In this paper, a robust adaptive grid method is developed for solving first-order nonlinear singularly perturbed Fredholm integro-differential equations (SPFIDEs). Firstly such SPFIDEs are discretized by the backward Euler formula for differential part and the composite numerical quadrature rule for integral part. Then both a prior and an a posterior error analysis in the maximum norm are derived. Based on the prior error bound and the mesh equidistribution principle, it is proved that there exists a mesh gives optimal first-order convergence which is robust with respect to the perturbation parameter. Finally, the posterior error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Numerical results are given to illustrate our theoretical result.

This study focused on the unsteady magnetohydrodynamic (MHD) flow and heat transfer of fractional viscoelastic nanofluids over an infinite vertical plate within a porous medium. Both ramped and isothermal wall temperature conditions were considered, along with the effects of heat injection and consumption. The momentum equation was formulated based on a dual-parameter fractional Maxwell constitutive relation, while the energy equation incorporated a fractional dual-phase-lag (DPL) model. The resulting fractional integrodifferential governing equations were solved numerically using a finite difference method combined with the L1 algorithm and the weighted-shifted Grünwald difference scheme. The accuracy of the proposed numerical scheme was verified through manufactured solutions. Numerical results show that increasing porous medium permeability enhances fluid flow, whereas a stronger magnetic field suppresses it. The effects of the phase-lag parameters on the thermal boundary layer differ under ramped and isothermal wall temperature conditions: the phase lag of the temperature gradient leads to a monotonic thickening in the former but a nonmonotonic variation in the latter, whereas the phase lag of the heat flux exhibits an opposite trend. This study provides valuable insights into the application of fractional integrodifferential models for the design and optimization of thermal systems involving nanofluids.