Numerical Solution of the Kolmogorov–Petrovsky–Piskunov Problem with Nonlocal Nonlinearity

This paper presents the results of a numerical solution of an initial-boundary value problem for the one-dimensional Kolmogorov–Petrovsky–Piskunov–Fisher (KPP-F) reaction-diffusion equation with a nonlocal integral term. The aim of the study was to demonstrate the effectiveness of an explicit finite-difference scheme for the numerical solution of a nonlocal integro-differential problem on a uniform spatio-temporal grid. The numerical results confirm that the explicit finite-difference scheme maintains stability and computational feasibility while strictly satisfying the Courant criterion. The simulation successfully reproduces characteristic nonlinear effects: propagation of traveling waves, interaction and merging of local maxima, and the formation of stable spatially periodic structures (self-oscillations).