In this paper, an effective numerical method for solving the variable-order(VO) fractional reaction diffusion equation with the Caputo fractional derivative is constructed and analyzed. Based on the generalized alternating numerical flux, we get a fully discrete local discontinuous Galerkin scheme for the problem. From a practical standpoint, the generalized alternating numerical flux, which is distinct from the purely alternating numerical flux, has a more extensive scope. For $0<\alpha (t)<1$, we prove that the method is unconditionally stable and the errors attain $(k+1)$-th order of accuracy for piecewise $P^k$ polynomials. Finally, some numerical experiments are performed to show the effectiveness and verify the accuracy of the method.

In this paper, an effective numerical method for the variable-order(VO) fourth-order problem with Caputo-Fabrizio derivative will be constructed and analyzed. Based on generalized alternating numerical flux, appropriate spatial and temporal discretization, we get a fully discrete local discontinuous Galerkin(LDG) scheme. The theoretic properties of the fully discrete LDG scheme are proved in detail by mathematical induction, and the method is proved to be unconditionally stable and convergent with $\mathrm{O}(\tau +h^{k+1})$, where $h$ is the spatial step, $\tau$ is the temporal step and $k$ is the degree of the piecewise $P^k$ polynomial. In order to show the efficiency of our method, some numerical examples are carried out by Matlab.

An adaptive moving grid method is developed to solve the time-fractional integrodifferential governing equations of viscoelastic nanofluid. The momentum equation is derived based on a dual-parameter fractional Maxwell constitutive relation, and the energy equation employs a generalized Cattaneo heat conduction relation. To improve solution accuracy, a monitor function based on the equidistribution principle is constructed, and an adaptive mesh redistribution strategy is developed in the spatial domain. The temporal fractional-order operators are approximated by the L1 algorithm and the weighted-shifted Grünwald difference method. Numerical experiments demonstrate that the adaptive grid achieves 77.6–88.4% higher accuracy compared to uniform grids at the same grid scale, along with enhanced stability in convergence. Parametric analysis indicates that increasing the fractional-order derivative in the energy equation results in a thickening of both the velocity and thermal boundary layers. Furthermore, the dual-fractional Maxwell model exhibits a thicker velocity boundary layer than its classical single-parameter counterpart. The proposed method offers an efficient and robust approach for simulating complex viscoelastic nanofluid systems with memory effects and multi-field coupling.