Fractal and Fractional

15 pages, 463 KiB

Open AccessArticle

On Extended Numerical Discretization Technique of Fractional Models with Caputo-Type Derivatives

by Reem Allogmany and S. S. Alzahrani

Fractal Fract. 2025, 9(5), 289; https://doi.org/10.3390/fractalfract9050289 - 28 Apr 2025

In this work, we investigate the extended numerical discretization technique for the solution of fractional Bernoulli equations and SIRD epidemic models under the Caputo fractional, which is accurate and versatile. We have demonstrated the method’s strength in examining complex systems; it is found [...] Read more.

In this work, we investigate the extended numerical discretization technique for the solution of fractional Bernoulli equations and SIRD epidemic models under the Caputo fractional, which is accurate and versatile. We have demonstrated the method’s strength in examining complex systems; it is found that the method produces solutions that are identical to the exact solution and approximate series solutions. The ENDT is its ability to proficiently handle complex systems governed by fractional differential equations while preserving memory and hereditary characteristics. Its simplicity, accuracy, and flexibility render it an effective instrument for replicating real-world phenomena in physics and biology. The ENDT method offers accuracy, stability, and efficiency compared to traditional methods. It effectively handles challenges in complex systems, supports any fractional order, is simple to implement, improves computing efficiency with sophisticated methodologies, and applies it to epidemic predictions and biological simulations. Full article

(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)

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19 pages, 619 KiB

Open AccessArticle

Existence and Ulam-Type Stability for Fractional Multi-Delay Differential Systems

by Xing Zhang and Mengmeng Li

Fractal Fract. 2025, 9(5), 288; https://doi.org/10.3390/fractalfract9050288 - 28 Apr 2025

Abstract

Fractional multi-delay differential systems, as an important mathematical model, can effectively describe viscoelastic materials and non-local delay responses in ecosystem population dynamics. It has a wide and profound application background in interdisciplinary fields such as physics, biomedicine, and intelligent control. Through literature review [...] Read more.

Fractional multi-delay differential systems, as an important mathematical model, can effectively describe viscoelastic materials and non-local delay responses in ecosystem population dynamics. It has a wide and profound application background in interdisciplinary fields such as physics, biomedicine, and intelligent control. Through literature review and classification, it is evident that for the fractional multi-delay differential system, the existence and uniqueness of the solution and Ulam-Hyers stability (UHS), Ulam-Hyers-Rassias stability (UHRS) of the fractional multi-delay differential system are rarely studied by using the multi-delayed perturbation of two parameter Mittag-Leffler typematrix function. In this paper, we first establish the existence and uniqueness of the solution for the Riemann-Liouville fractional multi-delay differential system on finite intervals using the Banach and Schauder fixed point theorems. Second, we demonstrate the existence and uniqueness of the solution for the system on the unbounded intervals in the weighted function space. Furthermore, we investigate UHS and UHRS for the nonlinear fractional multi-delay differential system in unbounded intervals. Finally, numerical examples are provided to validate the key theoretical results. Full article

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23 pages, 2348 KiB

Open AccessArticle

Chaotic Analysis and Wave Photon Dynamics of Fractional Whitham–Broer–Kaup Model with β Derivative

by Muhammad Idrees Afridi, Theodoros E. Karakasidis and Abdullah Alhushaybari

Fractal Fract. 2025, 9(5), 287; https://doi.org/10.3390/fractalfract9050287 - 27 Apr 2025

Abstract

This study uses a conformable derivative of order

β

to investigate a fractional Whitham–Broer–Kaup (

FWBK

) model. This model has significant uses in several scientific domains, such as plasma physics and nonlinear optics. The enhanced modified Sardar sub-equation

EMSSE

approach is applied [...] Read more.

