Well-posedness and asymptotic separation between solutions of stochastic fractional differential equations with doubly singular kernels

Hybrid fractional groupers and moray eels driven deep learning for pneumonia detection using multi-modal data in federated learning.

• How to generalize the integer order model into a time-fractional model and what are the specific benefits associated with it? • What are the differences in stability analysis for the integer and fractional order cases? • How to solve 2D time-fractional PDEs and What are the computational challenges associated with solving them specifically, when it comes to the Neumann boundary conditions to address boundary and corner points? • How do the results for the integer and fractional order differ from each other? • What are the effects of different fractional orders on the dynamics of the fairy circles and patterns? Vegetation ring formation is a spatial pattern observed in ecosystems influenced by plant–soil negative feedback. Classical integer-order models capture only instantaneous interactions and cannot represent the history-dependent processes shaping these dynamics. To overcome this limitation, we propose a new time-fractional reaction-diffusion problem, that extends the model of Cartenì et al. of 2012 [1], by incorporating memory effects through Caputo derivatives. The system, consisting of coupled fractional partial differential equations (FPDEs) for biomass and toxicity, is analyzed for existence and uniqueness of solutions, equilibrium states, and stability in both homogeneous and heterogeneous cases. Since the analytical solution is not available, a numerical approach has been proposed. The numerical experiments illustrate variations in ring formation throughout time and space. The study covers multiple aspects, such as the influence of the fractional derivation index κ on pattern formation, showing that when κ decreases, the biomass spreads over a larger area with fewer oscillations and amplitudes. Moreover, reducing κ has the effect of slowing down the dynamics, requiring more time to reach the equilibrium points, and causes the ring’s width to expand, which shrinks the internal ring diameter until only disks are visible as κ tends to 0.6.

Quantifying immune memory in colorectal cancer using fractional-order differential equations.

The role of fractional calculus in modeling vibrations of a Timoshenko beam influenced by low and high thermal conduction

An efficient hyperspectral classification model for foodborne pathogens based on fractional-order differentiation lightweight deep learning and membership broad learning

ABSTRACT Multidrug-resistant tuberculosis (MDR-TB) is a serious health concern and a global challenge due to its prolonged treatment duration and complex response to therapies. This study presents a novel computational modeling framework using fractional calculus and fractal theory to address the complex dynamics of MDR-TB by incorporating first and second-line treatment strategies. The Caputo fractional and fractal-fractional (FF) operators are employed to formulate the model that leverages the strength of these mathematical frameworks. The fundamental characteristics of both models, including existence and uniqueness of solutions, are rigorously investigated. The stability analysis is performed using Ulam-Hyers and Ulam-Hyers-Rassias stability criteria. The normalized sensitivity indices of model embedded parameters are evaluated to identify the influential parameters. Furthermore, computational schemes have been developed for both fractional and fractal-fractional models, utilizing interpolation techniques to perform simulations. Detailed simulation is conducted for various fractional and fractal orders demonstrating the stability of these models. This study aims to empower researchers by integrating advanced computational techniques into the modeling and control of infectious diseases, ultimately contributing to more effective interventions.

This research focused on computing eigenvalues and eigenfunctions of a class of fractional Sturm-Liouville boundary value problems with eigenparameter-dependent boundary conditions using the Adomian Decomposition Method (ADM). Fractional differential equations provide an effective mathematical tool for modeling and simulating non-classical dynamic phenomena in physics, engineering, and applied sciences. The main novelty of this work lies in the use of the Adomian Decomposition Method for computing both eigenvalues and eigenfunctions of fractional Sturm-Liouville boundary value problems, where the forcing term is an arbitrary function of 𝑥 and 𝑦. The results indicate that the suggested method provides a simple and efficient alternative for the spectral analysis of fractional differential operators. At the end of the paper, the representative example is presented to demonstrate the applicability of the method.

A physics-informed neural network for solving multi-term space-time fractional differential equations

Identifying the stratification expression of Ki67 is crucial for directing clinical treatment strategies in ER+/HER2- breast cancers. This diagnostic study investigated the value of first-order features extracted from conventional DWI, diffusion kurtosis imaging (DKI), fractional order calculus (FROC), and continuous-time random walk (CTRW) in discriminating Ki67 expression of ER+/HER2- invasive ductal breast cancer. This retrospective study included 121 patients who underwent DWI, DKI, FROC and CTRW and were pathologically categorized into the low (≤ 5%, n = 30), medium (> 5% to < 30%, n = 41), and high Ki67 expression group (≥ 30%, n = 50). Sixty-three diffusion parameters were computed and subsequently compared across different groups. The area under the receiver operating characteristic (ROC) curve (AUC) was used to quantify diagnostic efficacy. Multivariate logistic regression and bootstrap (1,000 samples) analyses were used to establish and evaluate, respectively, the optimal model to identify Ki67 expression. Twenty-three features showed statistically significant differences among the low, medium and high expression groups (all p values < 0.05). Further multivariable logistic regression analysis for discriminating the low Ki67 and non-low expression group showed that the FROC model constructed by D10%, µ90% and µSkewness had optimal diagnostic efficacy (AUC = 0.858; 95% confidence interval, 0.783-0.915), which was significantly better than that of DWI model (AUC = 0.677; p = 0.008). The validation model showed good accuracy (AUC = 0.850; 95%CI, 0.774-0.916). The FROC model could help identify the low Ki67 expression (≤ 5%) and prevent unnecessary chemotherapy.

