Fractal and Fractional

In this paper, we develop an analytical approximate solution for the nonlinear time-fractional Fisher’s equation using a right starting space function and a unique analytic-numeric technique referred to as the Laplace residual power series approach. The generalized Taylor’s formula and the Laplace transform operator are coupled in the aforementioned method, where the coefficients, obtained through fractional expansion in the Laplace space, are determined by applying the limit concept. In order to validate and illustrate the theoretical methodology of the LRPS technique, as well as to show its effectiveness, adaptability, and superiority in solving various types of nonlinear time and space fractional differential equations, numerical experiments are generated. The obtained analytical solutions are compatible with the precise solutions and concur with those proposed by the other approaches. The outcomes show that the Laplace residual power series strategy is incredibly successful, straightforward to implement, and well suited for handling the complexity of nonlinear problems. Full article

Viscoelastic materials are commonly used in civil engineering, biomedical sciences, and polymers, where understanding their creep and relaxation behaviors is essential for predicting long-term performance. This paper introduces an operator-based method for modeling viscoelastic materials, providing a unified framework to describe both creep and relaxation functions. The method utilizes stiffness and compliance operators, offering a systematic approach for analyzing viscoelastic problems. The operator-based method enhances the mathematical duality between the creep and relaxation functions, providing greater physical intuition and understanding of time-dependent material behavior. It directly reflects the intrinsic properties of materials, independent of input and output conditions. The method is extended to dynamic problems, with complex modulus and compliance derived through operator representations. The fractal tree model, with its constant loss factor across the frequency spectrum, demonstrates potential engineering applications. By incorporating a damage-based variable coefficient, the model now also accounts for the accelerated creep phase of rocks, capturing damage evolution under prolonged loading. While promising, the current method is limited to one-dimensional problems, and future research will aim to extend it to three-dimensional cases, integrate experimental validation, and explore broader applications. Full article

This article investigates non-fragile synchronization control and circuit implementation for incommensurate fractional-order (IFO) chaotic neural networks with parameter uncertainties. In this paper, we explore three aspects of the research challenges, i.e., theoretical limitations of uncertain IFO systems, the fragility of the synchronization controller, and the lack of circuit implementation. First, we establish an IFO chaotic neural network model incorporating parametric uncertainties, extending beyond conventional commensurate-order architectures. Second, a novel, non-fragile state-error feedback controller is designed. Through the formulation of FO Lyapunov functions and the application of inequality scaling techniques, sufficient conditions for asymptotic synchronization of master–slave systems are rigorously derived via the multi-order fractional comparison principle. Third, an analog circuit implementation scheme utilizing FO impedance units is developed to experimentally validate synchronization efficacy and accurately replicate the system’s dynamic behavior. Numerical simulations and circuit experiments substantiate the theoretical findings, demonstrating both robustness against parameter perturbations and the feasibility of circuit realization. Full article

To discuss the fractional order heat conduction based on Youssef’s model, a new mathematical model of a thermoelastic annular cylinder with variable thermal conductivity will be constructed in this work. The Moore–Gibson– Thompson theorem of generalized thermoelasticity will be considered and the governing equations will be derived in dimensionless forms. The Laplace transform technique will be used for a one-dimensional thermoelastic, isotropic, and homogeneous annular cylinder in which the interior surrounded surface is thermally shocked and there is an axial traction-free environment, while the outer surrounded surface has neither heat increment nor cubical deformation. The numerical results will be computed for the Laplace transform inversions by using Tzou’s iteration approach. The distributions of the cubical deformation, invariant average stress, axial stress, and temperature increment will be represented in figures to analyze and discuss. The results show that the fractional-order and variable thermal conductivity parameters have significant effects on all the studied functions. The physical behaviour of the thermal conductivity is closely aligned with the classification of thermal conductivity into weak, normal, and strong categories, which is essential. Full article

In this work, we explore the impact of integral operators such as the Libera and Alexander operators on specific families of analytic functions introduced in the literature and find some of their remarkable results. Using techniques from differential subordination and convolution theory, we establish results concerning the radius of convexity and convolution properties for these function classes. Additionally, we investigate how these integral operators influence the geometric properties of functions in $\mathcal{BS}\left(\mu \right)$ and $\mathcal{KS}\left(\mu \right)$, leading to new insights into their structural behavior. Full article

This study investigates fuzzy fixed points and fuzzy best proximity points for fuzzy mappings within the newly introduced framework $\theta$-fuzzy metric spaces that extends various existing fuzzy metric spaces. We establish novel fixed-point and best proximity-point theorems for both single-valued and multivalued mappings, thereby broadening the scope of fuzzy analysis. Furthermerefore, we have for aore, we apply one of our key results to derive conditions, ensuring the existence and uniqueness of a solution to Hadamard $\Psi$-Caputo tempered fuzzy fractional differential equations, particularly in the context of the SIR dynamics model. These theoretical advancements are expected to open new avenues for research in fuzzy fixed-point theory and its applications to hybrid models within $\theta$-fuzzy metric spaces. Full article

