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29 pages, 2180 KB

Open AccessArticle

Mathematical Perspectives of a Coupled System of Nonlinear Hybrid Stochastic Fractional Differential Equations

by Rabeb Sidaoui, Alnadhief H. A. Alfedeel, Jalil Ahmad, Khaled Aldwoah, Amjad Ali, Osman Osman and Ali H. Tedjani

Fractal Fract. 2025, 9(10), 622; https://doi.org/10.3390/fractalfract9100622 - 24 Sep 2025

Abstract

This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of [...] Read more.

This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of memory effects without singularities. Unlike existing approaches, which are limited to either neutral or hybrid stochastic structures, the proposed framework unifies both features within a fractional setting, capturing the joint influence of randomness, history, and abrupt transitions in real-world processes. We establish the existence and uniqueness of mild solutions via the Picard approximation method under generalized Carathéodory-type conditions, allowing for non-Lipschitz nonlinearities. In addition, mean-square Mittag–Leffler stability is analyzed to characterize the boundedness and decay properties of solutions under stochastic fluctuations. Several illustrative examples are provided to validate the theoretical findings and demonstrate their applicability. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

21 pages, 357 KB

Open AccessArticle

A New Study on the Approximate Controllability of Sobolev-Type Stochastic ABC-Fractional Impulsive Differential Inclusions with Clarke Sub-Differential and Poisson Jumps

by Yousef Alnafisah, Hamdy M. Ahmed and A. M. Sayed Ahmed

Fractal Fract. 2025, 9(9), 605; https://doi.org/10.3390/fractalfract9090605 - 18 Sep 2025

Viewed by 130

Abstract

This paper undertakes a rigorous analytical exposition of the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions, incorporating the Atangana–Baleanu fractional derivative in the Caputo configuration under the influence of Wiener process and Poissonian discontinuities. The system’s analytical landscape [...] Read more.

This paper undertakes a rigorous analytical exposition of the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions, incorporating the Atangana–Baleanu fractional derivative in the Caputo configuration under the influence of Wiener process and Poissonian discontinuities. The system’s analytical landscape is further enriched by the incorporation of Clarke sub-differentials, facilitating the treatment of nonsmooth, nonconvex, and multivalued dynamics. The inherent complexity arising from the confluence of fractional memory, stochastic perturbations, and impulsive phenomena necessitates the deployment of a sophisticated apparatus from variational analysis, measurable selection theory, and multivalued fixed point frameworks within infinite-dimensional Banach spaces. This study delineates rigorous sufficient conditions, ensuring controllability under such hybrid influences, thereby generalizing classical paradigms to encompass nonlocal and discontinuous dynamical regimes. A precisely articulated exemplar is included to validate the theoretical constructs and demonstrate the operational efficacy of the proposed analytical methodology. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

20 pages, 434 KB

Open AccessArticle

Large Deviation Principle for Hilfer Fractional Stochastic McKean–Vlasov Differential Equations

by Juan Chen, Haibo Gu, Yutao Yan and Lishan Liu

Fractal Fract. 2025, 9(8), 544; https://doi.org/10.3390/fractalfract9080544 - 19 Aug 2025

Viewed by 429

Abstract

This paper studies the large deviation principle (LDP) of a class of Hilfer fractional stochastic McKean–Vlasov differential equations with multiplicative noise. Firstly, by making use of the Laplace transform and its inverse transform, the solution of the equation is derived. Secondly, considering the [...] Read more.

This paper studies the large deviation principle (LDP) of a class of Hilfer fractional stochastic McKean–Vlasov differential equations with multiplicative noise. Firstly, by making use of the Laplace transform and its inverse transform, the solution of the equation is derived. Secondly, considering the equivalence between the LDP and the Laplace principle (LP), the weak convergence method is employed to prove that the equation satisfies the LDP. Finally, through specific example, it is elaborated how to utilize the LDP to analyze the behavioral characteristics of the system under small noise perturbation. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

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10 pages, 667 KB

Open AccessArticle

Finite-Time Stability of Equilibrium Points of Nonlinear Fractional Stochastic Differential Equations

by Guanli Xiao, Lulu Ren and Rui Liu

Fractal Fract. 2025, 9(8), 510; https://doi.org/10.3390/fractalfract9080510 - 5 Aug 2025

Cited by 1 | Viewed by 402

Abstract

This paper focuses on the problem, claimed in some works, of the non-existence of finite-time stable equilibria in nonlinear fractional differential equations. After dividing the equilibrium point into the initial equilibrium point and the finite-time equilibrium point, we provide sufficient conditions for the [...] Read more.

