Fractal and Fractional

This study investigated the fracture behavior of concrete beams with recycled coarse aggregate (RCA) and recycled fine aggregate (RFA) using the box-counting method to measure crack fractal dimensions under load. Beams with RCA showed higher fractal dimensions due to RCA’s lower elastic moduli and compressive strengths, resulting in reduced deformation resistance, ductility, and more late-stage crack propagation. A direct proportional relationship existed between RCA/RFA replacement ratios and crack fractal dimensions. Second-order and third-order polynomial trend surface-fitting techniques were applied to examine the complex relationships among RFA/RCA dosage, applied load, and crack fractal dimension. The results indicated that the RFA dosage had a negative quadratic influence, while load had a positive linear effect, with dosage impact increasing with load. A second-order functional relationship was found between mid-span deflection and crack fractal dimension, reflecting nonlinear behavior consistent with concrete mechanics. This study enhances the understanding of recycled aggregate concrete beam fracture behavior, with the crack fractal dimension serving as a valuable quantitative indicator for damage state and crack complexity assessment. These findings are crucial for engineering design and application, enabling better evaluation of structural performance under various conditions. Full article

This paper consists of various kinds of wave solitons to the mathematical model known as the truncated M-fractional FitzHugh–Nagumo model. This model explains the transmission of the electromechanical pulses in nerves. Through the application of the modified extended tanh function technique and the modified $(G^{\prime }/G^2)$-expansion technique, we are able to achieve the series of exact solitons. The results differ from the current solutions because of the fractional derivative. These solutions could be helpful in the telecommunication and bioscience domains. Contour plots, in two and three dimensions, are used to describe the results. Stability analysis is used to check the stability of the obtained solutions. Moreover, the stationary solutions of the focusing equation are studied through modulation instability. Future research on the focused model in question will benefit from the findings. The techniques used are simple and effective. Full article

The asymptotic behavior of discrete Riemann–Liouville fractional difference equations is a fundamental problem with important mathematical and physical implications. In this paper, we investigate a particular case of such an equation of the order $0.5$ subject to a given initial condition. We establish the existence of a unique solution expressed via a Mittag-Leffler-type function. The delta-asymptotic behavior of the solution is examined, and its convergence properties are rigorously analyzed. Numerical experiments are conducted to illustrate the qualitative features of the solution. Furthermore, a neural network-based approximation is employed to validate and compare with the analytical results, confirming the accuracy, stability, and sensitivity of the proposed method. Full article

In this paper, we propose a high-order compact difference scheme for a class of time-fractional convection–diffusion–reaction equations (CDREs) with variable coefficients. Using the Lagrange polynomial interpolation formula for the time-fractional derivative and a compact finite difference approximation for the spatial derivative, we establish an unconditionally stable compact difference method. The stability and convergence properties of the method are rigorously analyzed using the Fourier method. The convergence order of our discrete scheme is $\mathcal{O}({\tau }^{4-\alpha }+h^8)$, where $\tau$ and h represent the time step size and space step size, respectively. This work contributes to providing a better understanding of the dependability of the method by thoroughly examining convergence and conducting an error analysis. Numerical examples demonstrate the applicability, accuracy, and efficiency of the suggested technique, supplemented by comparisons with previous research. Full article

Existing theoretical models of wave-induced flow face challenges in coal applications due to the scarcity of experimental data in the seismic-frequency band. Additionally, traditional viscoelastic combination models exhibit inherent limitations in accurately capturing the attenuation characteristics of rocks. To overcome these constraints, we propose a novel binary classification physical fractal model, which provides a more robust framework for analyzing wave dispersion and attenuation in complex coal. The fractal cell was regarded as an element to re-establish the viscoelastic constitutive equation. In the new constitutive equation, three key fractional orders, $\alpha$, $\beta$, and $\gamma$, emerged. Among them, $\alpha$ mainly affects the attenuation at low frequencies; $\beta$ controls the attenuation in the middle-frequency band; and $\gamma$ dominates the attenuation in the tail-frequency band. After fitting with the measured attenuation data of partially saturated coal samples under variable confining pressures and variable temperature conditions, the results show that this model can effectively represent the attenuation characteristics of elastic wave propagation in coals with different coal structures. It provides a new theoretical model and analysis ideas for the study of elastic wave attenuation in tectonic coals and is of great significance for an in-depth understanding of the physical properties of coals and related geophysical prospecting. Full article