This study uses a conformable derivative of order

β

to investigate a fractional Whitham–Broer–Kaup (

FWBK

) model. This model has significant uses in several scientific domains, such as plasma physics and nonlinear optics. The enhanced modified Sardar sub-equation

EMSSE

approach is applied to achieve precise analytical solutions, demonstrating its effectiveness in resolving complex wave photons. Bright, solitary, trigonometric, dark, and plane waves are among the various wave dynamics that may be effectively and precisely determined using the

FWBK

model. Furthermore, the study explores the chaotic behaviour of both perturbed and unperturbed systems, revealing illumination on their dynamic characteristics. By demonstrating its validity in examining wave propagation in nonlinear fractional systems, the effectiveness and reliability of the suggested method in fractional modelling are confirmed through thorough investigation. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order, 4th Edition)

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18 pages, 552 KiB

Open AccessArticle

Error Analysis of the L1 Scheme on a Modified Graded Mesh for a Caputo–Hadamard Fractional Diffusion Equation

by Dan Liu, Libin Liu, Hongbin Chen and Xiongfa Mai

Fractal Fract. 2025, 9(5), 286; https://doi.org/10.3390/fractalfract9050286 - 27 Apr 2025

Abstract

The

L 1

scheme on a modified graded mesh is proposed to solve a Caputo–Hadamard fractional diffusion equation with order

α \in (0, 1)

. Firstly, an improved graded mesh frame is innovatively constructed, and its mathematical properties are verified. [...] Read more.

The

L 1

scheme on a modified graded mesh is proposed to solve a Caputo–Hadamard fractional diffusion equation with order

α \in (0, 1)

. Firstly, an improved graded mesh frame is innovatively constructed, and its mathematical properties are verified. Subsequently, a new truncation error bound for the L1 discretisation format of Caputo–Hadamard fractional-order derivatives is established by means of a Taylor cosine expansion of the integral form, and a second-order central difference method is used to achieve high-precision discretisation of spatial derivatives. Furthermore, a rigorous analysis of stability and convergence under the maximum norm is conducted, with special attention devoted to validating that the

L 1

approximation scheme manifests an optimal convergence order of

2 - α

when deployed on the modified graded mesh. Finally, the theoretical results are substantiated through a series of numerical experiments, which validate their accuracy and applicability. Full article

(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs, Second Edition)

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11 pages, 712 KiB

Open AccessArticle

Qualitative Analysis and Traveling Wave Solutions of a (3 + 1)- Dimensional Generalized Nonlinear Konopelchenko-Dubrovsky-Kaup-Kupershmidt System

by Zhao Li and Ejaz Hussain

Fractal Fract. 2025, 9(5), 285; https://doi.org/10.3390/fractalfract9050285 - 27 Apr 2025

Abstract

This article investigates the qualitative analysis and traveling wave solutions of a (3 + 1)-dimensional generalized nonlinear Konopelchenko-Dubrovsky-Kaup-Kupershmidt system. This equation is commonly used to simulate nonlinear wave problems in the fields of fluid mechanics, plasma physics, and nonlinear optics, as well as [...] Read more.

This article investigates the qualitative analysis and traveling wave solutions of a (3 + 1)-dimensional generalized nonlinear Konopelchenko-Dubrovsky-Kaup-Kupershmidt system. This equation is commonly used to simulate nonlinear wave problems in the fields of fluid mechanics, plasma physics, and nonlinear optics, as well as to transform nonlinear partial differential equations into nonlinear ordinary differential equations through wave transformations. Based on the analysis of planar dynamical systems, a nonlinear ordinary differential equation is transformed into a two-dimensional dynamical system, and the qualitative behavior of the two-dimensional dynamical system and its periodic disturbance system is studied. A two-dimensional phase portrait, three-dimensional phase portrait, sensitivity analysis diagrams, Poincaré section diagrams, and Lyapunov exponent diagrams are provided to illustrate the dynamic behavior of two-dimensional dynamical systems with disturbances. The traveling wave solution of a Konopelchenko-Dubrovsky-Kaup-Kupershmidt system is studied based on the complete discriminant system method, and its three-dimensional, two-dimensional graphs and contour plots are plotted. These works can provide a deeper understanding of the dynamic behavior of Konopelchenko-Dubrovsky-Kaup-Kupershmidt systems and the propagation process of waves. Full article

(This article belongs to the Special Issue Applications of General Fractional Calculus Models: Insights into Viscoelasticity and Wave Propagation)

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40 pages, 3297 KiB

Open AccessArticle

Numerical Analysis of a Fractional Cauchy Problem for the Laplace Equation in an Annular Circular Region

by José Julio Conde Mones, Julio Andrés Acevedo Vázquez, Eduardo Hernández Montero, María Monserrat Morín Castillo, Carlos Arturo Hernández Gracidas and José Jacobo Oliveros Oliveros

Fractal Fract. 2025, 9(5), 284; https://doi.org/10.3390/fractalfract9050284 - 27 Apr 2025

Abstract

The Cauchy problem for the Laplace equation in an annular bounded region consists of finding a harmonic function from the Dirichlet and Neumann data known on the exterior boundary. This work considers a fractional boundary condition instead of the Dirichlet condition in a [...] Read more.