This article presents a method for the approximate calculation of fractional Caputo derivatives, including a crucial aspect of the ability to handle arbitrary—even variable—terminals and order. The proposed method involves rearranging the fractional operator as a series of higher-order derivatives considered at a specific point. We demonstrate the effect of the number of terms included in the series expansion on the solution accuracy and error analysis. The advantage of the method is its simplicity and ease of implementation. Additionally, the method allows for a quick estimation of the fractional derivative by using a few first terms of the expansion. The elaborated algorithm is tested against a comprehensive series of illustrative examples, providing very good agreement with the exact/reference solutions. Furthermore, the application of the proposed method to fractional boundary/initial value problems is included. Full article

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. Full article

To address the limitations of traditional tumor diagnostic methods in image feature extraction and model generalization, this study innovatively proposes a synergistic diagnostic model that integrates fractional Fourier transform (FrFT) and error back-propagation (BP) neural networks. The model leverages the time–frequency analysis capability of FrFT and incorporates the fractal characteristics observed during tumor proliferation, effectively enhancing multi-scale feature extraction and representation. Experimental results show that the proposed model achieves an accuracy of 93.177% in classifying benign and malignant tumors, outperforming the support vector machine (SVM) method. The integration of FrFT improves feature distinguishability and reduces dependence on manual extraction. This study not only represents a breakthrough in tumor diagnostic technology but also paves new avenues for the application of fractional calculus and fractal geometry in medical image analysis. The findings show great potential for clinical application and future development. Full article

The successful exploration and development of shale oil in the clay-rich Gulong shale have sparked increased research into the influence of clay minerals on shale reservoirs. However, compared to chlorite in sandstones, limited studies have focused on the occurrence of chlorite in continental shales and its effects on shale reservoir properties. This study offers a comprehensive analysis of chlorite in Gulong shale samples from three wells at different diagenetic stages. Four primary chlorite occurrences are identified in the Gulong shale: Type I, which is chlorite filling dissolved pores in carbonate; Type II, which is isolated chlorite; Type III, which is chlorite filling organic matter; and Type IV, which is chlorite filling authigenic microquartz. Types I and III chlorites exhibit higher porosity, offering more storage space for shale reservoirs. Chlorites of Types I, III, and IV, filled with other substances, display higher fractal dimensions, indicating more complex pore structures. These complex pores are favorable for oil adsorption but hinder oil seepage. The processes of organic matter expulsion and dissolution, which intensify with increasing diagenesis, promote the development of Types I and III chlorites, thereby positively influencing the shale reservoir porosity of Gulong shale. This study underscores the influence of chlorite occurrences on shale reservoir properties, providing valuable insights for the future exploration and development of shale oil and gas. Full article

Pore structure plays a critical role in evaluating shale “sweet spots”. Compared to marine shale, lacustrine shale has more diverse lithofacies types and greater heterogeneity in pore structure due to frequently changing environmental conditions. Using methods such as mercury intrusion porosimetry (MIP), field emission scanning electron microscopy (FE-SEM), nuclear magnetic resonance (NMR), and X-ray diffraction (XRD), this study investigates the micropore structures and heterogeneity of different lithofacies in the Jurassic Dongyuemiao Member lacustrine shale. Image processing and multifractal theory were employed to identify the controlling factors of pore structure heterogeneity. The key findings are as follows. (1) Based on mineral content and laminae types, the lithofacies types of Dongyuemiao lacustrine shale are classified into four types: shell–laminae mixed shale (SLMS), silty–laminae clay shale (SLCS), clast–laminae clay shale (CLCS), and clay shale (CS). (2) Based on genesis, shale reservoirs’ pore and permeability space are categorized into inorganic pores, organic pores, and micro-fractures. Inorganic pores consist of inter-particle pores and intra-particle pores. Pore size distribution curves for all four lithofacies exhibit two main peaks, with pore sizes concentrated in the ranges of 2–10 nm and 50–80 nm. Mesopores and macropores dominate, accounting for over 80% of the total pore volume. Mesopores are most developed in CLCS, representing 56.3%. (3) Quartz content is positively correlated with the multifractal dimension, while clay content shows a negative correlation. Higher quartz content, coupled with lower clay content, weakens pore structure heterogeneity. A negative correlation exists between total organic carbon (TOC) and the multifractal dimension, indicating that higher organic matter content enhances organic pore development and increases microscopic heterogeneity. (4) Porosity heterogeneity in SLMS is effectively characterized by D₀-D_max, while in the other three lithofacies, it is characterized by D_min-D₀. Permeability across all lithofacies correlates with D₀-D_max. In CS, SLMS, and SLCS, permeability is positively correlated with D₀-D_max, with higher values indicating greater permeability heterogeneity. In CLCS, permeability is negatively correlated with D₀-D_max, such that lower values reflect stronger heterogeneity. Full article

