In this paper, the fractional-order delay differential equation (FDDE) model is used to explore the dynamics of nutrient transport in a vascular environment to delayed nutrient uptake and pulsatile blood flow. The external nutrient input and nonlinear interaction term and the product of nutrient concentration and blood flow over time are also studied. The delay is introduced to consider the physiological lags which are causes of metabolic processing and absorption of nutrients via the Caputo fractional derivative. The stability of the system is examined at steady and periodically variable blood flow conditions. To obtain constant flow, analytical expressions of the equilibrium points and the critical values of delay are examined. When the blood flow is pulsatile, and considered to be a sinusoidal function, the system has complex oscillatory dynamics, and numerical simulations are used to analyze when the system becomes unstable. A sensitivity analysis is examined to analysis the impact of the main model parameters. Findings show that longer delay times and larger amplitudes of pulsation enhance instability and steady oscillation of nutrient concentration. These results lend some understanding to the physiological connotation of the delayed interactions in nutrient transport and could be used to guide future studies on metabolic control and circulatory mechanisms.

Classical continuum theories cannot accurately describe thermoelastic behavior at the micro- and nano-scales due to scale-dependent effects. At these scales, long-range interactions significantly influence the thermomechanical response of structural systems. Fractional calculus has emerged as an effective framework for representing nonlocal interactions through space-fractional formulations within continuum mechanics. Although space-fractional structural models are capable of capturing scale-dependent mechanical behavior, thermoelastic effects in such formulations have not yet been adequately addressed. This study develops a space-fractional thermoelastic framework to analyse thermal effects in micro- and nano-scale truss structures. Thermal influences are incorporated through temperature-dependent material properties, allowing the effect of temperature variation on structural response to be examined within a continuum-based formulation. A parametric investigation is carried out to study the influence of Young’s modulus variation, material grain size, and material order, represented by the fractional order, under different temperature levels and boundary conditions. The analysis is performed within a thermal continuum-based space-fractional formulation, in which nonlocal interactions are described through a fractional-order representation embedded in the governing equations. The proposed framework advances thermomechanical modelling at small scales, and it supports the design and optimisation of micro- and nano-scale devices that account for the thermal effects, including MEMS and NEMS structural components.

The present study examines the numerical simulation of the generalized fractional-order wave-type Regularized Long Wave (RLW) equation, an advanced model for characterizing nonlinear dispersive wave phenomena in shallow water dynamics, plasma physics, and other fluid-like media. The incorporation of fractional-order derivatives enriches the RLW framework by capturing memory effects and nonlocal interaction features that classical integer-order formulations cannot adequately represent. The proposed model is of great significance for simulating the behavior of complex wave phenomena arising in real-world systems, such as anomalous diffusion and nonlocal elasticity.

For the numerical solution of the wave-type RLW model, the spatial domain is discretized using modified cubic B-spline basis functions to obtain smooth and accurate approximations. At the same time, the temporal fractional derivative is handled through a finite difference scheme. The method is assessed under various mesh sizes and time-step combinations to analyze its convergence properties, error behavior, and computational efficiency. The study further demonstrates how variations in the fractional order affect both the qualitative and quantitative characteristics of the resulting wave profiles. The obtained numerical results will provide deep insight into the underlying mechanisms that control wave propagation in nonlocal environments, which classical integer-order models cannot reveal. Overall, this combined spline-difference framework shows strong promise for efficiently solving fractional nonlinear dispersive wave equations.

This paper investigates the unsteady flow and heat transfer behaviors of viscoelastic fluids over a stretched plate. A novel fractional-order Maxwell constitutive relationship is proposed, which forms a coupled velocity–temperature–stress system and innovatively incorporates the temperature dependence of relaxation time. The governing partial differential equations, characterized by high nonlinearity and coupling effects, are numerically solved via a hybrid method combining the spectral method (for spatial discretization, ensuring high accuracy) and the finite difference method (for temporal discretization, improving stability). Analytical solutions under simplified conditions are derived to validate the reliability and accuracy of the numerical results. Comparative analyses of velocity, temperature, and shear stress fields are conducted across different constitutives, focusing on the influence of the Improved Variable Viscosity Model on flow and heat transfer characteristics. This work provides a reliable theoretical framework and numerical tool for optimizing fluid manipulation and process design in engineering applications involving stretched plate configurations. It enriches the theoretical system of fractional viscoelastic fluid mechanics and offers critical guidance for enhancing the performance of related engineering systems such as polymer processing and heat exchanger design. Moreover, the proposed hybrid numerical method and constitutive model can serve as a reference for subsequent studies on complex fluid flow and heat transfer problems in other similar configurations.

Over the past twenty years, the mathematical modeling of Fractional Differential Equations (FDEs) has attracted growing attention and expanded significantly across various scientific fields, including bioengineering, mathematical biology, physics, chemistry and computational medicine. In this context, finding analytical solutions of FDEs is often more challenging than for classical ordinary differential equations, while the derivation of accurate and reliable numerical methods suffers from the possible non-smoothness of the solution and/or the vector field at the starting time, not to mention that the efficient treatment of the persistent memory term can make long-time simulations computationally demanding, due to the non-locality of the operator. To mitigate the aforementioned issues, the class of Runge–Kutta-type methods, named Fractional HBVMs (FHBVMs) is presented, covering its design, development and analysis. In particular, a novel extension is proposed, allowing for a mixed graded/uniform stepsize strategy that gives rise to an updated version of the pre-existing fhbvm Matlab code. As confirmed by the numerical tests, the novel approach is mainly tailored to problems with a non-smooth vector field at the starting time, whose solution is both non-smooth at the origin and oscillatory, to concurrently gain accuracy in reproducing the initial non-smooth behaviour of the solution, while maintaining efficiency over extended time periods.

Fractional-order differential and integral operators, as well as fractional differential equations, play a significant role in modeling real-world phenomena across a wide range of scientific and engineering fields. Compared to classical integer-order differential equations, fractional differential equations provide a more effective framework for capturing hereditary and memory effects inherent in many materials and processes. In this talk, we investigate the existence of positive solutions for a system of two h-Riemann–Liouville fractional differential equations (S) in the unknown functions u and v, involving singular sign-changing nonlinearities and two positive parameters. This system is supplemented with general coupled boundary conditions (BCs) that incorporate various h-Riemann–Liouville fractional derivatives together with Riemann–Stieltjes integrals. The h-Riemann–Liouville fractional derivative extends both the classical Riemann–Liouville derivative (when h(t)=t) and the Hadamard derivative (when h(t)=ln t). We first derive the Green functions associated with problem (S)–(BC) and establish several upper and lower bounds for them. Then we make a change to the unknown functions (u,v) and (x,y), and the new problem (P) is written equivalently as a system of two integral equations (I). We then construct a suitable Banach space X and define an operator A in this space corresponding to system (I), noting that (x,y) is a solution of (I) (or problem (P)) if and only if (x,y) is a fixed point of operator A. Under appropriate assumptions of the nonlinearities of the system (S), we identify intervals of the parameters for which the operator A admits at least one fixed point or at least two fixed points, which correspond to positive solutions of the problem (P). The proof of our main results rely on the Leray–Schauder nonlinear alternative and the Guo–Krasnosel’skii fixed point theorem. Finally, we provide two examples that illustrate the applicability of the obtained results.