Accurate carbon market price prediction is crucial for promoting a low-carbon economy and sustainable engineering. Traditional models often face challenges in effectively capturing the multifractality inherent in carbon market prices. Inspired by the self-similarity and scale invariance inherent in fractal structures, this study proposes a novel multifractal-aware model, MF-Transformer-DEC, for carbon market price prediction. The multi-scale convolution (MSC) module employs multi-layer dilated convolutions constrained by shared convolution kernel weights to construct a scale-invariant convolutional network. By projecting and reconstructing time series data within a multi-scale fractal space, MSC enhances the model’s ability to adapt to complex nonlinear fluctuations while significantly suppressing noise interference. The fractal attention (FA) module calculates similarity matrices within a multi-scale feature space through multi-head attention, adaptively integrating multifractal market dynamics and implicit associations. The dynamic error correction (DEC) module models error commonality through variational autoencoder (VAE), and uncertainty-guided dynamic weighting achieves robust error correction. The proposed model achieved an average R² of 0.9777 and 0.9942 for 7-step ahead predictions on the Shanghai and Guangdong carbon price datasets, respectively. This study pioneers the interdisciplinary integration of fractal theory and artificial intelligence methods for complex engineering analysis, enhancing the accuracy of carbon market price prediction. The proposed technical pathway of “multi-scale deconstruction and similarity mining” offers a valuable reference for AI-driven fractal modeling. Full article

To alleviate environmental pressures, manufactured sand (MS) are increasingly being used in the production of ultra-high-performance concrete (UHPC) due to their consistent supply and environmental benefits. However, manufactured sand properties are critically influenced by processing and production techniques, resulting in substantial variations in fundamental characteristics that directly impact UHPC matrix pore structure and ultimately compromise performance. Traditional testing methods inadequately characterize UHPC’s pore structure, necessitating multifractal theory implementation to enhance pore structural interpretation capabilities. In this study, UHPC specimens were fabricated with five types of MS exhibiting distinct properties and at varying water to binder (w/b) ratios. The flowability, mechanical strength, and pore structure of the specimens were systematically evaluated. Additionally, multifractal analysis was conducted on each specimen group using mercury intrusion porosimetry (MIP) data to characterize pore complexity. SM-type sands have a more uniform distribution of pores of different scales, better pore structure and matrix homogeneity due to their finer particles, moderate stone powder content, and better cleanliness. Both excessively high and low stone powder content, as well as low cleanliness, will lead to pore aggregation and poor closure, degrading the pore structure. Full article

This paper addresses the design of nonfragile state estimators for memristor-based fractional-order neural networks that are subject to stochastic cyber-attacks and hybrid time delays. To mitigate the issue of limited bandwidth during signal transmission, quantitative processing is introduced to reduce network burden and prevent signal blocking. In real network environments, the outputs may be compromised by cyber-attacks, which can disrupt data transmission systems. To better reflect the actual conditions of fractional-order neural networks, a Bernoulli variable is utilized to describe the statistical properties. Additionally, novel conditions are presented to ensure the stochastic asymptotic stability of the augmented error system through a new fractional-order free-matrix-based integral inequality. Finally, the effectiveness of the proposed estimation methods is demonstrated through two numerical simulations. Full article

We investigate a generalized quantum Schrödinger equation in a comb-like structure that imposes geometric constraints on spatial variables. The model is extended by the introduction of nonlocal and fractional potentials to capture memory effects in both space and time. We consider four distinct scenarios: (i) a time-dependent nonlocal potential, (ii) a spatially nonlocal potential, (iii) a combined space–time nonlocal interaction with memory kernels, and (iv) a fractional spatial derivative, which is related to distributions asymptotically governed by power laws and to a position-dependent effective mass. For each scenario, we propose solutions based on the Green’s function for arbitrary initial conditions and analyze the resulting quantum dynamics. Our results reveal distinct spreading regimes, depending on the type of non-locality and the fractional operator applied to the spatial variable. These findings contribute to the broader generalization of comb models and open new questions for exploring quantum dynamics in backbone-like structures. Full article

We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative ($0<\alpha <1$) with mixed finite element methods (P1b–P1 and $P_2$–$P_1$) for spatial discretization of velocity and pressure. This approach addresses the key challenges of fractional models, including nonlocality and memory effects, while maintaining stability in the presence of the nonlinear damping term ${\gamma \left|\mathbf{u}\right|}^{r-2}\mathbf{u}$, for $r\ge 2$. We prove unconditional stability for both semi-discrete and fully discrete schemes and derive optimal error estimates for the velocity and pressure components. Numerical experiments validate the theoretical results. Convergence tests using exact solutions, along with benchmark problems such as backward-facing channel flow and lid-driven cavity flow, confirm the accuracy and reliability of the method. The computed velocity contours and streamlines show close agreement with analytical expectations. This scheme is particularly effective for capturing anomalous diffusion in Newtonian and turbulent flows, and it offers a strong foundation for future extensions to viscoelastic and biological fluid models. Full article