Fractional variable order systems with unusual dynamics in the order are a little-studied topic. In this study, we present three examples of very simple fractional systems with unusual dynamics in the derivative order. These cases involve different approaches to define the variable-order dynamics: (1) an integer-order differential equation that includes the state variable, (2) a differential equation that incorporates the state variable and features both integer- and fractional-order derivatives, and (3) fractional variable-order differential equations nested in the derivative orders. We prove a result that shows how the extended recursion of the last case is generalized. These examples illustrate the richness that simple dynamical systems can reveal through the order of their derivatives. Full article

The pore structure of reservoir rocks was a crucial factor affecting hydrocarbon production. Accurately characterized the micropore structure of different types of rock reservoirs was of great significance for unconventional natural gas exploration. In this study, multiple observation methods (field emission scanning electron microscope (FESEM) and low-field nuclear magnetic resonance (LFNMR)) and physical tests (mercury injection capillary pressure (MICP)) were employed, and double logarithmic plots for fractal fitting were illustrated. The fractal dimension of 15 samples was calculated using fractal theory to systematically investigate the pore–fracture structure and fractal characteristics of hydrocarbon source rock (limestone, mudstone, and sandstone) samples from the Late Carboniferous Taiyuan Formation in the Huainan coalfield. MICP experiments revealed that sandstone reservoirs had larger and more uniformly distributed pore throats compared to mudstone and limestone, exhibiting superior connectivity and permeability. The T₂ spectrum characteristic maps obtained using LFNMR were also consistent with the pore distribution patterns derived from MICP experiments, particularly showed that sandstone types exhibited excellent signal intensity across different relaxation time periods and had a broader T₂ spectrum width, which fully indicated that sandstone types possess superior pore structures and higher connectivity. FESEM experiments demonstrated that sandstone pores were highly developed and uniform, with sandstone fractures dominated by large fractures above the micrometer scale. Meanwhile, the FESEM fractal dimension results indicated that sandstone exhibits good fractal characteristics, validating its certain oil storage capacity. Furthermore, the FESEM fractal dimension exhibited a good correlation with the porosity and permeability of the hydrocarbon source rock reservoirs, suggesting that the FESEM fractal dimension can serve as an important parameter for evaluating the physical properties of hydrocarbon source rock reservoirs. This study enriched the basic geological theories for unconventional natural gas exploration in deep coal-bearing strata in the Huainan coalfield. Full article

High-tech industries are of strategic importance to the national economy, and Beijing has been designated as a science and technology innovation center by the State Council. Accurate analysis of its added value is crucial for technological development. While recent data enhance prediction accuracy, its limited volume poses challenges. The cumulative grey Lotka–Volterra model and grey differential dynamic multivariate model address this by leveraging short-term data effectively. This study applies these two models to analyze influencing factors and predict Beijing’s high-tech industry growth. Results show a competitive relationship with four systems, lacking synergy. In the next five years, a mutually beneficial trend is expected. The Mean Absolute Percentage Error (MAPE) remains within 10%, confirming the model’s reliability. Full article

In this article, we study a fractional lower-order differential equation, $-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)+\mathrm{a}\left(\xi \right){\rm Y}\left(\xi \right)=\mathrm{y}\left(\xi \right),\xi \in (0,1),\alpha \in (1,2),$ with a Dirichlet-type boundary condition, where $\mathrm{a}\left(\xi \right)\in L^1[0,1]$ permits singularity. When the coefficient of perturbation term $\mathrm{a}\left(\xi \right)$ is continuous on $[0,1]$, Graef et al. derived the associated Green’s function under certain conditions on $\mathrm{a}$, but failed to obtain its positivity. We obtain some positive properties of the Green’s function of this problem under certain conditions. The results will fill the gap in the above literature. As for application of the theoretical results, a singular fractional differential equation with a perturbation term is considered. The unique positive solution is obtained by using the fixed point theorem, and an iterative sequence is established to approximate it. In addition, a semipositone fractional differential equation, $-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)=\mu \mathcal{F}(\xi ,{\rm Y}\left(\xi \right)),\xi \in (0,1),\alpha \in (1,2),$ is also considered. The existence of positive solutions is determined under a more general condition, $\mathcal{F}(\xi ,\mathrm{x})\ge -\mathrm{b}\left(\xi \right)\mathrm{x}-\mathrm{e}\left(\xi \right)$, where $\mathrm{b}\left(\xi \right),\mathrm{e}\left(\xi \right)\in L^1[0,1]$ are non-negative functions. Relevant examples are listed to manifest the theoretical results. Full article