The Cauchy problem for the Laplace equation in an annular bounded region consists of finding a harmonic function from the Dirichlet and Neumann data known on the exterior boundary. This work considers a fractional boundary condition instead of the Dirichlet condition in a circular annular region. We found the solution to the fractional boundary problem using circular harmonics. Then, the Tikhonov regularization is used to handle the numerical instability of the fractional Cauchy problem. The regularization parameter was chosen using the L-curve method, Morozov’s discrepancy principle, and the Tikhonov criterion. From numerical tests, we found that the series expansion of the solution to the Cauchy problem can be truncated in

N = 20

,

N = 25

, or

N = 30

for smooth functions. For other functions, such as absolute value and the jump function, we have to choose other values of N. Thus, we found a stable method for finding the solution to the problem studied. To illustrate the proposed method, we elaborate on synthetic examples and MATLAB 2021 programs to implement it. The numerical results show the feasibility of the proposed stable algorithm. In almost all cases, the L-curve method gives better results than the Tikhonov Criterion and Morozov’s discrepancy principle. In all cases, the regularization using the L-curve method gives better results than without regularization. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)

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15 pages, 2574 KiB

Open AccessArticle

The Effect of Organic Acid Modification on the Pore Structure and Fractal Features of 1/3 Coking Coal

by Jiafeng Fan and Feng Cai

Fractal Fract. 2025, 9(5), 283; https://doi.org/10.3390/fractalfract9050283 - 26 Apr 2025

Abstract

The acidification modification of coal seams is a significant technical measure for transforming coalbed methane reservoirs and enhancing the permeability of coal seams, thereby improving the extractability of coalbed methane. However, the acids currently used in fracturing fluids are predominantly inorganic acids, which [...] Read more.

The acidification modification of coal seams is a significant technical measure for transforming coalbed methane reservoirs and enhancing the permeability of coal seams, thereby improving the extractability of coalbed methane. However, the acids currently used in fracturing fluids are predominantly inorganic acids, which are highly corrosive and can contaminate groundwater reservoirs. In contrast, organic acids are not only significantly less corrosive than inorganic acids but also readily bind with the coal matrix. Some organic acids even exhibit complexing and flocculating effects, thus avoiding groundwater contamination. This study focuses on the 1/3 coking coal from the Guqiao Coal Mine of Huainan Mining Group Co., Ltd., in China. It systematically investigates the fractal characteristics and chemical structure of coal samples before and after pore modification using four organic acids (acetic acid, glycolic acid, oxalic acid, and citric acid) and compares their effects with those of hydrochloric acid solutions at the same concentration. Following treatment with organic acids, the coal samples exhibit an increase in surface fractal dimension, a reduction in spatial fractal dimension, a decline in micropore volume proportion, and a rise in the proportions of transitional and mesopore volumes, and the structure of the hydroxyl group and oxygen-containing functional group decreased. This indicates that treating coal samples with organic acids enhances their pore structure and chemical structure. A comparative analysis reveals that hydrochloric acid is more effective than acetic acid in modifying coal pores, while oxalic acid and citric acid outperform hydrochloric acid, and citric acid shows the best results. The findings provide essential theoretical support for organic acidification modification technology in coalbed methane reservoirs and hydraulic fracturing techniques for coalbed methane extraction. Full article

(This article belongs to the Special Issue Applications of Fractal Analysis in Underground Engineering)

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20 pages, 7934 KiB

Open AccessArticle

Dynamical Behavior Analysis of Generalized Chen–Lee–Liu Equation via the Riemann–Hilbert Approach

by Wenxia Chen, Chaosheng Zhang and Lixin Tian

Fractal Fract. 2025, 9(5), 282; https://doi.org/10.3390/fractalfract9050282 - 26 Apr 2025

Abstract

In this paper, we investigate the dynamics of the generalized Chen–Lee–Liu (gCLL) equation utilizing the Riemann–Hilbert method to derive its

N

-soliton solution. We incorporate a self-steepening term and a Kerr nonlinear term to characterize nonlinear light propagation in optical fibers based on [...] Read more.

