Spectral Optimized Multiderivative Hybrid Block Method for Fitzhugh–Nagumo Equations

The Fitzhugh–Nagumo equation, a key model for excitable systems in biology and neuroscience, requires efficient numerical methods due to its nonlinear nature. A spectral optimized multiderivative hybrid block method is proposed, constructed using a multistep collocation and interpolation technique with an approximated power series as the basis function. Incorporating two optimal intra‐step points, the method demonstrates improved accuracy, with its consistency, convergence, and absolute stability rigorously analyzed. By combining the optimized multiderivative hybrid block method in time with a spectral collocation method in space, the approach demonstrates potency and flexibility in solving partial differential equations. Prior to using the spectral method, the partial differential equation is linearized using a linear partition technique. Numerical experiments confirm the accuracy and efficiency of the method compared to existing methods, demonstrating the potential of the method as a robust framework for solving partial differential equations requiring both high accuracy and stability.