Bayoud disease, caused by Fusarium oxysporum f. sp. albedinis, is a major threat to date palm trees. It leads to lower crop yields, financial losses, and decreased biodiversity. The complexity of the disease presents challenges to effective disease management. This study introduces a mathematical model comprising six compartments for palm trees: susceptible trees, resistant varieties, exposed trees, infected trees, isolated trees under treatment, and recovered trees, along with a contaminant water compartment. The model emphasizes the role of resistant varieties, contamination of irrigation water, and the treatment of infected trees in disease control. Theoretical analyses guarantee positivity, boundedness, and the existence of a unique solution. The existence of equilibrium points (disease-free and endemic) and the reproduction number (${\mathcal{R}}_0$) of the model are calculated analytically and validated through numerical simulations. Stability analysis at disease-free and endemic equilibrium points is conducted in terms of ${\mathcal{R}}_0$. Sensitivity analysis identifies key parameters influencing disease dynamics and is helpful to identify the potential control parameters. An optimal control problem is formulated to minimize infection spread and associated costs via preventive isolation and treatments, irrigation water treatment, and the promotion of resistant varieties. Numerical simulations demonstrate the efficacy of these strategies, highlighting the potential of resistant varieties and treatment measures in reducing infection rates and enhancing tree health. This research offers valuable insights into sustainable Bayoud disease management, underscoring the importance of mathematical modeling in addressing agricultural challenges. Full article

In this research article, we investigate a three-dimensional dynamical system governed by fractal-fractional-order evolution differential equations subject to terminal boundary conditions. We derive existence and uniqueness results using Schaefer’s and Banach’s fixed-point theorems, respectively. Additionally, the Hyers–Ulam stability approach is employed to analyze the system’s stability. We employ vector terminology for the proposed problem to make the analysis simple. To illustrate the practical relevance of our findings, we apply the derived results to a numerical example and graphically illustrate the solution for different fractal-fractional orders, emphasizing the effect of the derivative’s order on system behavior. Full article

This paper presents an advanced simulation-based investigation of tumor growth and chemical diffusion in biological tissues, using $\vartheta$-fractional stochastic integral equations. Based on the theoretical framework developed in [Fractal Fract. 2025, 9(1), 7], we develop an innovative computational model to explore the practical applications of these equations in the biological field. The model focuses on providing new insights into the dynamic interaction between stochastic effects of a fractional nature and complex biological tissue environments, contributing to a deeper understanding of the mechanisms of chemical diffusion within tissues and tumor growth under different conditions. The paper details the numerical techniques used to solve the $\vartheta$-fractional stochastic integral equations, focusing on the stability and accuracy of the solutions, while demonstrating their ability to accurately and effectively capture key biological phenomena. Through extensive computational experiments, the model demonstrates its ability to replicate realistic tumor growth patterns and complex chemical transport dynamics, providing a powerful and flexible tool for understanding tumor behavior and interaction with potential therapies. These results represent an important step toward improving biological models and enhancing biomedical applications, particularly in the areas of targeted drug design and analysis of tumor dynamics under chemotherapeutic influence. Full article

Fractal analysis was used to characterize the organic matter nanopore structure in tectonically deformed shales, providing insights into the heterogeneity and complexity of the pore network. Shale samples from different tectonic deformation styles (undeformed, brittle deformed, and ductile deformed) in the Lower Cambrian Niutitang Formation in western Hunan, South China, were collected. By comprehensively applying techniques such as low-temperature gaseous (CO₂ and N₂) adsorption (LTGA), scanning electron microscopy (SEM), and ImageJ analysis, we accurately obtained key parameters of the pore structure. The results show ductile deformation reduces fractal dimension (D_M) by ~0.2 compared to brittle deformed shale, reflecting the homogenization of organic nanopore structures. Brittle deformation leads to a more complex pore network, while ductile deformation reduces the complexity of the organic nanopore structure. The fractal dimensions are affected by various factors, with micropore development being crucial for undeformed shale, clay and pore length–width ratio dominating in brittle deformed shale, and all-scale pores being key for ductile deformed shale. This study provides the first comparative analysis of fractal dimensions across undeformed, brittle deformed, and ductile deformed shales, revealing distinct pore structure modifications linked to deformation styles. These findings not only enhance our understanding of the influence mechanism of tectonic deformation on shale pore structure and fractal characteristics but also provide a theoretical basis for optimizing shale gas exploration and production strategies. These findings offer a framework for predicting gas storage and flow dynamics in tectonically complex shale reservoirs. For instance, in areas with different tectonic deformation styles, we can better evaluate the gas storage capacity and production potential of shale reservoirs according to the obtained fractal characteristics, which is of great significance for efficient shale gas development. Full article