In this paper, we investigate the dynamics of the generalized Chen–Lee–Liu (gCLL) equation utilizing the Riemann–Hilbert method to derive its

N

-soliton solution. We incorporate a self-steepening term and a Kerr nonlinear term to characterize nonlinear light propagation in optical fibers based on the Chen–Lee–Liu (CLL) equation more accurately. The Riemann–Hilbert problem is addressed through spectral analysis derived from the Lax pair formulation, resulting in an

N

-soliton solution for a reflectance-less system. We also present explicit formulas for solutions involving one and two solitons, thereby providing theoretical support for stable long-distance signal transmission in optical fiber communication. Furthermore, by adjusting parameters and conducting comparative analyses, we generate three-dimensional soliton images that warrant further exploration. The stability of soliton solutions in optical fibers offers novel insights into the intricate propagation behavior of light pulses, and it is crucial for maintaining the integrity of communication signals. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics, 2nd Edition)

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23 pages, 3202 KiB

Open AccessArticle

Fractional Operator Approach and Hybrid Special Polynomials: The Generalized Gould–Hopper–Bell-Based Appell Polynomials and Their Characteristics

by Rabeb Sidaoui, E. I. Hassan, Abdulghani Muhyi, Khaled Aldwoah, A. H. A. Alfedeel, Khidir Shaib Mohamed and Alawia Adam

Fractal Fract. 2025, 9(5), 281; https://doi.org/10.3390/fractalfract9050281 - 25 Apr 2025

Abstract

This study introduces a novel generalized class of special polynomials using a fractional operator approach. These polynomials are referred to as the generalized Gould–Hopper–Bell-based Appell polynomials. In view of the operational method, we first introduce the operational representation of the Gould–Hopper–Bell-based Appell polynomials; [...] Read more.

This study introduces a novel generalized class of special polynomials using a fractional operator approach. These polynomials are referred to as the generalized Gould–Hopper–Bell-based Appell polynomials. In view of the operational method, we first introduce the operational representation of the Gould–Hopper–Bell-based Appell polynomials; then, using a fractional operator, we establish a new generalized form of these polynomials. The associated generating function, series representations, and summation formulas are also obtained. Additionally, certain operational identities, as well as determinant representation, are derived. The investigation further explores specific members of this generalized family, including the generalized Gould–Hopper–Bell-based Bernoulli polynomials, the generalized Gould–Hopper–Bell-based Euler polynomials, and the generalized Gould–Hopper–Bell-based Genocchi polynomials, revealing analogous results for each. Finally, the study employs Mathematica to present computational outcomes, zero distributions, and graphical representations associated with the special member, generalized Gould–Hopper–Bell-based Bernoulli polynomials. Full article

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11 pages, 256 KiB

Open AccessArticle

Improved High-Order Difference Scheme for the Conservation of Mass and Energy in the Two-Dimensional Spatial Fractional Schrödinger Equation

by Junhong Tian and Hengfei Ding

Fractal Fract. 2025, 9(5), 280; https://doi.org/10.3390/fractalfract9050280 - 25 Apr 2025

Abstract

In this paper, our primary objective is to develop a robust and efficient higher-order structure-preserving algorithm for the numerical solution of the two-dimensional nonlinear spatial fractional Schrödinger equation. This equation, which incorporates fractional derivatives, poses significant challenges due to its non-local nature and [...] Read more.

In this paper, our primary objective is to develop a robust and efficient higher-order structure-preserving algorithm for the numerical solution of the two-dimensional nonlinear spatial fractional Schrödinger equation. This equation, which incorporates fractional derivatives, poses significant challenges due to its non-local nature and nonlinearity, making it essential to design numerical methods that not only achieve high accuracy but also preserve the intrinsic physical and mathematical properties of the system. To address these challenges, we employ the scalar auxiliary variable (SAV) method, a powerful technique known for its ability to maintain energy stability and simplify the treatment of nonlinear terms. Combined with the composite Simpson’s formula for numerical integration, which ensures high precision in approximating integrals, and a fourth-order numerical differential formula for discretizing the Riesz derivative, we construct a highly effective finite difference scheme. This scheme is designed to balance computational efficiency with numerical accuracy, making it suitable for long-time simulations. Furthermore, we rigorously analyze the conserving properties of the numerical solution, including mass and energy conservation, which are critical for ensuring the physical relevance and stability of the results. Full article

