High-order hybrid computational method with adaptive mesh strategies for time-fractional nonlinear reaction–diffusion problems featuring weak singularity

This research focuses on developing a hybrid computational approach that leverages high-order finite difference techniques and non-polynomial spline methods to address time-fractional nonlinear reaction–diffusion equations with bounded and unbounded temporal derivatives at time t=0. In particular, we study the application of the Fitzhugh–Nagumo model in analyzing the behavior of electrical impulses, as well as the generalized Fisher’s reaction–diffusion equation in modeling abnormal diffusion in biological tissues. The temporal fractional operator is approximated through a high-order scheme constructed using various interpolation approaches including linear, quadratic, and cubic interpolation, achieving O(Nτ−(4−α)) precision on appropriately selected nonuniform meshes and O(Nτ−α) precision on uniformly spaced meshes, where α is the order of the time-fractional derivative and Nτ denotes the number of temporal discretization points. Further, we utilize a high precision stable parametric quintic spline (PQS) for spatial discretization. The obtained nonlinear equation system is resolved through an iterative computational algorithm. The stability analysis of the fully discrete scheme is conducted on uniform distributed meshes through Fourier analysis techniques. We also prove the convergence of the proposed method for the solution with bounded and unbounded temporal derivative in the L2-norm using the Fourier analysis method on uniformly distributed mesh. The convergence order is found to be (4−α) in the time direction and 4.5 in the space direction for bounded temporal derivative. Moreover, for the solutions with unbounded temporal derivatives, the accuracy achieves O(Nτ−α+h4.5) theoretically on uniform mesh, which numerically delivers superior temporal accuracy of O(Nτ−(4−α)) on nonuniform mesh distributions. Finally, numerical experiments are performed on two test cases: the Fitzhugh–Nagumo reaction–diffusion model and the Generalized Fisher’s reaction–diffusion model. The first case addresses solutions with bounded temporal derivatives, whereas the second case examines solutions with unbounded temporal derivatives.