In this paper we investigate the problem of limiting multistability in complex dynamical systems with fractional calculus. The study is based on the famous van der Pol-Duffing oscillator, which in the case of external excitation exhibits the co-existence of numerous attractors. We identify possible states of the system using the integer-order scheme and compare them to the ones obtained for fractional modelling. We show that depending on the order of derivatives, the number of solutions can be limited, leading to monostability of the system. The scenarios of attractors exclusion are discussed, including the study of their basin stabilities. Moreover, our results uncover the impact of fractional components on transient properties of the oscillator. Depending on the chosen order, the transient response can be majorly limited, allowing the trajectories to stabilize on the final state much earlier. We confirm our results in both periodic and chaotic dynamics, showing that the proposed methodology can be applied in various complex systems.

In this paper, we present an operational matrix method for integrating fractional Riccati differential equations (FRDEs) based on Haar wavelets. The fractional derivative is considered in the sense of Atangana’s beta derivative, which effectively captures the memory and nonlocal characteristics of complex dynamical systems. The proposed technique employs a truncated Haar wavelet series and an operational matrix of integration to convert the governing FRDEs into a system of algebraic equations. These equations are then formulated as objective functions, and the unknown Haar wavelet coefficients are determined using a random search optimization procedure. This transformation reduces the computational complexity and provides an efficient framework for handling nonlinear fractional-order problems. The convergence and validity of the proposed method are demonstrated using several illustrative examples. The numerical results obtained with the proposed approach are compared with those from the Adams–Bashforth method, and the results show that the present technique provides more accurate approximations. Furthermore, to assess the performance and reliability of the method, several error metrics were computed, including the mean absolute deviation, root mean square error, Theil’s inequality coefficient, Nash–Sutcliffe efficiency (NSE), and variance account for (VAF), for different numbers of collocation points. The results confirm that the Haar wavelet operational matrix method is simple to implement, computationally efficient, and highly accurate for solving fractional Riccati differential equations.

This paper investigates the existence, uniqueness, and Hyers–Ulam stability of a new class of higher-order coupled stochastic non-instantaneous impulsive Hilfer fractional switched differential equations with deviated arguments and Poisson jumps in finite-dimensional spaces. The well-posedness of the system is established using the method of integral contractors under suitable regularity assumptions and a weakened Lipschitz-type condition on the nonlinear operators. The proposed framework introduces a unified class of coupled Hilfer fractional stochastic switching systems that incorporates non-instantaneous impulses, switching dynamics, fractional integral initial conditions, and both continuous and jump-type stochastic perturbations. This structure provides a mathematically consistent and physically realistic model for hybrid dynamical systems with memory and random disturbances. The analysis employs techniques from fractional calculus, stochastic analysis, Laplace transforms, and Mittag–Leffler function theory. A numerical example and an application to electromechanical coupling and seeker stabilisation are presented to illustrate the effectiveness and practical relevance of the theoretical results.

The main aim of this research is to study the sufficient conditions for the existence of an integral-form mild solution and approximate controllability results for a new class of the nonlinear Ψ-Caputo fractional Sobolev-type delayed stochastic multivalued system with extended impulsive dynamics in a separable Hilbert space. The Ψ-Caputo fractional derivative offers a versatile and unifying framework that extends the classical fractional differential operators through an appropriate choice of the kernel function Ψ. This adaptability enables more accurate modelling of memory and hereditary characteristics inherent in complex dynamical systems. Firstly, the proposed control system is transferred into an equivalent fixed point problem using the Ψ-Riemann-Liouville fractional integral operator. Then, the Karlin fixed point theorem is implemented to establish the existence of mild solution. Furthermore, the approximate controllability result of the proposed control system is investigated under the consideration that the corresponding linear system is approximate controllable. The main results are derived using fractional calculus, theory of multivalued map, the concepts of stochastic analysis, and fixed point approach. At the end of the paper, a concrete example is provided to illustrate the theoretical findings.

Fractional dynamics of socio-ecological collapse: Nonlocal memory operators and hereditary stability in coupled human-Earth systems.

