Fractal and Fractional

Research

12 pages, 292 KB

Open AccessArticle

Existence and Uniqueness of Weak Solutions for the Stochastic Fractional Ginzburg–Landau Equation

by Jiaxin Li, Dongge Ma, Xinyao Li, Xiaoju Zhang and Lixu Yan

Fractal Fract. 2026, 10(5), 291; https://doi.org/10.3390/fractalfract10050291 - 24 Apr 2026

Viewed by 371

In this study, we investigate the existence and uniqueness of weak solutions for a stochastic Ginzburg–Landau equation involving the fractional Laplacian. The primary focus is on establishing a proper mathematical framework to handle the coexistence of the nonlocal fractional Laplacian and stochastic perturbations. [...] Read more.

In this study, we investigate the existence and uniqueness of weak solutions for a stochastic Ginzburg–Landau equation involving the fractional Laplacian. The primary focus is on establishing a proper mathematical framework to handle the coexistence of the nonlocal fractional Laplacian and stochastic perturbations. By employing the Galerkin method, we prove that the initial-boundary value problem admits a unique global weak solution for any

F_{0}

-measurable

L^{2} (I)

-valued random initial value with a finite second moment. We also utilize the properties of the fractional Laplacian and fractional Sobolev spaces to provide a proof of the existence of the uniqueness theorem. These results extend the analysis of the Ginzburg–Landau equations to models incorporating stochastic terms and the fractional Laplacian. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)

19 pages, 9151 KB

Open AccessArticle

On a Friction Oscillator of Integer and Fractional Order; Stick–Slip Attractors

by Marius-F. Danca

Fractal Fract. 2026, 10(1), 38; https://doi.org/10.3390/fractalfract10010038 - 7 Jan 2026

Cited by 1 | Viewed by 449

Abstract

This paper investigates a friction oscillator model in both its Integer-Order and Fractional-Order formulations. The lack of classical solutions for the governing differential equations with discontinuous right-hand sides is addressed by adopting a Differential Inclusion framework. Using Filippov regularization, the discontinuity is replaced [...] Read more.

This paper investigates a friction oscillator model in both its Integer-Order and Fractional-Order formulations. The lack of classical solutions for the governing differential equations with discontinuous right-hand sides is addressed by adopting a Differential Inclusion framework. Using Filippov regularization, the discontinuity is replaced by a set-valued map satisfying appropriate regularity conditions. Selection theory is then applied to construct a Lipschitz-continuous, single-valued function that approximates the set-valued map. This procedure reformulates the discontinuous initial value problem as a continuous, single-valued one, thereby providing a rigorous justification for the proposed approximation method. Numerical simulations are performed to study stick–slip attractors in both the Integer-Order and Fractional-Order cases. The results demonstrate that, in contrast to the Integer-Order system, periodic attractors cannot occur in the Fractional-Order regime. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)

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17 pages, 356 KB

Open AccessArticle

Pointwise-in-Time Error Analysis of the Nonuniform Alikhanov Scheme for the Fractional Wave Equation

by Mingze Sun and Chaobao Huang

Fractal Fract. 2026, 10(1), 34; https://doi.org/10.3390/fractalfract10010034 - 6 Jan 2026

Viewed by 515

Abstract

This paper develops a fully discrete finite element scheme for the fractional wave equation with order

α \in (1, 2)

, whose solution typically exhibits a weak singularity near the initial time

t = 0

. By introducing an auxiliary [...] Read more.

This paper develops a fully discrete finite element scheme for the fractional wave equation with order

α \in (1, 2)

, whose solution typically exhibits a weak singularity near the initial time

t = 0

. By introducing an auxiliary variable, we first reformulate the fractional wave problem into an equivalent coupled system of two fractional equations. The resulting coupled system is then discretized using the nonuniform Alikhanov formula in time and the standard finite element method on triangular meshes in space. Through rigorous analysis, we establish a pointwise-in-time error estimate for the proposed scheme in the

H^{1}

semi-norm. A key advantage of the proposed methodology is its ability to employ a sparser mesh near the initial time to achieve optimal convergence of local errors. In particular, our analysis shows that away from the initial time, the local rate of convergence reaches

O (N^{- 2})

in time for

r \geq 2

. Finally, numerical experiments are given to verify the sharpness of the theoretical convergence rates. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)

37 pages, 5538 KB

Open AccessArticle

Sustainable Water Treatment Through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation

by Kareem T. Elgindy

Fractal Fract. 2026, 10(1), 4; https://doi.org/10.3390/fractalfract10010004 - 19 Dec 2025

Cited by 2 | Viewed by 670

Abstract

This work develops and analyzes a novel fractional-order chemostat system (FOCS) with a Caputo fractional derivative (CFD) featuring a sliding memory window and periodic boundary conditions (PBCs), designed to model microbial pollutant degradation in sustainable water treatment. By incorporating the Caputo fractional derivative [...] Read more.

