Abstract

For a complex parameter c outside the unit disk and an integer

n \geq 2

, we examine the n-ary collinear fractal

E (c, n)

, defined as the attractor of the iterated function system [...] Read more.

For a complex parameter c outside the unit disk and an integer

n \geq 2

, we examine the n-ary collinear fractal

E (c, n)

, defined as the attractor of the iterated function system

{f_{k} : C ⟶ C}_{k = 1}^{n}

, where

f_{k} (z) : = 1 + n - 2 k + c^{- 1} z

. We investigate some topological features of the connectedness locus

M_{n}

defined as the set of those c for which

E (c, n)

is connected. In particular, we provide a detailed answer to an open question posed by Calegari, Koch, and Walker in 2017. We also extend and refine the technique of the “covering property” by Solomyak and Xu to any

n \geq 2

. We use it to show that a nontrivial portion of

M_{n}

is regular closed. When

n \geq 21

, we enhance this result by showing that in fact, the whole

M_{n} ∖ R

lies within the closure of its interior, thus proving that the generalized Bandt’s conjecture is true. Full article

(This article belongs to the Section Geometry)

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29 pages, 416 KiB

Open AccessArticle

New Studies for Dynamic Programming and Fractional Differential Equations in Partial Modular b-Metric Spaces

by Abdurrahman Büyükkaya, Dilek Kesik, Ülkü Yeşil and Mahpeyker Öztürk

Fractal Fract. 2024, 8(12), 724; https://doi.org/10.3390/fractalfract8120724 - 9 Dec 2024

Abstract

This study explores innovative insights into the realms of dynamic programming and fractional differential equations, situated explicitly within the framework of partial modular b-metric spaces enriched with a binary relation

R

, proposing a novel definition for a generalized

ℷ_{C}

-type [...] Read more.

This study explores innovative insights into the realms of dynamic programming and fractional differential equations, situated explicitly within the framework of partial modular b-metric spaces enriched with a binary relation

R

, proposing a novel definition for a generalized

ℷ_{C}

-type Suzuki

R

-contraction specific to these spaces. By doing so, we pave the way for a range of relation-theoretical common fixed-point theorems, highlighting the versatility of our approach. To illustrate the practical relevance of our findings, we present a compelling example. Ultimately, this work aims to enrich the existing academic discourse and stimulate further research and practical applications within the field. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)

9 pages, 678 KiB

Open AccessArticle

A Note on Exact Results for Burgers-like Equations Involving Laguerre Derivatives

by Roberto Garra, Giuseppe Dattoli and Riccardo Droghei

Fractal Fract. 2024, 8(12), 723; https://doi.org/10.3390/fractalfract8120723 - 9 Dec 2024

Abstract

In this work, we consider some Burgers-like equations involving Laguerre derivatives and demonstrate that it is possible to construct specific exact solutions using separation of variables. We prove that a general scheme exists for constructing exact solutions for these Burgers-like equations and extending [...] Read more.

In this work, we consider some Burgers-like equations involving Laguerre derivatives and demonstrate that it is possible to construct specific exact solutions using separation of variables. We prove that a general scheme exists for constructing exact solutions for these Burgers-like equations and extending to more general cases, including nonlinear time-fractional equations. Exact solutions can also be obtained for KDV-like equations involving Laguerre derivatives. We finally consider a particular class of Burgers equations with variable coefficients whose solutions can be obtained similarly. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

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12 pages, 549 KiB

Open AccessArticle

Generalized Dimensions of Self-Affine Sets with Overlaps

by Guanzhong Ma, Jun Luo and Xiao Zhou

Fractal Fract. 2024, 8(12), 722; https://doi.org/10.3390/fractalfract8120722 - 6 Dec 2024

Abstract

Two decades ago, Ngai and Wang introduced a well-known finite type condition (FTC) on the self-similar iterated function system (IFS) with overlaps and used it to calculate the Hausdorff dimension of self-similar sets. In this paper, inspired by Ngai and Wang’s idea, we [...] Read more.

Two decades ago, Ngai and Wang introduced a well-known finite type condition (FTC) on the self-similar iterated function system (IFS) with overlaps and used it to calculate the Hausdorff dimension of self-similar sets. In this paper, inspired by Ngai and Wang’s idea, we define a new FTC on self-affine IFS and obtain an analogous formula on the generalized dimensions of self-affine sets. The generalized dimensions raised by He and Lau are used to estimate the Hausdorff dimension of self-affine sets. Full article

(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)

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22 pages, 3483 KiB

Open AccessArticle

A Flexible Framework for Decentralized Composite Optimization with Compressed Communication

by Zhongyi Chang, Zhen Zhang, Shaofu Yang and Jinde Cao

Fractal Fract. 2024, 8(12), 721; https://doi.org/10.3390/fractalfract8120721 - 5 Dec 2024

Abstract

This paper addresses the decentralized composite optimization problem, where a network of agents cooperatively minimize the sum of their local objective functions with non-differentiable terms. We propose a novel communication-efficient decentralized ADMM framework, termed as CE-DADMM, by combining the ADMM framework with the [...] Read more.