Triangular shapes have been studied from different perspectives over a wide temporal frame since ancient times. Initially, fundamental theorems were formulated to demonstrate their geometrical properties. Philosophy and art leveraged the peculiar aspects of triangles as building blocks for more complex geometrical shapes. This paper will review triangles by adopting a multidisciplinary approach, recalling ancient science and Plato’s arguments in relation to their connection with philosophy. It will then consider the artistic utilization of triangles, particularly in compositions created during the medieval era, as exemplified by the Cosmati Italian family’s masterpieces. Various scientific environments have explored triangular 2D and 3D shapes for different purposes, which will be briefly reviewed here. After that, Sierpiński geometry and its properties will be introduced, focusing on the equilateral shape and its internal complexity generated by subdividing the entire triangle into smaller sub-triangles. Finally, examples of triangular planar shapes that fulfill the Sierpiński geometry will be presented as an application in signal processing for high-frequency signals in the microwave and millimeter-wave range. Full article

This paper introduces a new class of Newton-type inequalities (NTIs) within the framework of extended fractional integral operators. This study begins by establishing a fundamental identity for generalized fractional Riemann–Liouville (FR-L) operators, which forms the basis for deriving various inequalities under different assumptions on the integrand. In particular, fractional counterparts of the classical $1/3$ and $3/8$ Simpson rules are obtained when the modulus of the first derivative is convex. The analysis is further extended to include functions that satisfy a Lipschitz condition or have bounded first derivatives. Moreover, an additional NTI is presented for functions of bounded variation, expressed in terms of their total variation. In all scenarios, the proposed results reduce to classical inequalities when the fractional parameters are specified accordingly, thus offering a unified perspective on numerical integration through fractional operators. Full article

This article concerns a novel coupled implicit differential system under $\phi$–Riemann–Liouville ($\mathbb{RL}$) fractional derivatives with $\mathfrak{p}$-Laplacian operator and multi-point strip boundary conditions on unbounded domains. An applicable Banach space is introduced to define solutions on unbounded domains $[c,\infty )$. The explicit iterative solution’s existence and uniqueness ($\mathbb{E}a\mathbb{U}$) are established by employing the Banach fixed point strategy. The different types of Ulam–Hyers–Rassias ($\mathbb{UHR}$) stabilities are investigated. Ultimately, we provide a numerical application of a coupled $\phi$-$\mathbb{RL}$ fractional turbulent flow model to illustrate and test the effectiveness of our outcomes. Full article

This paper examines a family of operators that combine the features of fractional-order and classical operators. Our goal is to obtain results on their invertibility in function spaces, based on their inherent improving properties. The class of proportional operators we study is extensive and includes both fractional-order and classical operators. This leads to interesting function spaces in which we obtain the right- and left-handed properties of invertibility. Thus, we extend and unify results concerning fractional-order and proportional operators. To confirm the relevance of our results, we have supplemented the paper with a series of results on the equivalence of differential and integral forms for various problems, including terminal value problems. Full article

This paper addresses the synchronization problem in complex networks characterized by short-memory fractional dynamics and directed higher-order interactions. Sufficient conditions for global synchronization are rigorously derived using Lyapunov-based analysis and an effective pinning control strategy. To further enhance the adaptability and robustness of the network, an adaptive control law is constructed, accommodating uncertainties and time-varying coupling strengths. An improved predictor–corrector numerical algorithm is also proposed to efficiently solve the underlying short-memory systems. A numerical simulation is conducted to demonstrate the validity of the proposed theoretical results. This work deepens the theoretical understanding of synchronization in higher-order fractional networks and provides practical guidance for the design and control of complex systems with short-memory and higher-order effects. Full article