This paper focuses on the problem, claimed in some works, of the non-existence of finite-time stable equilibria in nonlinear fractional differential equations. After dividing the equilibrium point into the initial equilibrium point and the finite-time equilibrium point, we provide sufficient conditions for the equilibrium point of a fractional stochastic differential equation. Then the finite-time stability of the equilibrium points of nonlinear fractional stochastic differential equations is presented. Finally, the correctness of the theoretical analysis is illustrated through an example. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

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18 pages, 1301 KB

Open AccessArticle

Numerical Investigation for the Temporal Fractional Financial Option Pricing Partial Differential Equation Utilizing a Multiquadric Function

by Jia Li, Tao Liu, Jiaqi Xu, Xiaoxi Hu, Changan Xu and Yanlong Wei

Fractal Fract. 2025, 9(7), 414; https://doi.org/10.3390/fractalfract9070414 - 26 Jun 2025

Cited by 1 | Viewed by 677

Abstract

This paper proposes a computational procedure to resolve the temporal fractional financial option pricing partial differential equation (PDE) using a localized meshless approach via the multiquadric radial basis function (RBF). Given that financial market information is best characterized within a martingale framework, the [...] Read more.

This paper proposes a computational procedure to resolve the temporal fractional financial option pricing partial differential equation (PDE) using a localized meshless approach via the multiquadric radial basis function (RBF). Given that financial market information is best characterized within a martingale framework, the resulting option pricing model follows a modified Black–Sholes (BS) equation, requiring efficient numerical techniques for practical implementation. The key innovation in this study is the derivation of analytical weights for approximating first and second derivatives, ensuring improved numerical stability and accuracy. The construction of these weights is grounded in the second integration of a variant of the multiquadric RBF, which enhances smoothness and convergence properties. The performance of the presented solver is analyzed through computational tests, where the analytical weights exhibit superior accuracy and stability in comparison to conventional numerical weights. The results confirm that the new approach reduces absolute errors, demonstrating its effectiveness for financial option pricing problems. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

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11 pages, 261 KB

Open AccessArticle

A Result Regarding the Existence and Attractivity for a Class of Nonlinear Fractional Difference Equations with Time-Varying Delays

by Shihan Wang and Danfeng Luo

Fractal Fract. 2025, 9(6), 362; https://doi.org/10.3390/fractalfract9060362 - 31 May 2025

Viewed by 426

Abstract

In this paper, we are studying a class of nonlinear fractional difference equations with time-varying delays in Banach space. By means of mathematical induction and the Picard iteration method, we first obtain the existence result of this fractional difference system. Under some new [...] Read more.

In this paper, we are studying a class of nonlinear fractional difference equations with time-varying delays in Banach space. By means of mathematical induction and the Picard iteration method, we first obtain the existence result of this fractional difference system. Under some new criteria along with the Schauder’s fixed point theorem, we then derive the attractivity conclusions. Subsequently, with the aid of Grönwall’s inequality, we prove that the system is globally attractive. Finally, we give two examples to prove the validity of our theorems. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

27 pages, 392 KB

Open AccessArticle

L1 Scheme for Semilinear Stochastic Subdiffusion with Integrated Fractional Gaussian Noise

by Xiaolei Wu and Yubin Yan

Fractal Fract. 2025, 9(3), 173; https://doi.org/10.3390/fractalfract9030173 - 12 Mar 2025

Viewed by 754

Abstract

This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter

H \in (1 / 2, 1)

. The finite element method is employed for spatial [...] Read more.

This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter

H \in (1 / 2, 1)

. The finite element method is employed for spatial discretization, while the L1 scheme and Lubich’s first-order convolution quadrature formula are used to approximate the Caputo time-fractional derivative of order

α \in (0, 1)

and the Riemann–Liouville time-fractional integral of order

γ \in (0, 1)

, respectively. Using the semigroup approach, we establish the temporal and spatial regularity of the mild solution to the problem. The fully discrete solution is expressed as a convolution of a piecewise constant function with the inverse Laplace transform of a resolvent-related function. Based on the Laplace transform method and resolvent estimates, we prove that the proposed numerical scheme has the optimal convergence order

O (τ^{min {H + α + γ - 1 - ε, α}}), ε > 0

. Numerical experiments are presented to validate these theoretical convergence orders and demonstrate the effectiveness of this method. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