This work develops and analyzes a novel fractional-order chemostat system (FOCS) with a Caputo fractional derivative (CFD) featuring a sliding memory window and periodic boundary conditions (PBCs), designed to model microbial pollutant degradation in sustainable water treatment. By incorporating the Caputo fractional derivative with sliding memory (CFDS), the model captures time-dependent behaviors and memory effects in biological systems more realistically than classical integer-order formulations. We reduce the two-dimensional fractional differential equations (FDEs) governing substrate and biomass concentrations to a one-dimensional FDE by utilizing the PBCs. The existence and uniqueness of non-trivial, periodic solutions are established using the Carathéodory framework and fixed-point theorems, ensuring the system’s well-posedness. We prove the positivity and boundedness of solutions, demonstrating that substrate concentrations remain within physically meaningful bounds and biomass concentrations stay strictly positive, with solution trajectories confined to a biologically feasible invariant set. Additionally, we analyze non-trivial equilibria under constant dilution rates and derive their stability properties. The rigorous mathematical results confirm the viability of FOCS models for representing memory-driven, periodic bioprocesses, offering a foundation for advanced water treatment strategies that align with Sustainable Development Goal 6 (Clean Water and Sanitation). This work establishes a comprehensive mathematical framework that bridges fractional calculus with sustainable water treatment applications, providing both theoretical foundations and practical implications for optimizing bioreactor performance in environmental biotechnology. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)

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23 pages, 69858 KB

Open AccessArticle

The Fractional SI Reaction–Diffusion Model with Incommensurate Orders: Stability Analysis and Numerical Simulations

by Ali Aloui, Amel Hioual, Omar Kahouli, Adel Ouannas, Lilia El Amraoui and Mohamed Ayari

Fractal Fract. 2026, 10(1), 3; https://doi.org/10.3390/fractalfract10010003 - 19 Dec 2025

Viewed by 939

Abstract

In this work, we present a fractional-order reaction–diffusion model for the spread of infectious diseases, incorporating incommensurate Caputo derivatives to capture memory effects and heterogeneous temporal behavior across compartments. Focusing on a generalized SI model with nonlinear incidence, we explore the local asymptotic [...] Read more.

In this work, we present a fractional-order reaction–diffusion model for the spread of infectious diseases, incorporating incommensurate Caputo derivatives to capture memory effects and heterogeneous temporal behavior across compartments. Focusing on a generalized SI model with nonlinear incidence, we explore the local asymptotic stability of both disease-free and endemic equilibria. The model accommodates spatial diffusion, saturation effects, and varying fractional orders, yielding a more realistic depiction of epidemic propagation. Analytical techniques—ranging from linearization to spectral analysis—are employed to rigorously establish stability conditions. Numerical simulations support the theoretical findings, highlighting the impact of memory and spatial structure on long-term dynamics. This study offers a refined mathematical lens to understand the persistence or eradication of infectious diseases under memory-dependent and spatially heterogeneous environments. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)

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23 pages, 4068 KB

Open AccessArticle

Numerical Treatment of the Time-Fractional Kuramoto–Sivashinsky Equation Using a Combined Chebyshev-Collocation Approach

by Waleed Mohamed Abd-Elhameed, Mohamed A. Abdelkawy, Naher Mohammed A. Alsafri and Ahmed Gamal Atta

Fractal Fract. 2025, 9(11), 727; https://doi.org/10.3390/fractalfract9110727 - 10 Nov 2025

Cited by 3 | Viewed by 787

Abstract

In this paper, we present a collocation algorithm for numerically treating the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Certain orthogonal polynomials, which are expressed as combinations of Chebyshev polynomials, and their shifted polynomials are introduced. Some new theoretical formulas regarding these polynomials have been developed, [...] Read more.