To evaluate the diagnostic performance of multi-b-value DWI models for clinically significant prostate cancer (csPCa), identify zone-specific predictors in the peripheral (PZ) and transition zones (TZ), and validate the model's robustness across different MRI vendors. This retrospective study enrolled 238 patients, comprising a primary cohort (n = 162) and an independent cross-vendor validation cohort (n = 76). Seven diffusion models (mono-exponential model (MEM), intravoxel incoherent motion (IVIM), diffusion kurtosis imaging (DKI), stretched-exponential model (SEM), fractional order calculus (FROC), continuous-time random walk (CTRW), and IVIM-DKI model) were fitted to generate 18 parameters. LASSO regression and generalized estimating equations (GEE) identified independent predictors. Model performance was assessed using ROC curves and decision curve analysis (DCA). Subgroup analyses were performed in PZ/TZ. MEM_ADC and CTRW_alpha were identified as robust independent predictors of csPCa. In the test set of the primary cohort, the Clinical+Multib_DWI model achieved an AUC of 0.85. Although the improvement over the Clinical+ADC model (AUC = 0.80) was not statistically significant (p > 0.05), the multi-b-value model demonstrated superior clinical net benefit. Crucially, in the cross-vendor validation cohort, the model maintained robust diagnostic accuracy (AUC = 0.88). Subgroup analysis revealed that CTRW_alpha exhibited strong diagnostic value for TZ lesions (AUC = 0.86) and TZ PI-RADS 3 lesions (AUC = 0.82). MEM_ADC and CTRW_alpha are zone-specific predictors of csPCa. While the multi-b-value model did not significantly outperform the ADC model in AUC, it offered superior clinical utility through higher net benefit and demonstrated cross-vendor robustness, supporting the translational potential of advanced diffusion models. This study identifies zone-specific diffusion predictors for prostate cancer. By demonstrating robustness across different MRI vendors, the findings demonstrate that advanced diffusion models can be successfully translated from specialized protocols to clinical settings, providing superior decision-making utility regarding biopsy necessity. Advanced diffusion models lack cross-vendor validation for prostate cancer diagnosis. Selected diffusion parameters demonstrated robust cancer prediction across independent scanner vendors. Zone-specific evaluation offers superior clinical benefit for personalized biopsy decisions.

Applications of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si164.svg" display="inline" id="d1e117"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -Gamma function in fractional calculus and special functions

AI-enhanced analytical and numerical solutions for fractional cancer tumor models in older adults using healthcare technologies

Fractional differential equations (FDEs) have been used extensively to model systems with memory and non-local dynamics, but the high computational cost of evaluation of a fractional derivative means that their numerical solution is difficult to calculate. Classical methods like finite difference have the disadvantage of needing to store the entire history of the solutions, whereas Physics-Informed Neural Networks (PINNs) have the drawback of being unstable and high cost to train, and standalone Monte Carlo (MC) method exhibit slow convergence and large variance. To overcome these limitations, this study present a hybrid PINN-MC that will integrate MC sampling in the PINN architecture to effectively approximate the Caputo fractional derivative and minimize memory requirements and enhance computational efficiency. Numerical results of nonlinear fractional models such as decay or growth equation, damping equation, predator-prey type model, and Lotka-Volterra model indicate that the proposed method has high accuracy and convergence relative to the standard MC method, with low approximation error and high stability.

This study presents a novel numerical framework for simulating glucose-insulin regulatory dynamics using the Caputo-Fabrizio (CF) fractal-fractional operator with both constant and variable fractional orders. The model incorporates an exponential decay kernel to capture memory and hereditary effects in metabolic regulation. A Newton interpolation-based numerical scheme is developed to approximate the CF-FF derivatives, ensuring computational stability and accuracy. For the variable-order formulation, the fractional order dynamically evolves with time, reflecting physiological variability typically observed during intravenous glucose tolerance tests (IVGTT). Numerical experiments reproduce physiologically realistic glucose-insulin oscillations and demonstrate how feedback control stabilizes chaotic metabolic behavior. The results are based entirely on simulation evidence calibrated within clinically reported parameter ranges, providing conceptual validation rather than direct patient-data comparison. The proposed approach bridges mathematical fractional calculus with biomedical applications, offering new insights for personalized diabetes management and adaptive glucose control strategies.•Fractal-fractional model formulation capturing glucose-insulin memory and adaptation•Stable numerical scheme using Newton interpolation for accurate fractional integration•Linear feedback control applied to regulate chaotic glucose-insulin dynamics•Numerical Methodology for glucose-insulin dynamics. Our investigation of the fractal-fractional glucose-insulin system employs the following analytical framework:•Model Development: We formulate a fractal-fractional-order extension of the minimal glucose insulin model, incorporating an exponential decay type kernel to capture the system's memory effects and anomalous diffusion characteristics inherent in metabolic processes. The model accounts for both insulin-dependent and independent glucose utilization dynamics.•Computational Implementation: We develop a novel numerical solver based on Newton's interpolation polynomials, implementing the Atangana-Seda fractal-fractional derivative formulation. This method provides an efficient computational framework for solving the coupled nonlinear fractional differential equations while maintaining numerical stability across different fractional orders.•The purpose of this section is to define a mathematical model to study the dynamic behavior of glucose-insulin physiology.•With the Adams-Bashforth-Moulton numerical scheme, we compute the Lyapunov exponent of the system, which is useful for studying dissipative.•In a generalized numerical method, we simulate the solutions of the system using the time-fractal fractional derivative of Atangana-Seda.