This paper addresses the decentralized composite optimization problem, where a network of agents cooperatively minimize the sum of their local objective functions with non-differentiable terms. We propose a novel communication-efficient decentralized ADMM framework, termed as CE-DADMM, by combining the ADMM framework with the three-point compressed (3PC) communication mechanism. This framework not only covers existing mainstream communication-efficient algorithms but also introduces a series of new algorithms. One of the key features of the CE-DADMM framework is its flexibility, allowing it to adapt to different communication and computation needs, balancing communication efficiency and computational overhead. Notably, when employing quasi-Newton updates, CE-DADMM becomes the first communication-efficient second-order algorithm based on compression that can efficiently handle composite optimization problems. Theoretical analysis shows that, even in the presence of compression errors, the proposed algorithm maintains exact linear convergence when the local objective functions are strongly convex. Finally, numerical experiments demonstrate the algorithm’s impressive communication efficiency. Full article

(This article belongs to the Section Optimization, Big Data, and AI/ML)

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25 pages, 6353 KiB

Open AccessArticle

Fractional-Order Controller for the Course Tracking of Underactuated Surface Vessels Based on Dynamic Neural Fuzzy Model

by Guangyu Li, Yanxin Li, Xiang Li, Mutong Liu, Xuesong Zhang and Hua Jin

Fractal Fract. 2024, 8(12), 720; https://doi.org/10.3390/fractalfract8120720 - 5 Dec 2024

Abstract

Aiming at the uncertainty problem caused by the time-varying modeling parameters associated with ship speed in the course tracking control of underactuated surface vessels (USVs), this paper proposes a control algorithm based on the dynamic neural fuzzy model (DNFM). The DNFM simultaneously adjusts [...] Read more.

Aiming at the uncertainty problem caused by the time-varying modeling parameters associated with ship speed in the course tracking control of underactuated surface vessels (USVs), this paper proposes a control algorithm based on the dynamic neural fuzzy model (DNFM). The DNFM simultaneously adjusts the structure and parameters during learning and fully approximates the inverse dynamics of ships. Online identification and modeling lays the model foundation for ship motion control. The trained DNFM, serving as an inverse controller, is connected in parallel with the fractional-order PI^λD^μ controller to be used for the tracking control of the ship’s course. Moreover, the weights of the model can be further adjusted during the course tracking. Taking the actual ship data of a 5446 TEU large container ship, simulation experiments are conducted, respectively, for course tracking, course tracking under wind and wave interferences, and comparison with five different controllers. This proposed controller can overcome the influence of the uncertainty of modeling parameters, tracking the desired course quickly and effectively. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Systems to Automatic Control)

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23 pages, 11122 KiB

Open AccessArticle

Numerical Investigation of Water Transport and Effective Electrical Conductivity in Perforation of Gas Diffusion Layer Using Lattice Boltzmann Method

by Jae Yong Cho, Hee Min Lee, Muhammad Nasir Bashir and Joon Sang Lee

Fractal Fract. 2024, 8(12), 719; https://doi.org/10.3390/fractalfract8120719 - 5 Dec 2024

Abstract

In polymer electrolyte membrane fuel cells, the gas diffusion layer (GDL) is composed of porous media and serves a critical role as a mass transport layer, facilitating reactant gas diffusion, removal of water generated in the catalyst layer, and electron transport. Artificial spacings [...] Read more.

In polymer electrolyte membrane fuel cells, the gas diffusion layer (GDL) is composed of porous media and serves a critical role as a mass transport layer, facilitating reactant gas diffusion, removal of water generated in the catalyst layer, and electron transport. Artificial spacings known as perforations can be introduced to improve water management within this mass transport system. However, the impact of these perforations on the effective electrical conductivity has not been adequately studied. This study employs numerical methods to investigate water management and effective electrical conductivity in the presence of perforations, aiming to provide indicators for optimal design. The pseudopotential lattice Boltzmann method is utilized, which is particularly advantageous for modeling two-phase flow and electron transport in complex geometries. Using this numerical approach, we analyze water penetration in GDL structures and effective electrical conductivity based on electric potential fields focusing on geometric parameters such as the perforation size. Our results demonstrate a relationship between water management efficiency and effective electrical conductivity, suggesting the existence of an optimal perforation diameter. Moreover, when there is a water-induced penetration pattern due to the perforated structure, both the effective electrical conductivity and water management are enhanced at a lower porosity of the GDL structure. Full article

(This article belongs to the Topic Modeling, Stability, and Control of Dynamic Systems and Their Applications)

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15 pages, 1774 KiB

Open AccessArticle

A Fourier Series Technique for Approximate Solutions of Modified Anomalous Time-Fractional Sub-Diffusion Equations

by Samad Kheybari, Farzaneh Alizadeh, Mohammad Taghi Darvishi and Kamyar Hosseini

Fractal Fract. 2024, 8(12), 718; https://doi.org/10.3390/fractalfract8120718 - 5 Dec 2024

Abstract

This article presents an innovative approximating technique for addressing modified anomalous time-fractional sub-diffusion equations (MAFSDEs) of the Caputo type. These equations generalize classical diffusion equations, which involve fractional derivatives with respect to time, capturing the non-local and history-dependent behavior typical in sub-diffusion processes. [...] Read more.