In this paper, we present a collocation algorithm for numerically treating the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Certain orthogonal polynomials, which are expressed as combinations of Chebyshev polynomials, and their shifted polynomials are introduced. Some new theoretical formulas regarding these polynomials have been developed, including their operational matrices of both integer and fractional derivatives. The derived formulas will be the foundation for designing the proposed numerical algorithm, which relies on converting the governing problem with its underlying conditions into a nonlinear algebraic system, which can be solved using Newton’s iteration technique. A rigorous error analysis for the proposed combined Chebyshev expansion is presented. Some numerical examples are given to ensure the applicability and efficiency of the presented algorithm. These results demonstrate that the proposed algorithm attains superior accuracy with fewer expansion terms. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)

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32 pages, 1899 KB

Open AccessArticle

A Physics-Informed Neural Network Based on the Separation of Variables for Solving the Distributed-Order Time-Fractional Advection–Diffusion Equation

by Wenkai Liu and Yang Liu

Fractal Fract. 2025, 9(11), 712; https://doi.org/10.3390/fractalfract9110712 - 4 Nov 2025

Cited by 2 | Viewed by 1891

Abstract

In this work, we propose a new physics-informed neural network framework based on the method of separation of variables (SVPINN) to solve the distributed-order time-fractional advection–diffusion equation. We develop a new method for calculating the distributed-order derivative, which enables the fractional integral to [...] Read more.

In this work, we propose a new physics-informed neural network framework based on the method of separation of variables (SVPINN) to solve the distributed-order time-fractional advection–diffusion equation. We develop a new method for calculating the distributed-order derivative, which enables the fractional integral to be modeled by a network and directly solved by combining automatic differentiation technology. In this way, the approximation of the distributed-order derivative is integrated into the parameter training system of the network, and the data-driven adaptive learning mechanism is used to replace the numerical discretization scheme. In the SVPINN framework, we decompose the kernel function of the Caputo integral into three independent functions using the method of separation of variables, and apply a neural network as a surrogate model for the modified integral and the function related to the time variable. The new physical constraint generated by the modified integral serves as an extra supervised learning task for the network. We systematically evaluated the feasibility of the SVPINN on several numerical experiments and demonstrated its performance. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)

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28 pages, 4465 KB

Open AccessArticle

Neural Networks-Based Analytical Solver for Exact Solutions of Fractional Partial Differential Equations

by Shanhao Yuan, Yanqin Liu, Limei Yan, Runfa Zhang and Shunjun Wu

Fractal Fract. 2025, 9(8), 541; https://doi.org/10.3390/fractalfract9080541 - 16 Aug 2025

Cited by 4 | Viewed by 1798

Abstract

This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in [...] Read more.

This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in both computational speed and solution precision. The efficacy of the proposed method is validated through four numerical examples, with results visualized using three-dimensional surface plots, contour mappings, and density distributions. Numerical experiments demonstrate that the proposed framework successfully derives exact solutions for fPDEs without relying on data samples. This research provides a novel methodological framework for solving fPDEs, with broad applicability across scientific and engineering fields. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)

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20 pages, 873 KB

Open AccessArticle

A Mixed Finite Volume Element Method for Nonlinear Time Fractional Fourth-Order Reaction–Diffusion Models

by Jie Zhao, Min Cao and Zhichao Fang

Fractal Fract. 2025, 9(8), 481; https://doi.org/10.3390/fractalfract9080481 - 23 Jul 2025

Cited by 1 | Viewed by 809

Abstract

In this paper, a linearized mixed finite volume element (MFVE) scheme is proposed to solve the nonlinear time fractional fourth-order reaction–diffusion models with the Riemann–Liouville time fractional derivative. By introducing an auxiliary variable

σ = Δ u

, the original fourth-order model is [...] Read more.

In this paper, a linearized mixed finite volume element (MFVE) scheme is proposed to solve the nonlinear time fractional fourth-order reaction–diffusion models with the Riemann–Liouville time fractional derivative. By introducing an auxiliary variable

σ = Δ u

, the original fourth-order model is reformulated into a lower-order coupled system. The first-order time derivative and the time fractional derivative are discretized by using the BDF2 formula and the weighted and shifted Grünwald difference (WSGD) formula, respectively. Then, a fully discrete MFVE scheme is constructed by using the primal and dual grids. The existence and uniqueness of a solution for the MFVE scheme are proven based on the matrix theories. The scheme’s unconditional stability is rigorously derived by using the Gronwall inequality in detail. Moreover, the optimal error estimates for u in the discrete

L^{\infty} (L^{2} (Ω))

and

L^{2} (H^{1} (Ω))

norms and for

σ

in the discrete

L^{2} (L^{2} (Ω))

norm are obtained. Finally, three numerical examples are given to confirm its feasibility and effectiveness. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)

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