(This article belongs to the Special Issue Exploration and Analysis of Higher-Order Numerical Methods for Fractional Differential Equations)

24 pages, 2098 KiB

Open AccessArticle

Quasiparticle Solutions to the 1D Nonlocal Fisher–KPP Equation with a Fractal Time Derivative in the Weak Diffusion Approximation

by Alexander V. Shapovalov and Sergey A. Siniukov

Fractal Fract. 2025, 9(5), 279; https://doi.org/10.3390/fractalfract9050279 - 25 Apr 2025

Abstract

In this paper, we propose an approach for constructing quasiparticle-like asymptotic solutions within the weak diffusion approximation for the generalized population Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation, which incorporates nonlocal quadratic competitive losses and a fractal time derivative of non-integer order (

α

, where [...] Read more.

In this paper, we propose an approach for constructing quasiparticle-like asymptotic solutions within the weak diffusion approximation for the generalized population Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation, which incorporates nonlocal quadratic competitive losses and a fractal time derivative of non-integer order (

α

, where

0 < α \leq 1

). This approach is based on the semiclassical approximation and the principles of the Maslov method. The fractal time derivative is introduced in the framework of

F^{α}

calculus. The Fisher–KPP equation is decomposed into a system of nonlinear equations that describe the dynamics of interacting quasiparticles within classes of trajectory-concentrated functions. A key element in constructing approximate quasiparticle solutions is the interplay between the dynamical system of quasiparticle moments and an auxiliary linear system of equations, which is coupled with the nonlinear system. General constructions are illustrated through examples that examine the effect of the fractal parameter (

α

) on quasiparticle behavior. Full article

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14 pages, 2847 KiB

Open AccessArticle

Linear and Non-Linear Methods to Discriminate Cortical Parcels Based on Neurodynamics: Insights from sEEG Recordings

by Karolina Armonaite, Livio Conti, Luigi Laura, Michele Primavera and Franca Tecchio

Fractal Fract. 2025, 9(5), 278; https://doi.org/10.3390/fractalfract9050278 - 25 Apr 2025

Abstract

Understanding human cortical neurodynamics is increasingly important, as highlighted by the European Innovation Council, which prioritises tools for measuring and stimulating brain activity. Unravelling how cytoarchitecture, morphology, and connectivity shape neurodynamics is essential for developing technologies that target specific brain regions. Given the [...] Read more.

Understanding human cortical neurodynamics is increasingly important, as highlighted by the European Innovation Council, which prioritises tools for measuring and stimulating brain activity. Unravelling how cytoarchitecture, morphology, and connectivity shape neurodynamics is essential for developing technologies that target specific brain regions. Given the dynamic and non-stationary nature of neural interactions, there is an urgent need for non-linear signal analysis methods, in addition to the linear ones, to track local neurodynamics and differentiate cortical parcels. Here, we explore linear and non-linear methods using data from a public stereotactic intracranial EEG (sEEG) dataset, focusing on the superior temporal gyrus (STG), postcentral gyrus (postCG), and precentral gyrus (preCG) in 55 subjects during resting-state wakefulness. For this study, we used a linear Power Spectral Density (PSD) estimate and three non-linear measures: the Higuchi fractal dimension (HFD), a one-dimensional convolutional neural network (1D-CNN), and a one-shot learning model. The PSD was able to distinguish the regions in α, β, and γ frequency bands. The HFD showed a tendency of a higher value in the preCG than in the postCG, and both were higher in the STG. The 1D-CNN showed promise in identifying cortical parcels, with an 85% accuracy for the training set, although performance in the test phase indicates that further refinement is needed to integrate dynamic neural electrical activity patterns into neural networks for suitable classification. Full article

(This article belongs to the Special Issue Fractal Analysis in Biology and Medicine)

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17 pages, 22156 KiB

Open AccessArticle

Variable-Fractional-Order Nosé–Hoover System: Chaotic Dynamics and Numerical Simulations

by D. K. Almutairi, Dalal M. AlMutairi, Nidal E. Taha, Mohammed E. Dafaalla and Mohamed A. Abdoon

Fractal Fract. 2025, 9(5), 277; https://doi.org/10.3390/fractalfract9050277 - 25 Apr 2025

Abstract

This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s dynamics. The main objective is to examine these effects through numerical simulations, showcasing [...] Read more.

