A Novel Numerical Scheme and Analysis of a Nonlinear Nonlocal Equation Modeling Neurodynamics

Integral operators have been widely employed in biological, medical, and physical models. In this work, we investigate a nonlinear integrodifferential equation involving a convolutional nonlinearity, which represents an important class of models for describing neurodynamical processes. Higher order numerical schemes for nonlinear nonlocal equations of this type remain relatively limited in the literature. To overcome this limitation and to address the scarcity of higher order schemes for such models, we develop two piecewise polynomial methods based on B-splines combined with method-of-lines time integration. This formulation leads to fully discrete finite difference schemes as well as continuous Galerkin higher order schemes. A rigorous convergence analysis is presented, providing theoretical error estimates that are verified through extensive numerical experiments. The proposed methods are tested in both one- and two-dimensional spatial domains, where the simulations successfully capture the steady-state solution profiles, including the formation of sharp transition layers, in agreement with theoretical predictions. The numerical results show that the exponential decay kernel provides faster convergence compared with the Gaussian kernel. Furthermore, the formation and behavior of solution layers are strongly influenced by the choice of kernel function and some parameters. The developed framework provides an efficient, accurate, and reliable computational approach for solving nonlinear nonlocal models arising in a wide range of applications in science and engineering.