We investigate a fractional generalization of a one-dimensional scalar wave equation with a singular memory kernel, a model originally introduced to describe single-mode propagation in viscoelastic media exhibiting weakly singular hereditary effects. The fractional formulation is constructed by replacing the standard second-order time derivative with the Caputo fractional derivative of order α, where the order ranges from one to two, and by replacing the spatial Laplacian with the Riesz fractional pseudo-differential operator of order β, with the order likewise ranging from one to two.

An explicit solution is obtained in the form of a series involving Fox H-functions, which arise naturally from the combined influence of temporal and spatial fractional operators. This series representation provides a mathematically tractable structure that facilitates asymptotic analysis and numerical approximation. Furthermore, in the limiting cases α=2 and β=2, the formulation reduces to the classical viscoelastic wave equation, thereby illustrating the consistency of the fractional model with established theories of wave propagation in media with memory.

The results demonstrate that the fractional extension captures a broadened spectrum of propagation behaviors characterized by smoother wave fronts, delayed response, and modified dispersion patterns that depend sensitively on the fractional orders. These findings underscore the capacity of fractional operators to extend classical viscoelastic models and to describe complex dynamical effects associated with memory and spatial nonlocality in a unified mathematical setting.

With the rapid advancement of materials science and chemical engineering, engineering applications involving complex viscoelastic fluids—such as blood flow, asphalt paving, and colloidal processing—are increasingly prevalent. The traditional Oldroyd-B model remains one of the most classical constitutive models for viscoelastic fluids, yet it exhibits limitations when characterizing fluids with intricate microstructures. Caputo fractional derivatives demonstrate significant advantages in describing the complex rheological properties and long-time memory effects of viscoelastic fluids. Therefore, this study aims to investigate the flow behavior of Caputo fractional Oldroyd-B fluids in a top-driven square cavity through numerical simulation. Based on ANSYS Fluent software and employing the finite volume method, this study coupled the fractional Oldroyd-B fluid constitutive equations with the Navier-Stokes equations through user-defined functions and user-defined scalars. The effects of different fractional-derivative orders, Weissenberg numbers, and Reynolds numbers on flow-field structures, vorticity distributions, and vortex-center locations were investigated. To overcome numerical instability at high Weissenberg numbers, an artificial viscosity term was introduced into the transport equation, drawing on strategies from integer order models. Comparison with benchmark data from existing literature demonstrates good agreement in key metrics such as velocity distribution and vortex center location, validating the accuracy and effectiveness of the numerical model.

Diffusion within porous biological media, such as brain tissue, often exhibits anomalous behaviour that deviates from classical Fickian laws, motivating the use of space-fractional diffusion models. Analytically tractable space-fractional diffusion problems are typically solved using an integral transform. Bessel integrals arise in the scope of the radial Fourier transform on the plane, which is equivalent to the Hankel transform, and suffer from slow convergence.

This study focuses on the numerical evaluation of a space-fractional reaction-diffusion system with cylindrical symmetry that models the foreign body reaction around an implanted neural electrode. The system incorporates a space-fractional Riesz Laplacian to capture microscopic tissue heterogeneity, and its steady-state solution is derived via the Hankel transform, yielding expressions involving oscillatory Bessel integrals.

Numerical evaluation of these integrals is essential for practical application and parameter estimation. The foundational work implemented a Double-Exponential (DE) quadrature method, enhanced by a sine hyperbolic transformation for oscillatory kernels and accelerated through Wynn’s epsilon algorithm applied over intervals partitioned at Bessel function zeros. In this extended analysis, we compare the initial DE approach with two additional advanced quadrature methods: the Ogata quadrature and the sinc integration rule. Each method is systematically assessed for accuracy, convergence rate, and computational efficiency across a range of fractional orders and spatial ranges.

Our results demonstrate that fractional exponents produce heavy-tailed concentration profiles distinct from the exponential decay of integer-order solutions, and they provide clear performance benchmarks for these quadrature techniques. This comparative study not only validates the robustness of the numerical framework but also offers practical guidance for selecting efficient integration strategies when calibrating transport parameters from experimental data, thereby enhancing the reliability of anomalous diffusion models in biomedical applications.

The Brusselator chemical model is a well-known theoretical framework for describing autocatalytic reactions, oscillatory behavior, and pattern formation in chemical and biological systems. It has been widely employed to investigate nonlinear phenomena such as morphogen distribution, biological pattern selection, and periodic dynamics arising from reaction–diffusion mechanisms. Owing to its ability to capture complex temporal and spatial behaviors, the Brusselator model also finds applications in diverse scientific areas, including plasma physics and laser dynamics.

In this study, a fractional-order Brusselator chemical model is considered to incorporate memory and hereditary effects inherent in many real-world processes. The governing fractional differential equations are analyzed using two distinct approaches: Homotopy Analysis Method (HAM) and Euler’s Modified Method (EMM). HAM is employed as a semi-analytical technique to construct approximate solutions in the form of convergent polynomial series, facilitated by an appropriate selection of the convergence-control parameter. In parallel, EMM is used as a numerical scheme to obtain approximate solutions of the model for comparison purposes.