This article presents an innovative approximating technique for addressing modified anomalous time-fractional sub-diffusion equations (MAFSDEs) of the Caputo type. These equations generalize classical diffusion equations, which involve fractional derivatives with respect to time, capturing the non-local and history-dependent behavior typical in sub-diffusion processes. In such a model, the particle transports slower than in a standard diffusion, often due to obstacles or memory effects in the medium. The core of the proposed technique involves transforming the original problem into a family of independent fractional-order ordinary differential equations (FODEs). This transformation is achieved using the Fourier expansion method. Each of these resulting FODEs is defined under initial value conditions which are derived from the initial condition of the original problem. To solve them, for each resulting FODE, some secondary initial value problems are introduced. By solving these secondary initial value problems, some particular solutions are obtained and then we combine them linearly in an optimal manner. This combination is essential to estimate the solution of the original problem. To evaluate the accuracy and effectiveness of the proposed scheme, we conduct a various test problem. For each problem, we analyze the observed convergence order indicators and compare them with those from other methods. Our comparison demonstrates that the proposed technique provides enhanced precision and reliability in respect with the current numerical approaches in the literature. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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17 pages, 1782 KiB

Open AccessArticle

Subsurface Geological Profile Interpolation Using a Fractional Kriging Method Enhanced by Random Forest Regression

by Qile Ding, Yiren Wang, Yu Zheng, Fengyang Wang, Shudong Zhou, Donghui Pan, Yuchun Xiong and Yi Zhang

Fractal Fract. 2024, 8(12), 717; https://doi.org/10.3390/fractalfract8120717 - 5 Dec 2024

Abstract

Analyzing geological profiles is of great importance for various applications such as natural resource management, environmental assessment, and mining engineering projects. This study presents a novel geostatistical approach for subsurface geological profile interpolation using a fractional kriging method enhanced by random forest regression. [...] Read more.

Analyzing geological profiles is of great importance for various applications such as natural resource management, environmental assessment, and mining engineering projects. This study presents a novel geostatistical approach for subsurface geological profile interpolation using a fractional kriging method enhanced by random forest regression. Using bedrock elevation data from 49 boreholes in a study area in southeast China, we first use random forest regression to predict and optimize variogram parameters. We then use the fractional kriging method to interpolate the data and analyze the variability. We also compare the proposed model with traditional methods, including linear regression, K-nearest neighbors, ordinary kriging, and fractional kriging, using cross-validation metrics. The results indicate that the proposed model reduces prediction errors and enhances spatial prediction reliability compared to other models. The MSE of the proposed model is 25% lower than that of ordinary kriging and 10% lower than that of fractional kriging. In addition, the execution time of the proposed model is slightly higher than other models. The findings suggest that the proposed model effectively captures complex subsurface spatial relationships, offering a reliable and precise solution for performing spatial interpolation tasks. Full article

(This article belongs to the Section Engineering)

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16 pages, 1048 KiB

Open AccessArticle

Rich Dynamics Caused by a Fractional Diffusion Operator in Nonchaotic Rulkov Maps

by Huanqin Hu, Mingshu Peng and Yingfei Qi

Fractal Fract. 2024, 8(12), 716; https://doi.org/10.3390/fractalfract8120716 - 3 Dec 2024

Abstract

There are few works about Neimark–Sacker bifurcating analysis on discrete dynamical systems with linear diffusion and delayed coupling under periodic/Neumann-boundary conditions. In this paper, we build up the framework for Neimark–Sacker bifurcations caused by Turing instability on high-dimensional discrete-time dynamical systems with symmetrical [...] Read more.

There are few works about Neimark–Sacker bifurcating analysis on discrete dynamical systems with linear diffusion and delayed coupling under periodic/Neumann-boundary conditions. In this paper, we build up the framework for Neimark–Sacker bifurcations caused by Turing instability on high-dimensional discrete-time dynamical systems with symmetrical property in the linearized system. The fractional diffusion operator in higher-dimensional discrete dynamical systems is introduced and regular/chaotic Turing patterns are discovered by the computation of the largest Lyapunov exponents. Full article

(This article belongs to the Special Issue Fractional Diffusion, Multistability and Control in Complex Systems)

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