This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability into a system’s dynamics. The main objective is to examine these effects through numerical simulations, showcasing how changes in the order function influence a system’s behavior. The variable-order behavior is shown by phase space orbits and time series for various variable orders

α

. We look at how the system acts by using numerical solutions and numerical simulations. The phase space orbits and time series for different

α

show variable-order effects. The findings emphasize the role of variable-order derivatives in enhancing chaotic behavior, offering novel insights into their impact on dynamical systems. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)

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15 pages, 7159 KiB

Open AccessArticle

Discrete Memristive Hindmarsh-Rose Neural Model with Fractional-Order Differences

by Fatemeh Parastesh, Karthikeyan Rajagopal, Sajad Jafari and Matjaž Perc

Fractal Fract. 2025, 9(5), 276; https://doi.org/10.3390/fractalfract9050276 - 24 Apr 2025

Abstract

Discrete systems can offer advantages over continuous ones in certain contexts, particularly in terms of simplicity and reduced computational costs, though this may vary depending on the specific application and requirements. Recently, there has been growing interest in using fractional differences to enhance [...] Read more.

Discrete systems can offer advantages over continuous ones in certain contexts, particularly in terms of simplicity and reduced computational costs, though this may vary depending on the specific application and requirements. Recently, there has been growing interest in using fractional differences to enhance discrete models’ flexibility and incorporate memory effects. This paper examines the dynamics of the discrete memristive Hindmarsh-Rose model by integrating fractional-order differences. Our results highlight the complex dynamics of the fractional-order model, revealing that chaotic firing depends on both the fractional-order and magnetic strength. Notably, certain magnetic strengths induce a transition from periodic firing in the integer-order model to chaotic behavior in the fractional-order model. Additionally, we explore the dynamics of two coupled discrete systems, finding that electrical coupling leads to the synchronization of chaotic dynamics, while chemical coupling ultimately results in a quiescent state. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Chaotic and Complex Systems)

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12 pages, 679 KiB

Open AccessArticle

On the Laplace Residual Series Method and Its Application to Time-Fractional Fisher’s Equations

by Rawya Al-deiakeh, Sharifah Alhazmi, Shrideh Al-Omari, Mohammed Al-Smadi and Shaher Momani

Fractal Fract. 2025, 9(5), 275; https://doi.org/10.3390/fractalfract9050275 - 24 Apr 2025

Abstract

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 [...] Read more.

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

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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21 pages, 3946 KiB

Open AccessArticle

Duality Revelation and Operator-Based Method in Viscoelastic Problems

by Zelin Liu, Xiaobin Yu and Yajun Yin

Fractal Fract. 2025, 9(5), 274; https://doi.org/10.3390/fractalfract9050274 - 23 Apr 2025

Abstract

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 [...] Read more.

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 belongs to the Special Issue Fractal Analysis and Its Applications in Materials Science)

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16 pages, 1576 KiB

Open AccessArticle

Comparison Principle Based Synchronization Analysis of Fractional-Order Chaotic Neural Networks with Multi-Order and Its Circuit Implementation

by Rongbo Zhang, Kun Qiu, Chuang Liu, Hongli Ma and Zhaobi Chu

Fractal Fract. 2025, 9(5), 273; https://doi.org/10.3390/fractalfract9050273 - 23 Apr 2025

Abstract

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, [...] Read more.

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

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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21 pages, 3494 KiB

Open AccessArticle

Fractional Heat Conduction with Variable Thermal Conductivity of Infinite Annular Cylinder Under Thermoelasticity Theorem of Moore–Gibson–Thompson

by Eman A. N. Al-Lehaibi

Fractal Fract. 2025, 9(5), 272; https://doi.org/10.3390/fractalfract9050272 - 23 Apr 2025

Abstract

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 [...] Read more.

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

(This article belongs to the Special Issue Analysis of Heat Conduction and Anomalous Diffusion in Fractional Calculus)

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16 pages, 639 KiB

Open AccessArticle

Some Results Related to Booth Lemniscate and Integral Operators

by Bilal Khan, Zahra Orouji and Ali Ebadian

Fractal Fract. 2025, 9(5), 271; https://doi.org/10.3390/fractalfract9050271 - 22 Apr 2025

Abstract

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 [...] Read more.

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

BS (μ)

and

KS (μ)

, leading to new insights into their structural behavior. Full article

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