The obtained semi-analytical and numerical solutions are validated by comparing them with the solutions generated using the NDSolve routine in Mathematica. A detailed investigation of the model behavior is carried out for different values of the system parameters and fractional orders. The comparative analysis demonstrates excellent agreement among the results obtained by HAM, EMM, and NDSolve, confirming the accuracy and reliability of the proposed approaches. This comparative study highlights the rapid convergence and flexibility of HAM, compared to the numerical method EMM.

Finally, this work provides new insights into the fractional Brusselator model and establishes an effective framework for analyzing fractional nonlinear dynamical systems. The findings contribute to the numerical analysis of fractional chemical models and are relevant to broader applications in physics, biology, and engineering.

To address the issue of insufficient cooling efficiency in traditional thermal management solutions for high-heat-flux electronic devices, this study proposes a coupled model integrating a straight circular turbulent flow channel with a high-heat-flux chip. A carboxymethyl cellulose (CMC)-based Al₂O₃ nanofluid is selected as the working medium. The CMC base fluid exhibits typical linear viscoelastic behavior with notable viscoelastic memory decay characteristics. The addition of Al₂O₃ nanoparticles enhances the thermal conductivity of the fluid while preserving the viscoelastic properties of the base fluid, making it highly compatible with the fractional-order Maxwell constitutive model. Therefore, the Caputo fractional-order Maxwell fluid model is adopted to accurately characterize the viscoelastic memory effects of the CMC base fluid. Through user-defined functions (UDFs) in Fluent, this model is coupled with the k-ω SST turbulence model for numerical simulation. After validating the turbulence model, the influence of fractional-order parameters—namely the relaxation exponent (α) and relaxation time (λ)—on the turbulent flow structure, heat transfer performance, and temperature reduction of the electronic device is systematically investigated. The results demonstrate that the viscoelastic behavior of the fluid and turbulent fluctuations work synergistically to enhance heat transfer. Under optimized fractional-order parameters, a significant cooling effect on the electronic device is achieved. This research provides a novel cooling strategy for high-power electronic devices, and the established fractional-order modeling framework offers a valuable reference for similar thermal design and fluid dynamics studies.

Introduction: Variable-order fractional stochastic integro-differential equations (VO-FSIDEs) have attracted increasing attention due to their capability to model complex dynamical systems involving memory effects, hereditary properties, and stochastic perturbations. Such equations arise naturally in a wide range of applications, including anomalous diffusion, viscoelastic materials, population dynamics, and financial mathematics, where uncertainty and time-dependent fractional behavior play a significant role.

Method: In this study, an efficient numerical framework based on the operational matrix method is presented for solving VO-FSIDE . The unknown stochastic solution is approximated using Chelyshkov, Chebyshev, and Euler polynomial expansions. Operational matrices corresponding to the variable-order Caputo fractional differential operator are constructed for each polynomial basis. By employing Newton–Cotes nodes as collocation points, the proposed scheme transforms the original variable-order fractional stochastic integro-differential equations into a system of algebraic equations, which can be solved efficiently using standard numerical techniques such as the Gauss elimination method and Newton’s iterative method.

Results: A comparative numerical study is conducted to evaluate the performance of the proposed polynomial-based methods in terms of accuracy, convergence rate, and computational efficiency. Several numerical examples, including problems with known exact solutions, are presented to validate the proposed methodology. The numerical results demonstrate excellent agreement with the exact solutions and reveal that the Chelyshkov polynomial-based operational matrix method achieves higher accuracy with fewer basis functions compared to the Chebyshev and Euler polynomial approaches.

Conclusions: The proposed operational matrix framework provides a reliable, accurate, and computationally efficient tool for solving VO-FSIDEs. The comparative results highlight the effectiveness of the Chelyshkov polynomial approximation for capturing variable-order fractional dynamics, making the proposed approach well suited for practical applications in science and engineering.

Existing studies have confirmed that fluid–structure interactions (FSIs) between flexible structures and fluids can enhance heat transfer in enclosed and ventilated cavities, with relevant characteristics affected by structural elasticity, flow parameters, and other factors. However, current integer-order models fail to account for the viscoelastic memory effect of time-fractional Maxwell fluids and lack the application of fractional derivatives, making it difficult to accurately characterize the dynamic flow–deformation interaction. Taking the square cavity flow with flexible fins as the research object, and based on the advantages of Caputo fractional derivatives—including compatibility with physical initial conditions, simplicity of numerical discretization, and a superior ability to fit long-term memory characteristics—this paper introduces Caputo fractional Maxwell shear stress into the constitutive relation and incorporates time-fractional derivatives into the momentum and energy equations, respectively, to establish the governing equations, thereby constructing an FSI model of the fluid and flexible fins. Numerical simulations comparing the coupling results of integer-order and fractional-order fluids show that the fractional order (α) directly regulates the intensity of viscous memory; analysis of differences in velocity distribution, vortex structure, and wall shear stress reveals that the fractional-order model exhibits a more lagging flow response. The research indicates that fractional derivatives are a key tool for characterizing the long-term memory properties of Maxwell fluids, which can effectively improve the physical authenticity of FSI models and provide reliable theoretical support for engineering applications such as micro flexible fluid devices and fluid transportation.

