Piecewise linear emulation of propagating fronts as a method for determining their speeds

Area of Science:

• Mathematical modeling
• Nonlinear dynamics
• Chemical kinetics

Background:

• Reaction-diffusion equations model complex phenomena like pattern formation and population dynamics.
• Accurate determination of front propagation speed is crucial for understanding invasion dynamics.
• Existing methods for analyzing nonlinearities can be computationally intensive.

Purpose of the Study:

• To develop a simplified analytical method for approximating solutions to nonlinear reaction-diffusion equations.
• To accurately determine the selected speed of propagating fronts in reaction-diffusion systems.
• To provide a versatile and reliable technique applicable to various nonlinearities.

Main Methods:

• Emulating polynomial nonlinear terms with piecewise linear functions.
• Replacing nonlinear differential equations with a system of linear inhomogeneous differential equations.
• Deriving an analytic solution for the approximated system.

Main Results:

• The piecewise linear approximation yields an excellent analytic solution for propagating fronts.
• The method accurately predicts the physically selected speed of observable fronts for cubic and quintic nonlinearities.
• The approach demonstrates high reliability and accuracy across different nonlinearities.

Conclusions:

• Piecewise linear emulation offers a robust and accurate method for analyzing reaction-diffusion systems.
• This technique simplifies the analysis of front propagation and invasion dynamics.
• The method is particularly effective for determining the speed of pushed fronts invading unstable states.