The pore-throat structure of the extremely low-permeability sandstone reservoir in the Huachi area of the Ordos Basin is complex and highly heterogeneous. Currently, there are issues such as unclear understanding of the micro-pore-throat structural characteristics, primary controlling factors of reservoir quality, and classification boundaries of the reservoir in the study area, which seriously restricts the exploration and development effectiveness of the reservoir in this region. It is necessary to use a combination of various analytical techniques to comprehensively characterize the pore-throat structure and establish reservoir classification evaluation standards in order to better understand the reservoir. This study employs a suite of analytical and testing techniques, including cast thin sections (CTS), scanning electron microscopy (SEM), cathodoluminescence (CL), X-ray diffraction (XRD), as well as high-pressure mercury injection (HPMI) and constant-rate mercury injection (CRMI), and applies fractal theory for analysis. The research findings indicate that the extremely low-permeability sandstone reservoir of the Chang 3 section primarily consists of arkose and a minor amount of lithic arkose. The types of pore-throat are diverse, with intergranular pores, feldspar dissolution pores, and clay interstitial pores and microcracks being the most prevalent. The throat types are predominantly sheet-type, followed by pore shrinkage-type and tubular throats. The pore-throat network of low-permeability sandstone is primarily composed of nanopores (pore-throat radius r < 0.01 μm), micropores (0.01 < r < 0.1 μm), mesopores (0.1 < r < 1.0 μm), and macropores (r > 1.0 μm). The complexity of the reservoir pore-throat structure was quantitatively characterized by fractal theory. Nanopores do not exhibit ideal fractal characteristics. By splicing high-pressure mercury injection and constant-rate mercury injection at a pore-throat radius of 0.12 μm, a more detailed characterization of the full pore-throat size distribution can be achieved. The average fractal dimensions for micropores (Dh2), mesopores (Dc3), and macropores (Dc4) are 2.43, 2.75, and 2.95, respectively. This indicates that the larger the pore-throat size, the rougher the surface, and the more complex the structure. The degree of development and surface roughness of large pores significantly influence the heterogeneity and permeability of the reservoir in the study area. Dh2, Dc3, and Dc4 are primarily controlled by a combination of pore-throat structural parameters, sedimentary processes, and diagenetic processes. Underwater diversion channels and dissolution are key factors in the formation of effective storage space. Based on sedimentary processes, reservoir space types, pore-throat structural parameters, and the characteristics of mercury injection curves, the study area is divided into three categories. This classification provides a theoretical basis for predicting sweet spots in oil and gas exploration within the study area. Full article

In this study, we investigate the fractional Tzitzéica equation, a nonlinear evolution equation known for modeling complex phenomena in various scientific domains such as solid-state physics, crystal dislocation, electromagnetic waves, chemical kinetics, quantum field theory, and nonlinear optics. Using the (G′/G, 1/G)-expansion approach, we derive different categories of exact solutions, like hyperbolic, trigonometric, and rational functions. The beta fractional derivative is used here to generalize the classical idea of the derivative, which preserves important principles. The derived solutions with broader nonlinear wave structures are periodic waves, breathers, peakons, W-shaped solitons, and singular solitons, which enhance our understanding of nonlinear wave dynamics. In relation to these results, the findings are described by showing the solitons’ physical behaviors, their stabilities, and dispersions under fractional parameters in the form of contour plots and 2D and 3D graphs. Comparisons with earlier studies underscore the originality and consistency of the (G′/G, 1/G)-expansion approach in addressing fractional-order evolution equations. It contributes new solutions to analytical problems of fractional nonlinear integrable systems and helps understand the systems’ dynamic behavior in a wider scope of applications. Full article

This paper investigates the existence and uniqueness of solutions to a class of sequential fractional differential equations and inclusions involving the $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo derivatives under non-separated boundary conditions. By reformulating the problems into equivalent fixed-point systems, several classical fixed-point theorems, including those of Banach, Krasnosel’ski$\breve{\mathrm{i}}$’s, Schaefer, and the Leray–Schauder alternative, are employed to derive rigorous results. The study is further extended to the multi-valued setting, where existence results are established for both convex- and nonconvex-valued multi-functions using appropriate fixed-point techniques. Numerical examples are provided to illustrate the applicability and effectiveness of the theoretical findings. Full article

The primary aim of this research article is to investigate the soliton dynamics of the M-truncated derivative nonlinear Kodama equation, which is useful for optical solitons on nonlinear media, shallow water waves over complex media, nonlocal internal waves, and fractional viscoelastic wave propagation. We utilized two recently developed analytical techniques, the generalized Arnous method and the generalized Kudryashov method. First, the nonlinear Kodama equation is transformed into a nonlinear ordinary differential equation using the homogeneous balance principle and a traveling wave transformation. Next, various types of soliton solutions are constructed through the application of these effective methods. Finally, to visualize the behavior of the obtained solutions, three-dimensional, two-dimensional, and contour plots are generated using Maple (2023) mathematical software. Full article