Cooperative adaptive cruise control (CACC), an extension of adaptive cruise control (ACC), is an important intelligent transportation solution for future mobility. It leverages vehicle-to-vehicle information to improve longitudinal tracking, traffic throughput, energy consumption, and string stability. However, controller tuning remains sensitive to model uncertainties and communication delay. This paper presents an analytical parameter-space-based fractional-order PD (FOPD) controller tuning framework for the CACC problem. For the constant-time headway spacing policy, the fractional-order controller parameters are explored over the (kp, kd, µ) parameter space, accounting for plant uncertainties. To enable a tractable stability assessment for the commensurate fractional-order characteristic equation, a variable transformation is used to obtain an equivalent polynomial form, and stability is then verified using the Hurwitz criterion. The resulting parameter-space maps provide a transparent graphical approach to controller gain and fractional-order selection. Moreover, the CACC design is implemented by a feedforward controller that uses preceding vehicle acceleration information within the predecessor-vehicle-following communication topology. The proposed method is tested in a vehicle platoon simulation environment with respect to string stability and the time-domain responses of position, velocity, acceleration, and headway time. The fractional-order CACC is compared with integer-order controllers. Simulation studies under representative CACC maneuvers demonstrate that the proposed parameter-space approach enables systematic FOPD tuning and achieves improved performance requirements compared to integer-order PD control.

This study presents a theoretical investigation of the time-dependent electro-osmotic hemodynamics of a fractional second-grade fluid laden with tetra-hybrid nanoparticles (Au, Al₂O₃, TiO₂, and SWCNTs) flowing through a compliant bifurcated artery affected by multiple atherosclerotic stenoses. The mathematical model incorporates the coupled effects of time-varying electro-osmotic forcing, arterial wall elasticity, and complex multi-stenosed bifurcation geometry to realistically capture the dynamics of blood-based nanofluid transport and targeted drug delivery. A fractional constitutive framework is employed to account for the hereditary and memory-dependent viscoelastic behavior of blood, while the inclusion of tetra-hybrid nanoparticles significantly enhances the effective thermal and electrical conductivities, thereby improving flow regulation and nanoparticle dispersion. The resulting nonlinear fractional governing equations are solved numerically using Mathematica software. The parametric analysis demonstrates that an intensification of the time-dependent electro-osmotic field markedly suppresses axial velocity in both the parent and daughter arteries, while an increase in the fractional-order parameter further attenuates the velocity due to enhanced memory effects. The temperature field is observed to decrease with increasing nanoparticle volume fraction, whereas the absence of a nanolayer around nanoparticles leads to a pronounced enhancement in temperature, accompanied by dominant velocity modulation. The present model elucidates the synergistic interplay between electro-osmotic actuation, wall compliance, and fractional rheology in optimizing nanofluid-based drug transport, offering novel insights for the design of advanced electrohydrodynamic therapeutic strategies in cardiovascular disease management.

This study investigates the dynamic behaviour of coupled circular plates connected by viscoelastic layers exhibiting creep effects. Such multilayer systems are widely used in aerospace, civil engineering, and microsystems, where time-dependent material properties and long-term loading significantly affect structural stability and performance.

The proposed analytical framework applies fractional calculus to capture hereditary effects in viscoelastic materials, offering a more accurate representation of time-dependent behaviour than classical models. Starting from D’Alembert’s principle and the theory of hereditary viscoelasticity, we derive a system of partial integro-differential equations describing transverse oscillations of the plates. These equations are systematically reduced to ordinary integro-differential form using Bernoulli’s method and solved via Laplace transforms, enabling closed-form solutions for key dynamic characteristics.

The results provide explicit expressions for natural frequencies, mode shapes, and time-dependent response functions. Analysis reveals the emergence of multi-frequency vibration regimes, with modal families remaining temporally uncoupled. This property facilitates precise identification of resonance conditions and dynamic absorption phenomena. The fractional parameter acts as a tunable damping factor: lower values allow prolonged oscillations, while higher values induce rapid decay. Additionally, increasing the kinetic stiffness of coupling layers raises vibration frequencies and amplifies hereditary effects.

These findings deliver a robust tool for designing multilayered structures with tailored dynamic responses. By adjusting fractional order and stiffness ratios, engineers can optimise vibration control, enhance stability, and mitigate long-term deformation risks under complex loading conditions.