Finite difference approximations for the first-order hyperbolic partial differential equation with point-wise delay
Provides numerical schemes for a specific PDE model of neuronal firing intervals, but the methods are incremental extensions of existing finite difference techniques.
The paper constructs explicit finite difference methods (Lax-Friedrichs and Leap-Frog) for a first-order hyperbolic PDE with point-wise delay/advance, proving consistency, stability under CFL condition, and convergence. Numerical tests verify theoretical estimates and show the effect of delay.
Explicit numerical methods based on Lax-Friedrichs and Leap-Frog finite difference approximations are constructed to find the numerical solution of the first-order hyperbolic partial differential equation with point-wise delay or advance, i.e., shift in space. The differential equation involving point-wise delay and advance models the distribution of the time intervals between successive neuronal firings. We construct higher order numerical approximations and discuss their consistency, stability and convergence. The numerical approximations constructed in this paper are consistent, stable under CFL condition, and convergent. We also extend our methods to the higher space dimensions. Some test examples are included to illustrate our approach. These examples verify the theoretical estimates and shows the effect of point-wise delay on the solution.