24 pages, 362 KB

Open AccessArticle

Stability and Controllability Analysis of Stochastic Fractional Differential Equations Under Integral Boundary Conditions Driven by Rosenblatt Process with Impulses

by Mohamed S. Algolam, Sadam Hussain, Bakri A. I. Younis, Osman Osman, Blgys Muflh, Khaled Aldwoah and Nidal Eljaneid

Fractal Fract. 2025, 9(3), 146; https://doi.org/10.3390/fractalfract9030146 - 26 Feb 2025

Cited by 1 | Viewed by 1140

Abstract

Differential equations are frequently used to mathematically describe many problems in real life, but they are always subject to intrinsic phenomena that are neglected and could influence how the model behaves. In some cases like ecosystems, electrical circuits, or even economic models, the [...] Read more.

Differential equations are frequently used to mathematically describe many problems in real life, but they are always subject to intrinsic phenomena that are neglected and could influence how the model behaves. In some cases like ecosystems, electrical circuits, or even economic models, the model may suddenly change due to outside influences. Occasionally, such changes start off impulsively and continue to exist for specific amounts of time. Non-instantaneous impulses are used in the creation of the models for this kind of scenario. In this paper, a new class of non-instantaneous impulsive

ψ

-Caputo fractional stochastic differential equations under integral boundary conditions driven by the Rosenblatt process was examined. Semigroup theory, stochastic theory, the Banach fixed-point theorem, and fractional calculus were applied to investigating the existence of piecewise continuous mild solutions for the systems under consideration. The impulsive Gronwall’s inequality was employed to establish the unique stability conditions for the system under consideration. Furthermore, we examined the controllability results of the proposed system. Finally, some examples were provided to demonstrate the validity of the presented work. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

14 pages, 340 KB

Open AccessArticle

Successive Approximation and Stability Analysis of Fractional Stochastic Differential Systems with Non-Gaussian Process and Poisson Jumps

by Nidhi Asthana, Mohd Nadeem and Rajesh Dhayal

Fractal Fract. 2025, 9(2), 130; https://doi.org/10.3390/fractalfract9020130 - 19 Feb 2025

Viewed by 658

Abstract

This paper investigates a new class of fractional stochastic differential systems with non-Gaussian processes and Poisson jumps. Firstly, we examine the solvability results for the considered system. Furthermore, new stability results for the proposed system are derived. The findings are established through the [...] Read more.

This paper investigates a new class of fractional stochastic differential systems with non-Gaussian processes and Poisson jumps. Firstly, we examine the solvability results for the considered system. Furthermore, new stability results for the proposed system are derived. The findings are established through the application of Grönwall’s inequality, the successive approximation method, and the corollary of the Bihari inequality. Finally, the validity of the results is proved through an example. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

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17 pages, 410 KB

Open AccessArticle

Approximate Controllability of Hilfer Fractional Stochastic Evolution Inclusions of Order 1 < q < 2

by Anurag Shukla, Sumati Kumari Panda, Velusamy Vijayakumar, Kamalendra Kumar and Kothandabani Thilagavathi

Fractal Fract. 2024, 8(9), 499; https://doi.org/10.3390/fractalfract8090499 - 24 Aug 2024

Cited by 7 | Viewed by 1288

Abstract

This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order

1 < q < 2

. Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects [...] Read more.

This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order

1 < q < 2

. Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects of finding mild solutions for the Hilfer fractional stochastic differential equation. Subsequently, we determined that the specified system is approximately controllable. Finally, an example displays the theoretical application of the results. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

18 pages, 5746 KB

Open AccessArticle

Remaining Useful Life Prediction for Power Storage Electronic Components Based on Fractional Weibull Process and Shock Poisson Model

by Wanqing Song, Xianhua Yang, Wujin Deng, Piercarlo Cattani and Francesco Villecco

Fractal Fract. 2024, 8(8), 485; https://doi.org/10.3390/fractalfract8080485 - 19 Aug 2024

Cited by 5 | Viewed by 1528

Abstract

For lithium-ion batteries and supercapacitors in hybrid power storage facilities, both steady degradation and random shock contribute to their failure. To this end, in this paper, we propose to introduce the degradation-threshold-shock (DTS) model for their remaining useful life (RUL) prediction. Non-homogeneous compound [...] Read more.

For lithium-ion batteries and supercapacitors in hybrid power storage facilities, both steady degradation and random shock contribute to their failure. To this end, in this paper, we propose to introduce the degradation-threshold-shock (DTS) model for their remaining useful life (RUL) prediction. Non-homogeneous compound Poisson process (NHCP) is proposed to simulate the shock effect in the DTS model. Considering the long-range dependence and heavy-tailed characteristics of the degradation process, fractional Weibull process (fWp) is employed in the diffusion term of the stochastic degradation model. Furthermore, the drift and diffusion coefficients are constantly updated to describe the environmental interference. Prior to the model training, steady degradation and shock data must be separated, based on the three-sigma principle. Degradation data for the lithium-ion batteries (LIBs) and ultracapacitors are employed for model verification under different operation protocols in the power system. Recent deep learning models and stochastic process-based methods are utilized for model comparison, and the proposed model shows higher prediction accuracy. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

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15 pages, 315 KB

Open AccessArticle

Well-Posedness and Hyers–Ulam Stability of Fractional Stochastic Delay Systems Governed by the Rosenblatt Process

by Ghada AlNemer, Mohamed Hosny, Ramalingam Udhayakumar and Ahmed M. Elshenhab

Fractal Fract. 2024, 8(6), 342; https://doi.org/10.3390/fractalfract8060342 - 6 Jun 2024

Cited by 1 | Viewed by 1183

Abstract

Under the effect of the Rosenblatt process, the well-posedness and Hyers–Ulam stability of nonlinear fractional stochastic delay systems are considered. First, depending on fixed-point theory, the existence and uniqueness of solutions are proven. Next, utilizing the delayed Mittag–Leffler matrix functions and Grönwall’s inequality, [...] Read more.

Under the effect of the Rosenblatt process, the well-posedness and Hyers–Ulam stability of nonlinear fractional stochastic delay systems are considered. First, depending on fixed-point theory, the existence and uniqueness of solutions are proven. Next, utilizing the delayed Mittag–Leffler matrix functions and Grönwall’s inequality, sufficient criteria for Hyers–Ulam stability are established. Ultimately, an example is presented to demonstrate the effectiveness of the obtained findings. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

23 pages, 640 KB

Open AccessArticle

A Fractional Heston-Type Model as a Singular Stochastic Equation Driven by Fractional Brownian Motion

by Marc Mukendi Mpanda

Fractal Fract. 2024, 8(6), 330; https://doi.org/10.3390/fractalfract8060330 - 30 May 2024

Cited by 1 | Viewed by 1403

Abstract

This paper introduces the fractional Heston-type (fHt) model as a stochastic system comprising the stock price process modeled by a geometric Brownian motion. In this model, the infinitesimal return volatility is characterized by the square of a singular stochastic equation driven [...] Read more.

This paper introduces the fractional Heston-type (fHt) model as a stochastic system comprising the stock price process modeled by a geometric Brownian motion. In this model, the infinitesimal return volatility is characterized by the square of a singular stochastic equation driven by a fractional Brownian motion with a Hurst parameter

H \in (0, 1)

. We establish the Malliavin differentiability of the fHt model and derive an expression for the expected payoff function, revealing potential discontinuities. Simulation experiments are conducted to illustrate the dynamics of the stock price process and option prices. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

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9 pages, 340 KB

Open AccessBrief Report

Modeling Double Stochastic Opinion Dynamics with Fractional Inflow of New Opinions

by Vygintas Gontis

Fractal Fract. 2024, 8(9), 513; https://doi.org/10.3390/fractalfract8090513 - 29 Aug 2024

Cited by 1 | Viewed by 983

Abstract

Our recent analysis of empirical limit order flow data in financial markets reveals a power-law distribution in limit order cancellation times. These times are modeled using a discrete probability mass function derived from the Tsallis q-exponential distribution, closely aligned with the second [...] Read more.

Our recent analysis of empirical limit order flow data in financial markets reveals a power-law distribution in limit order cancellation times. These times are modeled using a discrete probability mass function derived from the Tsallis q-exponential distribution, closely aligned with the second form of the Pareto distribution. We elucidate this distinctive power-law statistical property through the lens of agent heterogeneity in trading activity and asset possession. Our study introduces a novel modeling approach that combines fractional Lévy stable motion for limit order inflow with this power-law distribution for cancellation times, significantly enhancing the prediction of order imbalances. This model not only addresses gaps in current financial market modeling but also extends to broader contexts such as opinion dynamics in social systems, capturing the finite lifespan of opinions. Characterized by stationary increments and a departure from self-similarity, our model provides a unique framework for exploring long-range dependencies in time series. This work paves the way for more precise financial market analyses and offers new insights into the dynamic nature of opinion formation in social systems. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

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